OOn the Unification ofRandom Matrix Theories By Rupert Small
University of BristolSchool of Mathematics
A dissertation submitted to the University of Bristol in accordance with the requirementsfor the degree of Doctor of Philosophy in the School of Mathematics
March 2015. a r X i v : . [ qu a n t - ph ] M a r Abstract
Random Matrix Theory (RMT) is the study of matrices with random variablesdetermining the entries, and various additional symmetry conditions imposed on thematrices. A comparitively young theory, it has its roots in Hungarian physicist andmathematician Eugene Paul Wigner’s work in the early 1950’s [Wig51a, Wig51b]. Inthe following years and continuing unabated to the present day, it has permeated nearlyevery area of modern physics and even number theory [Meh04, GMGW98, ABF11].In recent decades attempts have been made to further refine what has becomethe canonical random matrix theory with its associated symmetry conditions, byconsidering symmetries which allow the matrix representations of quantum potentialsto impose k -body forces on the particles in a system containing m particles ( k ≤ m )[BRW01a,BRW01b,Sre02,Kot14]. This is a generalisation of canonical RMT and is thetopic of this thesis. It will be refered to as the unification of random matrix theories ,because every randomised k -body potential gives rise to a new ensemble. For k = m these ensembles are exactly those known and studied already under the rubrik ofcanonical RMT, but for k < m the resulting ensembles are different and little is knownabout them. The phase space of random matrix ensembles generated by randomised k -body potentials represents an opportunity to unify all random matrix theories intoone, single mathematical theory of random matrices. The process involves embedding the k -body potential into the m -particle state space creating what has become knownas the embedded ensembles . The embedded ensembles, first introduced by Monand French [MF75] in 1975, gave physicists a framework for studying many-bodyinteractions using random matrix theory, but also presented practitioners with thechallenge of an incomplete theory, as it showed that the (canonical) form of randommatrix theory under consideration at the time was a single instance of a much largerphase space of random matrix theories. This meant that only a small subsection ofthe possible statistics were being calculated. Canonical RMT was already providingscientists, engineers and mathematicians insights into a variety of different patternsand physical phenomena in number theory, engineering, physics and computer science.However, these connections only became visible when they were uncovered, so theipotential that unification of random matrix theories offered presented both a bigopportunity and new mathematical problems [BRW01a, BRW01b, Sre02]. Alternativemathematical methods would be needed to manage the new complexities manifestedby studying the unification of random matrix theories.One of the greatest contributions to this area of study occured (in the author’sopinion) at the turn of the last decade with the publication of the breakthroughpaper by Benet, Rupp and Weidenmüller (BRW) which showed how a process ofeigenvector expansions could be used to calculate certain statistical properties of k -body potentials [BRW01a]. These methods were a great advance but were alsodifficult to implement, and it remains unclear if they can practically be used tocalculate moments of the level density of embedded ensembles which are higher thanthe fourth moment.This thesis presents a completely different approach to the method proposed by(BRW). A new method involving particle diagrams will be introduced and developedinto a rigourous framework for carrying out embedded random matrix calculations.Using particle diagrams and the attendent methodology including loop counting itbecomes possible to calculate the fourth, sixth and eighth moments of embeddedensembles in a straightforward way. The method, which will be called the methodof particle diagrams , proves useful firstly by providing a means of classifying thecomponents of moments into particle paths, or loops, and secondly by giving asimple algorithm for calculating the magnitude of combinatorial expressions prior tocalculating them explicitly. By confining calculations to the limit case m (cid:28) l → ∞ this in many cases provides a sufficient excuse not to calculate certain terms atall, since it can be foretold using the method of particle diagrams that they willnot survive in this asymptotic regime. Applying the method of particle diagramswashes out a great deal of the complexity intrinsic to the problem, with sufficientmathematical structure remaining to yield limiting statistics for the unified phasespace of random matrix theories.Finally, since the unified form of random matrix theory is essentially the set ofall randomised k -body potentials, it should be no surprise that the early statisticscalculated for the unified random matrix theories in some instances resemble theiistatistics currently being discovered for quantum spin hypergraphs and other ran-domised potentials on graphs [HMH05, ES14, KLW14]. This is just the beginning forstudies into the field of unified random matrix theories, or embedded ensembles, andthe applicability of the method of particle diagrams to a wide range of questions aswell as to the more exotic symmetry classes such as the symplectic ensembles, is stillan area of open-ended research. .....v Acknowledgements
I would like to give my gratitude and heartfelt thanks to Sara-Lea Small, GalePullen and Adam Zalcman for their role in my education, ultimately leading meto study Mathematics and Physics at the University of Bristol. I would also liketo thank Prof. John Hannay for his creative and inspirational teaching during myundergraduate studies. ..Thanks to my postgraduate supervisor Dr. Sebastian Müller of the School of Mathe-matics for pointing me towards such an interesting area of research and to Dr RémyDubertrand for his flawless lectures on Random Matrix Theory...Finally, many thanks to Leo Brennan, Kildare County Council (Ireland) and theEPSRC (United Kingdom) for their generous grants which enabled me to pursue mystudies to the level of postgraduate – something which appeared to me as a child asbeing far beyond the realms of possibility.
Author’s Declaration
I declare that the work in this thesis was carried out in accordance with the Regulationsof the University of Bristol. The work is original except where indicated by specialreference in the text. No part of the dissertation has been submitted for any otherdegree. Any views expressed in the dissertation are those of the author and do notnecessarily represent those of the University of Bristol. The thesis has not beenpresented to any other university for examination either in the United Kingdom oroverseas.
Rupert Small
March 2015 ontents k = m . . . . . . . . . . . . . . . . . . . . . . . . . 293.2 Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.2.1 Moments and Ensemble Averages . . . . . . . . . . . . . . . . 343.2.2 Partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.2.3 Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.2.4 Dyck Words . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.3 Supersymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.3.1 Anti-commutative Grassmann variables . . . . . . . . . . . . . 533.3.2 Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.3.3 The Generating Function . . . . . . . . . . . . . . . . . . . . . 563.3.4 Supertrace and Superdeterminant . . . . . . . . . . . . . . . . 593.3.5 Saddle-point Approximation . . . . . . . . . . . . . . . . . . . 61viONTENTS vii Embedded
RMT . . . . . . . . . . . . . . . . . . . . . . . . 634.1.1 Second Moments of the Random Variables . . . . . . . . . . . 654.1.2 Unitary Symmetry ( β = 2) . . . . . . . . . . . . . . . . . . . . 664.1.3 Orthogonal Symmetry ( β = 1) . . . . . . . . . . . . . . . . . . 684.1.4 Symplectic Symmetry ( β = 4) . . . . . . . . . . . . . . . . . . 694.2 The Embedded GUE . . . . . . . . . . . . . . . . . . . . . . . . . . . 744.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744.2.2 Particle Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . 764.2.3 Binomial Arguments . . . . . . . . . . . . . . . . . . . . . . . 784.2.4 Paths and Loops . . . . . . . . . . . . . . . . . . . . . . . . . 824.2.5 Example: The Fourth Moment with Loops . . . . . . . . . . . 85 k (cid:28) m (cid:28) l . . . . . . . . . . . . . . . . . . . . . . 1326.4 Bosonic states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1326.5 The Method of Particle Diagrams . . . . . . . . . . . . . . . . . . . . 1346.6 Related Literature and Wider Context . . . . . . . . . . . . . . . . . 1376.6.1 Many-Body Potentials Revisited . . . . . . . . . . . . . . . . . 1376.6.2 Other Unified RMT Results . . . . . . . . . . . . . . . . . . . 1386.6.3 Quantum Spin Graphs . . . . . . . . . . . . . . . . . . . . . . 1396.6.4 Neural Networks . . . . . . . . . . . . . . . . . . . . . . . . . 140 ist of Figures κ against k/m . . . . . . . . . . . . . . . . . . . 824.3 Loops Illustration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 844.4 Loops Illustration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 854.5 The Standard Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . 864.6 Loops Illustration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 875.1 Particle Siagrams for the 6’th moment . . . . . . . . . . . . . . . . . 935.2 Loops Illustration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 955.3 Octohedron particle diagram . . . . . . . . . . . . . . . . . . . . . . . 995.4 The sixth moment h against k/m . . . . . . . . . . . . . . . . . . . . 102viiiIST OF FIGURES ix5.5 Particle Diagram for the term A ttqq A qqpp A puvw A pwvu . . . . . . . . . . 1065.6 Particle Diagram for the term A eeww A wvup A upqt A vtqw . . . . . . . . . . 1075.7 Particle Diagram for the term A tucp A tpcu A cewv A cvwe . . . . . . . . . . 1075.8 Particle Diagram for the term A cewu A uwvt A ttvqp A pceq . . . . . . . . . . 1085.9 Loops Illustration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1085.10 Particle diagram of the term A tpce A evqc A puwq A uwvt . . . . . . . . . . . 1115.11 Particle diagram of A petq A qtuv A euwc A cwvp . . . . . . . . . . . . . . . . 1115.12 Particle Diagram for the term A uwvt A cqvt A pueq A pwec . . . . . . . . . . 1185.13 The particle diagram for A uwce A epqc A tqvu A twvp . . . . . . . . . . . . . 1185.13 Loops Illustration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1205.14 The particle diagram for A uvqt A twvc A cewp A pueq . . . . . . . . . . . . . 1245.15 The eighth moment τ against k/m . . . . . . . . . . . . . . . . . . . 1266.1 General form of the moments . . . . . . . . . . . . . . . . . . . . . . 1306.2 The Standard Diagram Review . . . . . . . . . . . . . . . . . . . . . 1366.3 Many Body Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 otation Symbol Definition . † Complex conjugate transpose. Ensemble average | . i Column vector ⊗ Tensor product O Growth rate C ∞ Infinitely differentiablex hapter 1Introduction
Physics is the pursuit of natural laws, or models, which characterise and thereby predictthe behaviour of events within our universe. This includes all measurable propertiesof the natural world, from how sub-atomic particles interact to the emergence ofcomplicated molecules, proteins, plants and animals. Scientific knowledge can be seenas a vast collection of these models and measurable properties, collected over centuriesfrom the efforts, insights and guesswork of all past scientists, both theoretical andexperimental. Each scientific discovery has its own unique history, interwoven into thefabric of previous discoveries. Sometimes a model discovered by a previous generationof scientists will be superseded by a new theory. Many physicists acknowledge thatthe cycle may never end [FLS05, Boh60, Wei92, Bar91]. That it is possible that thepursuit of knowledge is a never ending process of destruction and rediscovery, andthat our understanding of nature can always be improved, albeit never completed.The examples which epitomise this process of continual reformation is the replacementof Newtonian laws of motion by Relativistic laws, and separately the discovery thatNewtonian Mechanics is in fact a limiting case of the much more general theoryof Quantum Mechanics [FLS05]. It also occurs that sometimes two models whichpreviously were thought to be distinct are shown to be the result of a larger more1 CHAPTER 1. INTRODUCTIONgeneral law. This is called unification . Unification is one of the most importantaspects of scientific labour, since it reveals the connections between phenomena inthe natural world which were previously thought of as being different, but which arein fact a result of the same underlying patterns. A seminal example of unification isthe family of mathematical expressions called Maxwell’s equations , which explainthe relationship between electricity and magnetism, unifying two effects which werepreviously thought of as being distinct.Since the process of discovery often leads to different approaches to similarproblems [Fey67] it is not always obvious that the underlying patterns governingdifferent phenomena are the same. Not least of the problems is that seeminglydisparate scientific questions are often studied within the confines of their own specificjargon. To break down the barriers between problems and to reveal their inherentsimilarities it is essential to look uncompromisingly at the underlying mathematicswithout being swayed by the layer of human language which has been added “on top”.Another essential feature of unification theories is that they include mathematicalparameters which connect things which were previously thought of as being unrelated.In this sense unification requires an expansion of the mathematical expressions to“make room” for effects which were not given a voice in previous theories. This thesiswill look at a particular case of unfinished unification in physics and mathematics;the unification of random matrix theories. It will do this firstly by observing thatthe common, or canonical, form of random matrix theory is just a single point ina whole landscape of possible random matrix theories. This landscape of randommatrix theories is known as the embedded ensembles (see Chapter 2) which arethemselves a subclass of quantum k -body potentials. In Chapter 3 it will be shownheuristically how Wigner’s Semi-Circle law for canonical random matrices can becalculated using diagrammatic methods. In the same chapter an illustration of themethod of supersymmetry will show the sometimes stark difference in mathematicalapproaches to the same problem, in this case Wigner’s Semi-Circle Law. Finally in (i) ∇ · E = ρ(cid:15) (ii) ∇ · B = 0 (iii) ∇ × E = − ∂ B ∂t (iv) ∇ × B = µ (cid:0) J + (cid:15) ∂ E ∂t (cid:1) . Togetherequations (iii) and (iv) describe the electric field as a function of the magnetic field and vice versa,showing that E and B are two aspects of a single unified physical process. .1. PHYSICS AND UNIFICATION 3 Physics
Many-Body Potentials
Unified RMT
Canonical RMT
Figure 1.1: The canonical form of random matrix theory is just a single point in awhole landscape of possible random matrix theories (unified RMT). Unified RMT inturn represents the set of randomised quantum many-body potentials, also known asthe embedded ensembles.Chapters 4 and 5 it will be shown that a method of particle diagrams, involvinggraphs of the relationships between quantum states, is sufficient for calculating themoments of the level density of the embedded (unified) ensembles. The majority ofthe results presented here have been published in the rapid communication [SM14]and the article [SM15]. ..... hapter 2Random Matrix Theories
In the early 1960’s theoretical physicist Freeman Dyson published a trio of paperswith the heading
Statistical Theory of the Energy Levels of Complex Systems [Dys62a,Dys62b, Dys62c]. The results (which were also inspired by earlier works of EugeneWigner and Madan Lal Mehta [Wig51a, Wig51b, Meh04]) and the related field ofmathematics which grew around them, became known as random matrix theory(RMT). The basic tenet of random matrix theory as introduced by Dyson was theinvestigation of the statistics of “ all physical systems with equal probability ”. Toparaphrase events, a challenge which researchers were facing at the time was tocharacterise the behaviour of large complex nuclei. Wigner, Dyson and Mehtaenvisioned approaching the problem by assuming nothing, or at least as close tonothing as possible. From this position they began to investigate the quantities thatcan be known about the behaviour of a particular physical system. Their startingpoint was the hamiltonian of the physical system with the single constraint H = H † (2.1.1)4.1. CANONICAL RMT 5for physicality. Assuming nothing else, one would not be able to say anything abouta particular hamiltonian H . However, by allowing each H to occur with someprobability p ( H ) it becomes possible to study the whole space of probabilisticallyweighted hamiltonians. From this point onwards the investigation necessarily becomesstatistical, and these ensembles of hamiltonians are studied by asking questions ofthe form “ What is the probability that...(etc.) ”. The proposition which then needs tobe verified with experiment is that a natural system is some unknown hamiltonianobeying (2.1.1) and therefore the typical behaviour of the system may obey thebehaviour specified by the ensemble average of this. In other words, although theinformation extracted from these ensembles of hamiltonians takes the form “
What isthe probability that... ”, it is possible to use the probabilistic results to say somethingabout the properties of the “typical” (read average) of a hamiltonian, because f ( X ) = Z f ( X ) · (the probability that X) dX (2.1.2)where the horizontal line above f ( x ) indicates the average taken over the ensemble of X ’s. Investigating random hamiltonians in this way is what gave fruit to the field ofrandom matrix theory – a family of sophisticated mathematical tools and equationsfor extracting as much information as possible about an ensemble of hamiltoniansstarting from only a handful of very basic assumptions [Meh04]. The additionalassumptions would usually take the form1 . H = H † . H = S − HS (2.1.3)3 . Gaussian probability determining the matrix elements of H where the matrix S is some predefined matrix determined by the particular physicalsetup at hand. It imposes some symmetry on H additional to the hermitian symmetryit already satisfies by virtue of (2.1.1). The three classical examples for S as introducedby Dyson will be given in the next section (one of which simply takes S = 1). Theseessentially divide the physicial landscape into four statistical landscapes; three groups CHAPTER 2. RANDOM MATRIX THEORIESdefining a “threefold way” and a fourth being the complement of these (everything elsenot included in the three groups determing the threefold way). Although the initialdevelopment of a theory of random matrices focused on the first three symmetrygroups defined by Dyson, it turns out that this classification defines just a singlepoint in a much larger landscape of random matrix theories and a more nuancedapproach to symmetry is required to unify them all. Dyson’s initial development of random matrix theory was placed within the contextof a “threefold way”. These are three symmetry classes which determine a set ofthe additional symmetries (see (2.1.3)) obeyed by physical hamiltonians. Recreatingsome of Dyson’s steps, this section will detail the formulation of the three symmetryclasses determining the additional constraints on the matrix elements of H when thehamiltonian refers to a quantum system from one of three specific groups. Later itwill be shown why this analysis is incomplete, constituting just a single point in aphase space of additional symmetries. The three groups are distilled from the set ofall possible hamiltonians by asking some fundamental questions. For example it isreasonable to ask the question “What form does the potential take if the physicalsystem contains only fermions, or only bosons?” Similarly, it is natural to wonder“What form does the potential take if the system is time-reversal invariant comparedto the case when it is not?” These are questions of symmetry which are integral to theoriginal purpose of RMT, which is to assess the statistical properties of ensembles ofhamiltonians. Although random matrices do not have to be hamiltonians per se – theycan also be scattering matrices [ABF11], or any other matrix for that matter – theclassical categorisation of RMT into three symmetry classes as given by Dyson takesits starting point as the hamiltonian of a physically permitted quantum system. Inother words, the starting point is the hamiltonian satisfying the Schrödinger equationof the system Hψ = i (cid:126) ∂ψ∂t = Eψ. (2.1.4).1. CANONICAL RMT 7Even with a modest amount of complexity Schrödinger’s equation becomes toodifficult to solve analytically, which is why RMT was developed in the first place.In RMT it is assumed that the Hamiltonian H is highly complex and only somerudimentary symmetry properties of the physical system are known. After placingthese restrictions on the hamiltonian all remaining free parameters are randomised .Thereafter an attempt is made to calculate whatever statistical properties can beyielded as a result of the randomisation and symmetrisation process. Using this probabalistic view the problem becomes one of studying ensembles of matrices, eachgrouped into families depending on the set of minimum conditions imposed upon themand each weighted by some probability of occurring within the ensemble. Insteadof studying the eigenvalues of a single instance of H one calculates distributions ofeigenvalues defined by how likely they are to occur across the entire ensemble. Particle Wave Functions
The starting point for calculating the three symmetry classes proposed by Dyson is aremarkable law of nature which through repeated experimentation has establishedthat any quantum state ( a.k.a wave function or p.d.f ) describing a physical experimentcontaining elementary particles is either even-symmetric or odd-symmetric underparticle exchange. Particles with even wave functions (bosons) and particles withodd wave functions (fermions) can be represented mathematically with the respectiveexpressions ψ ( r P , . . . , r P m ) = ψ ( r , . . . , r m ) (bosons) ψ ( r P , . . . , r P m ) = ( − P ψ ( r , . . . , r m ) (fermions) (2.1.5)where { P , . . . , P m } is a permutation of { , . . . , m } and P is the number of pairwisepermutations between the single particle states required to bring { P , . . . , P m } backto the initial configuration. More concisely, for the wave function of both bosons andfermions the following symmetry condition holds ψ ( r P , . . . , r P m ) = ξ P ψ ( r , . . . , r m ) (2.1.6) CHAPTER 2. RANDOM MATRIX THEORIES Bosons ( ξ = 1) Fermions ( ξ = − He neutrinosquarks HeTable 2.1: Table of common bosons and fermions, the two classes of quantum particlesin nature. Bosons can always be represented by an even probability density functionwhile fermions, obeying the Pauli Exclusion Principle, can always be represented byan odd probability density function.where as before ξ = 1 for bosons and ξ = − Time Reversal Invariant Bosonic systems. Non Time Reversal Invariant Bosonic and Fermionic systems. Time Reversal Invariant Fermionic systems. None of these.
The first trio of systems define the three symmetry classes of canonical RMT. Onemay protest that all possible realities are taken into account by the first three groups.However, each group is defined in terms of the minimum set of restrictions on eachmember so systems requiring additional restrictions must go into the fourth group.What are the attendant symmetries that must be satisfied by a potential H for asystem in each group?.1. CANONICAL RMT 9 Invariance under Time Reversal
To form a meaningful mathematical picture of how time reversal invariance relatesto a physical measurement it is first necessary to look at unitary operators, or moreprecisely in the present context, unitary matrices . A good introduction to the followingideas can also be found in [Meh04, Haa10]. A unitary matrix U , is a matrix whichconserves the overlap between states h U ψ | U φ i = h ψ | φ i . (2.1.7)Rearrangment gives U U † = 1 . (2.1.8)Expressed in terms of a summation over the individual elements of the matrix this isequivalent to X k U ik U ∗ jk = δ ij . (2.1.9)In other words the rows (columns) of a unitary matrix are orthonormal. Replacing φ with ψ in (2.1.7) gives || h U ψ | U ψ i|| = ||h ψ | ψ i|| so that Unitary transforms alsoconserve the length of a vector. From the above it follows that U ∈ C n × n rigidlyrotates a vector in C n while conserving its length. Notably, a state transformed as ψ → U ψ is indistinguishable from the original state when the measurement operatoris also transformed as A → U AU † . (2.1.10)Additionally assuming a system of particles whose states are of the form ψ = ψ x e iω t ,it follows immediately that the overlap between two arbitrary quantum states is h ψ | φ i = h ψ x | φ x i · e i ( ω − ω ) t . (2.1.11)If time is flowing in the forward direction the coordinates of a particle will be givenby its position, momentum and t -variable { x , p , t } whereas if time runs backwardsthe particle’s momentum and time variable will reverse at every position so that the0 CHAPTER 2. RANDOM MATRIX THEORIEScoordinates become { x , − p , − t } and the overlap between two states will be h ψ R | φ R i = h ψ x | φ x i · e − i ( ω − ω ) t = h ψ | φ i ∗ (2.1.12)where | ψ R i is the wave function | ψ i under time reversal. Representing the timereversal of a state as an operator gives the tautological expression T ψ = ψ R (2.1.13)where by (2.1.12) T must obey h T ψ | T φ i = h ψ | φ i ∗ . (2.1.14)That is, the operator T is anti-unitary. Additionally, under time reversal the Hamil-tonian of the system becomes H R = T HT − . (2.1.15)Comparing (2.1.14) with (2.1.7) it can be seen that the only difference is complexconjugation, so T can be written as T = KC (2.1.16)where K is unitary and the operator C is defined by its action of taking the complexconjugate of the state on which it operates. Since double application of time reversalto a state should leave it in a state indistinguishable from itself i.e. differing only inphase, T must without exception obey the relation T = α (2.1.17)with | α | = 1, so KK ∗ = α. (2.1.18).1. CANONICAL RMT 11Additionally, by (2.1.10) T transforms as T → U T U − (2.1.19)under unitary transform of the wave-function by U , and the matrix K definedpreviously transforms as K → U KCU − = U KC (cid:16) U T (cid:17) ∗ = U KU T . (2.1.20)A system is called invariant under time reversal if H R = H (2.1.21)where H is the hamiltonian. A matrix satisfying (2.1.21) is called self-dual. A timereversal invariant system therefore satisfies the following sequence of equalities H = T HT − = KCH ( KC ) − = KCHCK − = KH T K − (2.1.22)where the fact that H is hermitian has been used, so that H ∗ = H T . Recalling that K is unitary, implying K ∗ K T = 1, and motivated by a stroke of insight to multiply(2.1.18) by K T it is seen that K = αK T = α K (2.1.23)which means that the value of α is restricted to satisfying α = ± . (2.1.24) Twofold Way
The insight that α can take two possible values reveals mathematically that thereare two distinct types of systems, both satisfying time-reversal invariance. Equation2 CHAPTER 2. RANDOM MATRIX THEORIES(2.1.24) implies two possible outcomes KK ∗ = 1 or (2.1.25) KK ∗ = − . (2.1.26)In addition to being unitary, if (2.1.25) holds K is symmetric and (2.1.26) implies K is anti-symmetric. Following, but not proven here, is that in the symmetric casethere is a unitary transform which by (2.1.20) gives K = 1 (2.1.27)which implies that the system is bosonic (even spin ) and in the anti-symmetric casethere is a unitary transform such that K becomes K = − o . . . o − (2.1.28)which implies that the system is fermionic (odd spin ) [Meh04,Haa10]. That is, a basiscan be chosen ψ → U ψ for each of the two possible time reversal invariant systemssuch that K has the above forms for each case respectively. After choosing sucha basis subsequent transforms on K can only be performed if the relevant relation(2.1.27) or (2.1.28) remains true. Hence for bosonic systems further transforms arerestricted by (2.1.20) and (2.1.27) to being of the form K → OKO T (2.1.29) This thesis will use bosonic and fermionic interchangeably with, respectively, even spin andodd spin throughout. .1. CANONICAL RMT 13where O must be orthogonal (real unitary) and hence by (2.1.22) H is real hermitiansymmetric. To summarise, the above developments gives justification for the followingdefinition Definition 1.
