The U(1) Lattice Gauge Theory Universally Connects All Classical Models with Continuous Variables, Including Background Gravity
Ying Xu, Gemma De las Cuevas, Wolfgang Dür, Hans J. Briegel, Miguel Angel Martin-Delgado
aa r X i v : . [ qu a n t - ph ] F e b The U(1) Lattice Gauge Theory Universally Connects AllClassical Models with Continuous Variables, IncludingBackground Gravity
Ying Xu , , Gemma De las Cuevas , , Wolfgang D ¨ur , Hans J. Briegel , and Miguel A. Martin–Delgado Institut f¨ur Quantenoptik und Quanteninformation,Technikerstraße 21a, 6020 Innsbruck, Austria Institut f¨ur Theoretische Physik, Universit¨at Innsbruck,Technikerstraße 25, A-6020 Innsbruck, Austria Departamento de F´ısica Te´orica I, Universidad Complutense, 28040 Madrid, SpainE-mail: [email protected], [email protected],[email protected], [email protected],[email protected]
Abstract.
We show that the partition function of many classical models with continuousdegrees of freedom, e.g. abelian lattice gauge theories and statistical mechanical models, canbe written as the partition function of an (enlarged) four–dimensional lattice gauge theory(LGT) with gauge group U ( ) . This result is very general that it includes models in differentdimensions with different symmetries. In particular, we show that a U ( ) LGT defined in acurved spacetime can be mapped to a U ( ) LGT with a flat background metric. The resultis achieved by expressing the U ( ) LGT partition function as an inner product between twoquantum states.PACS numbers: 03.67.-a, 11.15.Ha, 03.67.Lx, 75.10.Hk, 05.50.+q
Keywords : Lattice gauge theory, Statistical mechanics, Quantum information (1) LGT Connects All Classical Models with Continuous Variables
1. Introduction
The partition function Z is the keystone in both statistical mechanics [1] and quantum fieldtheory [2]. Thermodynamic quantities, like the free energy, the entropy, correlation functions, etc. can be evaluated once Z is known as a function of physical parameters (such as theinverse temperature b , couplings constants J , external fields and other order parameters).On the other hand, gauge theories have proven to be crucial in the description of nature. Inparticular, quantum electrodynamics (QED) is described by a U ( ) gauge theory.It has recently been shown [3, 4] that the partition function of any classical spin modelcan be mapped to that of an (enlarged) four–dimensional (4 D ) lattice gauge theory with gaugegroup Z (see also the original idea [5] and closely related works [6, 7, 8]). More precisely, ifone tunes the coupling strengths of the partition function of a (large enough) 4 D Z LGT, thiswould equal the partition function of a classical spin model in any dimension, with any typeof interaction pattern (including arbitrary many–body interactions), and thus also includesmodels with local and global symmetries. To obtain this, one expresses the partition functionof a large class of models as an inner product between two quantum states, and then relatesthe quantum states (see [9] for a detailed treatment of the quantum formulation). The resultis very general and unifies very different models. In this sense, the 4 D Z LGT is a complete model for this class of discrete models. In fact, these mappings from a general discrete modelto a Z LGT allow one to gain structural insight.In the present work, we show that the partition function of many continuous classicalmodels can be expressed approximately (to arbitrary precision) as the partition function of the4
D U ( ) LGT. The result holds exactly for a large class of models, namely, models whoseHamiltonian has a finite Fourier series and no constraints on the variables. This includesmodels of different geometry, and in arbitrary dimensions (1 D , 2 D , 3 D , etc. ) In the proofof the statement we generalize the quantum formulation developed previously [6, 9, 10, 5]and generate the (truncated) Fourier series of any target Hamiltonian. In parallel with the4 D Z LGT as the complete model for discrete models, the 4
D U ( ) LGT in this sense is complete for a certain class of continuous models. We also define and consider the U ( ) LGT defined in a curved spacetime background. We show that this class of models are alsoincluded in our completeness result, i.e. they can also be mapped to the flat 4
D U ( ) LGT.The physical content of this result is the following: as long as the gravity coupled to the QEDis a fixed background, it can be absorbed as if it were a flat spacetime as far as completenessis concerned. This situation has a physical counterpart in cosmology where one finds photonsin an approximate fixed curved spacetime.In this paper we will first present some basics on the U ( ) LGT in § 2 . Then we willprove the completeness of the U ( ) LGT in § 3 . Further illustrations of the completenessresult with some examples and applications will be given in § 4 . We will proceed to the U ( ) LGT in a curved spacetime and relate to the main result in § 5 . In § 6 we will generalize ourresult to a larger class of models. Finally, the conclusions will be drawn in § 7 . (1) LGT Connects All Classical Models with Continuous Variables
2. Basics on the U(1) Lattice Gauge Theory
LGTs are gauge theories on a lattice representing a discrete spacetime. Generically, LGTsare useful non–perturbative formulations of gauge theories which allow for numericalsimulations, e.g. using the Monte Carlo methods [11]. This allows one to go beyondperturbative calculations with Feynman diagrams. The abelian U ( ) LGT was introducedby Wilson [12] and Polyakov [13, 14] as a generalization of Wegner’s Ising gauge theories[15]. The U ( ) LGT can be considered a discretization of electrodynamics (a pure gaugetheory with an abelian gauge group U ( ) ) defined on discrete spacetime [16]. Non–abeliancontinuous LGTs are successful as having been proven to be asymptotically free [17] in theweak coupling limit, and important in the study of quark confinement at the strong couplinglimit [12].For abelian LGTs it is possible to simulate numerically the continuum limit, D →
0, in anappropriate way, and verify that the resulting theory is the actual QED without confinement.This verification is important since on the lattice, most LGTs show confinement which is anartifact. It so happens that in the strong coupling limit of an LGT, the property that the gaugegroup is compact is essential for observing confinement, regardless of whether it is abelian ornon–abelian. Related to this, the phase diagram of LGTs is very rich and relevant for takingthe continuum limit [16], and the U ( ) LGT serves as a test ground for this purpose. Unlikethe non–abelian SU ( N ) LGTs, which have a discrete center subgroup Z N , the abelian U ( ) LGT has a continuous center subgroup which is identical to the group itself. The role of the U ( ) group on the confinement to deconfinement transition is also of interest from the point ofview of the abelian dominance hypothesis for confinement which holds that a U ( ) subgroupcontrols the non–perturbative dynamics of non–abelian gauge theories [18].In the mean time LGTs have emerged as interesting theories by themselves. They areexamples of models with local symmetries and non–local order parameters. They exhibitphases which do not appear in the continuum limit [16]. The phases of the U ( ) LGT can becharacterized by the Wilson loop (see (1) below) which is an gauge invariant order parameterof the model. In the confined phase, this order parameter obeys an area law, whereas in theunconfined phase it obeys a perimeter law. The ’t Hooft loop is a dual variable to the Wilsonloop, and it constitutes another order parameter.There are compact and non–compact 4
