Ergodicity breaking in frustrated disordered systems: Replicas in mean-field spin-glass models

0.41 MB·English

by  V. Janis

Ergodicity breaking in frustrated disordered systems: Replicas in mean-field
spin-glass models
V. Janiˇs∗, A. Kauch, and A. Kl´ıˇc
Institute of Physics, Academy of Sciences of the Czech Republic, Na Slovance 2, CZ-18221 Praha 8, Czech Republic
We discuss ergodicity breaking in frustrated disordered systems with no apparent broken sym-
metry of the Hamiltonian and present a way how to amend it in the low-temperature phase. We
demonstrate this phenomenon on mean-field models of spin glasses. We use replicas of the spin
variablestotestthermodynamichomogeneityofergodicequilibriumsystems. Weshowthatreplica-
symmetry breaking reflects ergodicity breaking and is used to restore an ergodic state. We then
present explicit asymptotic solutions for the Ising, Potts and p-spin glasses. Each of the models
5 shows a different low-temperature behavior and the way the replica symmetry and ergodicity are
1 broken.
0
2 PACSnumbers: 05.50.+q,64.60.De,75.10.Nr

n
a
I. INTRODUCTION In this paper we discuss the peculiarities of ergodicity
J
breakinginphasetransitionsinfrustrateddisorderedsys-
7
tems described by random non-local interaction without
Ergodicity is one of the most important properties of
any directional preference in the phase space of the fun-
] largestatisticalsystems. Weusuallyassumethatmacro-
n damental variables. We use mean-field spin-glass models
n scopic systems in equilibrium are ergodic. Ergodicity, or with a Gaussian random spin-exchange and show how
- quasi-ergodicity, means the phase trajectory comes arbi-
s onecanrestoreergodicityviahierarchicalreplicatingthe
trarily close to any point of the space allowed by macro-
i spin variables of the original Hamiltonian. We present a
d scopic constraints. The physically most important form
general scheme of the real-replica method and apply it
.
t of the ergodic theorem due to Birkhoff tells us that time
a on mean-field Ising, Potts and p-spin glass models. The
average along the phase-space trajectory equals the sta-
m three generic models show different ways in which er-
tistical average over the whole phase volume.1 It means
godicity is broken and we demonstrate on them how the
- that the equilibrium state must span the whole avail-
d phase space of the equilibrium state can be constructed.
n ablephasespace. Althoughthefoundationsofstatistical Wemanifestthatergodicitybreakingisequivalenttothe
o mechanics are based on ergodicity, lack of ergodicity is
replica-symmetry breaking.
c widespreadinphysicalphenomena.2 Typicalexamplesof
[ ergodicity breaking are phase transitions with symme-

1 try breaking. Broken ergodicity is sometimes used as a
v generalization of symmetry breaking.3 Although broken II. ERGODICITY AND THERMODYNAMIC
3 global symmetry is always accompanied by broken er- HOMOGENEITY
5 godicity, the converse does not hold. There are systems
6
that break ergodicity without any apparent symmetry Birkhoffergodictheoremallowsonetointroduceequal
1
0 of the Hamiltonian being broken. The typical example `a priori probability of allowed states in the phase space.
. is structural glass with numerous metastable states that Itis,however,nontrivialtodetermineinmacroscopicpa-
1 prevent the macroscopic system from reaching the true rameterswhichtheallowedstatesindeedare. Thisisac-
0
equilibrium on experimental time scales. tually the most difficult task in constructing the proper
5
1 Broken ergodicity represents an obstruction in the ap- phase space in statistical models. That is, to find out
: plication of fundamental thermodynamic laws. It hence which points of the phase space are infinitesimally close
v
mustberecovered. Whenergodicityisbrokeninaphase to the trajectory of the many-body system extended to
i
X transition breaking a symmetry of the Hamiltonian, one infinite times. Since we never solve the equation of mo-
r introduces a symmetry breaking field into the Hamilto- tion of the statistical system, we have only static means
a
nian,beingtheLegendreconjugatetotheextensivevari- tocheckvalidityofergodicity. Wethentestconsequences
ablethatisnotconservedinthebrokensymmetrytrans- of the ergodic hypothesis on the behavior of the equilib-
formationinthelow-temperaturephase. Thesymmetry- rium state. The most important consequence of ergodic-
breaking field allows one to circumvent the critical point ity of statistical systems is the existence of the thermo-
of the symmetry-breaking phase transition and simulta- dynamic limit.