The ensemble of Hamiltonians of Bosonic time reversal invariantsystems are hermitian symmetric matrices statistically invariant under H → OHO T (2.1.30) where O is orthogonal. Here statistical invariance means that the probability of a given H occuring withinthe ensemble is the same as the probability of any other H found after applying thetransform, which in this case is (2.1.30). The ensemble defined by Definition 1 alongwith the condition that the p.d.f. of the elements of a member of the ensemble arestatistically independent defines the Gaussian Orthogonal Ensemble (GUE) which isdenoted by β = 1.Turning now to the fermionic case note that by (2.1.28) the number of rows(columns) of K is always even. Once a basis is chosen such that (2.1.28) holds, furthertransforms on K are restricted to satisfying K = ZKZ T (2.1.31)where Z is unitary. Matrices Z satisfying (2.1.31) form what in the literature is calledthe symplectic group. Notice that by substituting (2.1.28) into (2.1.22) it can be seenthat the hamiltonian of a time reversal invariant Fermionic system satisfies( T HT − ) ij = ( − i + j H ∗ i +( − i +1 , j +( − j +1 = H ij = H † ij (2.1.32)4 CHAPTER 2. RANDOM MATRIX THEORIESwhich can be written as H = e a b a b · · · a m b m e − b ∗ a ∗ − b ∗ a ∗ · · · − b ∗ m a ∗ m e a b · · · a m b m e − b ∗ a ∗ · · · − b ∗ m a ∗ m . . . ... .... . . ... ... * e m e m (2.1.33)where the * in the lower diagonal denotes that the matrix is hermitian so that thelower diagonal is defined by the upper diagonal elements. The diagonal of H is madeup of 2 × e e (2.1.34)where by hermitian symmetry e must be real. The off diagonal components consistof 2 × a b − b ∗ a ∗ (2.1.35)where a, b ∈ C . A useful shortcut to the same conclusion is to calculate the matrixelements of H relative to the basis ψ , T ψ , ψ , T ψ , . . . , ψ n , T ψ n . (2.1.36)Since T = KC and for this (symplectic) case T = − h ψ | H | T ψ i = −h T ψ | H | ψ i ∗ (2.1.37) h ψ | H | ψ i = h T ψ | H | T ψ i ∗ (2.1.38).1. CANONICAL RMT 15which is the equivalent of (2.1.35). Taking summary once again, the above discussiongives motivation for the following definition. Definition 2.
The ensemble of Hamiltonians of Fermionic time reversal invariantsystems are hermitian self-dual ( H R = H ) matrices statistically invariant under H → ZHZ − = ZH ( KZ T K − ) = ZHZ R (2.1.39) where Z is symplectic. Matrices which additionally have the property that entries not related by symmetryare statistically independent form the Gaussian Symplectic Ensemble (GSE), whichis denoted by β = 4. Non Time-Reversal Invariance
The Hamiltonian of non time-reversal invariant Bosonic and Fermionic systems isunrestricted other than by statistical invariance under a unitary transformation.Although this has been a guiding restriction for the cases β = 1 and β = 4, theadditional demands of time reversal invariance confined these to having the symmetriesdefined above. Relaxing these restrictions, which implies S = 1 in (2.1.3), gives thefollowing definition of non time-reversal invariant systems. Definition 3.
The ensemble of Hamiltonians of non time-reversal invariant Bosonicor Fermionic systems are hermitian matrices statistically invariant under H → U HU − (2.1.40) where U is unitary. If in addition the entries of the matrices not related by symmetry are statisticallyindependent Definition 3 defines the Gaussian Unitary Ensemble (GUE), which isdenoted by β = 2. Definitions 1, 2 and 3 form Dyson’s threefold way. They givethe restrictions on the matrix representation of the potential H depending on which6 CHAPTER 2. RANDOM MATRIX THEORIESclass of quantum system it describes. By classifying the physical landscape into theseclasses the matrix H is found to obey additional symmetry (on top of H = H † ), evenif everything else remains, for the present, unknown. The symmetrising matrices O , Z and U define the matrix S of (2.1.3). It should be emphasized that the additionalsymmetries satisfied by H in each of these three groups are the minimal constraintssatisfied by any H belonging to the set. Hence the set of non time-reversal invariantHamiltonians defined by the GUE do not exclude the time-reversal invariant systemsdefined by the GOE and GSE ensembles. Those matrices are present in the ensembleof GUE matrices as well, but occur with a lower probability. The set of non time-reversal invariant hamiltonians is then more strictly the set of “not necessarily butpossibly time-reversal invariant” hamiltonians. “None of These” – Embedded RMT While the canonical approach to random matrix theory places a single random variablein each cell of the matrix, it will be shown in subsequent sections that there are in factadditional ways to define the potential H , these being defined in terms of the order of the potential, k , which will be defined later under the framework of many-bodypotentials. In this way the canonical form of RMT will be extended and a new modelwill be proposed which has the canonical form as a special case. Under the new modelwhat is normally referred to as RMT will be shown to be just one of the possiblepoints in a phase space of random matrix theories , each with its own set of statisticalproperties. This unified phase space is sometimes referred to as Embedded RandomMatrix Theory .Instead of looking at classical, or canonical RMT hamiltonians, the purpose ofthis thesis will be to investigate the statistics of Embedded RMT hamiltonians. Thesedefine a unified form of RMT which is still in its infancy; each embedded hamiltonianrepresents a particular instance of a random matrix theory, and a unified theoryof random matrices must classify the statistical properties of each one. This thesiswill show how the symmetry restrictions imposed in any particular instance of arandom matrix theory can be represented in terms of particle diagrams and a new.1. CANONICAL RMT 17mathematical methodology will be introduced to make calculations of the statisticsof quantum many-body systems from a range of random matrix theories in the phasespace.
It will be seen that the hermitian matrices of canonical RMT with GOE, GSE or GUEsymmetry imposed on the matrix elements of the hamiltonian H represent only onepossible flavour of a random matrix theory. To address this problem and in an attemptto unify the field of random matrix theories, K. K. Mon and J. B. French introducedthe embedded RMT ensembles [MF75], which are hamiltonians written in second-quantised form and determined by a trio of parameters k , m and l (defined later).For each set of values { k, m, l } one attains a distinct random matrix theory. Thesesecond-quantised hamiltonians were already studied by physicists before the advent ofRMT in the context of Many-Body Quantum Mechanics where they appeared underthe rubric of Many-Body Potentials . In this section the intention is to describe in somedetail what a quantum many-body potential is, which will involve an introductionto the basic notational norms of second-quantisation. A good induction into thefollowing formalism can also be found in [NO88, DN05]. The preliminary aim will beto express an arbitrary many-body operator in terms of creation and annihilationoperators. The hamiltonian operator H expressed in this way will form the extensionto canonical RMT under consideration. It will then be seen that there is a particularset of values for the parameters { k, m, l } determining this hamiltonian where it returnsto the form used in classical RMT which, as noted, is a specific case of the moregeneral class.In the forthcoming model the number of quantum particles in the system ofinterest is a variable denoted by the letter m . The tensor product of m single-particlestates | α . . . α m ) ≡ | α i ⊗ . . . ⊗ | α m i (2.1.41)gives a natural way of describing an m -particle state. A simple list of the single-particle states is written in a single packet ( ket ) and defined as the quantum state.8 CHAPTER 2. RANDOM MATRIX THEORIESIn position representation this m -particle state can be rewritten as ψ α ...α m ( r , . . . , r m ) = ( r , . . . , r m | α . . . α m ) = h r | α ih r | α i . . . h α m | r m i = φ α ( r ) φ α ( r ) . . . φ α m ( r m ) . (2.1.42)Given an m -particle state ψ m the properly symmetrized and normalised bosonic ( B )and fermionic ( F ) state is therefore given by P { B,F } ψ ( r , . . . , r m ) = 1 m ! X P ξ P ψ ( r P , . . . , r P m ) (2.1.43)where the symbol ξ = 1 for states describing bosons and ξ = − P is the set of permutations on { , , . . . , m } and the same letter is alsoused to signify the parity of the permutation. To simplify notation later, define theproperly symmetrized (but not necessarily normalised) m -particle state by | α . . . α m } ≡ √ m ! P { B,F } | α . . . α m ) = 1 √ m ! X P ξ P | α P . . . α P m ) . (2.1.44)The closure relation for this space is then given as X α ...α m [ P { B,F } | α . . . α m )][ P { B,F } | α . . . α m )] ∗ = 1 m ! X α ...α m | α . . . α m }{ α . . . α m | = 1 . (2.1.45)Since m -particle states are regarded as describing a probability density function theyshould be normalized to unity. The odd symmetry of fermionic states prohibitsthem from containing any two particles with the same state, so the overlap of two m -particle Fermionic states when non-zero is { α . . . α m | α . . . α m } = ( − P . (2.1.46)If the two states do not contain the same number of particles there is certainly a zerooverlap. Bosonic m -particle states on the other hand – having even symmetry – are.1. CANONICAL RMT 19allowed to have single particle states equal to other single particle states. This meansthat when the overlap is taken between two bosonic states containing k unique stateswith n i single particle states in state i, ≤ i ≤ k the result is { α . . . α m | α . . . α m } = n ! n ! . . . n k ! (2.1.47)where the number of particles is m = P α n α . One can fortunately condense thisnotation further, expressing (2.1.46) and (2.1.47) as the single expression { α . . . α m | α . . . α m } = ξ P Y α n α ! (2.1.48)Hence for functions representing a collection of bosons the orthonormal states aregiven by | α . . . α m i = 1 qQ α n α ! | α . . . α m } (2.1.49)whereas for states representing a collection of fermions this simplifies to | α . . . α m i = | α . . . α m } . (2.1.50)To conclude, the orthonormal closure relation for bosons and fermions is1 m ! X α ...α m | α . . . α m }{ α . . . α m | = X α ...α m Q α n α ! m ! | α . . . α m ih α . . . α m | = 1 . (2.1.51) k -Body Operators The above formulates a way to describe states which represent collections of bosonsand fermions. How to mathematically describe the Hamiltonians which exert forceson these states? The hamiltonians will be expressed as k − body operators, where k isa variable determined by the particular system under consideration. An operator iscalled a k -body operator if its effect on an m -particle state is the sum of its effecton each of the (cid:16) mk (cid:17) single particle k -tuples contained in the m -particle state. For0 CHAPTER 2. RANDOM MATRIX THEORIESexample a 1-body operator is an operator satisfying H | α . . . α m ) = m X i =1 H i | α . . . α m ) (2.1.52)where H i is the effect of H hitting only the i ’th single-particle state. It can thereforebe seen that( α . . . α m | H | β . . . β m ) = X i ( α . . . α m | H i | β . . . β m ) = X i Y k = i h α k | β k ih α i | H i | β i i (2.1.53)and for non-orthogonal | α . . . α m ) and | β . . . β m ) one can conclude that( α . . . α m | H | β . . . β m )( α . . . α m | β . . . β m ) = X i h α i | H | β i ih α i | β i i . (2.1.54)The definition for 2-body operators is the natural extention of this idea. The effect of H on an m -particle state is the sum of its effect on each of the (cid:16) m (cid:17) single-particlepairs contained within the composite m -particle state so that H | α . . . α m ) = X i To describe H mathematically it is necessary to reformulate its properties as describedalready (which is to say, how it acts) in terms of creation and annihilation operators.These form an efficient way of talking about m -particle states because by usingcreation and annihilation operators it subsequently becomes possible to talk aboutthe existence or non-existence of the single-particle states comprising each m -particlestate. This binary way of thinking matters. It opens up a variety of shortcuts, forexample when taking overlaps between m -body states; the answer can be calculatedsimply by looking at whether certain single-particle states exist in the m -body states.This “superpower” will prove extremely useful later on.To begin, the m -particle creation operator a † λ adds a particle in the state | λ i tothe m -particle state on which it acts a † λ | α . . . α m } ≡ | λα . . . α m } . (2.1.60)This is consistent even if λ is already present in the m -particle state. By (2.1.49) it2 CHAPTER 2. RANDOM MATRIX THEORIESfollows that sY α n α ! a † λ | α . . . α m i = √ n λ + 1 sY α n α ! | λα . . . α m i (2.1.61)giving a † λ | α . . . α m i = √ n λ + 1 | λα . . . α m i (2.1.62)where n λ is the number of single-particle states equal to | λ i in the original m -bodystate | α . . . α m i . Defining | i as the vacuum state, any other state can then beexpressed in the form | λ . . . λ m i = 1 √ Q λ n λ ! a † λ . . . a † λ m | i . (2.1.63)To express an arbitrary k -body operator in terms of creation and annihilation operatorsin the standard form the commutation relations between them are needed. First itcan be noticed that since a † λ a † µ | λ . . . λ m } = | λµλ . . . λ m } = ξ | µλλ . . . λ m } = ξa † µ a † λ | λ . . . λ m } (2.1.64)one has a † λ a † µ ≡ ξa † µ a † λ which yields[ a † λ , a † µ ] − ξ := a † λ a † µ − ξa † µ a † λ = 0 . (2.1.65)Taking the complex conjugate gives[ a λ , a µ ] − ξ = 0 . (2.1.66)Equations (2.1.66) and (2.1.66) give the first two commutation relations. To find thethird requires an investigation into the operator a λ := ( a † λ ) † . Taking the overlap { α . . . α m | a λ | β . . . β n } = { λα . . . α m | β . . . β n } . (2.1.67).1. CANONICAL RMT 23one sees this can be non-zero only if m + 1 = n . Representing the action of a λ interms of the identity over all possible quantum states i.e. including states containingdifferent numbers of particles, and assuming that | λ i is in | β . . . β n } , gives a λ | β . . . β n } = ∞ X m =0 m ! X α ...α m { α . . . α m | a λ | β . . . β n }| α . . . α m } = ∞ X m =0 m ! X α ...α m { λα . . . α m | β . . . β n }| α . . . α m } = n X i ξ i − δ λβ i | β . . . β i − β i +1 . . . β n } = n X i ξ i − δ λβ i | β . . . ˆ β i . . . β n } (2.1.68)where ˆ β i denotes that the i ’th particle is removed from the set. Hence the effect of a λ on the state is to remove a particle in the state λ . The operator a λ is thereforereferred to as an annihilation operator , its normalized formulation being a λ | β . . . β n i = 1 qQ j n j ! n X i sY i n i ! ξ i − δ λβ i | β . . . ˆ β i . . . β n i = 1 √ n λ n X i ξ i − δ λβ i | β . . . ˆ β i . . . β n i . (2.1.69)As an aside note that for bosons ( ξ = 1) this becomes just a λ | n β . . . n λ . . . n β q i = √ n λ | n β . . . ( n λ − . . . n β q i (2.1.70)where | n β . . . n β q i denotes the state with n β i particles in the state β i . It should benoted that the final commutation relation [ a λ , a † µ ] − ξ is still unknown. It can be foundfirstly by noticing that a λ a † µ | α . . . α n } = a λ | µα . . . α n } = δ λµ | α . . . α n } + n X i =1 ξ i δ λα i | µα . . . ˆ α i . . . α n } (2.1.71)4 CHAPTER 2. RANDOM MATRIX THEORIESand likewise that a † µ a λ | α . . . α n } = a † µ n X i =1 ξ i − δ λα i | α . . . ˆ α i . . . α n } = n X i =1 ξ i − δ λα i | µα . . . ˆ α i . . . α n } . (2.1.72)Combining these two equations gives a λ a † µ = δ λµ | α . . . α n } + a † µ a λ | α . . . α n } = [ δ λµ + ξa † µ a λ ] | α . . . α n } (2.1.73)which is the final commutation relation sought. Namely[ a λ , a † µ ] − ξ = δ λµ . (2.1.74)Given (2.1.65), (2.1.66) and (2.1.74) an expression can now be found for an arbitrary k -body operator in terms of creation and annihilation operators, initially by workingin a basis where the operator is in its diagonal form, and finally by expressing theresult in any basis. To begin, following (2.1.58) and assuming that H k is diagonal H i ...i k = ( α i . . . α i k | H k | α i . . . α i k ) (2.1.75)so that for arbitrary m -body states | α i . . . α i m ) and | β i . . . β i m ) the following sequenceof equalities are attained( α i . . .α i m | H k | β i . . . β i m ) = ( α i . . . α i m | k ! m X i = i = ... = i k H i ...i k | β i . . . β i m )= 1 k ! X i = ... = i k Y j = i ,...,i k h α j | β j i ( α i . . . α i k | H i ...i k | β i . . . β i k )= 1 k ! X i = ... = i k Y j = i ,...,i k h α j | β j i ( β i . . . β i k | H k | β i . . . β i k )( α i . . . α i k | β i . . . β i k )= 1 k ! ( α i . . . α i m | β i . . . β i m ) X i = ... = i k ( β i . . . β i k | H k | β i . . . β i k ) (2.1.76).1. CANONICAL RMT 25the sum being over all k -tuples in the state β := | β i . . . β i m ). Tuple Counting Expressing (2.1.76) in terms of the matrix elements of the operator gives( α i . . . α i m | H k | β i . . . β i m ) = 1 k ! ( α i . . . α i n | X k u − tuples H i ...i k T k | β i . . . β i n ) (2.1.77)the sum now running over unique k -tuples k u (any permutation of a given k -tuple isconsidered non-unique) and the tuple-counting operator T k giving the total numberof k -tuples denoted by the labels k = { i . . . i k } in the state upon which it acts. For a1-body operator the number of particles in the state α is in fact given by the numberoperator n α = a † α a α so that T = a † α a α (2.1.78)and similarly the number of pairs consisting of the single-particle states | α i and | β i is n α n β if α = β and n α ( n α − 1) otherwise. These can be combined into the singlecondition T = n α ( n α − δ αβ ) = a † α a † β a β a α (2.1.79)Generalizing immediately for a k -body operator the number of k -tuples are given bythe operation T k = X i ...i k n i ( n i − δ i i )( n i − δ i i − δ i i ) . . . ( n i k − δ i i k − δ i i k − . . . − δ i k − i k ) . (2.1.80)Using the identity a p δ pq = a q δ pq (2.1.81)equation (2.1.80) reduces neatly to T k = a † i . . . a † i k a i k . . . a i (2.1.82)6 CHAPTER 2. RANDOM MATRIX THEORIESso that the second-quantized formulation of a diagonal k -body operator is H = 1 k ! X i ...i k H i ...i k a † i . . . a † i k a i k . . . a i . (2.1.83)To express this in terms of an arbitrary basis observe that in terms of some basis | λ i one can write a † λ i = X a h a | a λ i i a † a . (2.1.84)Taking the complex conjugate gives a λ i = X a h a λ i | a i a a (2.1.85)and substituting these expressions into (2.1.83) leads to the following H = 1 k ! X i ...i k H i ...i k X j h j | i i a † j . . . X j k h j k | i k i a † j k X i k h i k | i k i a i k . . . X i h i | i i a i = 1 k ! X i ...i k X j ...jki ...ik H i ...i k h j . . . j k | i . . . i k ih i . . . i k | i . . . i k i a † j . . . a † j k a i k . . . a i = 1 k ! X j ...jki ...ik H j ...j k ; i ...i k a † j . . . a † j k a i k . . . a i (2.1.86)which is the second-quantized form of a k -body operator in an arbitrary basis.This formula expresses the hamiltonian matrix H of (2.1.1) as a second-quantisedoperator i.e. an operator defined by a sequence of creation and annihilation operators.Implicit in the model are three important parameters. Firstly there is m , the variabledetermining the number of single particles in the system. In other words for any state | µ i one has | µ i = | α . . . α m i (2.1.87)for some set of single-particle states { α , . . . , α m } as seen in (2.1.50). Secondly thereis k ≤ m , the order of the interaction or in other words the “number of bodies”involved in each interaction under the force of the potential. Finally, there is the.1. CANONICAL RMT 27implicit parameter l which determines the number of energy levels available to each ofthe single-particle states in the compound m -body state. Hence l is the size of the setfrom which the single-particle states α , . . . , α m can take their values. As mentionedin section 2.1.3 the trio of values { k, m, l } together with the symmetry conditionsimposed on H form a single instance of a random matrix theory. As an unrestrictedphase space the hamiltonian of (2.1.86) represents the unification of these randommatrix theories. Next it will be shown that canonical RMT coincides with the case k = m with l → ∞ . To paraphrase section 2.1.1 Canonical RMT is the study of random matrices withsome predefined symmetry conditions and a single random variable determiningthe quantity in each cell of the matrix. The unified form of random matrix theorydetermined by the hamiltonian of (2.1.86) however, allows for the possibility thatmore than one p.d.f. determines any given cell of the matrix. For the special casewhere k = m this becomes H = 1 m ! X j ...jmi ...im H j ...j m ; i ...i m a † j . . . a † j m a i m . . . a i (2.1.88)so that for any two m -body states | µ i = | α , . . . , α m i (2.1.89) | ν i = | β , . . . , β m i (2.1.90)the matrix elements of H become H µν = 1 m ! X j ...jmi ...im H j ...j m ; i ...i m h µ | a † j . . . a † j m a i m . . . a i | ν i = H α ...α m ; β ...