Nature of the Spin Glass Phase in Finite Dimensional (Ising) Spin Glasses
JJune 24, 2020 0:38 ws-rv9x6 Book Title chapter˙ruiz˙lorenzo page 1
Chapter 1Nature of the Spin Glass Phase in Finite Dimensional(Ising) Spin Glasses
Juan J. Ruiz-Lorenzo ∗ Departamento de F´ısica,Universidad de Extremadura, 06006 Badajoz, SpainInstituto de Computaci´on Cient´ıfica Avanzada de Extremadura (ICCAEx),Universidad de Extremadura, 06006 Badajoz, SpainInstituto de Biocomputaci´on y F´ısica de los Sistemas Complejos (BIFI),50018 Zaragoza, Spain
Spin glasses are the paradigm of complex systems. These materialspresent really slow dynamics. However, the nature of the spin glass phasein finite dimensional systems is still controversial. Different theories de-scribing the low temperature phase have been proposed: droplet, replicasymmetry breaking and chaotic pairs. We present analytical studies ofcritical properties of spin glasses, in particular, critical exponents at andbelow the phase transition, existence of a phase transition in a magneticfield, computation of the lower critical dimension (in presence/absenceof a magnetic field). We also introduce some rigorous results based onthe concept of metastate. Finally, we report some numerical results re-garding the construction of the Aizenman-Wehr metastate, scaling ofthe correlation functions in the spin glass phase and existence of a phasetransition in a field, confronting these results with the predictions ofdifferent theories.
Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22. A brief tour of spin glasses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33. Some theoretical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.1. Mean-field solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2. Spin glasses in finite dimensions . . . . . . . . . . . . . . . . . . . . . . 153.3. The droplet model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 ∗ [email protected] 1 a r X i v : . [ c ond - m a t . d i s - nn ] J un une 24, 2020 0:38 ws-rv9x6 Book Title chapter˙ruiz˙lorenzo page 2 J.J. Ruiz-Lorenzo h = 0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.5. Field theory ( h (cid:54) = 0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.6. Metastate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194. Some numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.1. Phase transition at h = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . 224.2. Behavior of the correlation functions in the spin glass phase . . . . . . . 254.3. Metastate: Numerical results . . . . . . . . . . . . . . . . . . . . . . . . 294.4. Phase transition at h (cid:54) = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . 315. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36A.1. Characterization of a phase transition: Quotient and fixed coupling methods 38A.2. The Janus supercomputers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
1. Introduction
Spin glasses are often considered as the paradigm of complex systems. Theyshow frustration and randomness which are now ubiquitous characteristicsin nature. In addition, the role played by spin glasses in magnetism is huge:spin glass behavior jointly with ferrromagnetism and antiferromagnetismare the three most frequent forms of “magnetic order”. On the theoreticalside, radical approaches to describe them have been developed and in somecases are still needed. Finally, there is great interplay among spin glassesand other systems, from the molecular evolution to astrophysics. This chapter is devoted to the study of properties of spin glasses in finitedimensions (mainly in three dimensions) using analytical and numericalapproaches. The main goal is to determine existence of a spin glass phasein finite dimensions and, if it exists, to characterize its physical properties.We start by describing some basic properties of spin glasses in Sec. 2.Next, this book chapter continues in two main parts.In the first one, we report the main theoretical results (Sec. 3), startingwith the mean-field solution in Sec. 3.1, which already provides a com-plex picture even in infinite dimensions. Sec. 3.2 is devoted to study towhat extent this complex picture survives in finite dimensions. In Sec. 3.3we report the droplet theory, a phenomenological theory that can also beformulated in terms of the Migdal-Kadanoff approximation of the renormal-ization group. Hereafter, we resort to the “standard” approach based on afield theory approach built on the complex mean-field solution, describingits main findings both in absence (Sec. 3.4) and in presence of a magneticfield (Sec. 3.5). We finish this part by introducing in Sec. 3.6 an importantconcept and tool of the metastate.In the second part of the chapter, we describe numerical simulations atequilibrium (Sec 4). We start to report some important numerical facts in une 24, 2020 0:38 ws-rv9x6 Book Title chapter˙ruiz˙lorenzo page 3
Nature of the Spin Glass Phase in Finite Dimensional (Ising) Spin Glasses absence of a magnetic field. First, we show in Sec. 4.1 the existence of aphase transition in three dimensions and how its universality class has beencharacterized. Once we know there is a spin glass phase in three dimensions,in Sec. 4.2 we present numerical simulations that try to characterize theproperties of this phase, in particular, we focus on the behavior of theconditional correlation functions. Next, we continue by showing a numericalconstruction of the metastate and properties of the spin glass phase one candraw from this powerful tool (Sec. 4.3). We close this part with the studyof spin glasses in a field. We focus on the simulations performed in fourdimensions and the rationale of the new numerical approaches that havebeen useful to find the phase transition. However, the phase transition inthree dimensions in a field has been elusive even using these new numericaltools (Sec. 4.4).The book chapter finishes with the conclusions and two appendices. Inthe first one we report the finite size scaling tools needed to analyze the crit-ical behavior of these systems, namely the quotient method and the analysisat fixed coupling (Sec. A.1). Part of the numerical simulations presented inthis chapter have been obtained with the help of Janus I and II supercom-puters. In the last appendix, we have described the basic characteristics ofthese two dedicated computers (Sec. A.2).This book chapter is based on the lectures given by the author in Lvivduring the Ising Lectures 2019 and we have tried to report the contentslectured there during two days. In these two lectures, the focus was onequilibrium numerical simulations on finite dimensional Ising spin glasses.Hence, we have not discussed in this chapter important topics in spin glassphysics as experiments and out-of-equilibrium simulations.Finally, let us mention that it is also possible to study the propertiesof the low temperature phase with h = 0 and h (cid:54) = 0 by simulating D = 1dimensional Ising spin glass with the coupling decaying following a powerlaw and it has been used for the study of the spin glass phase inside andoutside the mean-field region.