D U ( ) LGTs [11, 19, 20, 21]. In the compact U ( ) LGT the degrees of freedom are exponentials of the edge degrees of freedom. Thus,they are directly elements of the U ( ) group which is compact, thereby the name. On theother hand in the non–compact U ( ) LGT the degrees of freedom are associated directly tothe edges of the lattice, which do not need to be elements of the U ( ) group. The formerone only has a deconfined phase with massless photons, thus one recovers QED in the limitof continuous spacetime. The latter one has two phases: a weak coupling phase, where themodel has massless photons (gapless excitations), and a strong coupling phase, where thereare massive photons and magnetic monopoles. The photons are screened by the monopoles,which corresponds to a mechanism of confinement of electrical charge. These phases arecharacterized with Wilson loops and ’t Hooft loops as mentioned above. In this paper, we will (1) LGT Connects All Classical Models with Continuous Variables U ( ) LGT.We briefly summarize the formulation of the U ( ) LGT as follows (with our notationclose to that of [16]). It is illustrative to derive the Lagrangian of classical electrodynamicsfrom that of U ( ) LGT in the limit of continuous spacetime [16]. Let us consider a 4 D squarelattice as the discretized spacetime. Let D denote the lattice spacing and ˆ a , ˆ b etc. the unit basisvectors. We denote vertices, edges and faces of the lattice by v , e and f , respectively. The setof all vertices, edges and faces is denoted by V , E , and F and the number of elements in eachset by | V | , | E | , and | F | accordingly. We choose a direction for each edge, see Fig. 1. The Figure 1. (a) A face of the square lattice. ˆ a and ˆ b are unit basis vectors and each edge isassigned a direction ± ˆ a or ± ˆ b . The four vertices of the face are labeled by P , P + ˆ a D , P + ˆ b D and P + ˆ a D + ˆ b D . (b) For notational convenience, we use a single labeling for the vertices,namely, n (the n th vertex), n + a , n + b and n + a + b , with a ≡ ˆ a D and b ≡ ˆ b D . gauge field A a ( n ) is defined along each edge incident to vertex n with direction ˆ a indicated bythe sub–index (see Fig. 2). The Wilson loop for an elementary face (such as in Fig. 1(a)) is U face = e i D G A a ( P ) e i D G A b ( P + D ˆ a ) e − i D G A a ( P + D ˆ b ) e − i D G A b ( P ) = e i D G ( [ A b ( P + D ˆ a ) − A b ( P )] − [ A a ( P + D ˆ b ) − A a ( P )] ) , (1)in which G is the coupling constant. The exponential of the second line in (1) resemblesthe field tensor F ab = ¶ a A b − ¶ b A a . The action is constructed from the Wilson loops. fornotational convenience, we switch to the simplified notation as in Fig. 1(b). We make thefollowing redefinition of the gauge field: q a ( n ) ≡ D G A a ( P ) , (2)as well as the following conventions: q − a ( n + a ) ≡ − q a ( n ) , Q ab ( n ) ≡ q a ( n ) + q b ( n + a ) + q − a ( n + a + b ) + q − b ( n + b ) . (3)We can take the redefined gauge field q a as defined on the edges instead of on the vertices,see Fig. 2. With these redefined gauge field notations, the exponential in (1) is equal to i Q ab (which is a sum over the four q e field variables along the edges of a face, e ∈ f ). Q ab can bethought of as defined for each face and denoted alternatively by Q f : Q f = q + q − q − q = (cid:229) e ∈ ¶ f c e q e . (4) (1) LGT Connects All Classical Models with Continuous Variables Figure 2. (a) The gauge fields q a as defined along the edges. (b) A further simplified notationwhere the fields along the four edges of a face are denoted by q , q , q and q , and q e ingeneral. where ¶ f denotes the boundary of face f . The signs of the four edge variables in (4)are chosen to be consistent with Fig. 2 (b). In general, a face can contain edges pointingclockwise and others counterclockwise. So that c e = [ − ] if e is oriented clockwise[counter–clockwise]. (Note that only the relative sign of the q e ’s is relevant for the action (5) below.)Now the Wilson–Kogut action of the U ( ) LGT can be written as S = G (cid:229) f ∈ F (cid:2) − cos Q f (cid:3) . (5)This (Euclidean) action in the naive continuum limit is seen to be G (cid:229) f ∈ F Q f ∼ R d x F ab ,consistent with the classical U ( ) theory. Also, the action takes the same form as in (5) in anyspacetime dimension. This action is invariant under the gauge rotation applied on any vertex v : g v = (cid:213) e : v ∈ ¶ e U c e e , e ∈ E (6)where U e is an element of the gauge group, in our case, U e ∈ U ( ) , and ¶ e denotes theboundary of e ( i.e. the product in (6) applies to all edges e incident to vertex v ). Finally, thepartition function (which corresponds to the Euclidean path integral) of this pure U ( ) gaugetheory on a lattice takes the following form: Z = Z p − p (cid:213) e ∈ E d q e ! exp ( − G (cid:229) f ∈ F (cid:2) − cos Q f (cid:3)) . (7)Note that the first constant term in the sum of the exponential in (7) only introduces anoverall constant factor which can be omitted.‡ In addition, the coupling constant G can bemade face–dependent. Accordingly, we will start with the partition function of the followinginhomogeneous model with local coupling constant J f for each face (The coupling constant ‡ Ignoring constant factors of the partition function corresponds to an overall shift in the action (or theHamiltonian) which never changes any observable nor the equations of motion. It only introduces an additionalconstant to the free energy as we take the logarithm. Observables only relate to differences (or derivatives) infree energy. (1) LGT Connects All Classical Models with Continuous Variables G and (possibly) an inverse temperature b are all contained in the local J f ’s): Z = Z p − p (cid:213) e ∈ E d q e ! exp ( (cid:229) f ∈ F J f cos Q f ) . (8)
3. 4D U(1) Lattice Gauge Theory as a Complete Model
In this section we show that the 4
D U ( ) LGT is complete in the sense that a large class ofclassical partition functions with continuous degrees of freedom can be represented as specialcases of the partition function of this model. In § 3.1 we define the set of models consideredin our completeness result. Then, we present some tools in § 3.2 and § 3.3 that we will requirefor the proof of the main result. Finally, our completeness result is proved in § 3.4 .
Our completeness results embrace all continuous classical ‘spin’ models, i.e. models with aHamiltonian satisfying the following conditions:(i) The Hamiltonian depends on a set of N continuous real variables { x j | j = , , . . . , N } andeach variable takes value within a finite interval, i.e.x j ∈ [ a j , b j ] , ∀ j . (9)(ii) The Hamiltonian is a sum over K -body interactions with 1 ≤ K ≤ N , i.e. H (cid:0) { x j } (cid:1) = N (cid:229) K = (cid:229) { K − body } H ( K ) (cid:0) { x j } (cid:1) . (10)(iii) There is no constraint on this set of variables, i.e.x j with j = , , . . . , N are independent variables . (11)Furthermore, we assume that each K -body Hamiltonian is a well–behaved function whichallows a Fourier series expansion over the x j variables. (Note that the spacial dimension ofmodels under consideration is arbitrary.)With the above conditions, we first normalize the ranges of variables by a linear changeof variables from { x j } to { q j } q j = p b j − a j (cid:18) x j − a j + b j (cid:19) , ∀ j (12)such that each q j ∈ [ − p , p ] . The relation (12) can be inverted x j = b j − a j p q j + a j + b j so that theHamiltonian is reexpressed in terms of these normalized variables and denoted by H (cid:0) { q j } (cid:1) .Next, we make a Fourier series expansion of each K -body Hamiltonian H ( K ) (cid:0) { q j } (cid:1) over the q j variables: H ( K ) (cid:0) { q j } (cid:1) = (cid:229) { m j } H ( K ) { m j } exp i K terms (cid:229) j m j q j ! = (cid:229) { m j } ´ H ( K ) { m j } cos K terms (cid:229) j m j q j ! − ` H ( K ) { m j } sin K terms (cid:229) j m j q j ! . (13) (1) LGT Connects All Classical Models with Continuous Variables m j ∈ Z , ∀ j and the Fourier coefficients ´ H ( K ) { m j } and − ` H ( K ) { m j } are all real ( ´ and ` denotes real and imaginary parts, respectively). Therefore the general Hamiltonian H in (10)is now written as a Fourier series over a set of basis functions { cos (cid:229) j m j q j ! , sin (cid:229) j m j q j ! (cid:12)(cid:12) | m j | = , , , . . . , j = , . . . , N } . (14) We present a quantum representation of the partition function Z in (8). First, assign aquantum state | Q f i to each face: | Q f i = | (cid:229) e ∈ f c e q e i . (15)Then we define the following quantum state | y i which contains the interaction pattern (By‘interaction pattern’ we mean the lattice or graph structure representing which variables areinteracting): | y i = Z p − p (cid:213) e ∈ E d q e ! O f ∈ F | Q f i . (16)Next we define another state | a i