neouslyrestoresergodicity. Systemswithnobrokensym- Thetrajectoryofthemany-bodysystemcoversalmost
metryoftheHamiltonianatthephasetransition,suchas the whole allowed phase space. It means that the space
spin glasses or some quantum phase transitions, do not coveredbysuchtrajectorydoesnotdependontheinitial
offer external symmetry-breaking fields and other tech- state in non-chaotic systems. In ergodic systems then
niques must be employed to find the proper portion of thethermodynamiclimitdoesnotdependonthespecific
the phase space covered by the phase-space trajectory. form of the volume in which the macroscopic state is
2
confined as well as on its surrounding environment. The ∆H(µ)=(cid:80) (cid:80)ν µabXaXb to the replicated Hamilto-
i a<b i i
macroscopic systems can either be isolated or embedded nianwithdynamicalvariablesX . Theoriginalsystemis
i
in a thermal bath. The thermodynamic equilibrium, the then stable with respect to fluctuations in the bath, rep-
equilibriumstateinthethermodynamiclimit,isthesame resented by the interaction with the replicated variables,
withvanishingrelativestatisticalfluctuations. Thether- if the linear response to perturbation µ is not broken. If
modynamic equilibrium can then be reached by limiting the linear response holds then the perturbed free energy
any partial volume of the whole to infinity. The ergodic per replica relaxes, after switching perturbation µ off, to
equilibrium state is homogeneous in the thermodynamic the original one in the thermodynamic limit
limit.
Thermodynamic homogeneity is usually expressed via 1 (cid:40) (cid:88)ν (cid:41)
Euler’s lemma4 −βFν(µ)= ν lnTrνexp −β Ha−β∆H(µ)
a=1
α S(U,V,N,...,X ,...)=S(αU,αV,αN,...,αX ,...) −−−→lnTrexp{−βH} , (3)
i i
(1) µ→0
telling us that entropy S is an extensive variable and is
where Tr refers to trace in the ν-times replicated phase
a first-order homogeneous function of all extensive vari- ν
space. If the linear response to the inter-replica inter-
ables, internal energy U, volume V number of particles
action does not hold, the thermodynamic limit of the
N, and model dependent other extensive variables X .
i original system is not uniquely defined and depends on
As a consequence of Euler’s lemma we obtain that ther-
properties of the thermal bath. If there are no appar-
modynamic equilibrium is attained as a one-parameter
ent physical fields breaking the symmetry of the Hamil-
scaling limit where we have only one independent large
tonian, the phase-space scaling represented by replicas
scale, extensive variable, be either volume or number of
of the dynamical variables introduces shadow or auxil-
particles,andtheotherextensivevariablesenterthermo-
iary symmetry-breaking fields, inter-replica interactions
dynamic potentials as volume or particle densities insen-
µab > 0. They induce new order parameters in the re-
sitive to changes of the scaling variable.