2. A brief tour of spin glasses
In this section we describe the main physical properties of these materials.The main ingredients to obtain materials with a spin glass behavior aremagnetic interaction, randomness, frustration and anisotropy. However,materials with spin glass behavior can be obtained in different ways, forexample magnetic interaction is not needed. une 24, 2020 0:38 ws-rv9x6 Book Title chapter˙ruiz˙lorenzo page 4 J.J. Ruiz-Lorenzo
Metals with very diluted magnetic impurities are considered canonicalspin glasses. We can mention, for example, CuMn and Ag:Mn at 2 . . IN . S and Fe . Mn . TiO . In these materials one can identifymagnetic interaction and randomness, via the dilution of the magnetic mo-ments.Since the characteristic times associated with the magnetic impuritiesare much bigger than the times associated with the electrons of the metal,we can assume that the impurities are quenched . This approximation issimilar to that performed in molecular physics called Born-Oppenheimerapproximation (in molecules the nuclei play the role of the magnetic impu-rities in spin glasses). It is possible to define another kind of disorder, theso-called annealed one, in which the components in the material (“normal”and “impurities”) have similar characteristic times and, thus, the statisticalmechanics considers all of them in the same way.In spin glasses, the magnetic impurities do not interact following thestandard exchange interaction, rather, they interact among them viathe electrons moving in the conduction band of the metal, the so-calledRKKY interaction following the work of Ruderman, Kittel, Kasuya andYosida. It has been shown that the strength of the interaction, J ( r ),between two magnetic moments (impurities) sited at distance r is givenby J ( r ) ∼ cos(2 k F r ) r , (1) k F being the Fermi momentum. In addition to a power law decay onthe distance of the interaction, an oscillatory factor (the cosine) appears:depending on distance, sometimes it will induce a positive interaction andother times a negative one (see Fig. 1).This change of the sign of the interaction produces the frustration in thesystem. In Fig. 2 we show a frustrated square. In this square the productof the four couplings ( J ij living in the links) is negative, and thus, there aredifferent spin configurations which provide the same energy: frustration.The joint effect of disorder and frustration usually produces a very com-plicated landscape of free energy, and in particular, a very slow dynamics.The landscape depicted in Fig. 3 is typical of glassy systems, showing agreat number of relative maxima and minima separated by high free en-ergy barriers.The last ingredient to build the Ising spin glass model is the anisotropy. For instance Ag:Mn and CdCr . IN . S are well described by Heisenbergspins, although the description of Fe . Mn . TiO is based on Ising spins. une 24, 2020 0:38 ws-rv9x6 Book Title chapter˙ruiz˙lorenzo page 5 Nature of the Spin Glass Phase in Finite Dimensional (Ising) Spin Glasses J ( r ) r Fig. 1. Dependence of coupling J ( r ) on distance in the RKKY interaction. Notice thedecay with distance, and most important fact, the oscillatory behavior, which inducesfrustration. However, some results obtained in experiments performed on films by theTexas group using Heisenberg spin glasses (as CuMn) have been con-fronted with numerical results simulating Ising spins showing a very goodquantitative agreement.
The previous discussion allows us to write the following Edwards-Anderson Hamiltonian which describes the Ising spin glass in a magneticfield h H J = − (cid:88) i,j J ij s i s j + h (cid:88) i s i , (2)where the quenched stochastic variables ( J ij ) can be drawn from a bimodaldistribution or from a Gaussian one, both with zero mean and unit varianceand s i = ± (cid:80)
Fig. 2. Frustrated square. For a given choice of couplings (living on the links and theirproduct being negative) and starting with S = 1 for the top left spin, the value of thespin lying on the bottom right corner can be +1 or −
1. Both values minimize the energy:frustration, the system has two options with the same “cost”.
First, we compute the free energy for a given instance, F J (realizationor sample) of the disorder, F J = − β log Z J , (3)with β ≡ / ( k B T ), where the partition function Z J for a given disorderrealization is given by Z J = (cid:88) [ s ] exp( − β H J ) , (4)where (cid:80) [ s ] denotes the trace on all the spins.Second, we take the average of the free energy (of a disorder instance)over the whole set of instances, distributed with the probability densityfunction p [ J ]: F = (cid:90) d[ J ] p [ J ] F J , (5) une 24, 2020 0:38 ws-rv9x6 Book Title chapter˙ruiz˙lorenzo page 7 Nature of the Spin Glass Phase in Finite Dimensional (Ising) Spin Glasses Configuration F r ee e n e r g y Fig. 3. Free energy landscape for a spin glass: notice the large number of minima,absolute and relative, and the diversity of free energy barriers separating them. with d[ J ] ≡ (cid:89) i 3. Some theoretical results In this section we discuss different analytical approaches to tackle spin glassbehavior. We start with the mean-field approximation, which is exact ininfinite dimensions, and then we study finite dimensional spin glasses bypresenting the droplet model and the approach based on field theory. Mean-field solution Let us summarize the solution of the Edwards-Anderson model (for h =0) in the mean-field approximation, the so-called Sherrington-Kirpatrickmodel. The starting point is to use the replica trick. In this trickwe replace the logarithm entering the quenched average by the followinglimit log x = lim n → x n − n . (21)Applying this trick to the computation of the quenched free energy weobtain log Z J = lim n → Z nJ − n . (22)The average on the disorder of the partition function of n non-interactingreplicas, { s ai } ( a = 1 , . . . , n ), can be written as Z n = Z nJ = (cid:88) { s a } (cid:90) d[ J ] exp (cid:18) β n (cid:88) a =1 (cid:88) i Now, we can compute the integral on the disorder, getting Z n = (cid:88) { s a } exp (cid:34) β N n + 12 β N n (cid:88) a
For example, in Fig. 4 is shown the 2-step level of the Parisi solution(which has infinite levels). At this 2-step level three real values of theoverlap q appear: q , q and q and two integer numbers which determinethe size of the sub-matrices (box and sub-box), m and m , such that0 < m < m < n . Notice that m must divide m and m must divide n . Fig. 4. 