which contains the (Euclidean) weight for each configurationof the q e variables of the whole lattice: | a i = Z p − p (cid:213) e ∈ E d q e ! (cid:213) f ∈ F e J f cos Q f ! O f ∈ F | Q f i . (17)In the following we show that h a | y i is proportional to the partition function Z in (8).Let us consider the linear transformation from the edge variables (gauge fields) q e to the facevariables (field tensors) Q f . We define two column vectors: q = (cid:0) q . . . q e . . . q | E | (cid:1) t Q = (cid:0) q . . . q f . . . q | F | (cid:1) t . (18)where t denotes transposition. They are related by a linear transformation Q = I · q where I isthe face–edge incidence matrix with matrix elements I f , e = e ∈ ¶ f and 0 otherwise.We always assume that all edge variables are independent. i.e. rank I = | E | . For a squarelattice with periodic boundary conditions in D -dimension ( D ≥ | E | and | F | such that | F | = D − | E | . If | F | ≤ | E | we add a set of linearly independent auxiliaryface variables F aux , where | F aux | = | E | − | F | . This results in an | E | × | E | incidence matrix ˜ I , viz. Q = (cid:0) Q . . . Q | F | . . . Q | E | (cid:1) t = ˜ I · q . (19)Geometrically, this amounts to saying that we are introducing new auxiliary faces to theoriginal lattice and these new faces need not be squares. If | F | ≥ | E | , we pick up a maximallyindependent set of face variables and eliminate linearly dependent rows until the incidencematrix has a size | E | × | E | . So that we may write Q = (cid:0) Q . . . Q | E | (cid:1) t = ˜ I · q , with again an (1) LGT Connects All Classical Models with Continuous Variables | E | × | E | incidence matrix ˜ I . Because not all face variables appearing in the quantum states | y i (16) and | a i (17) are independent, when we take the inner product of | y i with | a i , formalinfinities would arise. These infinities, e.g. d ( Q f − Q ′ f ) = d ( Q f − Q ′ f ) d ( ) , always take theform of powers of d ( ) . This problem can be treated in the following way. We introduce alarge ‘momentum’ truncation L in the Fourier space (inverse space) for each face variable. Sothat d ( Q = ) = p (cid:229) n ∈ Z e i n Q ∼ L p with L → ¥ . Therefore when taking the inner product weshall have an overall factor (cid:0) L p (cid:1) | F |−| E | which tends to infinity formally. However, this factordoes not affect any observable because it only introduces an additive constant to the freeenergy as we take the logarithm of the partition function. We shall call this a ‘regularization’method that allows us to make sense of the expressions that come out with harmless infinities.(A similar constant factor has been omitted going from (7) to (8), see also (25) below andthe argument there for comparison.) Finally we see that h a | y i is proportional to the partitionfunction Z in (8): h a | y i = Z p − p (cid:213) e ∈ E d q ′ e d q e ! (cid:213) f ∈ F e J f cos Q ′ f ! (cid:213) f ∈ F d ( Q ′ f − Q f )= (cid:20) | det ˜I | R p − p (cid:16) (cid:213) f ∈ F aux d Q ′ f (cid:17)(cid:21) Z = (cid:20) | det ˜I | ( p ) | E |−| F | (cid:21) Z , | F | ≤ | E | (cid:20) | det ˜I | (cid:0) L p (cid:1) | F |−| E | (cid:21) Z , | F | ≥ | E | = const . × Z . (20) Here we show how to generate Fourier series expansion (13) from a 4
D U ( ) LGT, that is,how to generate the basis functions of (14). In this way a subsystem of our complete modelwill behave as the target model; more precisely, the Hamiltonian of the complete model onthis subsystem will coincide with the (Fourier series of the) Hamiltonian of the target model.We remark, though, that we will generate the basis functions of (14) only for bounded valuesof m j , 0 ≤ m j ≤ M j for all j , where the M j ’s are large, positive integers. We will show this byfirst introducing the merge and deletion rules, and by then explaining how to obtain arbitrarymany–body interactions and build the Fourier basis from these fundamental rules (an explicitconstruction of many–body interactions with more technical details is given in Appendix A). Merge and deletion rules . For simplicity, we consider the state | a i of (17) defined onlyon two faces (see Fig. 3(a)): | a a , b i = Z p − p (cid:213) e = d q e ! e J a cos Q a e J b cos Q b | Q a i| Q b i . (21)We define the merge rule on, say, face a . Our aim is to obtain a delta function d ( Q a ) , in the (1) LGT Connects All Classical Models with Continuous Variables J a → ¥ . We consider a slight modification of (21): | ˜ a a , b i = N ( J a ) Z p − p (cid:213) e = , e = d q e d Q a ! e J a ( cos Q a − ) e J b cos Q b | Q a i| Q b i , (22)where N ( J a ) is defined as N ( J a ) = r J a p . (23)This implies that we will obtain the target partition function Z with the prefactors: M terms (cid:213) m e J m N ( J m ) ! Z (24)where M is the number of faces where the merge rule has been applied, and J m denotes thecoupling strength of the merged face, J m → ¥ . The usual quantity of interest, such as thefree energy per particle in the thermodynamic limit, is shifted by a known amount (again,thermodynamic quantities involving derivatives of the free energy are not altered using ourregularization method§):lim N → ¥ − FN = lim N → ¥ N ln Z + lim N → ¥ MN (cid:18) (cid:229) m J m −
12 ln (cid:229) m J m p (cid:19) (25)where N is the number of particles in the classical U ( ) LGT model.We have N ( J a ) Z p − p d Q a e J a ( cos Q a − ) = √ p J a e J a I ( J a ) , (26)where I ( J a ) is the modified Bessel function of the first kind. The asymptotic behavior of thisfunction is I ( J a ) ∼ e J a √ p J a ( + O ( J a )) , J a → ¥ . (27)Thus it follows thatlim J a → ¥ N ( J a ) e J a ( cos Q a − ) = d ( Q a ) , (28)as desired. Therefore,lim J a → ¥ Z p − p (cid:213) e = , e = d q e ! N ( J a ) e J a ( cos Q a − ) e J b cos Q b | Q a i| Q b i = Z p − p d q d q d q d q d q d q e J b cos ˜ Q b | i| ˜ Q b i . (29)where the variable Q b has become ˜ Q b after imposing the constraint of the delta function:˜ Q b = q + q + q − q − q − q . (30) § The prefactor of the partition function in (24), as well as the additive extra term to the free energy in (25),are all formally infinities. As far as these infinities appear in a controllable way, we can get them off by firstchoosing the coupling strength J m ’s large but still finite. So that a shift in the free energy has been introducedsuch as in (25). Then we can set J m ’s to infinity after calculation of observables. No observable will be affectedbecause they all involve a difference or derivative in the free energy. Since the additional term does not containany physical parameter, it will not contribute even being formally infinite. (1) LGT Connects All Classical Models with Continuous Variables === PSfrag replacements q q q q q q q q q q q q q J = ¥ J J = ¥ J = J (a)(b)(c) Figure 3.
A blue square represents an interaction in that face, whereas a shaded blue facerepresents an face with an infinite coupling strength. (a) Merge rule: by letting J f → ¥ of theleft face, this is merged with the face on its right. The resulting face only depends on the spinson its boundary. (b) The resulting face can be merged again with a neighboring face by letting J f → ¥ . Note the resulting direction of the arrows. (c) Deletion rule: (the interaction on) aface is deleted by setting J f = Note that this corresponds to a 6–body interaction of the same type, and with interactionstrength J b . Thus, letting J a → ¥ effectively merges face a and b . Note also that the variableswith opposite sign have opposite pointing directions in the resulting face, see Fig. 3(a).It is straightforward to see that the same derivation applies for | a i defined on all faces.In this case, one would only substitute the condition of the delta function, Q a =
0, on the facewith which it has to be merged. The process can be concatenated, that is, the face resultingfrom a merge rule can be merged again with a neighboring face, thereby becoming a 8–bodyinteraction (see Fig. 3(b)). Note that, whenever one face is merged to another, the new edgeshave the opposite direction than the original ones, as noted above.The deletion rule works by setting the J f =
0, which results in switching off theinteraction in that face (see Fig. 3(c)).
Construction of many–body interactions.