β m (2.1.91)8 CHAPTER 2. RANDOM MATRIX THEORIESHence the cells of the matrix H for the special case k = m contain only one element– the final line of (2.1.91). There is no summation as there would be for the case k < m . This single element, being defined in terms of a probability density functionand symmetrised by some condition on H will of course give back the correspondingcanonical RMT ensemble symmetrised under the same condition. This tells us that k = m is the point in the phase space { k, m, l } of the unified theory which coincideswith canonical random matrix theory. Moreover, since the random matrices of thecanonical theory are usually assumed to be infinite one takes l → ∞ , giving canonicalRMT as the theory coinciding with the point { m, m, ∞} in the unified phase space...... hapter 3Wigner’s Semi-Circle Law k = m Wigner’s Semi-Circle Law is one of the more iconic and widely known results to comeout of the field of random matrix theory. This is the rule, proven mathematically,which states that the average level density of Hamiltonians from the GUE, GOE andGSE ensembles take the form of a semi-circle [Meh04]. In other words, it says thaton average the p.d.f. of the energy values of these systems is a semi-circle.As shown in the previous section Wigner’s Semi-circle law is in fact a statementabout many-body hamiltonians, specifically those where the order of the interaction k is taken to be equal to the number of particles in the system, m . Assuming that allparticles interact simultaneously under the force of a potential was an assumption madeimplicitly in the canonical form of random matrix theory, because it led to importantsimplifications in a challenging and very technical new field of mathematical endeavor.There was a second rationale as well. Measurements of particular nuclear energy levelsin the 1950s and 1960s [RDRH60, DRHR60, DRHR64, HR51, MHR53] involved nuclearinteractions with a high degree of random mixing. This was caused by high excitationenergies due to the particles being accelerated within a synchrocyclotron. Theinteractions appeared to be so complex that a “black box” approach to studying thenuclei became a practical and reasonable way forward as it involved assuming the bare290 CHAPTER 3. WIGNER’S SEMI-CIRCLE LAW (a) N=10 (b) N=15(c) N=50 (d) N=200 Figure 3.1: Plots of the eigenvalue distributions for N × N matrices sampled fromthe GUE ensemble with N increasing in each simulation. The entries of the matrixare taken to be complex gaussian random variables with variance 1. The symmetrycondition placed on the matrices is H = H † ..1. WIGNER’S LAW FOR K = M H , beyond its symmetry.The most natural assumption was therefore to represent the physical hamiltonian asa hermitian matrix with some distribution (say a gaussian) determining the matrixelements; this was the black box. So apart from the mathematical simplification theinitial approach to studying nuclear resonances, which resulted in the foundationof canonical random matrix theory, was motivated by experiment. The fact that itimplicitly implied k = m did not detract from the possibility that nature did indeedappear to behave this way, statistically speaking, in some complex systems.However, it soon became apparent that many-body potentials taking values of k < m would need to be investigated as well. Indeed since the entire range 0 ≤ k ≤ m couldplausibly model some experiment it became an imperative to find the statistics ofthe energy values for all of these possibilities, as this would be the only way to checkwhether or not the resultant models predicted behaviour which is also measured inactual experiments. This unified approach to random matrix theory was proposed byK. K. Mon and J. B. French [MF75] and the unified ensembles determined by themodified hamiltonian (2.1.86) were called the Embedded Ensembles . Hence, theoreticalphysicists and mathematicians first solved the case k = m and only later turned tothe general case (still unsolved in its entirety) determined by allowing k to take anyvalue within the permitted range 0 ≤ k ≤ m . The next sections will explain howto calculate Wigners semi-circle law in two ways. One method will use a complextechnique called supersymmetry involving the liberal application of anti-commutingvariables and their properties, which will be discussed beforehand. The second waywill use a simple method involving some basic diagrams and combinatorics. Bothmethods will be used to derive the semi-circle law of canonical random matrix theory,i.e. for the special case k = m . An attempt to calculate Wigner’s law for the generalcase k ≤ m will be presented in later sections and will rely on new diagrammaticmethods which are the primary topic of this thesis. There it will be seen that theanalogue of Wigner’s Semi-Circle Law in the unified phase space of embedded RMTis, for many values of k , not even a semi-circle.2 CHAPTER 3. WIGNER’S SEMI-CIRCLE LAW In this section Wigner’s semi-circle law will be shown only for the case k = m with l → ∞ and just one restriction on the symmetry of the potential, namely H = H † .This is the canonical class of RMT known as the GUE ensemble (see section 2.1.2).For the purpose of the proof the quantum states containing m particles are assumedto consist entirely of fermions and these m -body states will be denoted by the greeksymbols | µ i , | ν i , | ρ i , | σ i et cetera. Each of the m particles in these states will beone of l single-particle occupation levels, with m (cid:28) l and the caveat that no twosingle-particle states in the same m -body state can be the same because the statesare fermionic. Hence there is a total of N = (cid:16) lm (cid:17) orthogonal m -body states in thebasis of the system. Notation The single-particle creation and annihilation operators are as before written as a † j and a j respectively with j = 1 , . . . , l . With the intention of simplifying the notationdefine the shorthand expression j = ( j , . . . , j k ) (3.2.1)and similarly for i . This furnishes a suitable abbreviation for the creation andannihilation operators which will now be written as a j = a j k . . . a j . (3.2.2)A useful corollary of this is the equality a † j = a † j . . . a † j k . (3.2.3)Each of the (cid:16) lm (cid:17) states in the basis can be written in the form a † j m . . . a † j | i (3.2.4).2. DIAGRAMS 33with | i denoting the vacuum state and the restriction 1 ≤ j < j < . . . < j m ≤ l since these are m -body states containing only fermions. From this it can be seen thatfor the special case k = m each m -body state can in fact be rewritten as a † j | i (3.2.5)for some particular set j . The k -body potential as derived in section 2.1.3 is writtenas H k = X ≤ j <... Using just (3.2.20) equation (3.2.18) can be written as an expression determined bysumming over only the product of the average of all possible pairs of the v . Specifically,let { σ } denote the set of (2 n − , , . . . , n with σ ( x ) taking the value of the integer paired with x for a given permutation σ fromthe set { σ } . Note that there is no x for which σ ( x ) = x . Then the average of theproduct of v ’s from (3.2.18) can be rewritten in terms of averages of pairs of vv j (1) i (1) v j (2) i (2) . . . v j (2 n ) i (2 n ) = X σ n Y x =1 v j ( x ) i ( x ) v j σ ( x ) i σ ( x ) . (3.2.21)This result is used frequently in statistical mechanics and is sometimes referred to as Wick’s Theorem (even when it doesn’t involve creation and annihilation operators).Hence (3.2.18) becomestr( H nk ) = "X σ n Y x =1 v j ( x ) i ( x ) v j σ ( x ) i σ ( x ) h µ | a † j (1) a i (1) a † j (2) a i (2) . . . a † j (2 n ) a i (2 n ) | µ i . (3.2.22)Of note is that since k = m , operating on a state | µ i with annihilation operator a i will give a i | µ i = δ i µ | i (3.2.23)This, and the fact that the state | µ i occurs on both the right and left hand side of thesequence of creation and annihilation operators of (3.2.22) means that this sequencemust follow some path in state-space, first removing | µ i to give the vacuum state | i then replacing it with some other state | t n i , removing this to give | i , replacing itwith some other state, | t n − i and so on, with the final step being a replacement ofa vacuum state with the initial state | µ i . This process of annihilation and creationoccurs 4 n times, with 2 n annihilation operations and 2 n creation operations. Theresult is the formation a path in state-space beginning and ending with the state | µ i which never removes a state which is not already there. Any terms in the sum of(3.2.22) which do not contain such a sequence will have a value of zero. Explicitly,.2. DIAGRAMS 37the rightmost component of the product given in (3.2.22) either takes the value zeroor can be written in the form h µ | a † j (1) a i (1) a † j (2) a i (2) . . . a † j (2 n ) a i (2 n ) | µ i = h µ | a † µ a t a † t . . . a t n a † t n a µ | µ i = 1 . (3.2.24)Hence h µ | a † j (1) a i (1) a † j (2) a i (2) . . . a † j (2 n ) a i (2 n ) | µ i = Y a δ j ( a ) t a Y b δ i ( b ) t b +1 (3.2.25)where t is identified with µ and t n +1 is identified with t . The above allows (3.2.22)to attain the final form tr( H nk ) = X σ n Y a =1 v t a t a +1 v t σ ( a ) t σ ( a )+1 . (3.2.26)Each component of the product can be expressed as v t m t m +1 v t n t n +1 (3.2.27)for some m, n . For uncorrelated v t m t m +1 and v t n t n +1 the ensemble average will yieldnull, since the gaussian determining the p.d.f. of these elements is an even function.However, for correlated v t m t m +1 and v t n t n +1 (3.2.27) will yield unity if and only if v t m t m +1 = v ∗ t n t n +1 (3.2.28)because the ensemble average will then yield the second moment of v which has beenfixed to take the value 1. By (3.2.13) this is equivalent to saying v t m t m +1 v t n t n +1 = δ t m t n +1 δ t m +1 t n (3.2.29)so that (3.2.26) becomes tr( H nk ) = X σ n Y a =1 δ t a t σ ( a )+1 δ t a +1 t σ ( a ) . (3.2.30)8 CHAPTER 3. WIGNER’S SEMI-CIRCLE LAWIt will be shown in subsequent sections how this can be represented by diagrams.These will reveal that the product of delta functions giving (3.2.30) leads to a sequenceof paths on the labels a = 1 , , . . . , n and each path will be identified with a partitionof the set 1 , , . . . , n . Each partition will in turn be identified with a diagram, andthese diagrams will be counted to give (3.2.30) which can then be inserted back into(3.2.17) to give the moments of the level density for the GUE. Equation (3.2.30), and through it the moments of the level density (3.2.17), arerelated to certain combinatorial diagrams. This will be shown by observing that eachinstance of the sum of (3.2.30) can be mapped to a partition on the set of integers { , , . . . , n } . Using diagrammatic methods for the k = m case of canonical randommatrix theory is not new [Kre72]. It will be seen that the moments of the leveldensity are in fact given by the Catalan Numbers , which occur in numerous othercombinatorial problems outside of the field of random matrix theory. For these –some of which are much older than random matrix theory itself and have their owndistinct literature – diagrams are frequently used. For 2 n = 4 (3.2.30) givestr( H k ) = v t t v t t v t t v t t = X σ Y a =1 δ t a t σ ( a )+1 δ t a +1 t σ ( a ) . (3.2.31)the right hand side of which can be read as the “product of the average of all uniquepairings of v ”. Hencetr( H k ) = v t t v t t v t t v t t = v t t v t t v t t v t t + v t t v t t v t t v t t + v t t v t t v t t v t t = δ t t + δ t t + δ t t δ t t δ t t δ t t (3.2.32) the sequence defined by C n = n +1 (cid:0) nn (cid:1) .2. DIAGRAMS 39 Partitions as Cycles Identifying t with 1, t with 2 and so on, the components of (3.2.32) can be representedas cycles in the following way δ t t ≡ (13)(2)(4) (3.2.33) δ t t ≡ (24)(1)(3) (3.2.34) δ t t δ t t δ t t δ t t ≡ (1234) . (3.2.35)Note that where c i denotes the number of orbits in a given cycle with length i thatfor each cycle of (3.2.33 – 3.2.35) it holds that X i ic i = 4 = 2 n. (3.2.36)Each orbit represents a degree of freedom in the variables t , t , . . . (etc) of (3.2.31).The value of tr( H k ) is given by implicitly summing over all values that each of thesefree variables can take, the result being that each orbit contributes a factor (cid:16) lm (cid:17) tothe final expression. Hence the cycle (13)(2)(4) of (3.2.33) is to be identified with (cid:16) lm (cid:17) and likewise for the cycle (24)(1)(3). The cycle (1234) however, consists only ofa single orbit, so that it is identified with the value (cid:16) lm (cid:17) . Hencetr( H k ) = 2 lm ! + lm ! (3.2.37)so that the fourth moment for the GUE with k = m given by (3.2.17) is β = tr( H m ) N = 2 (cid:16) lm (cid:17) + (cid:16) lm (cid:17)(cid:16) lm (cid:17) = 2 + O (cid:18) N (cid:19) . (3.2.38)For 2 n = 6 (3.2.30) readstr( H k ) = v t t v t t v t t v t t = v t t v t t v t t v t t v t t v t t v t t v t t v t t v t t v t t v t t + v t t v t t v t t v t t v t t v t t + v t t v t t v t t v t t v t t v t t + v t t v t t v t t v t t v t t v t t + v t t v t t v t t v t t v t t v t t + v t t v t t v t t v t t v t t v t t + v t t v t t v t t v t t v t t v t t + v t t v t t v t t v t t v t t v t t + v t t v t t v t t v t t v t t v t t + v t t v t t v t t v t t v t t v t t + v t t v t t v t t v t t v t t v t t + v t t v t t v t t v t t v t t v t t + v t t v t t v t t v t t v t t v t t + v t t v t t v t t v t t v t t v t t . (3.2.39)Again identifying t with 1, t with 2 (etc), the components of (3.2.39) can berepresented as the cycles δ t t δ t t δ t t ≡ (135)(2)(4)(6) δ t t δ t t δ t t δ t t δ t t ≡ (13654)(2) δ t t δ t t δ t t ≡ (13)(46)(2)(5) δ t t δ t t δ t t δ t t δ t t ≡ (14325)(6) δ t t δ t t δ t t δ t t δ t t ≡ (14)(2356) δ t t δ t t δ t t δ t t δ t t ≡ (14632)(5) δ t t δ t t ≡ (15)(24)(3)(6) δ t t δ t t δ t t δ t t δ t t δ t t ≡ (153)(246) δ t t δ t t δ t t δ t t δ t t ≡ (1542)(36).2. DIAGRAMS 41 δ t t δ t t δ t t δ t t δ t t ≡ (16524)(3) δ t t δ t t δ t t δ t t δ t t ≡ (1643)(25) δ t t δ t t δ t t δ t t δ t t ≡ (16352)(4) δ t t δ t t δ t t ≡ (246)(1)(3)(5) δ t t δ t t δ t t δ t t δ t t ≡ (26345)(1) δ t t δ t t ≡ (26)(35)(1)(4) . (3.2.40)This time the cycles obey X i ic i = 6 = 2 n (3.2.41)and since the value of each term is the number of orbits in the cycle times (cid:16) lm (cid:17) onegains tr( H m ) = 5 lm ! + 10 lm ! (3.2.42)so β = tr( H m ) N = 5 + O (cid:18) N (cid:19) . (3.2.43)For 2 n = 8 (3.2.30) readstr( H k ) = v t t v t t v t t v t t v t t v t t v t t v t t . (3.2.44)Now there are (2 n − v ’s, each giving rise to a uniquepartition on the integers 1 , , . . . , 8. This time there are just 14 cycles determiningthe leading behaviour of the 8-th moment as N → ∞ , given by(2468)(1)(3)(5)(7)(1357)(2)(4)(6)(8) (3.2.45)(15)(24)(68)(3)(7)(17)(26)(35)(4)(8)(28)(37)(46)(1)(5)2 CHAPTER 3. WIGNER’S SEMI-CIRCLE LAW(13)(48)(57)(2)(6) (3.2.46)(135)(68)(2)(4)(7)(248)(57)(1)(3)(6)(137)(46)(2)(5)(8)(246)(17)(3)(5)(8)(357)(28)(1)(4)(6)(268)(35)(1)(4)(7)(468)(13)(2)(5)(7)(157)(24)(3)(6)(8) . (3.2.47)Note that the cycles (1357)(2)(4)(6)(8) and (2468)(1)(3)(5)(7) of (3.2.45) are equiv-alent up to a constant mod 9. In other words the orbit (1357) with each elementincremented by 1 mod 9 is identical to the orbit (2468) and similarly the orbits(2)(4)(6)(8) incremented by 1 mod 9 are equal to (3)(5)(7)(1). The same applies tothe four cycles of (3.2.46), each being equivalent to eachother up to an increment of 1mod 9, 2 mod 9 or 3 mod 9. Finally, the eight cycles given by (3.2.47) are equivalentup to an increment of i mod 9 where 1 ≤ i ≤ 7. Assigning each orbit of each cyclethe value (cid:16) lm (cid:17) as before and dividing by the normalisation constant N n +1 gives theeighth moment for the GUE as β = tr( H m ) N = 14 + O (cid:18) N (cid:19) . (3.2.48)The process for calculating the moments could continue indefinitely in this way. Butwhat is the underlying pattern? And in what form can this pattern be codified inorder to find a general expression for all moments of the k = m scenario? In thenext section the cycles introduced above will be represented as diagrams. This willillustrate the possibility of transferring the problem (of calculating the moments)into a “diagram space” and using the new space to make calculations. The methodwill act as a precurser to a much more complex diagrammatic technique which will.2. DIAGRAMS 43be used in subsequent chapters to calculate moments for random matrix ensembleswhere k ≤ m . The first method for representing (3.2.30) will be to simply draw the cycles from theprevious section as segments of 2 n -sided polygons. It will be seen that the leadingorder behaviour of the moments, being the cycles with n + 1 orbits as shown insection 3.2.2, are given by those partitioned polygons where (i) no partitions intersect,(ii) no two isolated vertices neighbour eachother and (iii) any two vertices whichcan be connected by a non-intersecting edge are connected as such. For the 4-thmoment there are just two cycles with three orbits, namely δ t t ≡ (13)(2)(4) and δ t t ≡ (24)(1)(3) which are illustrated in Fig 3.2.123 4 123 4Figure 3.2: Plot of polygons for fourth moments with the left hand side giving anillustration of the cycle (13)(2)(4) on a 2 n = 4 sided polygon and the right hand sideillustrating the only unique rotation of this cycle, which is (24)(1)(3).1234 5 6 1234 5 6Figure 3.3: Illustration of the cycle (135)(2)(4)(6) and it’s unique rotation(246)(1)(3)(5) given by incrementing each element by 1 mod 7.4 CHAPTER 3. WIGNER’S SEMI-CIRCLE LAW1234 5 6 1234 5 6 1234 5 6Figure 3.4: Illustration of the cycles (13)(46)(2)(5) (left), (15)(24)(3)(6) (center) and(26)(35)(1)(4) (right) of (3.2.40).12345 6 7 8 12345 6 7 8Figure 3.5: The cycles (2468)(1)(3)(5)(7) (left) and (1357)(2)(4)(5)(8)..2. DIAGRAMS 4512345 6 7 8 12345 6 7 812345 6 7 8 12345 6 7 8Figure 3.6: The four equivalent cycles (up to addition modulo 9) given by(24)(15)(68)(3)(7) (top left), (17)(26)(35)(4)(8) (top right), (28)(37)(46)(1)(5) (bottomleft) and (13)(48)(57)(2)(6) (bottom right).6 CHAPTER 3. WIGNER’S SEMI-CIRCLE LAW12345 6 7 8 (a) (135)(68)(2)(4)(7) (b) (248)(57)(1)(3)(6) (c) (137)(46)(2)(5)(8) (d) (246)(17)(3)(5)(8) (e) (357)(28)(1)(4)(6) (f) (268)(35)(1)(4)(7) (g) (468)(13)(2)(5)(7) (h) (157)(24)(3)(6)(8) Figure 3.7: Illustration of the eight cycles of equation (3.2.47) each with the sameshape, but rigidly rotated about the origin..2. DIAGRAMS 47For the sixth moment the contributing cycles from (3.2.40) are illustrated in Figs3.3 and 3.4. Finally the 14 contributing cycles for the 8-th moment are illustratedin Figs 3.5 – 3.7. These illustrations prompt the following conclusion; the 2 n -thmoment of the level density of the GUE for the canonical case k = m is equal to thenumber of was of correctly partitioning a 2 n -sided poygon, where the conditions onthe partitions are (i) no partitions intersect, (ii) no two isolated vertices neighboureachother and (iii) any two vertices which can be connected by a non-intersectingedge are connected as such. It turns out this is an old problem and has been studiedbefore by the mathematician Germain Kreweras in the early 1970’s [Kre72]. In thenext section the problem of counting the number of correct partitians of a 2 n -sidedpolygon will be translated into another old problem; Dyck Words. In this way it willbe shown that the 2 n -th moment of the GUE with k = m is given by the sequence ofCatalan Numbers 1 , , , , , , , , , , . . . defined by β n = 1 n + 1 nn ! . (3.2.49) Each instance of the sum (3.2.26) gives a cycle and each cycle can be associated withthe partitioning of a 2 n -sided polygon. These in turn can be translated into DyckWords. Equations (3.2.33, 3.2.34) give the two cycles of the fourth moment as(13)(2)(4)(24)(1)(3) (3.2.50)These are translated as follows(i) Write the sequence 123 .. n in order without brackets(ii) A number with a left bracket in the partition notation is also given a left bracket(iii) A number with a right bracket in the partition notation is also given a rightbracket8 CHAPTER 3. WIGNER’S SEMI-CIRCLE LAW(iv) Remove all numbers, leaving only sequences of opening and closing brackets(v) Replace each opening bracket with a X and each closing bracket with a Y .The result is a Dyck Word. As an example take the cycle (13)(2)(4). For this 2 n = 4cycle the ordered sequence 1234 firstly becomes (1(2)3)(4) because the partition(13)(2)(4) has a left bracket for the integers 1 , , n + 2 = 6(13)(2)(4) ≡ XXY Y XY (3.2.51)by replacing each ( with X and each ) with Y . Similarly the cycle (24)(1)(3) becomes(1)(2(3)4) becomes ()(()) giving the 6 letter Dyck word(24)(1)(3) ≡ XY XXY Y. (3.2.52)It should be clear that this translation can be performed for every cycle (3.2.26)determining the moments of the level density. However the converse is not true. Notevery Dyck word of length 2 n refers to a permitted cycle of length 2 n + 2. The Dyckwords of length 6 for example are given by XXXY Y Y, XXY XY Y, XY XY XY, XXY Y XY, XY XXY Y (3.2.53)whereas the fourth moment of the level density results in the cycles (13)(2)(4) and(24)(1)(3) (see 3.2.33, 3.2.34) corresponding only to the last two Dyck words of (3.2.53).Hence the amount of information contained in a permitted partition of 2 n elementsis not equal to the information represented by a 2 n + 2 Dyck word. This can beappreciated by observing that the cycles (3.2.33– 3.2.34), (3.2.40) and (3.2.45–3.2.47)with polygon representations given in their respective diagrams Figs 3.2 – 3.7 are allcompletely determined by the cycles containing all the odd (or equivalently even)integers of the sequence. In other words, given the restrictions of section (3.2.3).2. DIAGRAMS 49each diagram can be reproduced knowing only the cycles involving odd integers, orconversely knowing only the cycles with even integers. For the fourth moment therelevant cycles (3.2.33– 3.2.34) become(13)(2)(4) ≡ (13)(24)(1)(3) ≡ (1)(3) . (3.2.54)In this way each polygon for the 2 n -th moment is associated to a partition of n oddnumbers as illustrated in (3.2.54). Because these sequences consist of odd numbers ithelps to make the identification 1 → , → , → , . . . , n − → n so that (3.2.54)ultimately is mapped to (13)(2)(4) ≡ (13) ≡ (12)(24)(1)(3) ≡ (1)(3) ≡ (1)(2) . (3.2.55)where the information given by the partitions on the l.h.s of the relation is equivalentto the information contained on the r.h.s and both can be used to reproduce acontributing partition for a 2 n -sided polygon. The cycles for the sixth moment cansimilarly be rewritten as (135)(2)(4)(6) ≡ (135) ≡ (123)(13)(46)(2)(5) ≡ (13)(5) ≡ (12)(3)(15)(24)(3)(6) ≡ (15)(3) ≡ (13)(2)(246)(1)(3)(5) ≡ (1)(3)(5) ≡ (1)(2)(3)(26)(35)(1)(4) ≡ (35)(1) ≡ (23)(1) . (3.2.56)It can be seen that each of the permitted cycles contributing to the leading orderterm of the 2 n -th moment is equivalently represented as a non-crossing partition of n integers, so that the number of cycles which give the value of the 2 n -th moment isequal to the number of non-crossing partitions of the set { , , . . . , n } . This impliesthat summation is identical to the summation of Dyck words of length 2 n , not 2 n + 2.0 CHAPTER 3. WIGNER’S SEMI-CIRCLE LAWThis is a very famous, beautiful and old result which has long been solved. Asillustrated above, it has numerous forms including the enumeration of non-crossingpartitions of { , , . . . , n } , as well as the partitioning of polygons in various ways. Catalan Numbers Every Dyck word of length 2 n can be represented as monotonic paths drawn in a n × n grid. The number of Dyck words of length 2 n is given by the n ’th Catalan (a) XYXYXY (b) XXYYXY (c) XYXXYY(d) XXYXYY (e) XXXYYY Figure 3.8: Plot of the Dyck Paths.number C n = 1 n + 1 nn ! (3.2.57)which is equivalent to saying that the level density of (4.2.5) is a semi-circle, sincethese are the moments of a semi-circular probability density function. In other words2 πr Z r − r x n √ r − x dx = C n . (3.2.58)We will briefly look at one way of showing that the number of Dyck words of length2 n is indeed given by 3.2.57. There are many cute proofs to this and the one presented.2. DIAGRAMS 51next is to illustrate the fact that a diagrammatic method can swiftly yield the semi-circle law. Proof The proof begins with some monotonic sequence involving n vertical steps and n horizontal steps which do indeed cross the horizontal line, and therefore do notconstitute an acceptable Dyck word. Identifying the first node lying above thediagonal, and drawing a line through this point which runs parrallel to the diagonalgives the picture illustrated in fig 3.9a where all steps taken after this point arecoloured in red. Defining the number of horizontal steps taken up to this point as k ,it follows that the number of vertical steps taken is k + 1, since there is one arrowpeaking up above the diagonal. Now reflect the remaining n − k horizontal steps, (a) A non-Dyck word on a n × n grid. (b) Monotonic path on ( n − × ( n + 1) grid.Figure 3.9: Illustration of the transformation showing that every non-Dyck word ona n × n grid can be mapped to a monotonic path on a ( n − × ( n + 1) grid.and the remaining n − k − n − k horizontal steps into vertical steps and n − k − 12 CHAPTER 3. WIGNER’S SEMI-CIRCLE LAWvertical steps into horizontal steps. Hence the resulting picture is a monotonic path ona ( n − k − k ) × ( n − k + k + 1) grid. In other words, any monotonic path from theinitial n × n grid which is not a Dyck word is a monotonic path on a ( n − × ( n + 1)grid. The number of Dyck words of length 2 n is then given by nn ! − nn + 1 ! = 1 n + 1 nn ! (3.2.59)since we first find all ways of making a monotonic path from n vertical steps and n horizontal steps, and then remove all monotonic paths making ( n − 1) horizontalsteps and ( n + 1) vertical steps.With relative simplicity it has been shown how to calculate a random matrixtheory result for the k = m case. But what about when k < m ? In later chapters