sponse of the system to these fields that need not vanish
Thermodynamic homogeneity allows us to use scal-
in the low-temperature phase, when the linear response
ing of the original phase space. Thermodynamic quanti-
breaks down. They offer a way to disclose a degeneracy
ties remain unchanged if we arbitrarily rescale the phase
when the thermodynamic limit is not uniquely defined
space and then divide the resulting thermodynamic po-
by a single extensive scale and densities of the other ex-
tential by the chosen scaling (geometric) factor. We can
tensive variables. The inter-replica interactions are not
do that by scaling energy E of the equilibrium state. If
measurable and hence to restore the physical situation
we use a scaling factor ν, that can be an arbitrary posi-
we have to switch off these fields at the end. If the sys-
tivenumber,thenthefollowingidentitiesholdforentropy
tem is thermodynamically homogeneous we must fulfill
S(E) of the microcanonical and free energy F(T) of the
the following identity
canonical ensemble with energy E and temperature T,
respectively (cid:20) (cid:21)
d
lim F (µ) ≡0 (4)
k k dν µ→0 ν
S(E)=k lnΓ(E)= B lnΓ(E)ν = B lnΓ(νE) ,
B ν ν
(2a) for arbitrary ν. This quantification of thermodynamic
homogeneity, thermodynamic independence of the scal-
F(T)=− kBT ln(cid:2)Tr e−βH(cid:3)ν =− kBT ln(cid:2)Tr e−βνH(cid:3) , ing parameter ν, will lead us in the construction of a
ν ν
stable solution of mean-field spin glass models. To use
(2b)
equation (4) in the replica approach we will need to con-
tinue analytically the replica-dependent free energy to
where we denoted Γ(E) the phase-space volume of the
arbitrary positive scaling factors ν ∈ R+. Specific as-
isolated system with energy E.
sumptions on the symmetry of matrix µab will have to
The scaling of the phase space with an integer scal-
be introduced. It is evident from Eq. (3) that the linear
ing factor ν can be simulated by replicating ν-times the
response to inter-replica interaction can be broken only
extensive variables. That is, we use instead of a single
if the replicas are mixed in the ν-times replicated free
phase space ν replicas of the original space. The reason
energy F .
tointroducereplicasoftheoriginalvariablesistoextend ν
the space of available states in the search for the allowed
spaceinequilibrium. Thereplicasareindependentwhen
introduced. Weusethereplicatedvariablestostudysta- III. FRUSTRATED DISORDERED SYSTEMS -
bility of the original system with respect to fluctuations MEAN-FIELD SPIN-GLASS MODELS
in the thermal bath. To this purpose we break indepen-
denceofthereplicavariablesbyswitchingona(homoge- We present models on which the replica approach to
neous)infinitesimalinteractionbetweenthereplicasthat the construction of the equilibrium state proves efficient.
we denote µab. We then add a small interacting part The linear response to a small inter-replica interaction
3
may be broken only if replicas are intermingled in ther- their scalar product
modynamic potentials. Mixing of replicas is achieved by
p p
randomness of an inter-particle interaction. We use lat- (cid:88)eα =0 , (cid:88)eαeβ =p δαβ , eαeα =p δ −1
tice spin models with random spin-exchange to study A A A A B AB
A=1 A=1
replica mixing thermodynamic potentials. To simplify
(7a)
the reasoningwe resortto mean-field models withno ex-
plicit spatial coherence. The mean-field approximations for α ∈ {1,...,p−1}. We use the Einstein summation
of lattice systems can either be introduced as models conventionforrepeatingGreekindicesofthevectorcom-
on fully connected graphs, models with long-range in- ponents indicating a scalar product of the Potts vectors.
teraction,orasthelimittoinfinitespatialdimensionson Using these properties we can construct an explicit rep-
hyper-cubic lattices. Thereby a scaling of the inter-site resentation of the Potts spin vectors
interactions is needed so that to keep the energy an ex-

tdeenmsiavnedvedarbiayblEe,q.li(n1e)a.rly proportional to the volume as eα =0(cid:113)p(p−α) AA=<αα (7b)
A p+1−α
 1 (cid:113)p(p−α) A>α .