The Fourier basis functions of the set (14) canbe generated by making repeated use of the fundamental merge and deletion rules presentedabove. More specifically, one first generates several “copies” of a certain variable, that is, onereplicates, say, m j times the variable q j , as required in (14). Then one generates the cosineand the sine of an arbitrary sum of such term (also with the corresponding signs). We refer thereader to Appendix A for technical details of these explicit constructions. Here we emphasizethat this construction is achieved by applying the merge and the deletion rule on specific faces (1) LGT Connects All Classical Models with Continuous Variables D lattice, and by fixing some variables using the gauge symmetry of the model (“gaugefixing”). The latter is a procedure that can be carried out so long as the edges whose variablehas been fixed do not form a closed loop [22]. This is precisely why we need to resort to a4 D lattice, since only then is our construction of the interactions (14) free of closed loops (thesame case as in [3]). We are now in the position of proving the main result of this paper. In § 3.2 we have expressedin general the U ( ) LGT partition function as an overlap between two quantum states. In § 3.3we have shown that through merge, deletion and gauge fixing in a 4
D U ( ) LGT, any basisfunctions of a Fourier series expansion can be generated (see also Appendix A). As a result,any partition function of the U ( ) LGT (with local coupling constants) of the following formcan be generated and expressed in terms of a quantum amplitude h a | y i : Z LGT (31) = Z p − p (cid:213) j d q j ! exp (cid:229) { m j } J cm j cos (cid:229) j m j q j ! + J sm j sin (cid:229) j m j q j ! with 0 ≤ m j ≤ M j , ∀ j . (32)Here all M j are (large) positive integers. Furthermore, in § 3.1 we have also expresseda general Hamiltonian with continuous variables in terms of a Fourier series with realcoefficients. Let us compare (31) with the partition function of a general classical modelsatisfying conditions (9), (10) and (11): Z classical (33) = (cid:213) j Z b j a j d x j ! e − b H { x j } = N Z p − p (cid:213) j d q j ! · exp (cid:229) { m j } − b ´ H { m j } cos (cid:229) j m j q j ! + b ` H { m j } sin (cid:229) j m j q j ! with N = (cid:213) j b j − a j p . We realize that if we choose the local coupling constants in (31) to beequal to the Fourier coefficients (times b ), i.e.J cm j = − b ´ H { m j } , J sm j = b ` H { m j } , (34)then (up to a constant factor N ) the general partition function Z classical can be approximatedby the partition function Z LGT . The approximation lies in the fact that in (33) the integers m j run from 0 to + ¥ , while in (31) it runs within a finite range, see (32). This approximationcan be made to any precision as we increase the repetition of the q j variables in the basis (1) LGT Connects All Classical Models with Continuous Variables D U ( ) LGT is complete such that thepartition function of any classical model of continuous variables (without constraints) can bemapped to that of the former. That is, by fixing some of the coupling strengthes of a 4
D U ( ) LGT, the remaining subsystem behaves as the continuous model to be simulated.We emphasize here that models satisfying conditions (9), (10) and (11) are all includedin the completeness result, regardless of their dimensions (can be larger, equal to, or smallerthan D =
4) and specific forms of interaction. Also, though the original 4
D U ( ) LGT isgauge invariant (6), this U ( ) gauge symmetry is broken after the mapping because the targetmodel no longer possesses the same gauge symmetry in general.Finally, we discuss the overhead in the system size of the complete model as a functionof the features of the target model. That is, we study how many variables are needed in thecomplete model in order to generate the Fourier series of (31) with a truncation at the M th mode (each m j ≤ M in (32)). We have seen in Appendix A that the generation of each Fouriercomponent (each element of (14)) requires a polynomial enlargement of our complete model. i.e. In order to generate a single Fourier term such as cos ( (cid:229) j m j q j ) , a number of Poly ( (cid:229) j m j ) edge variables are needed, where Poly ( · ) is a polynomial of its argument. (This fact alsoimplies that the same order Poly ( (cid:229) j m j ) of couplings need to be tuned to infinity or zero inproducing this Fourier term. See § 4.1 for a simple and explicit example.) Thus, the efficiencyis measured by the number of Fourier components to be generated in the expansion (31). Fora single K –body interaction term H K ( { q j } ) in the Hamiltonian (depending on a certain setof K variables { q j } ), we need to generate ∼ [ Poly ( M )] K number of Fourier components,where Poly ( M ) is a polynomial in M . Generally, all combinations of q j variables in K –body interactions, with 0 ≤ K ≤ N ( N is the total number of variables), may be present, thusresulting in the scaling ∼ N (cid:229) K = NK ! [ Poly ( M )] K ∼ exp ( N )[ Poly ( M )] K max (35)for the number of Fourier components. However, in most cases K does not scale with thesystem size, K = K max , and moreover only few–body interactions appear in the Hamiltonian( e.g. K = NK max ! [ Poly ( M )] K max ∼ Poly ( N )[ Poly ( M )] K max . (36)Note that this question of efficiency in generating Fourier components is unrelated to thequestion of the accuracy of the approximation of a finite Fourier series (see Appendix B forthe latter).
4. Examples and Applications of the Completeness Result
As an illustration of our completeness result, we give a few example of models whose partitionfunction can be reduced to our class of models. (1) LGT Connects All Classical Models with Continuous Variables In the XY model we have 2 D unit vectors (classical spins) ~ s i defined on a lattice. TheHamiltonian obeys O ( ) (or U ( ) ) symmetry: H XY = − (cid:229) < i j > J i j ~ s i · ~ s j = − (cid:229) < i j > J i j cos (cid:0) q i − q j (cid:1) , (37)The partition function of the XY model Z XY = Z p (cid:213) i d q i ! e b (cid:229) < i j > J i j cos ( q i − q j ) (38)is seen by itself to be of a U ( ) LGT type. Therefore, for this particular case, we do not needto generate the whole Fourier basis, but we just need to generate the 2–body interactions of(37) with the constructions explained in § 3.3 (more details in Appendix A). Here belowwe give an explicit pictorial construction of a 1
D XY model from a 4
D U ( ) LGT as atransparent example of our completeness result. In this particularly simple case we onlyneed a 3 D sublattice of the 4 D U ( ) LGT for the construction, as shown in Fig. 4 below.The thick black edges represent variables of the target XY model which eventually will builda 1 D chain (in terms of interactions). All the odd numbered variables q , q , q , etc. aredistributed along the same line. Even numbered ones q , q , etc. are distributed separatelyalong the two sides. This arrangement of variables guarantees the correct relative sign in theinteraction cos ( q i − q i + ) (to be generated after merge of faces) between every neighboringpair of variables. In the figure it is illustrated how to obtain interactions cos ( q − q ) andcos ( q − q ) by merging the blue faces and gauge fixing the red edge variables, and these arethe only two typical constructions we need. By direct repetitions all interaction terms of the XY Hamiltonian can be constructed. It is easy to see that in producing an interaction term likecos ( q − q ) , 12 couplings are taken to the infinity limit; similarly in producing cos ( q − q ) ,8 couplings are taken to infinity, etc.4.2. The Gaussian and Mean Spherical Models In both the Gaussian and the mean spherical models the Hamiltonian (also the ‘effective’Hamiltonian appearing in the expression of partition functions) is a quadratic form ofcontinuous unbounded spin variables s i ( i = , , . . . , N ): H = − (cid:229) < i j > J i j s i s j − (cid:229) i h i s i . (39)In the Gaussian model, each variable s i takes values in ( − ¥ , + ¥ ) and is assigned aprobability distribution of a Gaussian form: p ( s i ) d s i = (cid:18) A p (cid:19) e − A s i d s i , i = , , . . . , N , (40)so that the average value is h s i i = A . The partition function of the Gaussian model reads Z Gauss = (cid:18) A p (cid:19) N Z + ¥ − ¥ (cid:213) i d s i ! e − A (cid:229) i s i + b (cid:229) < i j > J i j s i s j + b (cid:229) i h i s i . (41) (1) LGT Connects All Classical Models with Continuous Variables q q q q q J J J J Figure 4.
Generation of the 1
D XY model from a 4
D U ( ) LGT. Red edges indicate edgeswhose variables have been gauge fixed, thick black edges indicate edges containing variablesthat are present in the target model (they will build the 1 D chain). Black arrow indicate cubeswhere the variable is replicated according to Fig. A1 (Appendix A). Interactions of the kindcos ( q i − q i + ) take place in blue prisms, and are a particular case of the interaction shownin Fig. A4 (Appendix A). Indication of merged faces (blue faces in the figures) has beensimplified here to avoid overloading. Alternatively, in the mean spherical model, instead of introducing a probabilitydistribution, we impose a constraint on the average value of (cid:229) i s i such that h N (cid:229) i = s i i = N . (42)This constraint can be incorporated in the Hamiltonian by introducing a spherical field l : H mean = − (cid:229) < i j > J i j s i s j − (cid:229) i h i s i + l (cid:229) i s i . (43)The partition function of the mean spherical model is now Z mean = Z + ¥ − ¥ (cid:213) i d s i ! e − bl (cid:229) i s i + b (cid:229) < i j > J i j s i s j + b (cid:229) i h i s i (44)subject to the constraint h ¶ H mean ¶l i = − b ¶ ln Z mean ¶l = N . (45)Comparing the form (44) with (41), we say that the Gaussian model is a mean spherical modelwith a prescribed spherical field A .For both the Gaussian and the mean spherical models, we can make a cut–off of the s i variables in order to satisfy our condition (9). That is to say, we assign a joint probabilitydistribution p { s i } with a compact support C to the set of s i variables (any s i vanishes outsidesome (large) radius in R N ): p { s i } = , ∀ s i / ∈ C , ∀ i . (46)As a result the integrals over s i ’s in (41) and (44) do not extend to infinity: Z + ¥ − ¥ (cid:213) i d s i ! → Z C (cid:213) i d s i ! (47) (1) LGT Connects All Classical Models with Continuous Variables D U ( ) LGT (the quadratic forms in (41) and(44) all have Fourier series expansions).