itwill be shown how diagrams can once again be utilised in calculations of the moments.Whereas for k < m these diagrams will appear and function differently to what hasbeen shown above, for k = m they become equivalent to the representation of cycleson polygons as shown in section 3.2.3. In this section it will be seen how an existing technique involving grassmannianvariables and an innovation by [BRW01a] can be used to show that the level densityof the embedded (a.k.a unified) GUE ensemble is a semi-circle, as may be expected.However, this result will only cover the domain m < k with l → ∞ . In the contextof unified RMT the result covers the volume of the phase space { k > m , m, ∞} . Inthe next chapter a new method will be introduced to calculate statistics of the unifiedRMT phase space for values of 2 k < m as well. It will be seen that in this portionof the volume of the phase space the analogue of Wigner’s semi-circle law is not asemi-circle!.3. SUPERSYMMETRY 53 The proof of Wigner’s semi-circle law for k -body potentials begins using the samesyperanalysis as used for the m -body case. A good account of this can be foundin [Haa10, AMPZJ94]. One begins with definitions which describe the form andbehaviour of anti-commutative variables, which are non-numerical “ things ” with thefollowing algebraic rules attached ξ p ξ q = − ξ q ξ p , p, q = 1 . . . d (3.3.1)which implies, among other things, that ξ p = 0. Defining ξ ∗ p := ( ξ p ) ∗ to be thecomplex conjugate of ξ p , where ξ ∗ p is another grassmannian variable which thereforemust also obey (3.3.1), and additionally enforcing the rules( ξ p ξ q ξ r ) ∗ = ξ ∗ p ξ ∗ q ξ ∗ r (3.3.2)and ξ ∗∗ p = − ξ p (3.3.3)it follows that ( ξ ∗ p ξ p ) ∗ = ξ ∗∗ p ξ ∗ p = ξ ∗ p ξ p . In other words, what has been called “complexconjugation” leaves the grassmannian ξ ∗ p ξ p unchanged. In this sense it can be thoughtof similarly to the length of a standard complex variable. Finding a complex analogyfor every grassmanian property, however, is not the goal. They are ultimately verydifferent, distinct objects. Another property of grassmannians which illustrates thiswell is that they do not have an inverse. Namely, assuming that ξ − p exists gives ξ − p ξ p ξ − p = − ξ − p ( ξ − p ξ p ) = − ξ − p and likewise ξ − p ξ p ξ − p = ( ξ − p ξ p ) ξ − p = ξ − p so that ξ − p = − ξ − p . Namely, it is not possible to abide by the original assumptions, (3.3.1),(3.3.2) and (3.3.3) without also contradicting the existence of an inverse. Analogouslyto the way in which a real or complex C ∞ function can be expressed in the form of aTaylor series f ( x ) = a + bx + cx + . . . any function of purely grassmannian variables4 CHAPTER 3. WIGNER’S SEMI-CIRCLE LAWcan be expressed in the form F ( ξ , . . . , ξ N ) = X m i =0 , f ( m , . . . , m N ) ξ m . . . ξ m N N (3.3.4)since ξ np = 0 for any n ≥ 2. Having defined the form of grassmanian functions it isnow possible to describe a formal meaning for differentiation and integration. Thatis, since grassmannian objects are different to numerical objects it is necessary toredefine what it means to differentiate and integrate. Moreover, there is no constraintto make these definitions in any way analogous to the numerical case, but rather thedefinitions are only required to be commensurate with (3.3.1), (3.3.2) and (3.3.3).The definitions are Z dξ i ξ = 1 (3.3.5)and Z dξ i = 0 (3.3.6)where the differential dξ i obeys the same commutation relations as a normal grass-mannian variable. Grassmannian differentiation is defined analogously to real dif-ferentiation, namely ∂/∂ξ i ( aξ i ) = a for any commuting variable a . Moreover, theoperator ∂/∂ξ i obeys the same commutation relations as grassmannian variables sothat ∂∂ξ i ∂∂ξ j = − ∂∂ξ j ∂∂ξ i (3.3.7)which yields ∂ /∂ξ i = 0 and ∂/∂ξ i ( ξ j ) = − ( ξ j ) ∂/∂ξ i for i = j . It follows immediatelythat ∂∂ξ N . . . ∂∂ξ F = f (1 , , . . . , 1) (3.3.8)and likewise Z dξ n . . . dξ F = f (1 , , . . . , . (3.3.9)That is, added to the already remarkable properties of anti-commuting variables isthe fact that the operation of differentiation is identical to that of integration..3. SUPERSYMMETRY 55 From this comes D = Z ...N Y j dξ ∗ j dξ j exp ( − ...N X ik ξ ∗ i A ik ξ k )= Z ...N Y j dξ ∗ j dξ j exp ( − ξ ∗ Aξ )= Z ...N Y j dξ ∗ j dξ j ( − N N ! ( ξ ∗ Aξ ) N = Z ...N Y j dξ j dξ ∗ j N ! ( ξ ∗ Aξ ) N (3.3.10)since by (3.3.6) the N ’th term is the only one contributing a non-zero term to theintegral. There are ( N !) terms in the product of the sum. N ! for the number ofways a non-zero term can be chosen, times N ! for the number of different orderingsfor such a term. Hence D = Z ...N Y j dξ ∗ j dξ j N ! X ij ξ ∗ i ξ j ξ ∗ i ξ j . . . ξ ∗ i N ξ j N A i j A i j . . . A i N j N (3.3.11)where i and j are permutations on the indices. Since each permutation on i givesthe same contribution it follows that D = Z ...N Y j dξ ∗ j dξ j X P ξ ∗ i ξ P ξ ∗ i ξ P . . . ξ ∗ i n ξ P n A i P A i P . . . A i N P N = X P ( − P A i P A i P . . . A i N P N = det A. (3.3.12)So using grassmannian variables the determinant of a matrix can be expressed asthe integral of an exponential function of anti-commuting variables. This will prove6 CHAPTER 3. WIGNER’S SEMI-CIRCLE LAWimportant later on. Another important is the observation that Z π N N Y k =1 d z k ! exp − X ij z ∗ i A ij z j = Z π N N Y k =1 d Q k ! exp − X k λ k Q ∗ k Q k ! (3.3.13)where d z = dRe { z } dIm { z } and it has been assumed that A is hermitian, andmaking the variable transformation Q k = ( U z ) k for A = U † Λ U . The determinant ofthe Jacobian relating the differentials d [ Z ] and d [ Q ] is subsequently unity. Hence Z π N N Y k =1 d z k ! exp − X ij z ∗ i A ij z j = 1 Q k λ k = 1det A . (3.3.14) It will now be shown how the Green’s function can be expressed in terms of agenerating function. The grassmannian variables and the resulting expressions fordeterminants found in the preceding section will then be used to express the generatingfunctions in terms of gaussian integrals. This process of changing the form of thedeterminant into a gaussian integral using functions of grassmannian and complexvariables is called the method of supersymmetry . The level density of the system is ρ ( E ) = π Im G ( E − ) where the Green’s function G ( E ) := N T r (cid:16) E − H (cid:17) . Introducingthe generating function Z ( E, j ) := det ( E − H )det ( E − H − j ) = det i ( E − H )det i ( E − H − j ) (3.3.15)and observing that ∂∂j " E − H − j ) j =0 = ∂∂j exp [ − T r ( ln ( E − H − j ))] j =0 = ∂∂j exp h − T r ( ln ( E − H )) + jT r ( E − H ) − + O ( j ) i j =0 .3. SUPERSYMMETRY 57= 1det ( E − H ) T r (cid:20) E − H (cid:21) (3.3.16)gives the Green’s function as G ( E ) = 1 N ∂∂j Z ( E, j ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) j =0 . (3.3.17)Given that ρ ( E , E ) = ρ ( E ) ρ ( E ) it is now possible to write the two point generatingfunction as Z ( E − , E +2 , E − , E +4 ) = det( E − − H )det( E +4 − H )det( E − − H )det( E +2 − H )= ( − N det[ i ( E − − H )]det[ i ( E +4 − H )]det[ i ( E − − H )]det[ − i ( E +2 − H )] (3.3.18)where using the notation of (3.3.15) gives E − = ( E − j ) − i(cid:15) , E +2 = ( E − k ) + i(cid:15) , E − = E − i(cid:15) and E +4 = E + i(cid:15) , implicitly with (cid:15) , → G ( E − ) G ( E +2 ) = 1 N ∂ Z ( E − , E +2 , E − , E +4 ) ∂E ∂E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E = E ,E = E (3.3.19)Plugging (3.3.12) and (3.3.14) into the above immediately yields Z = (cid:18) − π (cid:19) N Z d [ z , η ∗ , η ] d [ z , η ∗ , η ] × exp[ i ( z † ( H − E − ) z + η † ( H − E − ) η )] × exp[ i ( − z † ( H − E +2 ) z + η † ( H − E +4 ) η )] (3.3.20)8 CHAPTER 3. WIGNER’S SEMI-CIRCLE LAWwhere z , z are N -component complex variables and η , η are N -component grass-mannians. Now for brevity defineΦ = z z η η and L = − (3.3.21)so that d Φ ∗ d Φ = (cid:16) π (cid:17) N d [ z , η ∗ , η ] d [ z , η ∗ , η ], the matrix L is 4 N × N with L α := L αα and the generating function becomes Z = ( − N Z d Φ ∗ d Φ exp i X α Φ † α L α [ H − E α ]Φ α ! . (3.3.22)Hence, for the ensemble average the aim is to calculate W := exp( i X Φ α L α H Φ α )= Z d [ v ] exp i X Φ α L α H Φ α − X ji v ji v ij v o = Z d [ v ] exp − v o X ji h | v ji | − iv o X Φ † αµ L α h µ | a † j a i | ν i Φ αν v ji i(cid:17) . (3.3.23)Making the change of variables v ji → v ji − iv o P Φ † αµ L α h µ | a † i a j | ν i Φ αν (notice thechange in order of the creation and annihilation operators in the second term) andintegrating over v gives W asexp − v o X Φ † αµ Φ αν Φ † βρ Φ βσ L α L β h µ | a † i a j | ν ih ρ | a † j a i | σ i ! . (3.3.24)Using the method of expansion in the eigenvalues detailed in [BRW01a] one has A µνρσ := h µ | V | σ ih ρ | V | ν i = v o N P sa Λ s C saµσ C saρν . Substituting this into (3.3.24) andtaking caution with the anti-commutative properties of the grassmannian components.3. SUPERSYMMETRY 59of Φ grants the following equality W = exp − N X αβsa L α L β Q αβ Q βα (3.3.25)where L = − − − (3.3.26)and Q αβ = P µσ Φ † αµ ( v o λ s C saµσ )Φ βσ with λ s := √ Λ s . By writing out the 16 components of (3.3.25) explicitly it can be rewritten as W = exp − N X sa Str ˜ Q ! (3.3.27)with ˜ Q = L α Q αβ and the supertrace Str defined as Str F = Tr A − Tr D (3.3.28)where F = A BC D is a supermatrix, the matrices A and D consisting only ofcommuting variables (such as complex numbers), and the matrices B and C consistingonly of grassmanian variables. It is now possible to introduce an additional dummyvariable by noting that up to a normalization factorexp (cid:18) − N Str ˜ Q (cid:19) = Z exp (cid:18) − N Str σ + i Str σ ˜ Q (cid:19) d [ σ ] . (3.3.29)0 CHAPTER 3. WIGNER’S SEMI-CIRCLE LAWThen writing out the expression component-wise and rearranging it can be seen that Str σ ˜ Q = Φ † L ( sσ ) T ( v o λ s C sa I )Φ (3.3.30)where s = diag ( I , − I ) = s T , so that up to a normalisation factor the ensembleaverage of the generating function Z is given by Z exp (cid:18) − N Str σ + Φ † [ iL ( sσ ) T ( v o λ s C sa I ) − iLE ]Φ (cid:19) (3.3.31)to which the identity R d Φ † d Φ exp( − Φ † F Φ) = ( Sdet F ) − can now be applied, withthe superdeterminant Sdet being defined by Sdet = det( D − CA − B )det A . (3.3.32)Noting that this will be a 4 N × N superdeterminant (denoted in bold to distinguishit from the 4 × Z ( Sdet [ L ( sσ ) T ( v o λ s C sa ) − LE ]) − exp (cid:18) − N σ (cid:19) . (3.3.33)Using ln Sdet F = Str ln F and factoring out L in addition to noticing thatexp( Str ln L ) = exp Str iπ = exp( iN π ) (3.3.34)and ignoring the ± Z without normal-ization can be written as Z exp (cid:18) − N Str σ + Str ln (cid:16) E − ( sσ ) T ( v o λ s C sa ) (cid:17)(cid:19) . (3.3.35).3. SUPERSYMMETRY 61Finally, dropping the transpose due to the diagonal nature of the term v o λ s C sa in the4 × Str ln = tr Str lnyields Z ∝ Z exp (cid:18) − N [ 12 Str σ + 1 N tr Str ln( σsv o λ s C sa − E )] (cid:19) . (3.3.36) A saddle-point approximation will now be applied over the σ variables. Minimisingthe argument of the exponential yields σ sa = − N tr v o λ s C sa − Eχ − E ! (3.3.37)where χ = P sa v o σ sa λ s C sa . Multiplying across by v o λ s C saµν and summing over sa aswell as expressing the trace in component form gives χ µν = v o N X saρσ ( E − χ ) − ρσ Λ s C saρσ C saµν = v o X ρσ ( E − χ ) − ρσ h σ | V | ν ih µ | V | ρ i . (3.3.38)Assuming that χ is of the form r ˆ I for some constant r and given that all theeigenvectors C sa are traceless save for C = δ µν gives, self-consistent with the priorassumptions on χ , that χ = v o Λ E − χ (3.3.39)where it is shown in [BRW01a] thatΛ ( k ) = mk ! l − m + kk ! . (3.3.40)Finally, solving this for χ yields χ = v o λ τ ± (3.3.41)2 CHAPTER 3. WIGNER’S SEMI-CIRCLE LAWwith τ ± ( E ) = E v o λ ± i r − (cid:16) E v o λ (cid:17) . Plugging this into (3.3.37) implies σ = τ and σ sa = 0 for all s ≥ 1. Construct the supermatrix σ as σ = diag( τ + ( E ) , τ − ( E ) , τ + ( E ) , τ − ( E )) (3.3.42)so that the supertrace for σ doesn’t yield zero and the advanced and retardedcomponents of the generating function are matched. Then by setting E = E in (3.3.36) the two-point generating function reduces to a level density generatingfunction Z ( E , E ) proportional to Z exp − N " τ ( E )2 − τ ( E )2 exp (cid:16) − N h ln( λ v o τ + ( E ) − E ) − ln( λ v o τ + ( E ) − E ) i(cid:17) . (3.3.43)Taking the derivative with respect to E as per (3.3.19) and selecting the imaginarycomponent of the resulting Green’s function gives the embedded Wigner’s semi-circlelaw for the k -body EGUE aslim N →∞ ρ ( E ) ∝ s ( v o λ ) − (cid:18) E (cid:19) . (3.3.44)Due to a technicality involving the identity A µνρσ = v o N P sa Λ s C saµσ C saρν the saddle-pointapproximation converges to this result only for 2 k > m . In the next chapter a simplermethod will be introduced, which avoids the grassmannian gymnastics illustratedabove and can be used to find the moments of embedded RMT systems for all k ...... hapter 4Many-Body RMT Embedded RMT Many-body random matrix theory is the application of random matrix concepts tothe study of the k -body potentials (see section 2.1.3). This is a superset of randommatrix theory with hamiltonian H k = X ≤ j ≤ ... ≤ jk ≤ l ≤ i ≤ ... ≤ ik ≤ l v j ...j k ; i ...i k a † j . . . a † j k a i k . . . a i (4.1.1)and coefficients v j ...j k ; i ...i k taking the value of a random variable. By determining thecoefficients from a probability density function the static potential (4.1.1) generatesan ensemble of matrices which can be studied statistically, just as in the case ofcanonical RMT. As shown in 2.1.4 the canonical form of RMT coincides with justa single point k = m in the phase space of the unified random matrix theory, alsoknown as embedded RMT . One of the purposes of unification is to study all of theserandom matrix theories as a group, rather than creating new methods and theoremsfor each particular case. In this sense embedded RMT treats all randomised k -bodysystems equally, allowing k to take any value in the range 1 ≤ k ≤ m . While there isa vast literature for the statistics and theorems of canonical random matrix ensembles( k = m ) including Wigner’s semi-circle law described in chapter 3, little was known634 CHAPTER 4. MANY-BODY RMTabout the properties of unified RMT ensembles (1 ≤ k ≤ m ) including the analogueof Wigner’s Law for these embedded systems until relatively recently. In fact it turnsout that for some values of k < m the analogue of Wigner’s semi-circle law is noteven semi-circular.While some progress has previously been made towards calculating various statis-tics for embedded ensembles [MF75, BRW01a], the moments for the level densityfor these systems when k ≤ m were unknown beyond the fourth moment until themethod invented by the author was introduced. The current chapter will introducethis new method, called the method of particle diagrams , and illustrate how it can beused to calculate the fourth, sixth and eighth moments of the level density for theeGUE ensemble. Recall from see section 2.1.3 that the eGUE ensemble represents thehamiltonian of non time-reversal invariant quantum systems of m particles interactingunder the force of a k -body potential. Interestingly, the moments calculated inthis way point to a convergence in the statistical behaviours across a wide range ofmany-body hamiltonians of a similar form, albeit used independently to study thestatistics of quantum spin chains, graphs and hypergraphs [HMH05, ES14, KLW14].In the context of embedded RMT the three symmetry groups introduced by Dysonfor canonical RMT become the embedded GUE (eGUE), embedded GOE (eGOE)and embedded GSE (eGSE). For each of these classes the hamiltonian (4.1.1) obeysthe same symmetry rules as in the canonical case even though the resulting statisticscan be completely distinct due to the changing structure of H itself. The onlynon-deterministic components of the potential are the random variables v j ...j k ; i ...i k .With the foresight that their symmetry properties will be required in calculationslater, the next sections will investigate the second moments of v ji for the eGUE,eGOE and eGSE. This will also illustrate how the restrictions of symmetry fromeach of Dyson’s three groups affect the symmetry properties of an embedded k -bodyhamiltonian..1. RMT VS. EMBEDDED RMT 65 It was shown in section 2.1.3 that the hermitian potential describing the energyof a system of m fermions interacting under the influence of a k -body potential is H k = P ≤ j ≤ ... ≤ jk ≤ l ≤ i ≤ ... ≤ ik ≤ l v j ...j k ; i ...i k a † j . . . a † j k a i k . . . a i . The second moments of the randomvariables v j ...j k ; i ...i k = v ji are defined as v ji v j i . For the method proposed in thisthesis for calculating the moments of the level density, the socalled second momentsform the first of several essential ingredients. The normalised 2 n -th moments of thelevel density are given by β n = N tr( H nk ) (cid:16) N tr( H k ) (cid:17) n (4.1.2)for which the numerator can be re-written astr( H nk ) = v j (1) i (1) v j (2) i (2) . . . v j (2 n ) i (2 n ) h µ | a † j (1) a i (1) a † j (2) a i (2) . . . a † j (2 n ) a i (2 n ) | µ i . (4.1.3)Even before seeing the forthcoming calculations for the moments of the eGUE it canbe appreciated that in order to calculate (4.1.3) it is necessary to also calculate theensemble average v j (1) i (1) v j (2) i (2) . . . v j (2 n ) i (2 n ) . (4.1.4)It has been noted before in section 3.2.1 that the second moment v ji v j i = 0 foruncorrelated v ji , v j i whereas for correlated v ji , v j i with v ji = v ∗ j i the secondmoment is unity (by virtue of normalisation). This and the additional knowledgethat in general for a gaussian random variable v n = s π Z v n e − v / dv = (2 n − v j (1) i (1) v j (2) i (2) . . . v j (2 n ) i (2 n ) = X σ n Y x =1 v j ( x ) i ( x ) v j σ ( x ) i σ ( x ) . (4.1.6)6 CHAPTER 4. MANY-BODY RMTThis result is used frequently in statistical mechanics and is sometimes referred to as Wick’s Theorem (even when it doesn’t involve creation and annihilation operators).Since the first step for calculating the higher moments is to calculate these secondmoments the next sections will explain how to calculate the quantity v ji v j i for theeGUE, eGOE and eGSE. β = 2 ) It was shown in section 2.1.2 that the GUE ensemble refers to the set of randommatrices H k obeying hermitian symmetry H k = H † k (4.1.7)which are the class of hamiltonians referring to time-reversal invariant fermionicquantum systems. Since by the Pauli exclusion principle many-body fermionicquantum states cannot contain repeated single-particle states (4.1.1) can be rewrittenas the restricted sum H k = X ≤ j <... RMT 67space is N = (cid:16) lm (cid:17) . A matrix symmetrised with the condition (4.1.7) obeys h µ | H k | ν i = h ν | H k | µ i ∗ (4.1.10)for all quantum states | µ i and | ν i . This implies that X ji v ji h µ | a † j a i | ν i = X ji v ∗ ji h ν | a † j a i | µ i (4.1.11)and matching coefficients for each term in the sum gives v ∗ ji = v ij . (4.1.12)The coefficients v ji not related by (4.1.12) are then uncorrelated i.i.d. gaussian randomvariables with mean zero and variance 1. This means that for uncorrelated v ji and v j i v ji v j i = 12 π Z v ji v j i exp (cid:18) − v ji (cid:19) exp (cid:18) − v j i (cid:19) dv ji dv j i = 0 (4.1.13)since the gaussians are even functions, making the total argument of the integral oddwith respect to the integration variables. Likewise if v ji = v j i then v ji and v j i arecomplex and their product yields two even functions of opposite sign, again giving azero average. However in the final scenario with v ji = v ∗ j i (4.1.14)the second moment becomes v ji = 1 √ π Z v ji exp (cid:18) − v ji (cid:19) dv ji = (4.1.15)from which it can be directly concluded that v ji v j i = δ ji δ ij . (4.1.16)8 CHAPTER 4. MANY-BODY RMTThis is the same identity calculated in section 3.2 for the case k = m . It will be animportant identity in subsequent sections for calculations made in the space of unifiedrandom matrix theory. The reason it will play such a fundamental role in calculationsis based on the idea behind Wick’s Theorem; each term in certain summations ofproducts will be expressed in terms of summations of products of pairs, for which(4.1.16) will become an important aid. Analogous identities for the eGOE and eGSEwill be derived next. β = 1 ) The embedded gaussian orthogonal ensemble (eGOE) refers to the set of potentials ofbosonic time-reversal invariant quantum systems, as discussed in section 2.1.2. Thehamiltonian H k = X ≤ j ≤ ... ≤ j k ≤ l ≤ i ≤ ... ≤ i k ≤ l v j ... j k i ... i k a † j . . . a † j k a i k . . . a i (4.1.17)sums over m -body bosonic states which are permitted to have repeated single-particleindices. The number of bosonic m -particle states containing repeats of z ≤ m uniquesingle-particle states is (cid:16) lz (cid:17)(cid:16) m − z − (cid:17) . The dimension of the state space is then just thesum of all these N = m X z =0 lz ! m − z − ! = l + m − m ! . (4.1.18)As explained in section 2.1.2 the symmetry properties imposed on H k imply that itconsists only of real values, while also obeying the hermitian symmetry satisfied bymatrices in the eGUE (see previous section). Hence for β = 1 X v ji h µ | a † j a i | ν i = X (cid:16) v ji h ν | a † j a i | µ i (cid:17) ∗ = X v ∗ ji h µ | a † i a j | ν i = X v ji h ν | a † j a i | µ i . (4.1.19)Matching coefficients gives v ji = v ∗ ji = v ij (4.1.20).1. RMT VS. EMBEDDED RMT 69so that with v ji = v j i or v ji = v ∗ j i the value of v ji v j i will be 1 while for uncorrelated v ji , v j i the value of v ji v j i will be zero. There will be no cancellations in the integralsince these are real random variables, so the product of v ji with itself yields only asingle term. There are no cross terms as there would be if v ji were complex valued asin the case of the eGUE. Hence for the eGOE v ji v j i = δ ji δ ij + δ jj δ ii . (4.1.21)Analogously to the eGUE, this identity is important for calculations of the momentsof the level density for H k taken from the eGOE because the resulting expressionsinvolve summations of products of v ji which can be expressed as summations ofproducts of pairs of v ji , for which (4.1.21) then plays a central role. ( β = 4) For β = 4 the hamiltonian must satisfy both the hermitian symmetry of (4.1.7) andthe symmetry implied by the fact that the potential takes the form of (2.1.33). Tobegin, define the pairs of entries in the upper(lower) diagonal plane of (2.1.35) relatedby complex conjugation as blue pairs, and those related by complex conjugation times − red pairs. The additional symmetry enforced by (2.1.33) implies that for everyblue matrix element there is a map to the element with row number and columnnumber a distance +1 away, and related to it by complex conjugation, while for everyred element there is a map to the element with row number a distance +1 awayand column number a distance − − 1. Assuming that it is possible to use the same map σ to increase the columnnumber for each block of the matrix it follows, in addition to (4.1.7), that h µ | H k | ν i = h σ µ | H k | σ ν i ∗ (4.1.22)for a blue matrix cell ( µ, ν ) and h µ | H k | σ ν i = −h σ µ | H k | ν i ∗ (4.1.23)0 CHAPTER 4. MANY-BODY RMTfor a red matrix cell ( µ , ν ) where σ µ is the action of the map σ on the m -body state | µ i . Hence a blue cell corresponds to those given by (2.1.37) with h µ | H k | ν i = h T µ | H k | T ν i ∗ (4.1.24)for some µ, ν while a red cell corresponding to (2.1.38) can be written as h µ | H k | T ν i = −h T µ | H k | ν