α−p p+1−α
A. Ising spin glass
Potts model shows a glassy behavior if we introduce
randomnessinspinexchangeJ . Inthemean-fieldlimit
ij
The simplest lattice spin system consists of spins with
we chose the following Gaussian distribution
the lowest value (cid:126)/2 where only their projection to the
easy axis enters interaction. We set in this paper (cid:126) = 1 −(J −J )2
P(J )= exp ij 0 , (8)
1,kB =1. The spins can then be treated classically with ij (2πJ2/N)1/2 2J2/N
projections S = ±1. The resulting Hamiltonian of the
Ising model reiads where J0 = N−1(cid:80)jJ0j is the averaged (ferromagnetic)
interaction.
(cid:88) (cid:88)
H[J,S]= J S S +h S . (5)
ij i j i
i<j i
C. p-spin glass
To obtain a glassy behavior we introduce a randomness
into spin exchange J . We will explicitly consider only Potts model is not the only interesting extension of
ij
themean-fieldversionofthismodelwithaGaussiandis- the Ising model. Another generalization is the so-called
tribution of the spin-exchange. The energy remains an p-spin model. It describes a system of Ising spins where
extensive variable if we rescale the (long-range) interac- the spin exchange connects a cluster of p spins. The
tion in the man-field limit as follows Hamiltonian of such a model reads
(cid:88)
N N H [J,S]= J S S ...S . (9)
N(cid:104)J (cid:105) =(cid:88)J =0, N(cid:10)J2(cid:11) =(cid:88)J2 =J2 , p i1i2...ip i1 i2 ip
ij av ij ij av ij 1≤i1<i2<...<ip
j=1 j=1
RandomnessinthespinexchangeisagaintakenGaussian
where N is the number of lattice sites. so that the energy is an extensive variable in the large
volume limit N →∞:7
(cid:115)
B. Potts glass P (cid:0)J (cid:1)= Np−1 exp(cid:26)−Ji1i2...ip 2Np−1(cid:27) .
i1i2...ip πp! J 2p!
Thep-statePottsmodelisageneralizationoftheIsing
This model is interesting in that we can analytically
model to p > 2 spin components. The original formula-
study the limit p → ∞ for which we know an exact
tion of Potts5 with Hamiltonian H = −(cid:80) J δ
p i<j ij ni,nj solution.8,9
where n = 0,...,p−1 is an admissible value of the p-
i
state model on the lattice site R , is unsuitable for prac-
i
tical calculations. The Potts Hamiltonian can, however,
IV. ERGODICITY AND REPLICA-SYMMETRY
be represented via interacting spins6 BREAKING
1(cid:88) (cid:88)
H [J,S]=− J S ·S − h·S , (6) Randomness in the spin exchange causes mixing of
P 2 ij i j i
i,j i replicas of the spin variables. Frustration prevents selec-
tion of easy axes, even if inhomogeneously distributed,
where S = {s1,...sp−1} are Potts vector variables tak- thatcouldselectkindofregularspinordering. Thespin-
i i i
ingvaluesfromasetofstatevectors{e }p . Functions glassmodelsdonotprovideuswithapparentsymmetry-
A A=1
on vectors e are in equilibrium fully defined through breaking fields that could stabilize the low-temperature
A
4
glassy phase. There is, nevertheless, an ordered low- sum. The only difference is that there is no limit to
temperature phase with order parameters to be found. zeronumberofreplicasfortheannealedrandomness. In-
The determination of the proper phase space of homo- stead, independence of the replica index is demanded.
geneous order parameters is the most difficult part in The annealed and quenched free energies can then be
thesearchforthetrueequilibriumstateinthespin-glass represented in the ν-times replicated phase space:
models. Replicas proved to be the only available means
1
for reaching this goal. βF =− lim ln(cid:104)Zν(cid:105) , (11a)
an ν N→∞ N av
(cid:20) (cid:21)
1
βF =−lim lim ((cid:104)Zν(cid:105) −1) . (11b)
A. Annealed and quenched disorder qu ν→0 ν N→∞ N av
What is common for both cases is that we have to con-
Randomness in the spin exchange, introduced in
tinueanalyticallythefreeenergyofthereplicatedsystem
Sec.III,causesmixingofreplicasandmayleadtobreak-
to arbitrary positive multiplication factor ν. Either to
ingofergodicitymanifestedinareplica-symmetrybreak-
testquantitativelythermodynamichomogeneity,Eq.(4),
ing. We can, however, treat the disorder either dynam-
or to perform the limit ν →0 non-perturbatively.