Consider a number of planar pendulums with pairwise coupling (a discretization of the sine–Gordon model in 1 + Figure 5.
Coupled planar pendulums. The position ( x i , y i ) of each pendulum is determinedby the center coordinate ( x i , y i ) and an angle q i . Pendulums are interacting pairwise with theinteraction potential depending on the relative distances between their positions. between any pair of pendulums depends only on their relative distance and each pendulumhas a self–interaction depending only on its position ( e.g. a constant gravitational field). TheHamiltonian can be expressed as H = (cid:229) i u i ( cos q i , sin q i ) + (cid:229) < i j > u i j ( cos q i , cos q j , sin q i , sin q j , cos ( q i − q j )) (48)where in the simplest case the self–interaction u i (cid:181) J xi sin q i + J yi cos q i (49)and the pair interaction u i j (cid:181) q J xi j ( sin q i − sin q j ) + J yi j ( cos q i − cos q j ) + J i j cos ( q i − q j ) (50)with J i , J i j , etc . couplings. It is clear that this Hamiltonian (48) is another example in ourclass, whose partition function can be mapped to the partition function of the 4 D U ( ) LGT.
Consider the U ( ) LGT with a monopole term. This is described by an additional quadraticterm supplemented to the standard Wilson action. This system has been studied in connectionwith finding a confinement mechanism based on monopole condensation [23, 24, 25]. Theaction reads S mono = J (cid:229) a < b , n ( − cos Q ab ( n )) + l (cid:229) c , n | M c ( n ) | , (51) (1) LGT Connects All Classical Models with Continuous Variables M c ( n ) = p e cdab (cid:0) ¯ Q ab ( n + d ) − ¯ Q ab ( n ) (cid:1) (52)with e cdab the totally antisymmetric Levi–Civita symbol and n labels the face. The physicalflux ¯ Q ab ( n ) is related to the face variable Q ab ( n ) by¯ Q ab ( n ) = Q ab ( n ) − p N ab ( n ) (53)with N ab ( n ) the number of Dirac strings passing through the face [26]. The last term in (51)has a Fourier series expansion so that this model is included in the complete 4 D U ( ) LGT.
Our proof of the main result yields as a by–product the construction of a mean field theoryfor the 4
D U ( ) LGT. This corresponds to an interaction pattern where all pairwise 4-bodyinteractions between the edge variables are present. Formally, the action of the mean fieldtheory reads S m . f . = (cid:229) ∀ i < j < k < l J i jkl cos ( q i + q j + q k + q l ) . (54)Geometrically, all edges are connected with all edges. This action belongs to the class ofmodels whose action we can generate exactly. Therefore our completeness result also includesthe mean–field theory for the U ( ) LGT.
5. U(1) Lattice Gauge Theory in a Curved Spacetime
As a further development and also an important application of the completeness result of§ 3 , we study a U ( ) LGT defined in a curved background spacetime and show that it isincluded in our completeness result. Physically, this situation corresponds to photons in acurved spacetime, as it happens to cosmological photons that have propagated on a curvedspacetime during the cosmological evolution. In this section we use the following notation.Latin letters a , b etc. are used for flat indices and Greek letters m , n etc. for curved indices.We also adopt the convention that any minus sign before an index can be dragged out as anoverall ( − ) factor because it is abelian, e.g. A − a = − A a . The coupling constant is denotedby G and the lattice spacing by D as in previous sections. In order to formulate the U ( ) LGT in presence of an arbitrary background metric g mn , wefirst define a lattice in a curved spacetime manifold. In the formulation of the U ( ) LGTin § 2 we started with the Euclidean metric d ab = Diag ( , , , ) instead of the Minkowskimetric h ab = Diag ( − , , , ) , as the two are related by a Wick rotation. The Wick rotationmethod has been generalized to curve spacetime [27]. Given a background spacetime metric,a smooth family of local Wick rotations can be defined on each (co–)tangent vector spaceof the manifold (thus it is defined on the whole vector bundle). This is done by analytic (1) LGT Connects All Classical Models with Continuous Variables g mn to be Riemannian(positive definite) rather than Lorentzian for convenience without loss of generality.Now let us take a general Riemannian manifold with a given metric g mn . We could havedefined a lattice using the coordinate curves (with respect to the natural coordinates), i.e. wepick up a set of hypersurfaces defined by x m = const . such that the manifold is filled up byvolume cells formed by these hypersurfaces. The intersections of hypersurfaces (coordinatecurves) induces a ‘net’ (a graph structure) with ‘nods’ (vertices) linked by segments ofcoordinate curves (edges). This ‘net’ formed by coordinate curves might give rise to a latticestructure at first sight (once we specify a lattice constant). However, it turns out inconvenientto work with the coordinate basis. k Therefore, we switch to the non–coordinate basis definedby introducing the frame fields e a m such that g mn = e a m e b n h ab , g mn = e m a e n b h ab , (55)where the fields e a m are also referred to as tetrad or vierbein fields which bring the metric to aflat one locally. The determinant of the metric field and that of the vierbein are related suchthat g ≡ det [ g mn ] = ± (cid:16) det [ e a m ] (cid:17) with the sign determined by the signature of h ab . Note thatin our case we have chosen the flat metric to be Euclidean h ab = d ab . Therefore we have | det [ e a m ] | = q det [ g mn ] ≡ √ g . (56)Formally, we have switched from the general coordinate basis (cid:26) ¶ m ≡ ¶¶ x m (cid:27) for the local tangent space , { dx m } for the local dual space (57)with metric g mn to the orthonormal non–coordinate basis { ˆ e a ≡ e m a ¶ m } for the local tangent space , { ˆ w a ≡ e a m dx m } for the local dual space (58)with a flat metric h ab . With the introduction of the orthonormal non–coordinate basis, nowwe can make use of the integral curves of the basis vectors { ˆ e a } , a = , , , lattice by imposing a lattice constant on this ‘net’ formed byintegral curves of the non–coordinate basis. It is required that the integral curves are equallyseparated along each direction ˆ a , ˆ b , etc with a common distance D . The intersections (‘nods’of the ‘net’) of this set of integral curves are taken as vertices, so that every segment of integral k The natural coordinates may not be orthogonal and the coordinate differences D x m and D x n ( m = n ) mighthave different dimensions. It is in general invalid to identify the length of an edge in a lattice with a differencein coordinates such as D x m . (1) LGT Connects All Classical Models with Continuous Variables Figure 6.
The manifold is covered with a net of intersecting integral curves of the non-coordinate basis vectors. The curves are equally separated so that each small patch (face)is a curved rectangle with equal sides. curves in between two ‘nods’ becomes an edge. Each edge of the lattice has the same length D , see Fig. 7. In the figure and in the following we shall use the short notation a = e a m D x m and Figure 7. (a) A small patch (a curved quadrilateral) formed by coordinate curves. (b) A curvedrectangle with equal sides formed by integral curves of the orthonormal non-coordinate basis. | a | = | b | = D . With a lattice defined, now we can define the edge variables q a in a similar way as in the flatcase by q a ≡ D G A a (59)with q a = e m a q m and A a = e m a A m . Unlike in the flat case, here we have to be careful and specificon the position where exactly the edge variables are defined. It is convenient to put the edgevariables in the middle of each edge (see Fig. 8), e.g. , take the face with four vertices labeled (1) LGT Connects All Classical Models with Continuous Variables n , n + a , n + b and n + a + b , the edge variables are written as q a ( n + a ) , q b ( n + a + b ) , q − a ( n + a + b ) , q − b ( n + b ) (60)on the corresponding edges, respectively.Next is the face variable Q ab which can not be defined by simply taking summation overthe four edge variables belonging to the face because those vectors are defined at differenttangent spaces (of different points).¶ In order to be consistent with the continuum limit, thefour edge variables q e , e ∈ f for each face have to be parallelly transported to a common pointbefore summation. We choose the center of each face so that the expression of Q f can bewritten in a symmetrical way, see Fig. 8.We shall denote by ˜ q e the parallelly transported edge variable q e from the center of edge e to the center of face f ( e ∈ f ). For example, the parallelly transported q a ( n + a / ) is denotedby ˜ q a ( n + a / ) . + Then the face variable is defined by Q f = (cid:229) e ∈ f ˜ q e . (61)Take a face as the one depicted in Fig. 8. We have the face variable Figure 8. (a) The four edge variables each defined at the center of the corresponding edge. (b)The edge variables have to be parallelly transported to the center (or any other common point)before summing up. Q ab = ˜ q a ( n + a ) + ˜ q b ( n + a + b ) + ˜ q − a ( n + a + b ) + ˜ q − b ( n + b ) . (62)The transported ˜ q e is connected with the original q e by the following relation (taking q b ( n + b / ) transported to n + a / + b / q b ( n + b ) = q b ( n + b ) + a w cab q c ( n + b ) , a ≡ e a m D x m (63) ¶ The sum (cid:229) e ∈ f q e in the continuum limit D → D G ( F ab + g cab A c ) instead of the field tensor F ab alone. The extra term involves the structure factor g cab appearing in the commutation relation of the basis vectors [ ˆ e a , ˆ e b ] = g cab ˆ e c , which never vanishes identically in a non-coordinate basis unless within a finite globally flatregion. + Edge variables are always transported to the center of the face. Therefore for a given face, the location of thetransported face is omitted in notation. e.g.