i ∗ (4.1.25)for some ρ , ν . Setting µ = T ρ and comparing equations (4.1.22), (4.1.23) with(4.1.24) and (4.1.25) gives T | µ i = σ µ . (4.1.26)Equations (4.1.22) and (4.1.23) imply that the random variables of the potential aresubject to the condition v j k i k = v ∗ σ j k σ i k (4.1.27)when the condition is yielded by applying (4.1.22) and are subject to the condition v j k σ i k = − v ∗ σ j k i k (4.1.28)when the condition is yielded by applying (4.1.23). Hence for the eGSE the randomvariable v j k i k does not satisfy the same condition for all j k , i k . This distinctive featureof the eGSE indicates that it is necessary to count the number of sets of { j k , i k } for which the two conditions (4.1.27) and (4.1.28) overlap in order to calculate themoments of the level density of the eGSE. The overlapping of these conditions givesthe “ symplectic zeros ” which are the values of the random variable { j k , i k } for which v j k i k = − v j k i k implying v j k i k = 0. In other words, it may be that the sets of k single-particle states j , i are present in m -body states defining a blue cell as well asother distinct m -body states defining a red cell. This situation only arises for k < m .For k = m there can be no overlap between the conditions (4.1.27) and (4.1.28).When making the transition from studying the eGUE to studying the eGSE the firstdifference, and the most significant hurdle, is this overlap in the conditions on the.1. RMT VS. EMBEDDED RMT 71random variables v j k i k . Permutations As shown in section 2.1.2 the assumptions underlying the symplectic symmetry of H coincides with a fermionic time-reversal invariant quantum system. If there are m fermions in the system the state space will consist of m -body fermionic states.Defining the system by a k -body potential with l non-degenerate single-particle statesthere must be (cid:16) lm (cid:17) non-degenerate states in the basis describing the system and thematrix elements of the symplectic potential are given by H µν := h µ | H | ν i . Since K iseven (see section 2.1.2 ) it follows that H is even as well. If l = 2 n is even and m odd, Glaisher’s Rule gives lm ! ≡ l is even and m is odd; (cid:16) b l/ cb z/ c (cid:17) mod 2 otherwise . (4.1.29)which ensures that there are an even number (cid:16) lm (cid:17) of m -body states and therefore aneven number of rows and columns in the matrix representation of the potential H .This is just an example of the interesting dependency of m, l and (cid:16) lm (cid:17) that occursfor potentials of the eGSE. It is known that a basis can be chosen in which the timereversal operator K takes the form of (2.1.28). With K taking this block diagonalform, the operator T = KC defining time-reversal invariance, in addition to takingthe complex conjugate of a state, maps one basis vector to another and multipliesit by -1. The operator T being that which applies time reversal twice to a state,maps a basis vector to itself times a phase of -1 sans complex conjugation (see section2.1.2). By letting the state labels be the set { , , . . . , l } for the basis in which H takes the canonical symplectic form (2.1.33) it must be the case that the time reversaloperator T maps any given m -body state to another, times at most a phase. Hencethe action T | µ i = σ µ is limited to permuting the m single-particle labels in the set µ and multiplying the result by some phase e iφ . Assuming that this phase is the samefor all states it follows that T µ = σ µ = e iφ σ ( µ ) (4.1.30)2 CHAPTER 4. MANY-BODY RMTwhere σ ( · ) is a permutation on the set { , , . . . , l } . Applying T = − e iφ σ = − φ = 2 p − π (4.1.32)with p = 1 , , , etc. , and also σ = 1 (4.1.33)which implies that the permutation map σ is a pairwise permutation map. Thesepermutations can be thought of as single-particle state maps which exchange onesingle-particle state with another and vice versa as a function only of the initial state.For example, the pairwise permutation map σ = . . . n − n . . . n n − (4.1.34)exchanges the single-particle state 1 with 2, 2 with 1, 3 with 4, 4 with 3 and so on. The Zeros of the Random Variable The considerations so far for the second moments of the symplectic ensemble suggestthat the eGSE involves far more subtlety than the eGUE and eGOE. A summaryof the required approach to problems relying on the second moments, including allcalculations of the higher moments of the eGSE, can be expressed as follows:Fix T to canonical form → Define a map σ → Count Zeros → Calculate MomentsThe form of T = KC is fixed in order to predetermine the form of the symmetryrelations on H , namely (4.1.24) and (4.1.25) which are the analogue of the symmetryconditions of (2.1.33) written in terms of T . Next a map σ is chosen to determine.1. RMT VS. EMBEDDED RMT 73the action of T . Any resulting zeros from an overlap of (4.1.27) and (4.1.28) need tobe identified to completely characterise the second moments v ji v j i for all j , i , j , i .Hence a “zeros theorem” needs to be stated for each permutation map σ in order toidentify the k -sets { j , i } for which the random variable is not in fact random butinstead equal to zero. Given these additional steps for the eGSE however, the formof the equations do not change dramatically. As for the eGUE, when v ji and v j i arenot related by (4.1.27) or (4.1.28) v ji v j i = 12 π Z v ji v j i exp (cid:18) − v ji (cid:19) exp (cid:18) − v j i (cid:19) dv ji dv j i = 0 (4.1.35)since the gaussians are even functions, making the total argument of the integral oddwith respect to the integration variables. Likewise if v ji = v j i then v ji and v j i arecomplex and their product yields two even functions of opposite sign, again giving azero average. However, for v ji = ± v ∗ j i (4.1.36)the average becomes v ji = ± √ π Z v ji exp (cid:18) − v ji (cid:19) dv ji = ± σ . By substituting T | µ i = e iφ σ ( µ ) from (4.1.30) the socalled blue matrix elements with symmetrycondition given by (4.1.24) obey h µ | H | ν i = h e iφ σ ( µ ) | H | e iφ σ ( ν ) i ∗ = h σ ( µ ) | H | σ ( ν ) i ∗ (4.1.38)while red matrix elements (4.1.25) obey h e iφ σ ( µ ) | H | ν i = −h µ | H | e iφ σ ( ν ) i ∗ (4.1.39)implying h σ ( µ ) | H | ν i = −h µ | H | σ ( ν ) i ∗ . (4.1.40)4 CHAPTER 4. MANY-BODY RMTFrom these it follows that v ji = v ∗ σ ( j ) σ ( i ) (4.1.41)for k -tuples j , i which are elements of the m -particle states determining blue matrixelements and v ji = − v ∗ σ ( j ) σ ( i ) (4.1.42)for k -tuples j , i which are elements of the m -particle states determining red matrixelements. This finally yields the non-zero second moments as v ji v j i = sgn( ji ) · δ j σ ( j ) δ i σ ( i ) (4.1.43)Hence by the above argument and by (4.1.27) and (4.1.28) it can be concluded that v ji v j i = δ ji δ ij + δ β · sgn( ji ) · δ j σ ( j ) δ i σ ( i ) (4.1.44)where the amplitude of δ β gives the purely Symplectic component of the average.The sgn function determines the sign of the k -tuple, where the sign is positive fortuples resulting from (4.1.41), negative for those resulting from (4.1.42) and zero forthose tuples satisfying both. Given the results from sections 4.1.2, 4.1.3 and 4.1.4 it follows that the secondmoment for the embedded gaussian unitary, orthogonal and symplectic ensemblescan be written as the compound expression v ji v j i = δ ji δ ij + δ β δ jj δ ii + δ β · sgn( ji ) · δ j σ ( j ) δ i σ ( i ) . (4.2.1)This is just the linear combination of equations (4.1.16), (4.1.21) and (4.1.44). Thesecond moment is therefore greatly simplified in the case of the eGUE, reducing to.2. THE EMBEDDED GUE 75read just v ji v j i = δ ji δ ij . (4.2.2)It is for this reason that calculating the moments of the embedded gaussian unitaryensemble is the least difficult of the three cases. This thesis will therefore focusprimarily on the eGUE case, even though the methods introduced can be applied toother symmetry classes as well. To begin with, a known result for the normalisedfourth moment, or kurtosis, of the level density for the eGUE will be derived using asimpler alternate calculation to that found in [BRW01a]. This will serve as a platformfor introducing both the basic definitions and the new methodology involving particlediagrams which this thesis aims to explain. In subsequent sections the same methodswill be used to calculate the sixth and eighth moments as well, albeit with the additionof some extra methodological features. For the case of the eGUE we will consider m spinless fermions in a system with m (cid:28) l single-particle states, all interactingunder the action of a k -body potential ( k ≤ m ) with an identical gaussian p.d.f determining its independent entries. The single-particle creation and annihilationoperators are a † j and a j respectively with j = 1 , . . . , l . As introduced in prior chapters,a useful shorthand notation for products of these is the abbreviation j = ( j , . . . , j k ), a j = a j k . . . a j (similarly for i ). A corollary of this is a † j = a † j . . . a † j k (4.2.3)For the sake of continuity and comparison with extant literature, notational traditionswill be preserved for the calculation of the kurtosis by writing the orthonormal m -particle states as | µ i , | ν i , | ρ i , etc. where each state takes the form a † j m . . . a † j | i with | i denoting the vacuum state and the restriction 1 ≤ j < j < . . . < j m ≤ l . The k -body potential is given by H k = X ≤ j <... 12 showing how the kurtosis has a semi-circular value, κ = 2 for k/m > after which it transitions to a gaussian kurtosis, κ = 3 at k = 0. Higher values of m give faster convergence to the semi-circularmoment. Particle diagrams represent diagramatically the factors of A µνρσ (4.2.6) which formthe constituent parts of H n (see (4.2.12) for instance). What will be needed nextin order to calculate higher moments for embedded ensembles is an extension ofthe notation of particle diagrams to combine arguments with particle diagrams in amathematically robust way. In so doing, it becomes possible to calculate any diagram,no matter how complex. This will be done by using arguments (powers of l ) anddiagrams (representations of A µνρσ ) as before, with the additional concept of closed .2. THE EMBEDDED GUE 83 loops . It is a necessary to introduce closed loops on diagrams because as we lookto calculate higher and higher moments the resulting diagrams become increasinglycomplex and less responsive to such a simple analysis as that which yielded the fourthmoment. Closed loops form the unbreakable link between a diagram and it’s maximalargument.As explained in section 4.2.2 a particle diagram is a set of bonds drawn between m -body states indicating the minimal overlap between them. Each m -body state inthe diagram can be thought of as a node. In calculations of the moments of the leveldensity each node is connected by four bonds, two of size k and two of size m − k .See Fig 4.3 for a representation. This is a property that emerges from the applicationof Wick’s theorem to tr H n which can always be expressed in terms of sums of factorsof A µνρσ = h µ | H | σ ih ρ | H | ν i . Since from first principles the trace is simplytr H n = H a a H a a H a a . . . H a n a (4.2.19)every index appears twice, which directly implies that each index will appear twicein the resulting expansion of (4.2.19) in terms of A µνρσ . Hence apart from a singleexception (discussed next) every index will appear once in one factor of A and asecond time in another distinct factor of A . This in turn implies that every node inevery diagram will take the form of Fig 4.3 wherein the node (a set of m state labels)is completely determined by a k set h disjoint from the set f of m − k labels.That these must be disjoint is given by the fact that the nodes represent fermionicstates which are sets of m non-repeated labels. Morever, since there are two suchdisjoint sets connected to each node, one for each appearance of a state as an indexof A , these must overlap entirely in order to conserve the number of states in thenode as m . In other words, with reference to Fig 4.3, the state | µ i is the union of thetwo disjoint black bonds µ = h ∪ f (4.2.20)from the first factor of A containing µ as an index, as well as the union of the twodisjoint blue bonds µ = h ∪ f (4.2.21)4 CHAPTER 4. MANY-BODY RMTfrom the second factor of A containing the state µ as an index. Hence, for example,the m − k states determined by the blue solid bond f µ must themselves form adisjoint set; those which are shared with the black solid bond f µ and those sharedwith the black dashed bond h µ . In this way, the single-particle states which formthe set denoted by the blue solid bond will “travel” (read: be a member of) througheither the black dashed bond or the black solid bond, and onwards to the next node(see Fig 4.4). At the next node this process will be repeated and the single-particlestates will seperate again into disjoint sets and travel further, continuing along a patharound the diagram until forming a loop.For the singular case where a single factor of A µµρρ includes both instances of alabel the two bonds f connecting µ and ρ are equal and are represented by a socalled “tail” µ f ρ , and the k -bonds h are implicit in that they connect eachstate with itself, as opposed to the more common case where for example µ shares a k -bond with two distinct states, neither of which is equal to µ .These loops on the particle diagram indicate the single-particle states shared byall bonds and nodes (states) which participate in the loop. There are only ever afinite number of possible loops in any diagram. This relates directly to the argumentof the diagram, since the argument counts the degrees of freedom in a diagam, i.e. thenumber of labels which together determine, through bonds, all states in the diagram. µ Figure 4.3: Each node in a particle diagram takes the basic form shown. Since theblue bonds are disjoint and together completely determine the m single-particle labelsin | µ i they must overlap completely with the set defined by the black bonds, whichare also disjoint and completely determine the single-particle states in | µ i ..2. THE EMBEDDED GUE 85Hence the number of elements in a loop is the number which that loop contributes tothe total argument. In this way loops will form a fundamental tool for (a) expressingthe argument as a sum of loops and (b) maximising the arguments by optimising thenumber of elements in each loop. To illustrate the process of calculating arbitrarily high moments using arguments,particle diagrams and loops, an example will now be made with of fourth moment,which has already been calculated from first principles as (4.2.18). It has beenseen that the fourth moment, or kurtosis, κ = N tr( H k ) / (cid:16) N tr( H k ) (cid:17) where in thedenominator one has tr( H k ) = X µ h µ | H k | µ i = A µµρρ (4.2.22)and in the numerator tr( H k ) = 2 A σσρρ A σσµµ + A µνρσ A σµνρ . (4.2.23) µ νρσ ϕ Figure 4.4: The fact that each node in a diagram takes the form of Fig 4.3 implies thatthe combined diagram can always be “flattened” into the form of a cyclic lattice. Thisimplies that every single-particle state label in the particle diagram is the memberof a loop which travels around the lattice along those paths not excluded by thefermionic nature of the nodes (states).6 CHAPTER 4. MANY-BODY RMT µ ρνσ Figure 4.5: The particle diagram of the term A µνρσ A σµνρ is the standard diagram,as seen before in Fig 4.1a and 4.1b but now with both factors illustrated as a singlecombined particle diagram.As forecast in the previous section (i) both components of the fourth moment canbe expressed entirely in terms of the quantity A µνρσ and moreover (ii) every indexappears twice, either as an instance of a tail with repeated indices of the form A aabb or once each in two distinct factors of A such as the second term A µνρσ A σµνρ of(4.2.23). For terms of the form A aabb no further sophistication is necessary becausethese simply add a single factor of (cid:16) mk (cid:17)(cid:16) l − m + kk (cid:17) in cases where only b is a free choice,or alternatively (cid:16) lm (cid:17)(cid:16) mk (cid:17)(cid:16) l − m + kk (cid:17) in cases such as (4.2.13) where both a and b have notyet been determined by other bonds in the diagram. These considerations givetr( H k ) = lm ! mk ! l − m + kk ! (4.2.24)and tr( H k ) = 2 lm ! " mk ! l − m + kk ! + A µνρσ A σµνρ . (4.2.25)The term A µνρσ A σµνρ can be expressed as a particle diagram as in Fig 4.1a and 4.1b,however this time the diagram will be drawn as a single combined picture, Fig 4.5.This is the form in which it is possible to calculate the moments by identifying allloops within the diagram. Because of the fermionic nature of the states, loops cannotcontain two bonds from the same factor of A , but apart from this caveat all loops arepermitted. The loops for Fig 4.5 are illustrated in Fig 4.6. Lines curving outwardsindicate a path through a bond of the red component A σµνρ while curves bendinginwards indicate that the path is crossing a bond in the blue component A µνρσ . The.2. THE EMBEDDED GUE 87 σ ρµ ν (a) The loop −−→ µσµ defin-ing the set of single-particlestates shared by the bonds µ σ and µ σ ofFig 4.5. σ ρµ ν (b) The loop −−→ σρσ defin-ing the set of single-particlestates shared by the bonds σ ρ and σ ρ ofFig 4.5. σ ρµ ν (c) The loop −−→ µνµ defin-ing the set of single-particlestates shared by the bonds µ ν and µ ν ofFig 4.5. σ ρµ ν (d) The loop −−→ νρν defin-ing the set of single-particlestates shared by the bonds ν ρ and ν ρ ofFig 4.5. σ ρµ ν (e) The first loop −−−−→ µσρνµ defining the set of single-particle states shared by thebonds µ σ , σ ρ , ρ ν and ν µ ofFig 4.5. σ ρµ ν (f) The second loop −−−−→ µσρνµ defining the set of single-particle states shared by thebonds µ σ , σ ρ , ρ ν and ν µ ofFig 4.5. Figure 4.6: Illustration of the loops of the factor A µνρσ A σµνρ seen in (4.2.12) withthe factor A µνρσ being represented by the particle diagram of Fig 4.1a and the factor A σµνρ being represented by the particle diagram of Fig 4.1b. Each bond betweencompound states represents a set of single-particle states shared by both of thecompound states, and each loop represents the set of single-particle states shared byall of the compound quantum states which form the nodes of the loop.8 CHAPTER 4. MANY-BODY RMTsum of the single-particle labels in every loop in the diagram is necessarily the sumof all dinstinct single-particle labels contained in the diagram and hence is equal tothe argument. Labelling each loop gives α = −−→ µσµβ = −−→ σρσγ = −−→ µνµδ = −−→ νρν(cid:15) = −−−−→ µσρµνκ = −−−−→ µσρµν (4.2.26)so that arg ( A µνρσ A σµνρ ) = α + β + γ + δ + (cid:15) + κ. (4.2.27)Since the total number of single-particle states in each bond h is k and the sumof all single-particle labels in each of the solid bonds f must be m − k it becomespossible to write the following “conservation equations” wherein the sum of theelements (read single-particle labels) in all loops passing through a given bond mustequal the total number of elements in the bond (either k or m − k ). Now each loopis taken to denote the number of elements in that loop. The conservation equationsfor the bonds of the red component of the diagram read α + (cid:15) = m − k (4.2.28) β + κ = k (4.2.29) γ + κ = k (4.2.30) δ + (cid:15) = m − k (4.2.31)while the conservation equations for bonds belonging to the blue component of thediagram are α + κ = k (4.2.32).2. THE EMBEDDED GUE 89 β + (cid:15) = m − k (4.2.33) γ + (cid:15) = m − k (4.2.34) δ + κ = k. (4.2.35)Plugging equations (4.2.28) and (4.2.29) into (4.2.27) gives arg ( A µνρσ A σµνρ ) = m + γ + δ. (4.2.36)Now additionally substituting (4.2.30) and (4.2.35) gives arg ( A µνρσ A σµνρ ) = m + 2 k − κ (4.2.37)which is maximised when κ = 0, implying that to attain the maximum argument α = k − κ = k (4.2.38) β = k − κ = k (4.2.39) γ = k − κ = k (4.2.40) δ = k − κ = k (4.2.41) (cid:15) = m − k − γ = m − k. (4.2.42)The final step is simply selecting the single-particle states for each loop from the totalnumber of possible single-particle energy levels l to give the leading order term A µνρσ A σµνρ ∼ lm − k k k k k ! . (4.2.43)This shows, as found already by direct arguments, that the fourth moment islim N →∞ κ = 2 N A σσρρ A σσµµ + N A µνρσ A σµνρ (cid:16) N A µµρρ (cid:17) = 2 N (cid:16) lm (cid:17) h(cid:16) mk (cid:17)(cid:16) l − m + kk (cid:17)i + N (cid:16) lm − k k k k k (cid:17)(cid:16) N (cid:16) lm (cid:17)(cid:16) mk (cid:17)(cid:16) l − m + kk (cid:17)(cid:17) (cid:16) m − kk (cid:17)(cid:16) mk (cid:17) . (4.2.44)The above explanation illustrates the complete application of the method of particlediagrams. This involves firstly the notion of an argument which identifies the order ofmagnitude of terms in powers of l . Numerators in a quotient must necessarily attainthe same argument as the denominator or they will not survive in the limit l → ∞ .Those terms which will not survive can be ignored. Secondly, the method of particlediagrams maps every term into the form of a diagram of nodes and bonds.Finally, the method of particle diagrams is completed by identifying all loopswithin a diagram and maximising the argument by maximising the sum of the (numberof single-particle labels in) loops as seen in the example of (4.2.27). After this thepractitioner must simply select the components of each loop from the set of possiblesingle-particle state labels { , , . . . , l } as seen in the example of (4.2.43)...... hapter 5The 6’th and 8’th Moments If the fourth moment confirmed that the method of particle diagrams can greatlysimplify calculations, the sixth moment illustrates the flexibility of the method as wellas its ability to scale. For the sixth moment a transition is made from flat particlediagrams to three-dimensional graph-like diagrams. Indeed, the terminology usedwill be increasingly that of graphs and sets. The m -body states will be nodes onthe graph. Graphs will contain paths – sequences of neighbouring nodes. And theconcept of loops on the particle diagrams, where the first state in a path equals thelast state, will become an increasingly important tool for maximising the argumentof attendant binomial expressions as the complexity of the particle diagrams becomemore complex. The sixth moment of the level density is given by h = N tr( V k ) (cid:16) N tr( V k ) (cid:17) . (5.1.1)Using Wick’s theorem as before furnishestr( V k ) = 2 h tr V k V k V k V k V k V k i + 3 h tr V k V k V k V k V k V k i + 6 h tr V k V k V k V k V k V k i 912 CHAPTER 5. THE 6’TH AND 8’TH MOMENTS+ 3 h tr V k V k V k V k V k V k i + h tr V k V k V k V k V k V k i . (5.1.2)Here the prefactors indicate the number of equivalent diagrams that can be obtainedby cyclic permutation of the trace. Written in terms of A µνρσ the summands are1 N tr( V k ) = 1 N h A ptqq A tvuu A vpww + 3 A ptqq A uwvv ( A tpwu )+ 6 A ptqq ( A twvu A upwv ) + 3 A putq A qwvt A upwv + A pvuq A qwvt A tpwu i . (5.1.3)Terms involving A ’s with identical first and second (or third and fourth) indicessimplify as they give a contribution only if the two other indices coincide as well.For instance for A ptqq = h p | a † j a i | q ih q | a † i a j | t i to be nonzero the states a j | p i and a j | t i both have to coincide with a i | q i . Adding the single-particle states with indices in j then gives coinciding | p i and | t i . Using this idea as well as the reasoning leading to(4.2.13) the first two terms in (5.1.3) can be evaluated to give2 A ptqq A tvuu A vpww + 3 A ptqq A uwvv ( A tpwu ) = 5 lm ! " mk ! l − m + kk ! . (5.1.4) The term 6A ptqq ( A twvu A upwv )For the third component 6 A ptqq ( A twvu A upwv ) it is similarly required that | p i = | t i for a non-zero contribution. The particle diagram for this term is illustrated in Fig.5.1a. Note that it is nearly identical to the particle diagram of Fig 4.5 except for theaddition of a tail p f q which adds a factor (cid:16) mk (cid:17)(cid:16) l − m + kk (cid:17) obtained as for Eq. (4.2.13).This additional factor added to the expression we know already gives the leadingorder expression when l → ∞ as A ptqq ( A twvu A upwv ) ∼ mk ! l − m + kk ! ls r r r r ! . (5.1.5).1. SIXTH MOMENT 93 pµ ρνσ (a) v u tw p q (b) Figure 5.1: (a) illustrates the particle diagram for the term 6 A ptqq ( A twvu A upwv ). Thisis almost identical to Fig. 4.1 but with the equivalent of Fig. 4.1a and Fig. 4.1bjuxtaposed on the same diagram as in Fig 4.5. The additional tail adds a factor (cid:16) mk (cid:17)(cid:16) l − m + kk (cid:17) . (b) illustrates the particle diagram of 3 A putq A qwvt A upwv where all thebonds implied by the expression are illustrated in a single 3D triangular prism. Usingthe diagram it becomes simpler to identify the single-particle states which mustoverlap maximally in order to maximise the argument of the whole sum.The argument of this term is k + 2 r where r = min ( k, m − k ) is defined as before, soit will only survive in the limit of h as l → ∞ for k ≤ m − k . The term A putq A qwvt A upwv The particle diagram for the fourth term A putq A qwvt A upwv of (5.1.3) is illustrated inFig. 