2-step of the overlap matrix Q ab . Notice that the matrix has been broken intotwo main submatrices of sizes m and m , taking values q , q and q . From Ref. [45]. This scheme can be generalized, assuming that n , the number of replicas,is large enough to allow a k -step level of breaking the symmetry of thereplicas, where k could be arbitrarily large. Finally, we need to do ananalytic continuation to n = 0 (to comply with the replica trick).Given matrix Q ab with the Parisi breaking scheme, one can computewhat is the probability to find a given value of q , denoted as p ( q ), assumingthat all the matrix elements have the same probability: p ( q ) = 1 n ( n − (cid:88) a (cid:54) = b δ ( Q ab − q )= nn ( n − (cid:2) ( n − m ) δ ( q − q ) + ( m − m ) δ ( q − q )+ ( m − m ) δ ( q − q ) + . . . (cid:3) , (36)with n > m > m > . . . > n → one obtains, p ( q ) = m δ ( q − q ) + ( m − m ) δ ( q − q ) + ( m − m ) δ ( q − q ) + . . . (37) une 24, 2020 0:38 ws-rv9x6 Book Title chapter˙ruiz˙lorenzo page 13 Nature of the Spin Glass Phase in Finite Dimensional (Ising) Spin Glasses Notice that p ( q ) is a probability density function composed by sum ofDirac’s deltas with different weights, hence, all these weights must be pos-itive and so 0 < m < m < . . . < 1. Notice that the limiting process( n → 0) has inverted the order of the different m ’s.At this point we can connect p ( q ) with the pdf of the overlap defined inEq. (15), denoted as P ( q ).The overlap defined in Eqs. (9) and (10) can be written in the frameworkof the replica theory as ( a (cid:54) = b ) q = (cid:104) s i (cid:105) = (cid:104) s ai s bi (cid:105) = (cid:34) (cid:80) { s a ,s b } s ai s bi exp (cid:0) − β (cid:80) k 1] and that the function x ( q ) (inverse of q ( x )) is relatedwith P ( q ) via d x d q = P ( q ) . (42)Before finishing this section we summarize some of the most importantphysical and mathematical properties of the Parisi solution: une 24, 2020 0:38 ws-rv9x6 Book Title chapter˙ruiz˙lorenzo page 14 J.J. Ruiz-Lorenzo • It is exact in infinite dimensions. This has rigorously been shownin Refs. [52–54]. • It shows an infinite number of pure states not related by any sym-metry. • These infinite pure states are organized in a ultrametric way . Wecan recall at this point, the definition of an ultrametric space. Aspace is ultrametric if all the triplets of elements belonging to thisspace ( A , B , C ) satisfy the ultrametric inequality: d ( A ; B ) ≤ max( d ( A, C ) , d ( B, C )) . In spin glasses we can introduce a distance by using the overlaps(now the elements of this space are the pure states, see Sec. 3.6) d ( α, β ) = 12 ( q EA − q αβ ) . (43)In Fig. 5 we have drawn the ultrametic organization of the purestates: the end of the leaves are the pure states, having q EA as theiroverlap. • The spin glass phase is stable under small magnetic fields. A transi-tion line in the temperature-magnetic field plane separates a para-magnetic phase from a spin glass one (the de Almeida-Thoulessline ). • The excitations of the ground state are space filling, i.e. the di-mension of the excitations is just that of the space, D . • Overlap equivalence. All the possible definitions of the overlap, e.g.spin overlap or link overlap, provide the same information on thephysical properties of the system. For example the link overlap isdefined as q l = 1 N D (cid:88) Nature of the Spin Glass Phase in Finite Dimensional (Ising) Spin Glasses pure states Fig. 5. Ultrametric organization of the pure states in the Parisi solution. Spin glasses in finite dimensions Once we have characterized the behavior of spin glasses in infinite dimen-sions, the fully connected model, we want to understand what are the prop-erties of spin glasses in finite dimensions. Different theories have been devel-oped to describe the behavior of spin glass in finite dimensions. Hereafter,in the next three sections, we report the two most important approaches:the droplet model and the RSB theory. The droplet model The droplet theory predicts that the spin glasses above the lower criticaldimension ( D l ) show only two pure states in the spin glass phase and thebehavior of this spin glass phase is determined by compact excitations onthe ground state. The droplet model can be formulated in terms of theMigdal-Kadanoff approximation of the renormalization group (whichis exact in D = 1) or by means a phenomenological theory, both approaches being equivalent.The most important properties of this phenomenological theory are: • The droplets are compact excitations with fractal dimension D F .The energy of a droplet of linear dimension L grows as L θ with une 24, 2020 0:38 ws-rv9x6 Book Title chapter˙ruiz˙lorenzo page 16 J.J. Ruiz-Lorenzo θ < ( D − / < D − < D F < D . • In the dynamics, the free energy barriers behave as L ψ , with ψ ≥ θ . • The spin glass phase is unstable against the presence of a magneticfield. • There are two pure states (related by spin flip), and thus, theprobability distribution of the overlap is trivial: sum of two Dirac’sdeltas.Finally, there is a variation of the droplet model, known as the chaoticpairs model. In this picture the system has two pure states (as in thedroplet model), but these two states vary chaotically with the size of thesystem, see Sec. 3.6. Field theory ( h = 0 ) In this section, we address the problem of how to build a field theory usingthe RSB solution as starting point in absence of a magnetic field. For atheoretical description of the theory in presence of a magnetic field, seeSec. 3.5.We start considering the theory in infinite dimensions (using the mean-field approximation, which is exact in D = ∞ ). Next, the upper criticaldimension D u is computed. Above it, the predictions of the mean-fieldapproximation hold and below, infrared divergences appear and we need toresort to the renormalization group to tackle them. This is the standard approach and it has been applied (with a hugesuccess) in the study of a large number of models, for example, the Isingmodel or models with O ( N ) symmetry.The upper critical dimension is determined by the dimension of thecubic coupling, g , in the effective Hamiltonian, see Eq. (35), and it turnsto be D u = 6. Below the upper critical dimension τ and g are the relevantparameters (using the terminology of the renormalization group ) and g and λ are irrelevant, hence, we need to study the field theory of a φ theory with tensor couplings. By using this theory, it is possible to computeanalytically the critical exponents using the (cid:15) -expansion (where (cid:15) = 6 − D ),see Refs. [18,65–67]. Moreover, by using this theoretical framework, it hasbeen possible to compute the logarithmic corrections at the upper criticaldimension in these models, see Refs. [68–72].To complete this discussion, let us remark that the lower critical dimen-sions in absence of a magnetic field seems to be D l = 2 . This issuehas been studied experimentally by studying spin glasses in film geometries, une 24, 2020 0:38 ws-rv9x6 Book Title chapter˙ruiz˙lorenzo page 17 Nature of the Spin Glass Phase in Finite Dimensional (Ising) Spin Glasses finding a strong evidence that 2 < D l < The low temperature spin glass phase is critical (in this model the T = 0critical point has infinite correlation length, as in the O ( N ) model, N > D > 2) and to perform a field theoretical analysis we also need to considerthe g and λ couplings. The mean field solution, on which is based thisapproach, is very complicated mathematically, as we have shown in Sec. 3.1.These computations have been partially performed and the behaviorof the different correlation functions (propagators) which depend on theoverlap q was obtained. All the connected correlation functions presentan algebraic decay (since the low temperature is critical) as in the dropletmodel.Thereupon, we summarize the main results. Firstly, we definethe (connected) correlation function C ( r | q ) ≡ V (cid:88) x (cid:104) s (1) x s (1) x + r s (2) x s (2) x + r (cid:105) . (45) (cid:104) ( · · · ) (cid:105) denotes the thermal average, ( · · · ) is the average over the disor-der, V is the volume of the system, { s (1) i } and { s (2) i } are, as usual, twonon-interacting real copies of the system (real replicas) and the system isconstrained to have a fixed overlap qq = 1 V (cid:88) x (cid:104) s (1) x s (2) x (cid:105) . (46)This conditional