In ˜ q a ( n + a / ) , the argument n + a / (1) LGT Connects All Classical Models with Continuous Variables w cab given by w cab = e c n e m a ( ¶ m e n b + e r b G nmr ) . (64)Here G rmn is the Christoffel connection which is symmetric with respect to the two lowerindices, i.e. G rmn = G rnm (torsion free). We emphasize two points here:(i) In expressions involving a parallel transport such as (63), there is no summation overindex a if we write e a m D x m in replacing a in the last term of the right hand side. Itrepresents a particular direction rather than a dummy index. Therefore we prefer to writedirectly ˜ q b ( n + b ) = q b ( n + b ) + D w cab q c , | a | ≡ | e a m D x m | = D (65)in following parts of the paper (assuming indices a , b etc. not containing a minus signfor simplicity).(ii) The functions w cab as well as q c appearing in the last term are position dependent.They could be defined either at the point n + b / n + a / + b /
2. These different choices have the same continuum limit and are equivalentup to O ( D ) .Therefore, by plugging (63) into (62), we obtain Q ab = q a ( n + a ) + q b ( n + a + b ) + q − a ( n + a + b ) + q − b ( n + b ) − D w cab q c + D w cba q c . (66)Let us look at the continuum limit. The sum of the first four terms in (66) gives (cid:229) e ∈ f q e D → −→ D e m a ¶ m q b − D e n b ¶ n q a = D e m a ¶ m ( e n b q n ) − D e n b ¶ n ( e m a q m )= D e m a e n b ( ¶ m q n − ¶ n q m ) + D ( e m a ¶ m e n b − e m b ¶ m e n a ) q n . (67)On the other hand, using (64), we see that the last two terms in (66) equal − D ( w cab − w cba ) q c = − D (cid:16) e m a ( ¶ m e n b + e l b G nml ) − e m b ( ¶ m e n a + e l a G nml ) (cid:17) e c n q c = − D ( e m a ¶ m e n b − e m b ¶ m e n a ) q n − D ( e m a e l b G nml − e m b e l a G nml ) q n = − D ( e m a ¶ m e n b − e m b ¶ m e n a ) q n . (68)In obtaining (68) we have used the symmetry properties of the connection G rmn with respect tothe two lower indices. As a consequence of (67) and (68), the face variable in the continuumlimit approaches Q ab = (cid:229) e ∈ f q e − D ( w cab − w cba ) q c D → −→ D e m a e n b ( ¶ m q n − ¶ n q m )= D G e m a e n b ( ¶ m A n − ¶ n A m )= D G e m a e n b F mn = D G F ab . (69) (1) LGT Connects All Classical Models with Continuous Variables U ( ) LGT with a background metric as follows: S = G (cid:229) f ∈ F [ − cos Q f ] . (70)This expression resembles the action in (5) for the flat metric case (with different definitionsof the faces and face variables). We shall prove that (70) leads to the correct continuum limit.As D →
0, we have each Q f being small, so that S D → −→ G (cid:229) f ∈ F Q f = G (cid:229) ab D G F ab = (cid:229) ab D F ab . (71)Furthermore, we have D = | ( e a m D x m ) ∧ ( e b n D x n ) ∧ ( e c r D x r ) ∧ ( e d s D x s ) | D → −→ | ˆ w a ∧ ˆ w b ∧ ˆ w c ∧ ˆ w d | = | det [ e a m ] | d x = √ g d x (72)being the invariant volume. In obtaining (72) we have used (56) with ∧ the wedge (exterior)product and ˆ w a etc. the dual basis. On the other hand, F ab = F ab F cd h ac h bd = F ab F cd e a m e c r g mr e b n e d s g ns = F mn F rs g mr g ns = F mn F mn . (73)As a result, by combining (72) and (73), we prove that S D → −→ Z d x √ g F mn F mn . (74)That is, the discrete U ( ) LGT action defined in (70) approaches the pure U ( ) gauge actionwith a background metric in the continuum limit. ∗ The expression of the action defined in (70) with the face variable given in (61) involvesexplicitly the lattice constant D because of (65) and (66). It is possible to rewrite the explicitexpression of the face variable Q ab (66) in a more similar way to that in the flat case (4).Let us look at the last two terms in (66). As shown in (68), the Christoffel connections (see(64)) cancel out; each remaining term depends on the lattice constant and a derivative of thevierbein field in such a way that it is identified as a first order differential of the vierbein field.Therefore it can be rewritten as a difference in the vierbein up to the order O ( D ) , e.g. D e m a ( ¶ m e n b ) q n ∼ [ e n b ( n + a ) − e n b ( n )] q n . (75)Therefore, the last two terms in (66) can be split into four terms (to be more symmetric) eachinvolving a difference of two vierbeins, one defined at the center of the face, the other at oneedge. i.e. − D ( w cab q c − w cba ) q c = − D ( e m a ¶ m e n b − e m b ¶ m e n a ) q n ∗ We should have instead of the field tensor F mn = ¶ m A n − ¶ n A m , the covariant field tensor F mn = D m A n − D n A m with the covariant derivative D m A n = ¶ m A n − G rmn A r appearing in (20). However, since the connection G rmn issymmetric with respect to the two lower indices while the field tensor is antisymmetric, we have F mn = F mn identically. (1) LGT Connects All Classical Models with Continuous Variables ∼ [ e m a ( n + a + b ) − e m a ( n + a )] q m ( n + a )+ [ e m b ( n + a + b ) − e m b ( n + a + b )] q m ( n + a + b ) − [ e m a ( n + a + b ) − e m a ( n + a + b )] q m ( n + a + b ) − [ e m b ( n + a + b ) − e m b ( n + b )] q m ( n + b ) . (76)Now with this new form (76), the face variable can be rewritten in the following form: Q f ≡ Q ab = (cid:229) e ∈ f q e − D ( w cab − w cba ) q c ∼ q m ( n + a ) e m a ( n + a + b )+ q m ( n + a + b ) e m b ( n + a + b ) − q m ( n + a + b ) e m a ( n + a + b ) − q m ( n + b ) e m b ( n + a + b )= (cid:229) e ∈ f q edge · e face . (77)In the last line of (77) we have used a concise notation to indicate that the face variable Q f is asum over the edge variables q e with each edge variable coupled to the vierbein e m a at the centerof the face. The form (77) resembles (4) in the flat case. We also realize that in this form thevierbein fields need only to be defined for each face. Moreover, by rearranging terms, thisform can also be written in the following way Q ab = [ q m ( n + a ) − q m ( n + a + b )] e m a ( n + a + b )+ [ q m ( n + a + b ) − q m ( n + b )] e m b ( n + a + b ) . (78)The expression (78) is transparent in taking the continuum limit (reducing to F ab ) as well asthe ‘flat’ limit (e m a = d m a ).We emphasize here that a consistent theory of the U ( ) LGT with a background metricis not unique. (There are two requirements only: 1 ) The recovery of a U ( ) pure gaugetheory with a metric in the continuum limit; 2 ) The recovery of a flat U ( ) LGT in the limit g mn = h mn , and these do not defined the theory uniquely.) We are making a particular choiceso that the resulting theory takes a form similar to the flat theory and the form of the facevariable looks more symmetric with respect to the edge variables involved. ♯ ♯ Had we made another choice, e.g. had we preferred the edge variable q n in (75) and q m in (76) be defined atthe center of the face rather than on the edge, the face variable could have taken a different form such as Q ab = [ q m ( n + a ) − q m ( n + a + b )] e m a ( n + a )+ [ q m ( n + a + b ) − q m ( n + a + b )] e m b ( n + a + b ) (1) LGT Connects All Classical Models with Continuous Variables We have defined the action of the U ( ) LGT coupled to a background gravitational field (abackground metric). The action is given by (70) which takes a similar form as in the flatcase (5). The face variables are defined in (61) (by summing over edge variables parallellytransported to the center of each face) or in (77) equivalently (by summing over edge variablescoupled with the vierbein field at the center of each face). As a result, the partition functionof the model reads Z curved = Z (cid:213) e ∈ E d q e ! exp ( − G (cid:229) f ∈ F " − cos (cid:229) e ∈ f ˜ q e ! = Z (cid:213) e ∈ E d q edge ! exp ( − G (cid:229) f ∈ F " − cos (cid:229) e ∈ f q edge · e face ! . (79)The action (70) of the U ( ) LGT with a curved background metric satisfies the threeconditions (9), (10) and (11) in § 3.1 . Therefore, we conclude that our completeness resultcontains this model, i.e. , that the U ( ) LGT with a curved background metric can be mappedto a 4
D U ( ) LGT with a flat metric.