5.1b as a single three dimensional triangular prism denoting the interrelatedconditions on the states that must be satisfied for the expression to be non-zero.The three faces correspond to the three factors. Since these are fermionic states asbefore, adjacent bonds on the same face cannot share single-particle states. In orderto maximise the number of participating single-particle states in the sum, and hencethe argument, the four overlaps of k states between | v i , | t i , | q i and | w i are chosen tobe disjoint. This imperative will be shown later using the full method involving loops,but for now it will be momentarily assumed in order to complete the calculation of theresult sans loops. Next choose the m − k additional states participating in the bond v f w but not in v h w . Note that the many-particle state | v i is given by thecombination of the bonds v f w and v h t , and equivalently by the combination4 CHAPTER 5. THE 6’TH AND 8’TH MOMENTSof v h w and v f u . These combinations can coincide only if v f u involves theoverlap v h t as well as the aforementioned m − k states. Hence the m − k statesparticipate not only in | v i and | w i but also in | u i . An analogous argument showsthat they also have to be included in | p i , and thus in all many-particle states of the“left” face in Fig. 5.1b. To fully determine | u i and | p i it is still necessary to choosethe k states participating in the “left” of the two bonds u h p , which altogethergives m − k = m − k + k states in addition to the original four sets of k states.Considering the “right” face in an analogous way one obtains the same choice of m − k states but now broken down differently into a choice of m − k and a choice of k states. Hence our choice is restricted to selecting four sets of k states and one setof m − k states from the l available states, and then splitting the latter into sets of m − k and k states in two independent ways. This gives A putq A qwvt A upwv ∼ lk k k k m − k ! m − kk ! . (5.1.6)How to calculate this utilising loops? As before the first step is simply to identify allpossible loops in the diagram, in this case the prism-like diagram of Fig 5.1b. Theseloops are illustrated in Fig 5.2. Giving each loop a label indicating the number ofsingle-particle state labels contained in the loop gives α = −−−→ utqpu η = −−−−−→ qputvwqβ = −−−−→ uvwpu φ = −−−−−−→ wpuvtqwγ = −−−−−→ vutqpwv ϕ = −→ tqtδ = −−→ vutv π = −−→ vwv(cid:15) = −−−→ qpwq ω = −→ pupκ = −−−→ vtqwv (5.1.7)Since the number of labels passing through any bond h must sum to k and thenumber of labels passing through any solid bond f must sum to m − k it isstraightforward to write down the following conservation equations which restrict the.1. SIXTH MOMENT 95 (a) The loop −−−→ utqpu (b) The loop −−−−→ uvwpu (c) The loop −−−−−→ vutqpwv (d) The loop −−→ vutv (e) The loop −−−→ qpwq (f) The loop −−−→ vtqwv (g) The loop −−−−−→ qputvwq (h) The loop −−−−−−→ wpuvtqw (i) The loop −→ tqt (j) The loop −−→ vwv (k) The loop −→ pup Figure 5.2: Illustration of all possible loops within the prism-like particle diagram ofFig 5.1b corresponding to the term A putq A qwvt A upwv .6 CHAPTER 5. THE 6’TH AND 8’TH MOMENTSvalues of the loops to α + η + ω = k (5.1.8) α + γ + δ + η = m − k (5.1.9) α + γ + ϕ = m − k (5.1.10) α + γ + (cid:15) + η = m − k (5.1.11) β + φ + ω = k (5.1.12) β + γ + δ + φ = m − k (5.1.13) β + γ + π = m − k (5.1.14) β + γ + (cid:15) + φ = m − k (5.1.15) (cid:15) + κ + η + φ = k. (5.1.16)The argument of the diagram is simply the sum of all degrees of freedom, which isthe sum of the number of labels in each loop arg ( A putq A qwvt A upwv ) = α + β + γ + δ + (cid:15) + κ + η + φ + ϕ + π + ω. (5.1.17)Substituting (5.1.9) and (5.1.12) yields arg ( A putq A qwvt A upwv ) = m + (cid:15) + κ + ϕ + π (5.1.18)and finally inserting (5.1.16) gives arg ( A putq A qwvt A upwv ) = m + k + ϕ + π − η − φ (5.1.19)so that the maximal argument m + 3 k is achieved when ϕ = π = k and η = φ = 0.Plugging these values back into (5.1.32 – 5.1.16) reveals that ϕ = π = δ = (cid:15) = k and.1. SIXTH MOMENT 97 κ = η = φ = 0. Additionally α + γ = m − kβ + γ = m − kα + ω = kβ + ω = k (5.1.20)so that α, β and γ are functions of ω and A putq A qwvt A upwv ∼ X ω lk k k k ω α ( ω ) β ( ω ) γ ( ω ) ! = X ω lk k k k ω k − ω k − ω m − k + ω ! = X ω lk k k k m − k ! m − kk ! kω ! m − kk − ω ! = lk k k k m − k ! m − kk ! (5.1.21)where the final step uses Vandermonde’s identity P ω = (cid:16) kω (cid:17)(cid:16) m − kk − ω (cid:17) = (cid:16) m − kk (cid:17) . This isthe same result claimed earlier in (5.1.6). The term A pvuq A qwvt A tpwu For the final term A pvuq A qwvt A tpwu of (5.1.3) illustrated in Fig. 5.3 it is useful first tonotice that the argument never exceeds 2 m . This can be observed on sight since thestates | w i and | t i together define all the single-particle states in | u i , | v i , | p i and | q i .And with an argument never exceeding 2 m it can be concluded that this term will onlycontribute to the limit value of h for 3 k ≤ m . Since | w i and | t i together determine allthe single-particle states in the diagram and hence all the m -body compound states,there will be maximal degrees of freedom in the choice of the single-particle statesin | w i and | t i when | w i and | t i overlap minimally. This also determines when theargument for the entire sum (diagram) is maximal. Single-particle states shared by8 CHAPTER 5. THE 6’TH AND 8’TH MOMENTSboth | w i and | t i must necessarily be manifested in the bonds which make up thediagram, so for instance if a single-particle state α is in both | w i and | t i , then it musthave “travelled upwards” from the bottom node of the diagram at | w i , proceeded“through” (read: be a member of) the bonds w f p , w f v , w h q or w h u andcontinued towards the uppermost node of the diagram | t i “through” bonds q f t , u f t , p h t or v h t . More concretely, taking the bond w f p to represent theset of m − k single-particle states in both | w i and | p i but not in | u i , the bond w h u to represent the set of k single-particle states in both | w i and | u i but not in | p i , andso on, it can be seen that for any single-particle state α in both | w i and | t i it followsboth that α ∈ w f p ∪ w f v ∪ w h q ∪ w h u (5.1.22)and α ∈ q f t ∪ u f t ∪ p h t ∪ v h t (5.1.23)whereas for a given single-particle state ˜ α in | w i but not in | t i it follows both that˜ α ∈ w f p ∪ w f v ∪ w h q ∪ w h u (5.1.24)and ˜ α / ∈ q f t ∪ u f t ∪ p h t ∪ v h t. (5.1.25)This implies that states such as α in both | w i and | t i can be found in a connectednon-repeating sequence of neighbouring nodes in the diagram starting at | w i passingthrough | t i and looping back to | w i . Likewise for a single-particle state ˜ α which isin | w i but not in | t i one can trace a non self-crossing loop of bonds in the diagramstarting from | w i and returning back to | w i without passing through the node definedby the state | t i . In conclusion, since it is imperative to maximise the number ofsingle-particle states that are in | w i but not | t i and vice versa, and since the numberof single-particle states which are an element of a loop excluding the node | t i is thecomplement of the number of loops passing through the states (nodes) | w i and | t i , ofall the particle diagrams taking the form of Fig. 5.3 those with the largest argumentare the ones consisting of the maximum possible number of loops from | w i back to.1. SIXTH MOMENT 99 | w i without passing through | t i , and from | t i back to | t i without passing through | w i . Any diagram with even one less single-particle state than the maximal numberpartaking in these loops will have an argument less than the diagrams containingthe maximal number of loops, so these do not have to be calculated. As beforeit is prohibited to include a path containing bonds which are mutually exclusive;neighbouring bonds from the same factor of A . In Fig. 5.3 for example the bonds w h q and w f v are mutually exclusive as they both appear in the diagram dueto the same factor A qwvt in A pvuq A qwvt A tpwu . The k single-particle states in w h q are therefore the complement of the m − k single-particle states in w f v , theunion being the set of single-particle states contained in | w i . On the other handthe bonds w h q and w f p appear in the diagram due to separate factors A qwvt and A tpwu respectively so these are allowed to share single-particle states. Hence theloop −−−−→ wquvw is not a valid loop for the diagram of Fig. 5.3 as it contains mutually tp u vq w Figure 5.3: The particle diagram for the term A pvuq A qwvt A tpwu which takes the formof a regular octahedron, or two square pyramids with a shared base determined bythe plane on which the sub-diagram for A pvuq is illustrated. The states | t i and | w i together determing the states | v i and | q i through the bonds defined by A qwvt just asthey determine the states | u i and | p i through the bonds defined by A tpwu . Hence themaximal degrees of freedom in our choice of single-particle states from the set of l possibilities is bounded by 2 m .00 CHAPTER 5. THE 6’TH AND 8’TH MOMENTSexclusive bonds, whereas −−−→ wqpw is a valid loop. To summarise, a single-particle statewhich is an element of both | w i and | t i must necessarily be an element of every bondin a loop passing through | w i and | t i . A single-particle state in | w i and not in | t i must necessarily be an element of every bond in a loop from | w i to | w i not passingthrough | t i and conversely for a single-particle state in | t i but not in | w i . Since theargument of the diagram is maximised by maximising the number of single-particlestates contained in every bond of a loop passing through only one of | w i or | t i , theonly diagrams which will contribute in the asymptotic regime are those where k single-particle states from each of the bonds in the loop p = −−−→ wqpw (5.1.26)are equal to the k single-partical states in the bond w h q , each of the bonds in theloop p = −−−→ wuvw (5.1.27)contain k single-particle states identical to the single-particle states in the bond w h u , and so on for all the loops p = −−−→ wpvw with the bond p h v , p = −−→ tpqt with the bond t h p , p = −−→ tuvt with the bond v h t and p = −−→ tuqt with the bond u h q . Borrowing set notation again one has for the case of (5.1.26) for example,that w h q ∩ p f q = w h q (5.1.28)and w h q ∩ w f p = w h q. (5.1.29)In this way the 3 k single-particle states from | w i which participate in the loops p , p and p do not overlap with | t i and the 3 k single-particle states from | t i whichparticipate in the loops p , p and p do not overlap with | w i . The remaining m − k single-particle states from | w i and | t i which cannot be included in loopsmust necessarily be included in a loop through | w i and | t i . That is, the number ofoverlapping single-particle states between the m -body states | w i and | t i is m − k .This has profound implications for the value of the sixth moment and we will see a.1. SIXTH MOMENT 101related feature appear in the eighth moment as well. To make the comparison withthe corresponding calculation (4.2.17) for the fourth moment more explicit, define r = min ( k, m − k ) as before and ˜ s = m − r . We then have A pvuq A qwvt A tpwu ∼ l ˜ s r r r r r r ! (5.1.30)It should be noted that this time a relatively complex particle diagram has beenevaluated with loops, albeit not by identifying all loops and maximising the argumentas before, but instead by using arguments unique to the particular diagram at handto determine what the size of each loop must be. It is useful to keep in mind that thisis sometimes possible for larger particle diagrams, as it offers a shortcut to a resultwhich would otherwise require a potentially far more laborious calculation. From(5.1.30) it follows that lim N →∞ N A pvuq A qwvt A tpwu (cid:16) N tr( V k ) (cid:17) = (cid:16) m − kk (cid:17)(cid:16) m − kk (cid:17)(cid:16) mk (cid:17) . (5.1.31)Taking the quotient for h using the above expressions (5.1.4), (5.1.5) (5.1.21) and(5.1.31) gives the final result for the sixth momentlim N →∞ h = 5 + (cid:16) m − kk (cid:17)(cid:16) m − kk (cid:17)(cid:16) mk (cid:17) + 6 (cid:16) m − kk (cid:17)(cid:16) mk (cid:17) + 3 (cid:16) m − kk (cid:17) (cid:16) mk (cid:17) . (5.1.32)02 CHAPTER 5. THE 6’TH AND 8’TH MOMENTSFigure 5.4: h against k/m for m = 6 , , , 12. Once again we see a domain k/m > for which the sixth moment takes the semi-circular value h = 5 and a transitionthereafter towards a gaussian moment, h = 15 at k = 0. Higher values of m givefaster convergence to the semi-circular moment. The eighth moment will exhibit the same features as the lower moments; a transitionstarting immediately after 2 k = m from a semi-circular moment τ = 14, to a gaussianmoment τ = 105 and a single term of tr V k taking the form l ˜ s r r r r r r r r ! (5.2.1).2. THE EIGHTH MOMENT 103analogous to (4.2.17) and (5.1.30) and with ˜ s := m − n k = m − k . Of course we mustagain calculate products of ensemble averaged pairs of the matrix elements of H k for which there are now / ( )! = 105 components in the sum. As may be expected,the method of particle diagrams can be applied analogously to the lower momentsthough with the diagrams now increasing in size and complexity. Some of these willbe identical to particle diagrams which have already been seen in calculations of thelower moments, but with the addition of tails leading to extra factors of (cid:16) l − m + kk (cid:17)(cid:16) mk (cid:17) .This feature was seen in the transition from figure 4.1 to figure 5.1a for example.For progressively higher powers of H k the particle diagrams become progressivelylarger and, as for the diagrams illustrated in Fig 5.1b and Fig. 5.3 one can think ofsome of these as graphs in three dimensions, rather than as a flat structure. Thelarger structure of the subsequent diagrams makes it more difficult to identify justtwo states which determine all others (as was the case in Fig. 4.1 and Fig. 5.3).Instead we will see diagrams where three or more states determine all others (as wasthe case in Fig. 5.1b). In these cases calculations will rely more on the analyticalmethod of solving the diagrams utilising loops as first introduced in section 4.2.5.Calculating all loops in a diagram and maximising the argument over these paths isa method which works for all diagrams. However, using the method blindly can leadto unecessary work, as there are frequently shortcuts to determining the answer, aswas seen in the case of Fig 5.3.The normalised eighth moment is given by τ = N tr( V k ) (cid:16) N tr( V k ) (cid:17) . (5.2.2)In this case Wick’s theorem gives (dropping the indices k )tr( V ) = 2 h tr V V V V V V V V i + 8 h tr V V V V V V V V i + 4 h tr V V V V V V V V i (5.2.3)04 CHAPTER 5. THE 6’TH AND 8’TH MOMENTS+ 8 h tr V V V V V V V V i + 8 h tr V V V V V V V V i + 8 h tr V V V V V V V V i + 4 h tr V V V V V V V V i (5.2.4)+ 8 h tr V V V V V V V V i (5.2.5)+ 8 h tr V V V V V V V V i + 16 h tr V V V V V V V V i (5.2.6)+ 4 h tr V V V V V V V V i (5.2.7)+ 4 h tr V V V V V V V V i + 2 h tr V V V V V V V V i + 4 h tr V V V V V V V V i + 8 h tr V V V V V V V V i + 8 h tr V V V V V V V V i + h tr V V V V V V V V i (5.2.8)where the prefactors indicate the number of equivalent contributions that can beobtained by cyclic permutation of the trace. The prefactor 16 in the second termof (5.2.6) also incorporates equivalent contributions obtained by reverting the orderof V ’s. Reassuringly the prefactors sum to 105 = (8 − V k ) = 14 A ppuu A uuww A wwvv A vvqq + 28 A ttqq A qqpp A puvw A pwvu + 24 A eeww A wvup A upqt A vtqw + 4 A cewu A uwvt A ttvqp A pceq + 2 A petq A qtuv A euwc A cwvp + 8 A tpce A evqc A puwq A uwvt .2. THE EIGHTH MOMENT 105+ 4 A tucp A tpcu A cewv A cvwe + 8 A qwvt A uwpt A pvuq A ccpp + 4 A uwvt A cqvt A pueq A pwec + 8 A uwce A epqc A tqvu A twvp + A uvqt A twvc A cewp A pueq . (5.2.9)The particle diagrams for each of the terms for τ are illustrated in Figures 5.5 – 5.14and the process for evaluating them is the same as for the fourth and sixth moments;(1) diagrams which are otherwise the same as diagrams for lower order moments butwith the addition of tails simply gain additional factors of (cid:16) l − m + kk (cid:17)(cid:16) mk (cid:17) for each tail.(2) Some diagrams, although initially appearing distinct, actually “collapse” – takingthe same value as a known diagram – when the condition is imposed on the bonds ofthe diagram that it’s argument be maximal, ie., when we look only at those instancesof the diagram which attain the maximal argument of the diagram. This processwill become clearer when looking at some actual calculations. (3) In the absence oftails the argument of the particle diagram will be maximised by using the method ofparticle diagrams. This involves determining all loops in the particle diagram andidentifying which of these, by containing more single-particle states, increases theargument of the particle diagram. These methods do not need to be used in isolation.For example loop counting can be used initially, followed by the observation that adiagram will “collapse”, just as it is perfectly legitimate to observe a diagram collapsealmost fully, except for the addition of a remaining tail. A Single Chain For the first term A ppuu A uuww A wwvv A vvqq of τ the particle diagram is simply a chainof bonds p f u f w f v f q (5.2.10)so that we first choose m states to determine | p i , then add factors of (cid:16) mk (cid:17)(cid:16) l − m + kk (cid:17) foreach additional tail, giving A ppuu A uuww A wwvv A vvqq = lm ! " mk ! l − m + kk ! . (5.2.11)06 CHAPTER 5. THE 6’TH AND 8’TH MOMENTS Diagrams of Lower Order Terms + Tails The next term A ttqq A qqpp A puvw A pwvu illustrated in Fig 5.5 is identical in form to thestandard diagram but with the addition of two tails, or alternatively Fig 5.1a withthe addition of a single tail so that A ttqq A qqpp A puvw A pwvu ∼ mk ! l − m + kk ! ls r r r r ! . (5.2.12)Likewise the term A eeww A wvup A upqt A vtqw with particle diagram as in Fig. 5.6 isidentical to the particle diagram of Fig. 5.1b but with the addition of a single tail,giving q tv puw Figure 5.5: Particle diagram for the term A ttqq A qqpp A puvw A pwvu which is identical tothe particle diagram of Fig. 5.1a with the addition of a tail, which is the same as thestandard diagram illustrated in Fig. 4.5 but with two tails. A eeww A wvup A upqt A vtqw ∼ mk ! l − m + kk ! lk k k k m − k ! m − kk ! . (5.2.13)The same applies to A qwvt A uwpt A pvuq A ccpp . Since this is equivalent to the particlediagram of Fig. 5.3 but with the addition of a tail we have simply A qwvt A uwpt A pvuq A ccpp ∼ mk ! l − m + kk ! l ˜ s r r r r r r ! . (5.2.14).2. THE EIGHTH MOMENT 107 e v u tw p q Figure 5.6: The particle diagram for the term A eeww A wvup A upqt A vtqw which is identicalto the particle diagram of the term A putq A qwvt A upwv illustrated in Fig 5.1b givingthis term the same value but with an additional factor of (cid:16) mk (cid:17)(cid:16) l − m + kk (cid:17) with the resultgiven by (5.2.13). t pu c wve Figure 5.7: The particle diagram for A tucp A tpcu A cewv A cvwe , although not an extentionof a previously seen diagram with a tail, is simply the product of two standarddiagrams with a single shared center node given by the state | c i . The term A tucp A tpcu A cewv A cvwe Finally, although we do not have simply a tail extention of a previously calculateddiagram for the term A tucp A tpcu A cewv A cvwe illustrated in Fig 5.7, this is a compositionof two copies of the standard diagram with each copy sharing the state | c i . Choosingthe state | c i will give a factor of (cid:16) lm (cid:17) , after which the reasoning follows identically forthe left and right instances as it does for the standard diagram so that the binomialterms for each are the same as in (4.2.17), giving A tucp A tpcu A cewv A cvwe ∼ lm ! " l − mr ! l − m − rr ! mr ! m − rr ! . (5.2.15)08 CHAPTER 5. THE 6’TH AND 8’TH MOMENTS qc p tue vw Figure 5.8: Particle diagram for the term A cewu A uwvt A ttvqp A pceq of τ which takes theform of a cuboid structure. The contributing sum is calculated by determining allpermitted loops in the particle diagram. Then, maximising the argument over allpermitted loops reveals the structure exclusively of those loops which maximise theargument of the particle diagram (see Figs 5.9a – 5.9f) (a) The loop −−−→ cutpc (b) The loop −−→ uwu (c) The loop −−−−→ utvwu (d) The loop −−−−→ wecuw (e) The loop −−−−−→ cutvwec (f) The loop −−−−−−−→ cutpqvwec Figure 5.9: Illustration of a selection of the loops within the particle diagram for A cewu A uwvt A ttvqp A pceq illustrated in Fig 5.8..2. THE EIGHTH MOMENT 109 The Cuboid Diagram The term A cewu A uwvt A ttvqp A pceq of (5.2.9) has a cuboid particle diagram, Fig. 5.8.One begines by identifying all the permitted loops in the diagram and ends bymaximising the argument by tuning the number of single-particle states participatingin each path. The permitted paths for the cuboid diagram are −−−→ cutpc , −−−→ ewvqe , −−−→ cuwec , −−−→ vtpqv , −−−−−→ utpqvwu , −−−−−→ cutvwec , −−−−−→ pcuweqp , −−−−→ cptvqec , −−−→ cpqec , −−−−→ utvwu , −−−−−−−→ cuweqvtpc , −−−−−−−→ cutpqvwec , −−−−−−−→ utvweqpcu , −−−−−−−→ ptuwvqecp , −→ cec , −−→ uwu , −→ tvt and −→ pqp . However, if we check the loops whichmaximise the argument of the diagram we find the contributing loops which, unlike thepermitted non-contributing loops, can contain a non-zero number of single-particlestates without decreasing the argument of the diagram from its maximal value m − k .The contributing loops for the cuboid diagram are −−−→ ewvqe −−−→ cutpc (see Fig . . −→ cec −−−−−−−→ cutpqvwec (see Fig . . −→ pqp −−→ wuw (see Fig . . −→ tvt −−−−→ utvwu (see Fig . . −−−→ vtpqv −−−→ cuwec (see Fig . . −−−−−→ utpqvwu −−−−−→ cutvwec (see Fig . . A cewu A uwvt A ttvqp A pceq ∼ X αβ lm k k α β k − α k − β ! mk ! m − kk ! sα ! sβ ! . (5.2.16) The term A tpce A evqc A puwq A uwvt We next look at the particle diagram of the term A tpce A evqc A puwq A uwvt of (5.2.9),illustrated in Fig. 5.10. To leading order it is the standard diagram in disguise, withthe addition of 2 k degrees of freedom which require explaining. In fact the overlappingof the bond t f e with e f c and the overlapping of the bond e f c with c f p 10 CHAPTER 5. THE 6’TH AND 8’TH MOMENTSon the left sub-diagram must necessarily be minimised in order to maximise theargument of the combined diagram. The same applies to the corresponding statesin the right sub-diagram of Fig. 5.10. The minimum overlap is of course m − k asbefore, giving e h v ⊂ e f t (5.2.17) c h q ⊂ c f p (5.2.18) e h c ⊂ e f c (5.2.19)The additional coincidence in m − k states between say | t i and | p i which alreadyshare k single-particle states through the bond t h p brings the total overlap betweenthem to m − k . This gives an additional factor (cid:16) m − kk (cid:17) , one factor each for the leftand right sub-diagrams which together form Fig. 5.10. One also gains the doublefactor (cid:16) l − k − sk (cid:17) from the choice of single-particles contained in e h c ∩ e f c andcorrespondingly u h w ∩ u f w . In this way we find A tpce A evqc A puwq A uwvt ∼ ls k k k k ! m − kk ! l − k − sk ! . (5.2.20)This is an instance of a particle diagram “collapsing” partially to a diagram seenbefore in the calculation of a lower moment, in this case the standard diagram of Fig4.5 which has been a presence in calculations from the fourth moment onwards. The term A petq A qtuv A euwc A cwvp The calculation for the term A petq A qtuv A euwc A cwvp of (5.2.9) with diagram shown inFig. 