correlation function decays algebraically as C ( r | q ) (cid:39) q + A ( qr θ ( q ) . (47)The following values for the exponent θ ( q ) were obtained: • θ ( q EA ) = D − 2. This results could be exact, a sort of Goldstonetheorem. • θ ( q ) = D − q EA > q > 0. This exponent could be modifiedbelow D u = 6. • θ ( q m ) = D − q m = 0. This mode is calledreplicon.The full (non-constrained) correlation function can be recovered with thehelp of P ( q ), the probability to find a given overlap q , see Eq. (37), as C ( r ) = (cid:90) d q P ( q ) C ( r | q ) . (48) une 24, 2020 0:38 ws-rv9x6 Book Title chapter˙ruiz˙lorenzo page 18 J.J. Ruiz-Lorenzo We can refer that the prediction of the droplet model is C ( r ) (cid:39) q + Ar θ , (49)where θ is the exponent which controls the thermal excitations of the system(see Sec. 3.3).In the droplet model there is only one correlation function since there isonly a pure state having an equilibrium overlap q EA , see Sec. 3.3, whereas inthe RSB theory it is possible to find two states with overlap in the interval[0 , q EA ], and therefore it is possible to obtain a correlation function in whichone replica belongs to the state α and the other to the β one having mutualoverlap q αβ = q , measuring C ( r | q ).Finally, let us remark that it is possible to show in a rigorous way that ifthe spin glasses in finite dimensions present ultrametricity, the mathemat-ical properties of this ultrametricity are the same as of the ultrametricityfound in the Parisi theory (RSB) which holds in infinite dimensions. Furthermore, stochastic stability and replica equivalence or overlapequivalence imply ultrametricity and numerical simulations have providedstrong numerical support for both stochastic stability and overlap equiva-lence in three dimensional spin glasses. Field theory ( h (cid:54) = 0 ) The analytical study of spin glasses in presence of a magnetic field belowits upper critical dimensions has been and still is a challenge. In Fig. 6we show the renormalization group flow for spin glasses below the uppercritical dimension and in presence of a magnetic field assuming RSB ordroplet. Despite all the difficulties, the following facts are known: • The upper critical dimensions is 6. • Due to the appearance of a dangerous irrelevant variable the criticalexponents for some observables change already in D = 8. • One can project the full theory on its most divergent components.Working with this projected theory no fixed points were foundat the first order in perturbation theory. The no existence of fixedpoint of the renormalization group in a theory is usually interpretedas(1) it is needed to work at higher order in perturbation theory inorder to find it (or them); (2) the fixed point is of non-perturbative nature and one needs toresort to non-perturbative methods; une 24, 2020 0:38 ws-rv9x6 Book Title chapter˙ruiz˙lorenzo page 19 Nature of the Spin Glass Phase in Finite Dimensional (Ising) Spin Glasses (3) the phase transition is first-order (runaway trajectories). • By using the most general Hamiltonian in the replica symmetricphase and by relaxing the replica trick condition n = 0, a fixedpoint below six dimensions was found. • In Refs. [91,92] the de Almeida-Thouless transition was found belowthe upper critical dimension ( D U = 6) using analytical prolonga-tion. Th Th T c T c h c RSB RS RS * * AT-line(a) (b) Fig. 6. Renormalization group flow for spin glasses in presence of a magnetic field fordimension D l < D < D u . On the left panel, we show the flow assuming replicasymmetry breaking (phase transition in field). On the right the renormalization groupflow assuming the spin glass phase is unstable under magnetic field (droplet model). Thefixed points of the renormalization group equations for temperature and magnetic fieldare marked with a star. The continuous lines, with an arrow, mark the critical surfaces,in which the correlation length is infinite, and the arrows point to the directions in whichthe flow is moving as the scale (in space) of the renormalization group transformationgrows. Finally, in Ref. [93] analytical computations were performed suggestingthe possible existence of a quasi-first-order phase transition below the uppercritical dimension. Metastate Pure phases (or pure states) are macroscopic homogeneous states of matter,for example, liquid, gaseous and solid phases. In principle, it is not difficultto determine if a lump of matter is in a given phase or not. However, todefine states in the thermodynamic limit (infinite volume) is not easy.Rigorously, a state is a probability distribution and with it we can com-pute averages: hence, we can see this probability distribution as a linear une 24, 2020 0:38 ws-rv9x6 Book Title chapter˙ruiz˙lorenzo page 20 J.J. Ruiz-Lorenzo functional. For example, in the two dimensional Ising model (with no disorder)one can define three (states) phases: paramagnetic, ferromagnetic (withpositive magnetization) and ferromagnetic (with negative magnetization)phase. The two ferromagnetic phases can be defined using the followinglimits (cid:104) ( · · · ) (cid:105) + = lim h → lim L →∞ (cid:104) ( · · · ) (cid:105) ( L,h ) , (50) (cid:104) ( · · · ) (cid:105) − = lim h → − lim L →∞ (cid:104) ( · · · ) (cid:105) ( L,h ) . (51)In general, both in experiments and in numerical simulations, we obtainmixed states: i.e. numerical configurations or lumps with interfaces whichsplit different pure states (or pure phases). These mixed states form aconvex set in which its extremal points define the pure states. The mixedstates can be written in terms of pure states in the following way (weparticularize for the Ising model above one dimension and below the criticaltemperature): (cid:104) ( · · · ) (cid:105) = w (cid:104) ( · · · ) (cid:105) + + (1 − w ) (cid:104) ( · · · ) (cid:105) − , (52)where 0 < w < α Γ = (cid:88) α w α Γ α , (53)with (cid:80) α w α = 1, w α > L,J , J denotes the quenched disorder, usually does not converge(chaotic dependence on system size). Some models in which the sequence of states does not converge are: (1) Ising model ( D > 1) without disorder with spins on the fixed boundaryconditions which change as it changes the size of the system.(2) Ising model ( D > 2) in presence of a random magnetic field with zeromean and unit variance. The magnetization of the ferromagnetic phasedoes not converge since it is given by the sign of (cid:80) i h i , which is astochastic variable. une 24, 2020 0:38 ws-rv9x6 Book Title chapter˙ruiz˙lorenzo page 21 Nature of the Spin Glass Phase in Finite Dimensional (Ising) Spin Glasses (3) Ising spin glasses in the chaotic pairs scenario. For each size the systempresents two pure states (related by a global spin flip symmetry), how-ever, these two states vary in a chaotic way as the lattice size grows. The concept of metastate was introduced to tackle the intrinsic lack ofconvergence that show these models. Despite the lack of convergenceof the sequence Γ L,J , it is possible to compute the frequency a given stateappears in that sequence as L → ∞ . The set of these frequencies definesthe Newman-Stein metastate. Fig. 7. Construction of the Aizenman-Wehr metastate. Notice the different boxes(squares in the figure) defined in order to compute the metastate. Before the introduction of the Newman-Stein metastate, another defi-nition of the metastate was given by Aizenman and Wehr. Although, aproof of the equivalence of both metastates is still lacking.The definition of the Aizenman-Wehr allows for a numerical implemen-tation. The construction of the Aizenman-Wehr is as follows:(1) For simplicity, let us consider a two-dimensional system, see Fig. 7. Inthis figure we draw three different regions: Λ W , Λ R and Λ L .