6. Generalizations
The completeness result obtained in § 3.4 can be generalized to include an even broader classof classical models. The generalization is achieved by relaxing the third condition (11) in§ 3.1 . In other words, we shall consider classical physical systems with dynamical variablessubject to constraints in this section. A very important set of physical models are classicalHeisenberg models. For these models the spin variables s are classical but subject to theconstraint s + s + ... + s n = i.e. they live in an internal space which is an S n − sphere. Let us take a classical system satisfying only conditions (9) and (10). Assume that there are aset of M ( M < N ) independent unsolvable constraints†† over the variable x j ’s such that F l { x j } = , l = , , . . . , M . (80)The functions F l are also assumed to have Fourier series expansions. After normalization ofthe variables x j → q j , we write the Hamiltonian as H { q j } and the constraints as F { q j } . So − [ q m ( n + a + b ) − q m ( n + a + b )] e m b ( n + a + b ) − [ q m ( n + b ) − q m ( n + a + b )] e m b ( n + b ) . Also, had we parallelly transported the four edge variables to a common point other than the center of the face,the face variable Q ab might have taken a less symmetric form.††If a constraint is solvable, then we could release the constraint by a change of variables such that there will beno constraint over this new set of independent variables. (1) LGT Connects All Classical Models with Continuous Variables Z classical = N Z p − p (cid:213) j d q j ! (cid:213) l d (cid:0) F l { q j } (cid:1)! e − b H { q j } . (81)Now for each constraint F l we associate an additional variable k l and re-write each deltafunction as a Fourier transform. Then (81) becomes Z classical = N Z p − p (cid:213) j d q j ! Z + ¥ − ¥ (cid:213) l d k l p ! · exp ( − b H { q j } + i (cid:229) l k l F l { q j } ) . (82)If we make a (large ultraviolet) cut–off on each of the k l variables such that k l ∈ [ − L l , + L l ] ,then the integrals R + ¥ − ¥ become R + L l − L l . Now we can normalize these variables by f l = pL l k l .Then the partition function (82) takes the following form approximately Z classical ∼ N N ′ Z p − p (cid:213) j , l d q j d f l ! e H { q j , f l } (83)with N ′ = (cid:213) l L l p and an effective complex Hamiltonian H { q j , f l } defined by H { q j , f l } = − b H { q j } + i (cid:229) l L l p f l F l { q j } (84)This Hamiltonian in (84) with additional variables f l has a Fourier series expansion H { q j , f l } = (cid:229) { m j , m l } H { m j , m l } cos (cid:229) j , l m j q j + m l f l ! + ˜H { m j , m l } sin (cid:229) j , l m j q j + m l f l ! (85)with complex Fourier coefficients H { m j , m l } and ˜H { m j , m l } . Therefore, if we allow for complexcoupling constants J { m j , m l } = H { m j , m l } and ˜ J { m j , m l } = ˜H { m j , m l } , the partition function Z classical in (83) is of a U ( ) LGT type. As a result, if we assume that the cut–off (a regularization)in the delta function representation above works universally, then the 4
D U ( ) LGT withgeneral complex coupling constants is a complete model for classical statistical models withcontinuous variables subject to constraints.
Another possible generalization is to consider classical models with discrete degrees offreedom, such as the Ising model, the Potts model, the vertex models, and the Z q LGT wherethe dynamical variables take values only in a discrete set. These models can be considered aspecial case of continuous models with constraints.Let us again consider a classical system satisfying only conditions (9) and (10). Each ofthe variables x j can assume only finite discrete values x j = X l j , a j ≤ X l j ≤ b j , l j = , , . . . , L j . (86) (1) LGT Connects All Classical Models with Continuous Variables X l j and the number of possible values L j may depend on j . It is equivalentto express these constraints by the following expression with d -functions (cid:213) j " L j (cid:229) l j = d (cid:0) x j − X l j (cid:1) (87)such that for an arbitrary function of a set of x j ’s, we have (cid:213) j Z b j a j d x j ! (cid:213) j " L j (cid:229) l j = d (cid:0) x j − X l j (cid:1) f { x j } = L N (cid:229) l N = · · · L (cid:229) l = f { x j = X l j } . (88)In the last summation of (88) we are summing over all possible configurations (all possiblecombination of x j values). Moreover, after normalization of the variables x j → q j , H { x j } → H { q j } , the delta functions become d (cid:0) x j − X l j (cid:1) → p b j − a j d (cid:18) q j − p b j − a j (cid:18) X l j − a j + b j (cid:19)(cid:19) . (89)We define Q l j = p b j − a j (cid:18) X l j − a j + b j (cid:19) , ∀ l j , j (90)for notational convenience. Consequently, the partition function of a classical model withdiscrete degrees of freedom can be expressed as Z discrete = (cid:229) all { l j } e − b H { x j = X l j } (91) = Z p − p (cid:213) j d q j ! (cid:213) j " (cid:229) l j d (cid:0) q j − Q l j (cid:1) e − b H { q j } . Now, with these particular forms of d -functions appearing in (91), it is not necessary to go tothe region of a complex Hamiltonian. Let us represent each d -function by a limit d (cid:0) q j − Q l j (cid:1) = lim e j → d e j (cid:0) q j − Q l j (cid:1) (92)with each d e j (cid:0) q j − Q l j (cid:1) being a positive–definite function on [ − p , p ] depending on theparameter e j . (This function could be a Gaussian or a ‘square’ function.) If we make acut–off approximation on these e j parameters, then Z discrete can be rewritten as Z discrete { e j } (93) ∼ ( p ) N Z p − p (cid:213) j d q j ! exp ( − b H { q j } + (cid:229) j ln " (cid:229) l j d e j (cid:0) q j − Q l j (cid:1) . The expression on the exponential of (93) can be interpreted as an effective (real) Hamiltonianwhich allows a Fourier series expansion over q j ’s with real coefficients. Therefore weconclude that U ( ) LGT is complete also for certain classical discrete models, again assumingthat the cut–off (regularization) (92) works universally. (1) LGT Connects All Classical Models with Continuous Variables
7. Conclusion
We have proven that any classical partition function depending on continuous variables subjectto conditions (9), (10) and (11) can be approximated (to an arbitrary precision) by the partitionfunction of a 4
D U ( ) LGT. In the proof we first introduced a quantum representation of the U ( ) LGT partition function. Then through merging and deletion of gauge field variablesand proper choices of local coupling constants, a mapping from a 4
D U ( ) LGT partitionfunction to a more general partition function is established. In this sense the 4
D U ( ) LGT is a complete model for a large class of classical models. The completeness result isalso generalized to include continuous models with constraints (if we are allowing complexcoupling constants) and discrete models. As a further development and important applicationof the completeness result, we have developed a consistent theory of the U ( ) LGT coupled toa background metric. The action is defined in a form (70) very close to that of the model in flatspacetime. Our completeness result holds for this model such that its partition function canbe mapped (approximated to an arbitrary precision) to the partition function of a U ( ) LGTin flat spacetime. This is the first time that a completeness result is established for continuousstatistical models.We believe that our completeness result cannot be proven with a 3
D U ( ) LGT. Anotheropen question is whether a similar result can be found for non–Abelian LGTs (e.g. for SU ( ) or SU ( ) LGTs). We envisage that these theories may require a new approach since a directgeneralization of our construction seems not to be possible.