5.11 will be calculated using the method of particle diagrams by utilising loopsummation. Explicitly, the number of single-particle states contained in each loop isoptimised in order to maximise the argument. The contributing loops are those whichdo not necessarily have to contain zero elements in order to maximise the argumentgiven by the sum arg = P i n i where n i is the number of elements in loop i . The loops.2. THE EIGHTH MOMENT 111 t e vp c q p u tq w v Figure 5.10: The A tpce A evqc A puwq A uwvt which to leading order is the same as thestandard diagram, but with an additional k degrees of freedom. This is analogousbut not exactly the same as the case where we have a known diagram with an extratail. For k ≤ m − k the overlap of the bonds linking the states | e i , | c i and | u i , | w i respectively implies that the states | t i and | p i overlap by an additional s = m − k states, bringing their total overlap to m − k and likewise for the states | t i and | v i . p q ve t u e c pu w v Figure 5.11: Illustrated is the particle diagram of A petq A qtuv A euwc A cwvp which hassome similarities to the standard diagram. For the purpose of evaluation however,it is not similar enough. To calculate the leading order summation for the termssatisfying this diagram one needs to use the method of particle diagrams, availing ofloop summation.on the particle diagram of the term A petq A qtuv A euwc A cwvp are α = −→ pvq ξ = −−−−→ qvpeutβ = −→ pec θ = −−−−→ qpvuetγ = −→ eut λ = −−−−→ cpeuvwδ = −−→ uvw ν = −−−−→ cepvuw(cid:15) = −−→ qtuv π = −−−−−−→ pqvwutecη = −−→ qtep σ = −→ qtω = −−−→ cwvp τ = −→ cwµ = −−−→ cwue ρ = −−→ pvue 12 CHAPTER 5. THE 6’TH AND 8’TH MOMENTSBecause the number of single-particle states contained in all paths which pass througha bond h must sum to k and similarly the number of single-particle states containedin all paths which pass through a bond f must sum to m − k one can immediatelyread off the following conservation equations n α + n ξ + n θ + n ν + n ρ = kn β + n ξ + n λ + n ν + n ρ = kn γ + n ξ + n θ + n λ + n ρ = kn δ + n θ + n λ + n ν + n ρ = k (5.2.21) n (cid:15) + n ξ + n σ = kn η + n θ + n σ = kn ω + n λ + n τ = kn µ + n ν + n τ = k (5.2.22) n (cid:15) + n ξ + n ω + n π + n α = m − kn (cid:15) + n ξ + n µ + n π + n γ = m − kn (cid:15) + n ω + n λ + n π + n δ = m − kn (cid:15) + n µ + n ν + n π + n δ = m − k (5.2.23) n η + n θ + n ω + n π + n α = m − kn η + n ω + n λ + n π + n β = m − kn η + n µ + n ν + n π + n β = m − kn η + n θ + n µ + n π + n γ = m − k (5.2.24)For a particular loop to contribute the argument must additionally satisfyarg = X i n i = n α + n β + n γ + n δ + n (cid:15) + n η + n ω + n µ + n ξ + n θ + n λ + n ν + n π + n ρ + n σ + n τ .2. THE EIGHTH MOMENT 113= m + 4 k (5.2.25)since m +4 k is the maximal argument that this diagram can attain and therefore the n i satisfying this equality are the only terms of interested in the limit l → ∞ . Imposingthis contraint on the conservation equations yields n α = n β = n γ = n δ = k and n ξ = n θ = n λ = n ν = n ρ = 0. Moreover n (cid:15) = n η , n π = n π n ω = n µ , n σ = n σ n τ = n τ . Inserting these back into the conservation equations gives n (cid:15) = k − n σ (5.2.26) n ω = k − n τ (5.2.27) n (cid:15) + n ω + n π = m − k (5.2.28)so that in the limit l → ∞ the surviving term reads A petq A qtuv A euwc A cwvp ∼ X ln α n β n γ n δ n σ n τ n (cid:15) n η n ω n µ n π ! = X n α n β lk k k k n σ n τ k − n σ k − n σ k − n τ k − n τ m − k + n σ + n τ ! . (5.2.29)Dividing (5.2.29) by the normalisation term (cid:16) N tr( V k ) (cid:17) and summing out τ oneextraordinarily attains the following Hahn polynomial lim N →∞ N A petq A qtuv A euwc A cwvp (cid:16) N tr( V k ) (cid:17) = (cid:16) m − kk (cid:17)(cid:16) mk (cid:17) X n σ m − k − n σ k ! m − kn σ ! kn σ ! . (5.2.30)This is unusual because no previous term has explicitly required a Hahn polynomialin order to be expressed properly. Additionally, this is an interesting developmentbecause it hints that it may be possible to express all previous terms as Hahn14 CHAPTER 5. THE 6’TH AND 8’TH MOMENTSpolynomials and perhaps even find a set of tools with which to calculate the momentsusing a “Hahn Method” as apposed to using the method of particle diagrams whichthis thesis sets out to present and explain. The author’s proof of the following lemma will briefly be highlighted to express (5.2.29)as a quotient of binomials in m and k . The proof uses the formalism of the class of Hahn polynomials for which definitions can be found in the compendium [KRLS10]. Lemma. (cid:16) m − kk (cid:17)(cid:16) mk (cid:17) X α m − k − αk ! kα ! m − kα ! = (cid:16) m − kk (cid:17) (cid:16) mk (cid:17) X p (cid:16) kp (cid:17) (cid:16) m − kk − p (cid:17)(cid:16) m − kp (cid:17) (5.2.31) Proof: Writing the l.h.s as an hypergeometric function X m − k − αk ! kα ! m − kα ! = m − kk ! F − k, k − m, k − m , k − m ; 1 (5.2.32)and recalling the definition of a Hahn polynomial Q n ( x ; α, β, N ) := F − n, n + α + β + 1 , − xα + 1 , − N ; 1 (5.2.33)we have n = k , α = 0, β = k − m − x = m − k . To express this as a series itshould then be noted that F − xα + 1 ; − t F x − Nβ + 1 ; t = X ( − N ) n ( β + 1) n n ! Q n ( x ; α, β, N ) t n . (5.2.34).2. THE EIGHTH MOMENT 115(see [KRLS10] chapter 9). Furthermore, by definition F ab ; t := ∞ X n =0 ( a ) n ( b ) n t n n ! (5.2.35)where ( a ) n := Γ( a + n )Γ( a ) = ( a + 1)( a + 2) . . . ( a + n ) (5.2.36)and the symmetric extention of this is( − x ) n = ( − n ( x − n + 1) n . (5.2.37)Using the definition it then follows straightforwardly that F k − m − t = ∞ X n =0 Γ( m − k + 1)Γ( m − k + 1 − n ) n ! t n n ! = ∞ X n =0 m − kn ! t n n ! (5.2.38)and similarly F − k − ( m − k ) ; t = ∞ X n =0 Γ( k + 1)Γ( m − k − n + 1)Γ( k − n + 1)Γ( m − k + 1) t n n ! = ∞ X n =0 (cid:16) kn (cid:17)(cid:16) m − kn (cid:17) t n n ! (5.2.39)so that the complete series can be written as F k − m − t F − k − ( m − k ) ; t = ∞ X n,p =0 (cid:16) m − kn (cid:17)(cid:16) kp (cid:17)(cid:16) m − kp (cid:17) t n + p n ! p ! . (5.2.40)Finally, comparing the coefficients of t k gives Q k ( x ; α, β, N ) = k X p =0 (cid:16) kp (cid:17) (cid:16) m − kk − p (cid:17)(cid:16) m − kp (cid:17) (5.2.41)which completes the proof. (cid:3) 16 CHAPTER 5. THE 6’TH AND 8’TH MOMENTS There are just three remaining terms left in the calculation of (5.2.9) for which we havealready fully covered the techniques required; (1) comparison to diagrams representinglower order terms, (2) the “collapse” of diagrams to others of familiar form such as inthe case of Fig 5.10 which collapses to take the form of the standard diagram with atail when we just look at those components of the diagram which attain the maximalargument, and finally (3) the method of particle diagrams in which we maximise theargument ( = degrees of freedom) of the diagram by identifying all permitted loops in the diagram and from these finding all contributing loops . The path summationmethod can be used in every case, even those where (1) and (2) do not apply. The term A uwvt A cqvt A pueq A pwec Identifying paths and maximising the argument for A uwvt A cqvt A pueq A pwec of (5.2.9)with corresponding diagram Fig. 5.12 in the same manner as described previously weyields the following paths α = −−−→ utvwe σ = −−−−−−→ ewvtupcqβ = −−−→ tcpqv τ = −−−−−−→ pqvtceuwγ = −−→ uew φ = −−−−−−→ pctvqewuδ = −→ cpq θ = −−−→ ectvq(cid:15) = −−→ utce λ = −−−→ putvwη = −−−→ wvqe ν = −→ ecqω = −−−→ qvwp π = −−→ puwµ = −−→ ctup ψ = −−−−→ euwpqcξ = −−−−−−→ eutcpqvw κ = −−−−→ pcqewuρ = −−−−−−→ eutvwpqc χ = −→ tv and the following loop conservation equations n α + n λ + n σ + n ρ + n χ = kn β + n θ + n τ + n φ + n χ = k (5.2.42).2. THE EIGHTH MOMENT 117 n (cid:15) + n θ + n ν + n ρ + n τ + n ψ = kn η + n θ + n ν + n σ + n φ + n κ = kn ω + n λ + n π + n ρ + n τ + n ψ = kn µ + n λ + n π + n σ + n φ + n κ = kn γ + n π + n τ + n φ + n ψ + n κ = kn δ + n ν + n ρ + n σ + n ψ + n κ = k (5.2.43) n α + n γ + n (cid:15) + n ξ + n ρ + n τ + n ψ = m − kn α + n γ + n η + n ξ + n σ + n φ + n κ = m − kn α + n (cid:15) + n µ + n ξ + n λ + n ρ + n σ = m − kn α + n η + n ω + n ξ + n λ + n ρ + n σ = m − k (5.2.44) n β + n (cid:15) + n µ + n ξ + n θ + n τ + n φ = m − kn β + n δ + n µ + n ξ + n σ + n φ + n κ = m − kn β + n δ + n ω + n ξ + n ρ + n τ + n ψ = m − kn β + n η + n ω + n ξ + n θ + n τ + n φ = m − k. (5.2.45)Solving forarg = X i n i = n α + n β + n γ + n δ + n (cid:15) + n η + n ω + n µ + n ξ + n θ + n λ + n ν + n π + n ρ + n σ + n τ + n φ + n ψ + n κ + n χ = m + 4 k (5.2.46)gives the following restrictions on the number of single-particles in each of these loops n θ = n λ = n ν = n π = n ρ = n σ = n τ = n φ = n ψ = n κ = 0 (5.2.47) n γ = n δ = n (cid:15) = n η = n ω = n µ = k (5.2.48)18 CHAPTER 5. THE 6’TH AND 8’TH MOMENTS u t cw v q p euq p ecw Figure 5.12: Illustration of the particle diagram for the term A uwvt A cqvt A pueq A pwec . w c qu e p t vqu t vpw Figure 5.13: The particle diagram for A uwce A epqc A tqvu A twvp . By acknowledging theoverlaps between the bonds of the left sub-diagram we can conclude that the completediagram is identical to Fig 5.3 which we have already calculated for the sixth moment,with an additional factor (cid:16) m − kk (cid:17)(cid:16) l − m − kk (cid:17) due to the degrees of freedom between | c i and | e i .with n α = n β , as well as the following identities n χ = k − n α (5.2.49) n ξ = m − k − n α . (5.2.50)Hence A uwvt A cqvt A pueq A pwec ∼ X n α lk k k k k k n α n β n χ n ξ ! = X n α lk k k k k k n α n α k − n α m − k − n α ! (5.2.51).2. THE EIGHTH MOMENT 119 w c qt vu pe (a) The loop −−−−−−−→ wtqcepvuw w c qt vu pe (b) The loop −−−−−−−→ wvqptuecw w c qt vu pe (c) The loop −−−−−−−→ wvqceptuw w c qt vu pe (d) The loop −−−−−−−→ tqpvuecwt w c qt vu pe (e) The loop −−−→ tqcwt w c qt vu pe (f) The loop −−−−→ wvqcw w c qt vu pe (g) The loop −−−→ tpeut w c qt vu pe (h) The loop −−−→ uvpeu w c qt vu pe (i) The loop −−−−−−−→ twcqvpeut w c qt vu pe (j) The loop −−→ twut w c qt vu pe (k) The loop −−→ qvpq w c qt vu pe (l) The loop −−−−→ twceut 20 CHAPTER 5. THE 6’TH AND 8’TH MOMENTS w c qt vu pe (a) The loop −−−−→ cqvpec w c qt vu pe (b) The loop −−−→ wvuw w c qt vu pe (c) The loop −−→ qtpq w c qt vu pe (d) The loop −−−→ tqcept w c qt vu pe (e) The loop −−−−−→ wvuecw w c qt vu pe (f) The loop −−−−−→ tqpvuwt w c qt vu pe (g) The loop −−−−−−→ wvqptuw w c qt vu pe (h) The loop −→ cec Figure 5.13: Illustration of all the loops in the particle diagram for the term A uwce A epqc A tqvu A twvp represented in Fig 5.13..2. THE EIGHTH MOMENT 121giving lim N →∞ N A uwvt A cqvt A pueq A pwec (cid:16) N tr( V k ) (cid:17) = (cid:16) m − kk (cid:17)(cid:16) m − kk (cid:17) (cid:16) mk (cid:17) . (5.2.52) The term A uwce A epqc A tqvu A twvp The penultimate term A uwce A epqc A tqvu A twvp of (5.2.9) yields the particle diagram ofFig. 5.13 which has the particle loops α = −−−−−−→ wtqcepvu λ = −→ qvpβ = −−−−−−→ wceutpqv ρ = −−→ wvuγ = −−−−−−→ wutpecqv σ = −→ tqpδ = −−−−−−→ wceuvpqt ν = −−−→ wtuec(cid:15) = −−→ wtqc π = −−−→ cqvpeη = −−−→ wvqc τ = −−−→ tqcepω = −−→ tuep φ = −−−−→ wceuvµ = −−→ uepv ψ = −−−−→ wtqpvuξ = −−−−−−→ twcqvpeu κ = −−−−→ wvqptuθ = −−→ wtu χ = −→ ce and the following particle conservation equations n α + n δ + n (cid:15) + n ξ + n θ + n ν + n ψ = k (5.2.53) n α + n δ + n µ + n ξ + n λ + n π + n ψ = k (5.2.54) n α + n γ + n (cid:15) + n η + n ξ + n π + n τ = k (5.2.55) n α + n γ + n ω + n µ + n ξ + n π + n τ = k (5.2.56) n α + n δ + n (cid:15) + n τ + n ψ + n σ = k (5.2.57) n α + n δ + n µ + n ρ + n φ + n ψ = k (5.2.58) n α + n γ + n θ + n ρ + n ψ + n κ = k (5.2.59) n α + n γ + n π + n τ + n χ = k (5.2.60)22 CHAPTER 5. THE 6’TH AND 8’TH MOMENTS n β + n γ + n ω + n ξ + n θ + n ν + n κ = m − k (5.2.61) n β + n γ + n η + n ξ + n λ + n π + n κ = m − k (5.2.62) n β + n δ + n (cid:15) + n η + n ξ + n ν + n φ = m − k (5.2.63) n β + n δ + n ω + n µ + n ξ + n ν + n φ = m − k (5.2.64) n β + n γ + n η + n ρ + n φ + n κ = m − k (5.2.65) n β + n γ + n ω + n σ + n τ + n κ = m − k (5.2.66) n β + n δ + n λ + n σ + n ψ + n κ = m − k (5.2.67) n β + n δ + n ν + n φ + n χ = m − k. (5.2.68)Solving forarg = X i n i = n α + n β + n γ + n δ + n (cid:15) + n η + n ω + n µ + n ξ + n θ + n λ + n ν + n π + n ρ + n σ + n τ + n φ + n ψ + n κ + n χ = m + 4 k (5.2.69)gives the following identities n β + n κ = m − k (5.2.70) n θ + n ν = k (5.2.71) n θ + n κ = n φ (5.2.72) n ρ + n κ = n ν (5.2.73)as well as the restrictions n α = n γ = n δ = n (cid:15) = n µ = n ξ = n π = n τ = n ψ = 0and n η = n ω = n λ = n σ = n χ = k on the number of single-particles in each of.2. THE EIGHTH MOMENT 123these loops. In addition one finds n β = m − k − n κ n θ = n φ − n κ = k − n ρ − n κ n ν = n ρ + n κ n φ = k − n ρ (5.2.74) n ρ = n ρ n κ = n κ (5.2.75)so that the final expression takes the value A uwce A epqc A tqvu A twvp ∼ X κρ lk k k k k κ m − k − κ ρ ρ + κ k − ρ − κ ! (5.2.76)which then gives lim N →∞ N A uwce A epqc A tqvu A twvp (cid:16) N tr( V k ) (cid:17) = (cid:16) m − kk (cid:17) (cid:16) m − kk (cid:17)(cid:16) mk (cid:17) (5.2.77) The final term A uvqt A twvc A cewp A pueq For the evaluation of the final term A uvqt A twvc A cewp A pueq of (5.2.9) one can use thesame method as outlined in the evaluation of A pvuq A qwvt A tpwu (Fig. 5.3). That is,note that the states | u i and | v i together determine all other states in the diagram,Fig 5.14. This means that to maximise the argument of the diagram we minimise theoverlap between | u i and | v i . As before this is done by identifying as many loops aspossible within the diagram from | u i to | u i without passing through | v i and from | v i to | v i without passing through | u i . The single-particle states which cannot beincorporated into these loops must necessarily be in the set that is shared by both | u i and | v i . And whereas for the case of Fig 5.3 each loop is composed of three bonds,for this analogous term of the eighth moment each loop consists of n = 4 bonds. The24 CHAPTER 5. THE 6’TH AND 8’TH MOMENTS et w vcu qp e pcw q uvt Figure 5.14: The particle diagram for A uvqt A twvc A cewp A pueq which takes the form (cid:16) l ˆ s k k k k k k k k (cid:17) with ˆ s = m − k .loops passing through the state | u i without passing through | v i are p = −−−→ utcpu (5.2.78) p = −−−→ utweu (5.2.79) p = −−−−→ uewvu (5.2.80) p = −−−−−→ utcpweu (5.2.81)and similarly the loops passing through the state | u i without passing through | v i are p = −−−→ qpcvq (5.2.82) p = −−−→ qewvq (5.2.83) p = −−−→ qpctq (5.2.84) p = −−−−−→ qvwecpq. (5.2.85)The single-particle states which are not an element of each of the bonds in a loopare necessarily an element of all the bonds in a loop through | u i and | v i and thereare ˆ s := m − k of these. The final expression then reads A uvqt A twvc A cewp A pueq ∼ l ˆ s k k k k k k k k ! . (5.2.86).2. THE EIGHTH MOMENT 125Bringing all terms (5.2.11 – 5.2.16), (5.2.20), (5.2.30), (5.2.51), (5.2.76) and (5.2.86)together gives lim N →∞ τ = 14 + (cid:16) m − kk (cid:17)(cid:16) m − kk (cid:17)(cid:16) m − kk (cid:17)(cid:16) mk (cid:17) + 4 (cid:16) m − kk (cid:17)(cid:16) m − kk (cid:17) (cid:16) mk (cid:17) + 8 (cid:16) m − kk (cid:17)(cid:16) m − kk (cid:17)(cid:16) mk (cid:17) + 8 (cid:16) m − kk (cid:17) (cid:16) m − kk (cid:17)(cid:16) mk (cid:17) + 8 (cid:16) m − kk (cid:17) (cid:16) mk (cid:17) + 4 (cid:16) m − kk (cid:17) (cid:16) mk (cid:17) + 28 (cid:16) m − kk (cid:17)(cid:16) mk (cid:17) + 24 (cid:16) m − kk (cid:17) (cid:16) mk (cid:17) + 4 (cid:16) m − kk (cid:17) (cid:16) mk (cid:17) + 2 (cid:16) m − k − αk (cid:17)(cid:16) mk (cid:17) X α m − k − αk ! m − kα ! kα ! . (5.2.87)Illustrated in Fig 5.15 the eighth moment takes the same form as the fourth andsixth moments, transitioning from a gaussian value (2 n − k = 0 to asemi-circular moment for all k > m . Unlike a the lower moments the eighth momentcontains a Hahn term. Fortunately the method of particle diagrams can be used toyield all such terms, and as we have seen the procedure is the same, independentof how complex a particular diagram is. These results and the method of particlediagrams will be discussed in greater detail in the next chapter.26 CHAPTER 5. THE 6’TH AND 8’TH MOMENTSFigure 5.15: The eighth moment τ of the level density against k/m for m = 6 , , , τ = 14for k/m > converging to a gaussian value, τ = 105 for k = 0. Higher values of m give faster convergence to the semi-circular moment...... hapter 6Conclusions This thesis has focused on the problem of characterising the statistics of unified RMT,more commonly known as embedded many-body random matrix theory. As notedin chapter 1 this is a relatively young area of research, with few practitioners and amodest body of literature. The approach established here has involved the methodof particle diagrams, a mathematical toolkit invented by the author, with which wehave investigated the calculation of moments of the level density ρ ( E ) of the randompotential derived in chapter 2 given by H k = X ji v ji a † j a i . (6.1.1)These potentials are characterised by the parameter set { k, m, l, S } where k is theorder of the potential, m is the number of particles in the experiment, l is the numberof single-particle energy levels and S represents any additional symmetry conditionsplaced on the potential H k . Together these determine the state space of the potential,and thereby the state space of unified random matrix theory. This is a superspace ofcanonical random matrix theory. Canonical RMT places a single random variable intoeach matrix cell and coincides with the point { m, m, ∞ , S } , with S usually enforcing12728 CHAPTER 6. CONCLUSIONSorthogonal ( β = 1), unitary β = 2 or symplectic ( β = 4) symmetry on H k . Here wehave made calculations of the fourth, sixth and eighth moments for all k ≤ m forthe embedded GUE (see section 4.2). It was noted in (4.1.5) that the moments of aGaussian are given by v n = s π Z v n e − v / dv = (2 n − πr Z r − r x n √ r − x dx = C n (6.1.3)where the Catalan numbers (3.2.57) are C n = 1 n + 1 nn ! . (6.1.4)Either through direct calculation or by using the method of particle diagrams wehave seen in section 4.2 that the fourth moment of the level density islim N →∞ κ = 2 + lim N →∞ N A µνρσ A σµνρ h N P A µµρρ i = 2 + (cid:16) m − kk (cid:17)(cid:16) mk (cid:17) (6.1.5)which is illustrated in Fig 4.2. The figure reveals a transition from a Gaussian fourthmoment (2 n − k = 0 to a semi-circular moment C n = 2 for all k > m .This includes k = m which we know already from Wigner’s semi-circle law shouldyield κ = 2. Hence there is a portion of the domain in the unified theory ( m , m ] inwhich k varies (and therefore the action of the potential H k changes) but the fourthmoment remains fixed to a semi-circular value.It is shown in chapter 5 that the sixth moment is given by the combinatorial sumlim N →∞ h = 5 + (cid:16) m − kk (cid:17)(cid:16) m − kk (cid:17)(cid:16) mk (cid:17) + 6 (cid:16) m − kk (cid:17)(cid:16) mk (cid:17) + 3 (cid:16) m − kk (cid:17) (cid:16) mk (cid:17) . (6.1.6).2. INTERNAL STRUCTURE OF THE MOMENTS 129This was illustrated previously in Fig 5.4. Again we can observe a gradual transitionfrom a Gaussian valued moment (2 n − k = 0 to a semi-circular moment C n = 5 not just for the canonical case k = m , but for all k > m .The eighth moment, also derived in chapter 5, was shown to be equal to therambling expression lim N →∞ τ = 14 + (cid:16) m − kk (cid:17)(cid:16) m − kk (cid:17)(cid:16) m − kk (cid:17)(cid:16) mk (cid:17) + 4 (cid:16) m − kk (cid:17)(cid:16) m − kk (cid:17) (cid:16) mk (cid:17) + 8 (cid:16) m − kk (cid:17)(cid:16) m − kk (cid:17)(cid:16) mk (cid:17) + 8 (cid:16) m − kk (cid:17) (cid:16) m − kk (cid:17)(cid:16) mk (cid:17) + 8 (cid:16) m − kk (cid:17) (cid:16) mk (cid:17) + 4 (cid:16) m − kk (cid:17) (cid:16) mk (cid:17) + 28 (cid:16) m − kk (cid:17)(cid:16) mk (cid:17) + 24 (cid:16) m − kk (cid:17) (cid:16) mk (cid:17) + 4 (cid:16) m − kk (cid:17) (cid:16) mk (cid:17) + 2 (cid:16) m − kk (cid:17)(cid:16) mk (cid:17) X α m − k − αk ! m − kα ! kα ! (6.1.7)which has been illustrated in Fig 5.15. From the figure we can immediately note theway in which the value of the moment transitions from a gaussian value (2 n − k = 0 to a semi-circular moment C n = 14 forall values m < k ≤ m where despite the potential H k being fundamentally different,the eighth moment remains constant. The results shown above suggest that the unified RMT state space for the eGUEconsists of two distinct domains in the order k . Firstly, each of the moments reveala canonical domain defined by m < k ≤ m (see Fig 6.1). In this region of thephase space the action of the potential is changing, but the moments of the leveldensity remain fixed to semi-circular values. Hence, we may expect that in this regionWigner’s semi-circle law applies for all values of k .Secondly, again with reference to Fig 6.1, there is a critical domain defined by0 ≤ k ≤ m . Here the value of k is crucial for determining the value of the moments,which vary for all k within this region of the state space.30 CHAPTER 6. CONCLUSIONSFigure 6.1: Illustration of the general form of all the moments, showing a transitionfrom a Gaussian value to a semi-circular value for all k in the canonical domain, andsome additional structure in the critical domain at the discrete points k = m , m , . . . , mn .Finally, the critical domain for each of the moments contains an internal struc-ture. For the 2 n ’th moment this additional detail is defined by the discrete points m , m , . . . , mn at which the combinatorial expressions (6.1.5–6.1.7) contain terms whichbecome zero for all values of k thereafter. For example the quotient (cid:16) m − kk (cid:17)(cid:16) mk (cid:17) (6.2.1).2. INTERNAL STRUCTURE OF THE MOMENTS 131giving the contribution of the standard diagram to the fourth moment (6.1.5) will bezero for all k > m − k while the quotient (cid:16) m − kk (cid:17)(cid:16) m − kk (cid:17)(cid:16) mk (cid:17) (6.2.2)from (6.1.6) will be zero for all k > m − k . The gradual removal of these terms at thediscrete values of k = m , m , . . . , mn is what produces the decline in the moments fromgaussian values to semi-circular values and explaining the physical ramifications ofthis (if any) is an open question which would make for some very interesting reading.Related to this, but offering perhaps less detail, is a physical explanation for why therandom potential, being different for each value of k , should nonetheless have thesame level density for all k in the canonical domain k > m .A fundamental component of the internal structure of the critical domain is theexistence, in each of the moments, of a single diagram which takes the value ls k k . . . k | {z } n times ! (6.2.3)where for the 2 n ’th moment s := m − nk and the sequence of k ’s repeats 2 n times sothat the sum is s + P n k = m + nk . This is the argument attained by any diagramcontributing to the limit value of the 2 n ’th moment as l → ∞ . For the fourth momentthe diagram which takes the form of (6.2.3) is the standard diagram, Fig 4.5, whilefor the sixth and eighth moments it is Figures 5.3 and 5.14 respectively. The fact thatthese diagrams take the form of (6.2.3) is due to the existence in each case of justtwo m -body states which together determine all other single-particle labels present inthe diagram.32 CHAPTER 6. CONCLUSIONS k (cid:28) m (cid:28) l Another result of the analysis is the conclusion that in the limit case k/m → m → ∞ and applying Stirling’s approximation tothe resulting binomials, yielding the moments for the special case k (cid:28) m (cid:28) l whichis also known as the dilute limit . In this limit each binomial expression contributesjust a factor 1, so that the moments become equal to the sum of the coefficients.Since the coefficients sum to the number of unique pairwise partitions of a set of 2 n objects the result is always (2 n − m in the numerator is always equal to thenumber of combinatorial terms in m in the denominator. For example the component (cid:16) m − kk (cid:17)(cid:16) m − kk (cid:17)(cid:16) m − kk (cid:17)(cid:16) mk (cid:17) (6.3.1)of the eighth moment (6.1.7) has three binomial terms (cid:16) m − kk (cid:17)(cid:16) m − kk (cid:17)(cid:16) m − kk (cid:17) in m forming the numerator and three binomial terms (cid:16) mk (cid:17) = (cid:16) mk (cid:17)(cid:16) mk (cid:17)(cid:16) mk (cid:17) in m formingthe denominator. Likewise, the Hahn expression2 (cid:16) m − k − αk (cid:17)(cid:16) mk (cid:17) X α m − k − αk ! m − kα ! kα ! (6.3.2)has the three factors (cid:16) m − kk (cid:17)(cid:16) m − k − αk (cid:17)(cid:16) m − kα (cid:17) in m forming the numerator and the threefactors (cid:16) mk (cid:17) in the denominator. Using the method of particle diagrams it is possible to make a final remark regardingthe statistics of the hermitian eGUE potential (4.2.4) given a bosonic state space..4. BOSONIC STATES 133From section 2.1.3 it is known that bosonic many-particle states may contain multiplecopies of the same single-particle states. Retaining the same potential (4.2.4), but nowassuming a bosonic state space, it is possible to use what has been learned alreadyfor the eGUE embedded in a fermionic state space to make far-reaching conclusionsabout the bosonic case. Namely, it is straightforward to see that in the limit l → ∞ the results for all moments agree with fermionic systems. Intuitively, this happensbecause in this limit contributions arise only for those choices of many-particle stateswhich maximise the number of participating single-particle states. This means thatin the bosonic case multiple occupancy of the same single-particle states is penalised,such that for the terms which survive in the limit l → ∞ there is no difference fromthe fermionic case.For a formal derivation consider bosonic m -particle states containing repeats of z ≤ m unique single-particle states. There are (cid:16) lz (cid:17) ways to select the participatingsingle-particle states, and (cid:16) m − z − (cid:17) ways to select their multiplicities to obtain altogether z particles. As stated before in section 4.1.3 the number of many-particle states isthus N = m X z =1 lz ! m − z − ! = l + m − m ! (6.4.1)where the overall sum as well as the summand z = m have the argument m whereasall other summands have lower arguments. As a consequence the asymptotic form of N coincides with the fermionic case where N = (cid:16) lm (cid:17) .The same logic applies when evaluating the particle diagrams, where the dashedbonds h contain only unique sets of k single-particle state labels whereas incontrast to the fermionic case the solid bonds f contain m − k single-particle statelabels with repeats now permitted. In the fermionic case, the contributions to theparticle diagrams with maximal argument m + nk can always be factorized into aterm (cid:16) lm + nk (cid:17) counting the number of ways in which the m + nk participating statescan be chosen, and further l -independent factors counting the number of ways inwhich these m + nk states can be distributed among different sets while obeying theconditions implied by the diagram. In the bosonic case we instead have to select34 CHAPTER 6. CONCLUSIONS1 ≤ z ≤ m + nk states with multiplicities summing to m + nk . The contribution ofeach diagram turns into a sum over z , and for each z a single l -dependent term (cid:16) lz (cid:17) is obtained, together with further l -independent finite factors counting the numberof ways to distribute these states among sets containing m particles. The latterchoices have to be compatible both with the restrictions implied by the diagram andthe requirement on the multiplicities. However, the only summand attaining themaximal argument will be the last, for which z = m + nk . For this summand allmultiplicities are 1 meaning that we only have to consider bosonic states composedentirely of distinct single-particle states. Since these are mathematically equivalent tofermionic states it can be concluded without any further calculations that all bosonicand fermionic moments are equal in the limit l → ∞ . The focus of this thesis has been the study of quantum many-body potentials takingthe second quantised form of (6.1.1). This involved the study of a unified phasespace of random matrix theories composed of the parameters k, m, l which are theorder of interaction, number of particles and number of available states respectively.These, in addition to the symmetry constraints on H k , determine each point inthe phase space. The canonical form of random matrix theory corresponds to thesingle point { m, m, ∞} . In their seminal paper Benet, Rupp and Weidenmüllerwere able to calculate the fourth moment of the level density of the eGUE in thedomain { ≤ k ≤ m, m, ∞} [BRW01a]. In so doing they also illustrated the difficultyinvolved in studying the more general embedded RMT state space, suggesting thatnew methods would be needed beyond those which are specialised for the canonical k = m case. The author’s response to this was the creation of the method ofparticle diagrams which has been illustrated in the previous chapters to calculate thefourth, sixth and eighth moments of the embedded GUE. This is a modest successin itself, since the sixth and eighth moments were not known before in the criticaldomain k ≤ m . Moreover the method is unmatched in it’s simplicity, involving nosuperalgebra and no complex mathematics beyond basic combinatorics. The basic.5. THE METHOD OF PARTICLE DIAGRAMS 135ingredients are as follows.1. Arguments (order of magnitude). The first ingredient of the method of particlediagrams is the quantity called the argument which is the power in l of a binomialexpression and can be shown using Stirling’s formula lim n →∞ n ! = √ πn (cid:16) ne (cid:17) n to be given by arg Y n l − a n b n ! i n = X n i n b n . (6.5.1)Arguments are a simplifying feature of the method of particle diagrams, workingto seperate those terms which will not contribute to the limit value of themoments as l → ∞ but simultaneously leaving enough mathematical structurebehind so that the limiting result can be calculated. Arguments also providethe constraints neccessary to optimise loops.2. Particle Diagrams (A graph structure to represent factors of A µνρσ ). Theparticle diagrams, although themselves not essential to the calculations, aid usin visualising “what is going on” as we proceed to calculate the arguments andidentify loops in the diagram for each term of the higher moments. Particlediagrams form the bridge between arguments and loops. They also provide away of identifying visually, as opposed to analytically, all the loops in a giventerm. Particle diagrams found in calculations for lower moments re-appear incalculations for higher moments so that calculating successive moments makeslater work easier. It also means that the standard diagram, Fig 6.2, is found incalculations for all the higher moments, even though it’s maiden appearance isin the calculation of the fourth moment.3. Loops (Intersections between states). Loops represent intersections betweenstates. These are the single-particle state labels shared between the many-bodystates (which are sets of m state labels). Loops also represent the sets whichmust be summed over in order to calculate the trace which yields the moments β n = N tr( H nk ) (cid:16) N tr( H k ) (cid:17) n . (6.5.2)36 CHAPTER 6. CONCLUSIONS µ ρνσ Figure 6.2: The particle diagram of the term A µνρσ A σµνρ is the standard diagram.Since diagrams from lower order moments appear again in calculations of highermoments the standard diagram appears in calculations of all the moments. Thestandard diagram is also an example of a diagram whose value takes the form of (6.2.3)since the states | µ i and | ρ i together determine the single-particle states contained in | σ i and | ν i .Finally and most importantly, loops determine the argument of a diagram, sothat the sum of loops in a diagram must be maximised.Together these three ingredients form the method of particle diagrams. Although themethod has been used here to calculate moments for the eGUE it can be used tocalculate statistics for other symmetry classes as well. The most interesting of theseis the embedded gaussian symplectic ensemble (eGSE) about which virtually nothingis currently known. The author has also calculated the fourth and sixth moments forthe case where the potential satisfies the eGOE symmetry of (4.1.21) and moreover isembedded in a fermionic state space. In this instance it is seen that, conditional ontaking the limit l → ∞ , these moments are identical for the eGUE and eGOE.To end, it should be noted that although particle diagrams are used here tocalculate moments for all 0 ≤ k ≤ m , the results coincide with Wigner’s semi-circlelaw as expected for the canonical case k = m as well. In fact, for k = m the dashedbonds h must necessarily contain k = m elements, whereas the solid bonds f contain m − k = 0 elements. This in turn results in diagrams which are equivalentto those shown in section 3.2.3 where cycles are represented as partitions of regularpolygons, or equivalently Dyck paths..6. RELATED LITERATURE AND WIDER CONTEXT 137 The unification of random matrix theory offers the possibility that the landscapeof statistical quantities of random matrices can be studied en masse , with a singleequation for each quantity. Each statistical quantity found in canonical randommatrix theory would then be just a special case ( k = m ) of an equation determinedby the parameters k, m and l as well as the symmetry conditions placed on thematrix elements. There is another paradigm, however, which is larger still thanthe embedded state space and which amalgamates and connects a vast array ofphysical and mathematical models. This is the paradigm of many-body potentials .The set of many-body potentials includes random matrix theory, both its canonicaland unified form (6.1.1), as well as countless other models. These models do notshare the same hamiltonian but are unified by the similarities in their mathematicalstructure. Together they explain a diverse array of disparate phenomena from thestudy of quantum spin hypergraphs to the application of neural networks implementedalgorithmically by computers (essentially “artificial brains”) which underly mostmodern online services from Google, Amazon, Netflix and countless other technology-driven enterprises. The fundamental form underlying all many-body potentials isgiven by the equation H = X k k v ( k , k ) A † k B k (6.6.1)where | k | = | k | = k are k -tuples. Here A k and B k are any k -tuple operators.They can be complex and multidimensional so, for example, they can be matrices.The coefficients v ( k , k ) are numbers, possibly complex and if they are randomvariables one refers to (6.6.1) as a random many-body problem. The defining featureof many-body potentials is the fact that the operators A k and B k are k -tupleoperators.38 CHAPTER 6. CONCLUSIONS m -tuples k -tuples Figure 6.3: Illustration of a many-body problem, given by (6.6.1). The hamiltonian is asum over k -tuples in an m -tuple with k ≤ m . The m -tuples usually represent quantummany-body states, and the k -tuple operators A † k B k with coefficient v ( k , k ) givethe interaction terms between k -tuples of single-particle states within the compound m -body state. The embedded form of RMT (6.1.1) introduced by Mon and French [MF75] coincideswith the general form (6.6.1) when the k -tuple operators A k = a † j B k = a i (6.6.2)and v ( k , k ) = v ji where v ji is a Gaussian random variable and as before j =( j , j , . . . , j k ). These ensembles have been studied before by [MF75, FW70, BRW01a,Kot03, NO14, Sre02] among others. For the limit case k/m → ∞ it was found.6. RELATED LITERATURE AND WIDER CONTEXT 139by [MF75] that the distribution is a Gaussian. This is corroborated by the results asdiscussed in section 6.3 above. Additionally, before the method of particle diagramswas used to calculate the same results Benet, Rupp and Weidenmüller (BRW) hadalready shown using their eigenvector expansion method that the fourth moment ofthe eGUE is given by (6.1.5). They also showed using a mixture of supersymmetry andeigenvector expansions that the moments are semi-circular in the canonical domain k > m [BRW01a]. The method of particle diagrams, which is the latest addition tothe family of embedded many-body methods, reproduces the fourth moment of theeGUE as shown by BRW and adds the sixth and eighth moments for all 0 ≤ k ≤ m . Italso reveals the internal structure of the moments in the critical domain as discussedin section 6.2. An area of current research which yields fascinating similarities to the embeddedmany-body state space is the study of quantum spin graphs. These are many-bodypotentials taking the form H n = n X ( j , ··· ,j k ) ∈ Γ n X a , ··· ,a k =1 α a , ··· ,a k ( j , ··· ,j k ) σ ( a ) j · · · σ ( a k ) j k (6.6.3)Comparing this with (6.6.1) shows that the k -tuple operator is A k B k = σ ( a ) j · · · α ( a k ) j k (6.6.4)and the coefficients are v ( k , k ) = α a , ··· ,a k ( j , ··· ,j k ) . (6.6.5)In this context k is the number of vertices in every hyperedge of the graph Γ. Viewing(6.6.3) in the context of the general form (6.6.1) shows us that this is a genericmany-body hamiltonian with Pauli spin tuples acting as the k -tuple operators. It isalso known as a quantum spin glass model. If k = 2 the potential coincides with aquantum spin chain model. These hamiltonians, particularly the form with k = 2, are40 CHAPTER 6. CONCLUSIONSthe focus of a growing body of research which includes [ES14,HMH05,GFS12,KLW14].The work of [ES14] is particularly interesting in the context of this thesis, since it wasproved there that the level density of a random hypergraph (6.6.3) takes the formof a Gaussian distribution for k (cid:28) √ m , a semi-circle for k (cid:29) √ m and a compactlysupported density function for values of k and m satisfying lim m →∞ k √ m = λ > k = m , the results of [ES14] suggest it is at k ∼ √ m for thequantum spin hypergraph potential. Finally, whereas for the eGUE one takes l → ∞ ,for spin graphs there are only two available states, up and down, so that l = 2. Hence,although the embedded RMT potential (6.1.1) and the quantum spin graph potential(6.6.3) are clearly different, they also share many statistical features and both modelstake the form of a generic many-body potential (6.6.1). Exploring their statisticalsimilarities further could yield some remarkable insights into many-body potentials. Many-body potentials of the form (6.6.1) are also present in research into artificialintelligence, notably machine learning neural network models which attempt tosimulate some basic functionality of a generic brain. These machine learning networksare used by the vast majority of modern technology companies such as Google (search),Amazon (online shopping), Netflix (movie streaming) and IBM (most famously init’s Watson machine which is now the world champion of the popular quiz showJeopardy). Neural network models taking the form of a many-body potential (6.6.1)are referred to variously as higher-order Boltzmann machines [Sej86], higher-orderneural networks [GM87] or simply spin-glass models [AGS85]. The general form ofthe quantity analogous to the potential in neural networks is given by H = 1 k X i i ...i k ∈ Γ v i i ...i k s i s i . . . s i k (6.6.6).6. RELATED LITERATURE AND WIDER CONTEXT 141where Γ is the graph of the neural network, the v j are weights and the s i take binaryvalues [Sej86]. Models used in most applications are optimised in order to makedecisions, and so are deterministic. However, by optimising the paramaters of (6.6.6)over a state space of inputs and outputs, s i , the optimisation phase of these modelsinvolves the artificial brain exploring a subset of the randomised phase space of themodel. ..... ibliography [ABF11] Akemann, Baik, and Di Francesco. The Oxford Handbook of RandomMatrix Theory . Oxford University Press, 2011.[AGS85] Daniel J Amit, Hanoch Gutfreund, and Haim Sompolinsky. Spin-glassmodels of neural networks. Physical Review A , 32(2):1007, 1985.[AMPZJ94] E. Akkermans, G. Montambaux, J. L. Pichard, and J. Zinn-Justin,editors. Les Houches, Session LXI . Elsevier, 1994.[Bar91] John. D. Barrow. Theories of Everything . Oxford University Press, 1991.[Bee94] C.W.J. Beenakker. Universality of Brezin and Zee’s spectral correlator. Nuclear Physics B , 422(3):515 – 520, 1994.[Boh60] Niels Bohr. The Unity of Human Knowledge . TO ADD, 1960.[BRW01a] L. Benet, T. Rupp, and H. A. Weidenmueller. Nonuniversal behaviorof the k-body embedded gaussian unitary ensemble of random matrices. Physical Review Letters , 87(1), 2001.[BRW01b] L. Benet, T. Rupp, and H. A. Weidenmueller. Nonuniversal behaviorof the k-body embedded gaussian unitary ensemble of random matrices. Annals of Physics , 292(1), 2001.[DN05] Willem H Dickhoff and Dimitri Van Neck. Many-Body Theory Exposed! World Scientific, 2005. 142IBLIOGRAPHY 143[DRHR60] J. S. Desjardins, J. L. Rosen, W. W. Havens, and J. Rainwater. Slow neu-tron resonance spectroscopy. II. Ag, Au, Ta. Physical Review , 120(2214),December 1960.[DRHR64] J. S. Desjardins, J. L. Rosen, W. W. Havens, and J. Rainwater. Neutronresonance spectroscopy. III. Th and U . Physical Review , 134(B985),June 1964.[Dys62a] F. J. Dyson. Statistical theory of the energy levels of complex systems.I. Journal of Mathematical Physics , 3(140), 1962.[Dys62b] F. J. Dyson. Statistical theory of the energy levels of complex systems.II. Journal of Mathematical Physics , 3(157), 1962.[Dys62c] F. J. Dyson. Statistical theory of the energy levels of complex systems.III. Journal of Mathematical Physics , 3(166), 1962.[ES14] Laszlo Erdos and Dominik Schroder. Phase transition in the densityof states of quantum spin glasses. Mathematical Physics, Analysis andGeometry , 17(3-4), 2014.[Fey67] Richard. P. Feynman. The Character of Physical Law . MIT Press, 1967.[FLS05] Richard. P. Feynman, Robert B. Leighton, and Matthew Sands. TheFeynman Lectures on Physics . Addison Wesley, 2005.[FW70] J. B. French and S. S. M. Wong. Validity of random matrix theories formany-particle systems. Physics Letters B , 33(7):449 – 452, 1970.[GFS12] Aviva Gubin and Lea F. Santos. Quantum chaos: An introduction viachains of interacting spins 1/2. American Journal of Physics , 80(3):246–251, 2012.[GM87] C Lee Giles and Tom Maxwell. Learning, invariance, and generalizationin high-order neural networks. Applied optics , 26(23):4972–4978, 1987.44 BIBLIOGRAPHY[GMGW98] T. Guhr, A. Mueller-Groeling, and H. A. Weidenmueller. Randommatrix theory in quantum physics: Common concepts. Annals of Physics ,299:189–425, 1998.[Haa10] Fritz Haake. Quantum Signatures of Chaos . Springer, 2010.[HMH05] M. Hartmann, G. Mahler, and O. Hess. Spectral densities and partitionfunctions of modular quantum systems as derived from a central limittheorem. Journal of Statistical Physics , 119(5–6):1139–1151, 2005.[HR51] W. W. Havens and L. J. Rainwater. Slow neutron velocity spectrometerstudies. IV. Au, Ag, Br, Fe, Co, Ni, Zn. Physical Review , 83(1123),September 1951.[HW95] G. Hackenbroich and H. A. Weidenmüller. Universality of random-matrixresults for non-gaussian ensembles. Phys. Rev. Lett. , 74:4118–4121, May1995.[KLW14] J. Keating, N. Linden, and H. Wells. Random matrices and quantumspin chains. arXiv:1403.1114 [math-ph] , 2014.[Kot03] V.K.B. Kota. Convergence of moment expansions for expectation valueswith embedded random matrix ensembles and quantum chaos. Annalsof Physics , 306(1):58 – 77, 2003.[Kot14] Venkata Krishna Brahmam Kota. Embedded Random Matrix Ensemblesin Quantum Physics . Springer, 2014.[Kre72] G. Kreweras. Sur les partitions non croisees d’un cycle. Discrete Mathe-matics , 1(4):333–350, 1972.[KRLS10] Koekoek, Roelof, Lesky, and Peter A. Swarttouw. Hypergeometric Or-thogonal Polynomials and Their q-Analogues . Springer, 2010.[Meh04] Madan Lal Mehta. Random Matrices . Elsevier, 2004.IBLIOGRAPHY 145[MF75] K. K. Mon and J. B. French. Statistical properties of many-particlespectra. Annals of Physics , 95(1), 1975.[MHR53] E. Melkonian, W. W. Havens, and L. J. Rainwater. Slow neutron velocityspectrometer studies. V. Re, Ta, Ru, Cr, Ga. Physical Review , 92(702),November 1953.[NO88] John W. Negele and Henri Orland. Quantum Many-Particle Systems .Addison-Wesley, 1988.[NO14] Yoshifumi Nakata and Tobias J. Osborne. Thermal states of randomquantum many-body systems. Physical Review A , 90(050304(R)), 2014.[RDRH60] J. L. Rosen, J. S. Desjardins, J. Rainwater, and W. W. Havens. Slowneutron resonance spectroscopy. I. U . Physical Review , 118(687), May1960.[Sej86] Terrence J Sejnowski. Higher-order Boltzmann machines. In AIP Con-ference Proceedings , volume 151, pages 398–403, 1986.[SM14] Rupert A. Small and Sebastian Mueller. Particle diagrams and embeddedmany-body random matrix theory. Physical Review E , 90(010102(R)),2014.[SM15] Rupert A. Small and Sebastian Müller. Particle diagrams and statisticsof many-body random potentials. Annals of Physics , (0):–, 2015.[Sre02] M. Srednicki. Spectral statistics of the k -body random-interaction model. Physical Review E , 66(046138), 2002.[Wei92] Steven Weinberg. Dreams of a Final Theory . Random House, 1992.[Wig51a] Eugene. P. Wigner. On a class of analytic functions from the quantumtheory of collisions. Annals of mathematics , 53(1), 1951.46 BIBLIOGRAPHY[Wig51b] Eugene. P. Wigner. On the statistical distribution of the widths andspacings of nuclear resonance levels.