(2) We call the disorder inside the inner region Λ R , inner disorder, denotedas I .(3) The disorder in the outer region Λ L \ Λ R is called an outer disorder anddenoted as O .(4) The measurements will be done in the region Λ W . To avoid influencesof the transition region between the inner and outer regions, we willtry to implement the limit Λ W << Λ R << Λ L . une 24, 2020 0:38 ws-rv9x6 Book Title chapter˙ruiz˙lorenzo page 22 J.J. Ruiz-Lorenzo In this framework we are interested to compute the following limit κ I ,R (Γ) = lim L →∞ E O (cid:104) δ ( F ) (Γ − Γ L,J ) (cid:105) , (54)where E O [ · · · ] denotes the average over the outer disorder.Assuming that the following limit exits κ (Γ) = lim R →∞ κ I ,R (Γ) , (55)then it will not depend on the inner disorder I and will provide us theAizenman-Wehr metastate.We can compute averages over the metastate by using (cid:104)· · · (cid:105) ρ ≡ [ (cid:104)· · · (cid:105) Γ ] κ . (56)Therefore, the metataste is also a state.To characterize the metastate, it is very useful to compute the followingcorrelation function C ρ ( x ) = [ (cid:104) s s x (cid:105) Γ ] κ ∼ | x | − ( D − ζ ) , (57)where ζ is a new exponent introduced by Read. The ζ -exponent provides important information about the structure ofthe states of the spin glass phase: • log N states ( W ) ∼ W D − ζ (with ζ ≥ N states ( W ) is the numberof states in a system with size W . • If ζ < D then metastate is disperse (therefore, the droplet theory doesnot hold). • Read conjectured ζ = D − θ (0), where, we recall, θ (0) is the exponentof the replicon mode, see Eq. (47) of Sec. 3.4. 4. Some numerical results After the discussion of some aspects of the theories aimed to explain theproperties of spin glasses in infinite and finite dimensions, we report in thefollowing sections numerical simulations at equilibrium in presence/absenceof the magnetic field. Phase transition at h = 0 Let us present numerical results showing the existence of a phase transitionin the three dimensional Ising spin glass at h = 0. une 24, 2020 0:38 ws-rv9x6 Book Title chapter˙ruiz˙lorenzo page 23 Nature of the Spin Glass Phase in Finite Dimensional (Ising) Spin Glasses Usually, the numerical simulations are performed using the MonteCarlo exchange method (also known as parallel tempering). An-other popular approach in the last years has been the population annealingmethod. In this section, we closely follow Ref. [109] where the supercomputerJanus I was used, see Sec. A.2 for a description of the most importantcharacteristic of this computer.As it has been stated in this chapter, the spin glass order parameteris based on the overlap. In numerical simulations, one can run in paralleldifferent non-interacting real replicas, denoted, as usual, as s a x . Usually,the number of simulated real replicas is between two and six. We can recallagain the general definition of the overlap ( a (cid:54) = b ) q ab x = s a x s b x . (58)In the simulations reported in this section four copies of the system weresimulated with the same disorder (real replicas), therefore, one can computesix different overlaps q ab .One can define the overlap-overlap correlation function given by G ( r ) = 1 V (cid:88) x (cid:104) q ab x + r q ab x (cid:105) , (59)and finally the total overlap per volume of the system ( V ) q ab = 1 V (cid:88) x q ab x . (60)Next, we compute the Fourier transform of the overlap-overlap correla-tion function (cid:101) G ( k ) = 1 V (cid:88) r G ( r ) e i k · r , (61)which allows us to compute the second-moment correlation length ξ = 12 sin( k min / (cid:115) (cid:101) G (0) (cid:101) G ( k min ) − , (62)where k min = (2 π/L, , 0) or the other two possible permutations. The totalsusceptibility is given by the correlation function computed in the Fourierspace at zero momentum χ SG = (cid:101) G (0) = V (cid:104) q (cid:105) . (63)Finally, the correlation length in units of system size ξ/L is R ξ = ξ/L . (64) une 24, 2020 0:38 ws-rv9x6 Book Title chapter˙ruiz˙lorenzo page 24 J.J. Ruiz-Lorenzo We recall that ξ/L is universal at the critical point.In addition to ξ/L we can define the following observables which alsotake universal values at the critical point: U = (cid:104) q (cid:105)(cid:104) q (cid:105) , (65) U = (cid:104) q (cid:105) (cid:104) q (cid:105) , (66) U = (cid:104) q q q (cid:105) / (cid:104) q (cid:105) , (67) U = (cid:104) q q q q (cid:105)(cid:104) q (cid:105) , (68) R = (cid:101) G (2 π/L, , (cid:101) G (2 π/L, π/L, , (69) B χ = 3 V (cid:104)| (cid:101) q (2 π/L, , | (cid:105) [ (cid:101) G (2 π/L, , , (70)where (cid:101) q ab ( k ) = 1 V (cid:88) x q ab x e i k · x . (71)We show in Fig. 8 the different crossing points of the R ξ as a functionof the temperature and the size of the system. The study of these cross-ing points by means of the quotient method (see Sec. A.1 for a detaileddescription of this method) allows to compute the critical exponents andother universal quantities. The results of this analysis can be read inTable 1. In this part of the book chapter, we have only focused on the Isingspin glass with binary couplings. The field theory of Ising spin glass onlydepends on the first two cumulants of the probability distribution of thecouplings: the other (infinite) cumulants of the probability distributionof the couplings J ij only induce irrelevant couplings in the field theory,and following the general theory of the renormalization group, they only induce scaling corrections without changing the values of thecritical exponents (universality). This issue has been checked in Refs. [111–114]. However, some references, see for example Ref. [115], have reportedviolation of universality in Ising spin glass models. une 24, 2020 0:38 ws-rv9x6 Book Title chapter˙ruiz˙lorenzo page 25 Nature of the Spin Glass Phase in Finite Dimensional (Ising) Spin Glasses ξ / L T L = 6 L = 8 L = 10 L = 12 L = 16 L = 20 L = 24 L = 32 L = 400.600.65 1.1 1.12 1.14 Fig. 8. Behavior of the second-moment correlation length as a function of the tem-perature for different values of the size of the system. In the inset, we show a zoom ofthe region in which ξ/L curves cross. We mark with two vertical lines the final valueof the critical temperature, and taking into account the statistical error, one can quote: T c = 1 . Behavior of the correlation functions in the spin glassphase Once we have characterized the critical point, we try to characterize theproperties of the low temperature spin glass phase.Let us consider the conditional spatial correlation function C ( r | q ), seeEq. (45). We focus on the real space behavior of this conditional correlationfunction, a detailed study of its Fourier transform can be found in Ref. [118].The first numerical computation of C ( r | q = 0) was performed inRef. [116] working in the out-of-equilibrium regime (with an extrapolationto infinite time). Subsequently, this out-equilibrium correlation functionwas confronted with an equilibrium computation of the same observable. In Fig. 9 we show these two C ( r | q = 0) (lower curves of this figure) to-gether with the full correlation one (the upper curve of the figure): noticethe