Acknowledgments
The authors would like to thank Maarten Van den Nest for fruitful discussions and valuablesuggestions. The work is supported by the FWF (SFB-F40) and the European Union (QICS,NAMEQUAM), the Spanish MICINN grant FIS2009-10061, CAM research consortiumQUITEMAD S2009-ESP-1594, European Commission PICC: FP7 2007-2013. grant no.249958, UCM-BS grant GICC-910758.
Appendix A. Construction of Many–Body Interactions
To generate the functions of the set (14) we will proceed similarly as in [3, 4], namely wewill make use of the merge and the deletion rule, and of the gauge fixing of edges. The latteris a procedure for fixing the values of the variables in the lattice, which results in a theorythat is physically equivalent to the original one. This can be carried out by virtue of thegauge symmetry of the model. The only restriction in this procedure is that the edges whosevariables are fixed by the gauge cannot form a closed loop [22].To construct the many–body interactions, we first need to “propagate” the variablesinside the lattice in order to bring them close together to interact. This propagation isachieved with the following construction (see Fig. A1). On a cube (on the left of Fig. A1),we merge the face at the front, at the bottom, and at the back to generate the interaction (1) LGT Connects All Classical Models with Continuous Variables J f cos ( a − a + q + q ) . Then, we let this coupling strength go to infinity, J f → ¥ , whichimposes the constraint q + q =
0. The same process is repeated for the cube on the right ofFig. A1, where the constraint reads q + q =
0. Thus, we have set q = q , that is, q has“propagated” two sites to the right. Note that if q were propagated an odd number of times,the resulting variable would equal − q . This can be circumvented by letting q participatein the final interaction with the opposite sign (as explained below). In order to see how toturn the propagation path, we refer the reader to the explanations on Fig. 11 of [4], since theconstruction is analogous.PSfrag replacements q q q a a a Figure A1.
Propagation of the variable q across the 4 D square lattice (the figure shows onlya 3 D projection of this space). In all figures, red edges denote edges whose variable has beenfixed by the gauge. By means of the merge rule, q is propagated into q , since q = − q and q = − q . Now we focus on the replication of the (classical, continuous) variables, that is, on thegeneration of several copies of a given variable. This is achieved by applying the propagationprocedure explained above into the fourth dimension, as shown in Fig. A2. The interaction inthe yellow cube is of the form J f cos ( q + a + q − a ) , on which we let J f → ¥ and therebyimpose q = − q (i.e. we apply the merge rule on this cube as well). The rest works exactly asthe propagation explained in Fig. A1. We note that the reason for using a fourth dimension inthe replication of edge variables is that all schemes we have found in three dimensions involveclosed loops of variables fixed by the gauge [3].Next we show how to generate interactions of the type (14). The generation of theinteractions cos ( (cid:229) Ki = q i ) for the specific case K = q , . . . , q inside the lattice in order to distribute them on the edges ofthe rectangular prism shown in Fig. A3. Then we merge all faces on the exterior surface ofthis prism into one large, blue face. As shown above, this face only depends on the spins atits boundaries, that is, it has the form J f cos ( q − a + q − a + a − a + a + a − a + a + a + q − a + q − a + q + a ) . (A.1)Since the dependence on every auxiliary variable a i cancels, it takes the desired form J f cos ( (cid:229) i = q i ) . The generalization to any K is straightforward. For odd K , we construct alonger or shorter prism than that of Fig. A3, arranging ( K − ) / ( K − ) / q and q are arranged in front of q and q ). The remaining variablewould be unpaired, and the vertical edge on the corner of the prism fixed by the gauge (justas q ). For even K the construction is simpler. For K = q , . . . , q as (1) LGT Connects All Classical Models with Continuous Variables q q q q q q a a a q q q q q a a a a Figure A2.
Yellow faces are in the direction of the fourth dimension and have the samemeaning as blue faces ( i.e. merged faces). Replication of the variable q into q , q and q .This replication is essentially a propagation (as the one of Fig. A1) in the fourth dimension,i.e. the variable lives now in another 3 D space. in Fig. A3, and we merge the large, blue face over the face joining the red u–shapes of q and q on the right. For any other even K , we similarly arrange K / ( q ) isachieved by first replicating the variable q , and then letting its two copies participate in atwo–body interaction as explained above.PSfrag replacements q q q q q a a a a a a a J f s Figure A3.
Bold, black edges contain variables that participate in the final interaction q , . . . , q . Red edges (except for the one marked with s ) stand for edges whose variableshave been fixed to zero using the gauge symmetry, and blue faces stand for merged faces, asin Fig. 3. If s =
0, as the other spins, the large blue face contains the five–body interaction J f cos ( q + q + q + q + q ) , whereas if s = − p /
2, it depends on J f sin ( q + q + q + q + q ) . Now we only need to show how to generate interactions involving variables with differentsigns. The generation of J f cos ( q + q + q − q − q ) is shown in Fig. A4. The variableswhich have the same relative sign are arranged as explained for the cosine of the sum ofvariables (see Fig. A3). The new element here is that the two sets of variables which havethe opposite relative sign must arranged perpendicularly to each other. Then a large, blueface is merged over the external faces, where the desired interaction takes place. One canalso verify that the dependency on the auxiliary variables a i cancels out, and that variablesperpendicular to each other have the opposite sign. The generalization to an interaction (1) LGT Connects All Classical Models with Continuous Variables J f cos ( (cid:229) K i = q i − (cid:229) K j = q j ) is also straightforward. One only has to arrange the first set ofvariables as explained for the case J f cos ( (cid:229) K i = q i ) , with K either odd or even. The otherset of variables is arranged also as explained above (with K being odd or even), and setperpendicular to the first set.PSfrag replacements q q q q q a a a a a a a a a J f Figure A4.
Five–body interaction J f cos ( q + q − q − q − q ) . The meaning of the symbolsis the same as in Fig. A3. Finally, we point out that the sine functions are generated by making use of the relationsin ( g ) = cos ( g − p / ) . This phase amounts to gauge fixing one of the spins to − p / ( q + q + q + q + q ) we construct the interactionof Fig. A3, and we fix s = − p / D U ( ) LGT. The construction also shows that each Fourier basis function (i.e. each term in the set(14)) requires a polynomial enlargement in the number of variables of the 4
D U ( ) LGT.We shall return to this fact in § 3.4 , where we specify the overhead in the system size of thecomplete model as a function of the features of the target model.
Appendix B. Accuracy of the Finite Fourier Series
In the completeness result of § 3.4 , we have made a truncation in the Fourier series basis (see(31) and (32)) so as to approximate a general Hamiltonian as an expansion. Here comes aquestion of accuracy. That is, given a truncation − M ≤ m j ≤ M , ∀ j in the Fourier modes,how close is the following finite Fourier series F M h H ( K ) i ≡ (cid:229) { m j } H ( K ) { m j } exp ( i (cid:229) j m j q j ) (B.1)to the original Hamiltonian function H ( K ) ( { q j } ) of a K -body interaction term? Accordingto [30], for a smooth enough (usually at least differentiable) single variable function f ( q ) ∈ C a [ − p , p ] with a >
0, we have | F M [ f ] − f ( q ) | ≤ A ( f ) (cid:229) P ( a ) M P ( a ) (B.2)where F M [ f ] is a finite Fourier series of f ( q ) with 2 M + M th mode asin our case), A ( f ) is a finite factor depending on the function form of f ( q ) on the domain only, (1) LGT Connects All Classical Models with Continuous Variables P ( a ) of a . Therefore we see that in order to havean accuracy ∼ N for the finite Fourier series of a single variable, we need a polynomiallylarge truncation Poly ( N ) in the Fourier modes. We could say that we have a polynomialaccuracy in this case. Now with a K -body interaction term which is a function of K variables,in order to have an accuracy | F M h H ( K ) i − H ( K ) ( { q j } ) | ≤ N , (B.3)we shall need M ∼ [ Poly ( N )] K terms in the finite Fourier series. i.e. the K th power of somepolynomial of N . Therefore it is still efficient if the model has only few–body interactions( e.g. nearest neighbor interactions only) or if K scales polynomially with the system size. References [1] Pathria R K 1996
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