very good agreement between the out-of-equilibrium and equilibriumcomputation of C ( r | q = 0).The conditional correlation function C ( r | q = 0) corresponds with the une 24, 2020 0:38 ws-rv9x6 Book Title chapter˙ruiz˙lorenzo page 26 J.J. Ruiz-Lorenzo Table 1. Summary of the cri-tical exponents and the values(universal) of different cumu-lants at the critical point. Exponent/Observable ω = 1 . η = − . ν = 2 . R ∗ ξ = 0 . U ∗ = 1 . α = − . β = 0 . γ = 6 . T c = 1 . U ∗ = 0 . U ∗ = 0 . U ∗ = 0 . B ∗ χ = 2 . R ∗ = 2 . ± . replicon correlation function in the framework of RSB theory and it shoulddecay algebraically with the exponent θ (0), see Eq. (47) of Sec. 3.4. Thispower law decay was corroborated in Refs. [116,117] (see Fig. 9). The repli-con exponent was estimated to be θ (0) = 0 . Most recent out-of-equilibrium numerical simulations performed on theJanus I supercomputer have provided with an accurate value of the repliconexponent: θ (0) = 0 . In the next paragraphs, we report some results obtained at equilibriumin the three dimensional Ising spin glass model with the help of Janus I. Inthese numerical simulations, we were able to proceed up to L ≤ 32 insidethe thermalized low temperature phase. First, we study the q = 0 sector in which the prediction of the dropletmodel and the RSB theory are fully different. In left panel of Fig. 10 weshow C ( r | q = 0) for T = 0 . ∼ . T c which goes to zero for large r .In the right panel of this figure we show the scaling plot of this observableby using the replicon exponent θ = 0 . y = θ ( q EA ) = 0 . Note that by using θ = 0 . r = L/ r = L/ 2, which following RSB should une 24, 2020 0:38 ws-rv9x6 Book Title chapter˙ruiz˙lorenzo page 27 Nature of the Spin Glass Phase in Finite Dimensional (Ising) Spin Glasses C ≡ C ( x | q ) for the three-dimensional Ising spin glass with Gaussian couplings. T = 0 . (cid:39) . T c . The lower curveis the infinite time extrapolation of the non-equilibrium correlation function C ( x | q = 0)obtained by a sudden quench ( L = 64). The second curve from the bottom is C ( x | q = 0)obtained by a slow annealing ( L = 64). The third curve is the equilibrium correlationfunction computed with the constraint | q | < . 01 ( L = 16). The upper curve is the fullequilibrium correlation function, including all configurations ( L = 16). From Refs. [116,117]. behave as (see Eq. (47)) L θ ( q ) (cid:16) C ( r = L/ | q ) − C ( r = L/ | q ) (cid:17) ∼ , (72)in order to remove the background, for large distances, in the C ( r | q ) corre-lation function (finite volume effect). This rescaled difference as a functionof q is shown in Fig. 11. Notice that a good scaling was found in the re- une 24, 2020 0:38 ws-rv9x6 Book Title chapter˙ruiz˙lorenzo page 28 J.J. Ruiz-Lorenzo Fig. 10. C ( r | q = 0) versus r for T = 0 . (cid:39) . T c (left panel). We show in the rightpanel the scaling collapse of L θ (0) C ( r/L | q = 0) using the replicon exponent θ = 0 . 38 asa function of r/L . Fig. 11. Rescaled difference of the correlation function, Eq. (72), as function of q . Wehave used θ (0) = 0 . gion q < . 2, this means that the correlation functions C ( r | q ) − q decayfollowing a power law in the overlap interval (see also Ref. [119]).Moreover, for q > . 2, the scaling breaks down pointing to a value of the θ -exponent used to rescale in Eq. (72) that is larger than θ (0). A detailedanalysis suggests that θ ( q EA ) ∼ . q can be analyzed using finite size scaling. une 24, 2020 0:38 ws-rv9x6 Book Title chapter˙ruiz˙lorenzo page 29 Nature of the Spin Glass Phase in Finite Dimensional (Ising) Spin Glasses To close this section, we report the final values of these exponents ob-tained in these numerical simulations at equilibrium: θ (0) = 0 . and θ ( q EA ) (cid:39) . Finally we quote that an out-of-equilibriumanalysis provided θ ( q EA ) (cid:39) . Metastate: Numerical results In this section, we build numerically the metastate following the Aizenmanand Wehr and we present the main results obtained in Ref. [99].We start with some definitions. The average over the Gibbs state (cid:104)· · · (cid:105) Γ is estimated via Monte Carlo thermal averages (cid:104)· · · (cid:105) at fixed disorder J .The average over the metastate is given by[ · · · ] κ = 1 N O (cid:88) o ( · · · ) , (73)and the one over the internal disorder by( · · · ) = 1 N I (cid:88) i ( · · · ) . (74)The indices i and o run over the number of inner and outer disorder real-izations, denoted as N I and N O , respectively.For example, the metastate spin correlation function (see Eq. (57)) canbe explicitly computed as C ρ ( | x | ) = [ (cid:104) s s x (cid:105) Γ ] κ = 1 N I (cid:88) i (cid:32) N O (cid:88) o (cid:104) s i ; o s i ; o x (cid:105) (cid:33) = 1 N I (cid:88) i N O (cid:88) o , o (cid:48) (cid:104) s i ; o s i ; o x s i ; o (cid:48) s i ; o (cid:48) x (cid:105) . (75)Inside Λ W , see Fig. 7, we can compute a generalized overlap using twonon-interacting real replicas (Ising spins), denoted as { σ } and { τ } . Thesetwo real replicas share the same disorder, denoted by i , with different orthe same outer disorder, denoted as o , for σ , and o (cid:48) , for τ : q i ; o , o (cid:48) ≡ W (cid:88) x ∈ Λ W σ i ; o x τ i ; o (cid:48) x . (76)Notice the possible cases: o = o (cid:48) and o (cid:54) = o (cid:48) . Therefore, we can computethe following probability distributions of the generalized overlap q i ; o , o (cid:48) : P i ( q ) = 1 N O (cid:88) o (cid:104) δ ( q − q i ; o , o ) (cid:105) , (77) une 24, 2020 0:38 ws-rv9x6 Book Title chapter˙ruiz˙lorenzo page 30 J.J. Ruiz-Lorenzo P ( q ) = 1 N I (cid:88) i P i ( q ) , (78) P ρ, i ( q ) = 1 N O (cid:88) o , o (cid:48) (cid:104) δ ( q − q i ; o , o (cid:48) ) (cid:105) , (79) P ρ ( q ) = 1 N I (cid:88) i P ρ, i ( q ) . (80) L =24, T =0.698 W =4 W =8 W =12 qP ρ ( q ) P ( q ) Fig. 12. The overlap probability distributions P ρ ( q ) and P ( q ) against the overlap for L = 24, R = L/ T = 0 . 698 and for different values of the window size, W = 4 , P ( q ) is just the standard overlap probability distribution, but P ρ ( q ) isthe probability distribution of the overlap averaged over the metastate.Regarding P ρ ( q ), despite having a trivial limit P ρ ( q ) → δ ( q ) for W → ∞ , its variance for finite values of W provides with very usefulinformation χ ρ = (cid:88) x ∈ Λ W C ρ ( x ) = W d (cid:90) q P ρ ( q ) dq ∼ W ζ , (81)which allows us to write the following scaling behavior χ ρ ( W, R ) = R ζ f ( W/R ) = W ζ g ( W/R ) . (82)Finally, it is possible to show that the re-scaled metastate-averagedprobability distribution is Gaussian: P ρ ( q/ ( W − ( ζ − D ) / ) . In Fig. 12, we present the behavior of the standard probability distri-bution of the overlap, P ( q ), and the metastate one. The first point is thatboth probabilities are completely different. Moreover, for large values of W the Gaussian shape of P ρ ( q ) starts to be emerging.The scaling of the metastate susceptibility allows to compute the ζ -exponent obtained, ζ = 2 . une 24, 2020 0:38 ws-rv9x6 Book Title chapter˙ruiz˙lorenzo page 31 Nature of the Spin Glass Phase in Finite Dimensional (Ising) Spin Glasses R = L /2, T=0.698 χ ρ / R . W / RL =8 L =12 L =16 L =24 40 80 120 0 2 4 6 8 10 12 14 R =12 χ ρ W L =14 L =16 L =24 Fig. 13. Scaling behavior of the metastate susceptibility χ ρ as a function of W/R for R = L/ T (cid:39) . T c . In the small panel we show the breakdown of the scalingbehavior (asymptotic) which happens as R/L > / replicon exponent ζ = 2 . ζ -exponenton the space dimension, marking the mean-field value (the horizontal valuefor D ≥ D U = 6), and the values obtained in numerical works. Therefore, the numerical implementation of the metastate approach un-veil a structure of the low temperature phase with a disperse metastate (i.e. D > ζ ). Only chaotic pairs and the RSB theory satisfy the property to havea dispersed metastate and not the droplets. Phase transition at h (cid:54) = 0 Since the work of de Almeida and Thouless forty years ago, a large amountof work has been devoted to understand the behavior of spin glasses inpresence of a magnetic field (see Sec. 3.5). In particular during this longperiod of time a huge number of numerical simulations have been per-formed, though, a complete understanding of the model in a fieldis still lacking. One of the reasons is that strong finite size corrections arepresent in these systems which mask the infinite volume behavior.As in the case of the h = 0 numerical simulations, we have initiallybased our analysis on the study of the behavior of the correlation lengthin units of the lattice size R ξ . To do that, we need to compute a critical une 24, 2020 0:38 ws-rv9x6 Book Title chapter˙ruiz˙lorenzo page 32 J.J. Ruiz-Lorenzo d U = 6 d L ≈ ζ = d d ζζ q =0 Fig. 14. The ζ -exponent against dimensionality d ≡ D . In blue we have marked theknown values (mean-field and D = 3 computed in Ref. [99]). In red we have marked thereplicon exponent computed in numerical simulations at different dimensionalities (theRead conjecture is ζ = D − θ (0)). In green the droplet prediction for a non dispersemetastate D = ζ . Finally, we have drawn vertical lines at the lower and upper criticaldimensions ( D l (cid:39) . D u = 6). correlation function. In particular, in presence of a magnetic field, one canextract the critical behavior from two different correlation functions G ( r ) = 1 L (cid:88) x (cid:0) (cid:104) s x s x + r (cid:105) − (cid:104) s x (cid:105)(cid:104) s x + r (cid:105) (cid:1) , (83) G ( r ) = 1 L (cid:88) x (cid:0) (cid:104) s x s x + r (cid:105) − (cid:104) s x (cid:105) (cid:104) s x + r (cid:105) (cid:1) . (84)Both correlation functions have the same critical behavior. The associatedcorrelation lengths are computed in the usual way by calling Eq. (62).This correlation length in units of the lattice size has worked pretty wellin characterizing the phase transition at h = 0 (see Sec. 4.1). However, inpresence of a magnetic field, it fails to identify a phase transition via theusual crossing point of the different curves computed with different latticesizes, see the top panels of Fig. 15 and the left panels of Fig. 16. This lackof crossing on the ξ/L -curves has been interpreted in the past as a clearsignal for a stable paramagnetic phase for all positive temperatures (i.e.there is no phase transition).However, in some models based on random graphs, the phase transitionhas not been found numerically even in models where the phase transitionhas already been characterized analytically. In order to analyze the origin of these strong finite size effects, which une 24, 2020 0:38 ws-rv9x6 Book Title chapter˙ruiz˙lorenzo page 33 Nature of the Spin Glass Phase in Finite Dimensional (Ising) Spin Glasses R TL = 5 L = 6 L = 8 L = 10 L = 12 L = 160.20.40.6 ξ / L Fig. 15. In the top panel we show ξ /L for the simulated lattice sizes at h = 0 . R cumulant, which avoidsthe zero mode in its definition. This observable presents crossing points for the differentlattices, therefore, it is possible to characterize a second order phase transition, includingthe critical temperature and exponents. could spoil the crossing of the cumulants, we can compare the mean-fieldprobability density function of the overlap, P ( q ), with the one computedin a numerical simulation working on finite systems. In Fig. 17-top weshow the numerical P ( q ) for different values of a magnetic field, and in thebottom panel the mean-field prediction. Notice that the support of theanalytical P ( q ) is fully contained in the positive overlap axis and the samehappens for the P ( q ) in the droplet theory. Instead, the numerical P ( q ) stillshows large tails in the negative overlap region. These tails in the negativeoverlap region bias the correlation length, mainly via the zero mode usedin its definition. Furthermore, the spin glass susceptibility, which is theFourier transform of the correlation function computed a zero momentum,strongly suffers from the existence of these tails in P ( q ). Another way tounderstand this phenomena is, following Refs. [136,139], realizing that thefinal results are dominated by atypical measurements. Focusing on typical une 24, 2020 0:38 ws-rv9x6 Book Title chapter˙ruiz˙lorenzo page 34 J.J. Ruiz-Lorenzo ξ L / L h =0.1 L =32 L =24 L =16 L =12 L = 8 L = 6 L = ∞ , h =0 R h =0.1 ξ L / L Th =0.2 R Th =0.2 Fig. 16. We plot in the left panels ξ/L (and R in the right panels) as a function ofthe temperature for h = 0 . h = 0 . 2. As happens for h = 0 . 15, see Fig. 15, the R -curves cross but those of ξ/L ratio do not. measurements will improve the final description of these systems. Table 2. Summary of the critical exponents andthe critical temperatures for three magnetic fields. Parameters h = 0 . h = 0 . h = 0 . T c ( h ) 0.906(40)[3] 1.229(30)[2] 1.50(7) ν η − . To avoid the strong effects induced by the zero mode we defined a newcumulant using the two smallest (and non-zero) momenta. The R hasbeen used and defined in the section devoted to h = 0 (see Eq. (69)), butfor the commodity of the reader we repeat here its definition particularizingin four dimensions R = (cid:101) G ( k ) (cid:101) G ( k ) , (85)where k = (2 π/L, , , 0) and k = (2 π/L, π/L, , 0) (plus permutations)are the two non-zero smallest momenta allowed by the periodic boundaryconditions imposed at the system. une 24, 2020 0:38 ws-rv9x6 Book Title chapter˙ruiz˙lorenzo page 35 Nature of the Spin Glass Phase in Finite Dimensional (Ising) Spin Glasses -3 -2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 P ( q ) qh = 0.05 h = 0.1 h = 0.2 h = 0.4 Fig. 17. P ( q ) from numerical simulations (top) and RSB prediction (bottom) for theprobability distribution of the overlap in the spin glass phase in presence of a magneticfield. The big arrows in the bottom panel represent delta functions. The RSB P ( q )function is different from zero only for positive overlaps. Fig. 15-bottom shows that the cumulant R signals the phase transi-tion. Its value at the critical point, as for other cumulants, is universal.Using these R crossing points it has been possible to characterize thephase transition in four dimension. Furthermore, it is possible to showthat the critical exponents are independent of the strength of the magneticfield (taking into account corrections to scaling). In Table 2 we report thevalues computed for the critical exponents and the critical temperatures. une 24, 2020 0:38 ws-rv9x6 Book Title chapter˙ruiz˙lorenzo page 36 J.J. Ruiz-Lorenzo The analysis of the numerical data was performed working at constant cou-pling, see Sec. A.1 for a description of this method.We can analyze the scaling behavior of these critical temperature per-forming a test of the Fisher-Sompolinsky relation: h ( T c ) (cid:39) A | T c ( h ) − T c (0) | β (0) + γ (0) , (86)where T c ( h ) is the critical temperature in a field, and the symbols witha zero as superscript refer to the critical properties of the model in zeromagnetic field: critical temperature and exponents.In the inset of Fig. 18 we report this analysis finding a very good agree-ment between the numerical results (for the computed critical tempera-tures) and the previous relation. In addition, the fit has only one freeparameter.However, using this methodology, no traces of a phase transition in thethee dimensional model in a field has been found. The simplest explanationis that the lower critical dimension of the model in a magnetic field (3 4) is different of that in h = 0 ( D l = 2 . h = 0 is one and inpresence of a random magnetic field is two). 5. Conclusions We have presented an overview of different analytical approaches to finitedimensional Ising spin glasses. The analytical approach based on field the-ory is still incomplete both for h = 0 and h (cid:54) = 0. In presence of a magneticfield, recent analytical work points out to a complicated structure of thecritical behavior below the upper critical dimension.Moreover, we have presented numerical evidence for existence of thespin glass phase in absence of a magnetic field. The properties of thislow temperature phase fit very well in the framework of the RSB theory.Nextly, we have studied in detailed the replicon propagator and we haveshown how a disperse metastate has been found. The situation in presenceof a magnetic field is not clear. In particular, there is no evidence of aphase transition in three dimensions, though in four dimensions it has beenfound. Acknowledgments This work was partially supported by Ministerio de Econom´ıa y Com-petitividad (Spain) through Grant No. FIS2016-76359-P, by Junta de Ex- une 24, 2020 0:38 ws-rv9x6 Book Title chapter˙ruiz˙lorenzo page 37 Nature of the Spin Glass Phase in Finite Dimensional (Ising) Spin Glasses ξ ( T , h = . ) TL = 5 L = 6 L = 8 L = 10 L = 12 L = 16 h T PARAMAGNETICPHASESPIN-GLASSPHASE Fig. 18. In the inset we plot the Fisher-Sompolinsky relation for a four-dimensionalspin glass with binary couplings: h ( T c ) (cid:39) A | T c ( h ) − T c (0) | β (0) + γ (0) . Notice that thereis only one free parameter in the fit. In the main plot, we present behavior of thecorrelation length for different lattice sizes with the infinite volume extrapolation at h = 0 . tremadura (Spain) through Grants No. GRU18079 and IB16013 (partiallyfunded by FEDER).I have enjoyed interesting and fruitful discussions on spin glasses andnumerical simulations with M. Baity-Jesi, A. Billoire, A. Cruz, L.A. Fernan-dez, A. Gordillo-Guerrero, D. I˜niguez, R. Kenna, A. Lasanta, L. Leuzzi, A.Maiorano, E. Marinari, V. Martin-Mayor, J. Monforte, A. Mu˜noz-Sudupe,D. Navarro, G. Parisi, S. Perez-Gaviro, F. Ricci-Tersenghi, B. Seoane, A.Tarancon, R. Tripiccione and D. Yllanes.I warmly thank Yu. Holovatch and E. Ruiz Espejo for a careful readingof the manuscript.Finally, I would like to thank M. Dudka and Yu. Holovatch for invitingme to present these results in the 2019 Lviv Ising Lectures. une 24, 2020 0:38 ws-rv9x6 Book Title chapter˙ruiz˙lorenzo page 38 J.J. Ruiz-Lorenzo A.1. Characterization of a phase transition: Quotient andfixed coupling methods Let us consider a quantity O ( β, L ) which scales in the thermodynamic limitas ξ x O /ν . We can study the behavior of this observable by computing it at L and 2 L , Q O = O L /O L , at the crossing point β cross ( L, L ) of R ξ or U .In the case of a dimensionless observable the exponent x O = 0. We willdenote as g all the dimensionless quantities. Hence, one gets Q cross O = 2 x O /ν + O ( L − ω ) , (A.1)or g cross = g ∗ + O ( L − ω ) , (A.2)where x O /ν , g ∗ and the correction-to-scaling exponent ω are universalquantities. Examples of dimensionless quantities are R ξ and the cumu-lants (e.g. U and U ). We could also consider dimensionful observablesas the the susceptibility ( x χ = ν (2 − η )) and the β -derivatives of R ξ and U ( x = 1 for both).The behavior of the crossing points of the inverse temperature( β cross ( L, L )) are given by β cross ( L, L ) = β c + A β c ,g L − ω − /ν + . . . , (A.3)where in our case g = R ξ or U .In order to study the leading correction-to-scaling exponent we can buildthe quotient of a given dimensionless quantity g Q g = g L /g L at β cross ( L, L ). This quotient behaves as Q cross g ( L ) = 1 + A g L − ω + B g L − ω + . . . . (A.4)In the fixed coupling method the analysis is slightly different. For in-stance we have a fixed value of a dimensionless observable g = g f near theuniversal one (for example a given value of R ξ ) and we compute the valueof β ( L ) at which g f = g ( β ( g f , L ) , L ) . (A.5)At this value of the inverse temperature we can study scaling of the deriva-tives of different observables (e.g. susceptibility, derivatives of R ξ andBinder cumulant, etc.) which allows to extract the critical exponents via O ( β ( g f , L ) , L ) = A ( g f ) L x O /ν (cid:18) O (cid:18) L ω (cid:19)(cid:19) . (A.6) une 24, 2020 0:38 ws-rv9x6 Book Title chapter˙ruiz˙lorenzo page 39 Nature of the Spin Glass Phase in Finite Dimensional (Ising) Spin Glasses A.2. The Janus supercomputers Most of the numerical simulations presented in this chapter were obtainedusing the Janus I and II supercomputers. These computers were built totake advantage of the powerful integer arithmetic and a high number ofprocessor units of the FPGA (Fast Programmable Devices). They weredesigned and used by a scientific collaboration composed by researchersof two Italian Universities, Ferrara and Roma I “La Sapienza” and threeSpanish ones: Complutense de Madrid, Zaragoza and Extremadura.The physics obtained with the help of the Janus supercomput-ers has covered a wide variety of spin models,mainly Ising spin glass but also Potts glass models. These discrete modelsallow the supercomputers to achieve their maximum performance. Fig. A.1. External view of the Janus I supercomputer during its presentation in Italy. The Janus I computer, see Fig. A.1, entered in production mode in2008. Its most important features are: • It is composed by 16 boards of 16 FPGAs each (Virtex 4). • For Ising models, Janus I is equivalent to 10000 PC. • High degree of parallelization inside the boards. une 24, 2020 0:38 ws-rv9x6 Book Title chapter˙ruiz˙lorenzo page 40 J.J. Ruiz-Lorenzo • Janus allows us to simulate in the 0.1 s time region. Usually, experimen-tal times range from 1 s to 3000 s and previous numerical simulationssimulated the 10 − s region (SSUE).The next generation of Janus computer, the Janus II one, was built in2015 and its principal characteristic are: • Janus II is 5 times most powerful than Janus. • It is still a dedicated computer optimized to simulate a wide variety ofspin models. • It presents a more flexible topology. • It is formed by 16 boards of 16 FPGAs each (one IOP and PC integratedon each board) (Virtex 7). • Janus II allows to simulate in the 1 second time region. References 1. J. A. Mydosh, Spin Glasses: an Experimental Introduction . Taylor and Fran-cis, London (1993).2. G. Kotliar, P. W. Anderson, and D. L. Stein, One-dimensional spin-glassmodel with long-range random interactions, Phys. Rev. B . , 602 (1983).doi: 10.1103/PhysRevB.27.602.3. L. Leuzzi, Critical behaviour and ultrametricity of Ising spin-glass with long-range interactions, Journal of Physics A: Mathematical and General . 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