ically or statically. That is, we can prepare the system
so that the random configurations are thermally equi-
librated, annealed disorder, and contribute to a single
B. Replica-symmetry breaking
equilibrium state. In this situation we average over ran-
dom configurations of the partition sum. If we have ν
replicas of the spin variables we have to calculate the To find the equilibrium state for the spin glass mod-
following configurationally averaged partition function els we will test validity of the linear-response to a small
inter-replica interaction. Although ergodicity of the
(cid:90) ν N original model may be broken, we expect that it will
(cid:89)(cid:89)
(cid:104)Zν(cid:105) = D[J]µ[J] d[Sa]ρ[Sa] be restored in an appropriately replicated space. The
N av i i
a=1i=1 inter-replica interaction generates new order parameters
(cid:40) ν (cid:41) χab =(cid:104)(cid:104)SaSb(cid:105) (cid:105) −q that will not vanish if ergodicity,
(cid:88) T av
×exp −β H[J,Sa] , (10a) replica-symmetry is broken. We denoted q = (cid:104)(cid:104)Sa(cid:105)2(cid:105)
T av
a=1 that is the order parameter in the non-replicated (en-
larged) space. A ν-times replicated free energy density
where µ(J) and ρ(S) are distribution functions for the
of the Ising spin glass (quenched disorder) in the mean-
spin exchange and spins, respectively. Or we can cool
field limit can be represented as11
the macroscopic system down very fast so that there is
notenoughrelaxationtimetoreachthelow-temperature
 
ethqeurimlibordiuymna,mqiucenpcohteedntdiailssoridnert,heantdhewrmeohdayvneatmoiacvleirmaigte, fν = β4J2 ν1 (cid:88)ν (cid:110)(cid:0)χab(cid:1)2+2qχab(cid:111)−(1−q)2
e. g. free energy, a(cid:54)=b
(cid:90) (cid:90) (cid:89)ν (cid:89)N − 1 (cid:90)∞ √dη e−η2/2lnTrexp(cid:40)β2J2(cid:88)ν χabSaSb
−β(cid:104)Fν(cid:105) = D[J]µ[J]ln d[Sa]ρ[Sa] βν 2π
N av i i −∞ a<b
×exp(cid:40)−β(cid:88)aν=1Hi=[J1,Sa](cid:41) . (10b) +βh¯(cid:88)ν Sa(cid:41) , (12)
a=1
a=1
Spinglassesareassumedtobequenched,henceaveraging where we denoted the fluctuating magnetic field h¯ =
√
over spin couplings from Eq. (10b) is to be used. h+η q. Before we investigate validity of he linear re-
From the mathematical point of view it is much more sponse, we analytically continue the expression for the
complicated to evaluate the quenched averaging, since freeenergytoarbitrary(non-integer)positivereplication
logarithmisdifficulttohandle. EdwardsandAnderson10 indices ν ∈R+. Parisi found restrictions on the symme-
introduced the replica trick, lnx = limn→0(xn−1)/n try of the matrix of overlap susceptibilities χab to make
to convert quenched into the annealed averaging. The fν an analytic function of ν.12–14 They are
replica trick was originally the reason to introduce repli-
ν
cas. Hence, to distinguish the replicas from the replica χaa =0 , χab =χba , (cid:88)(cid:0)χac−χbc(cid:1)=0 . (13)
trickfromthosetestingthermodynamichomogeneity,we
c=1
call the former mathematical replicas and the latter real
replicas. These restrictions reduce the number of independent
At the end, there is no big difference in the annealed overlap susceptibilities to K ≤ ν. Let each indepen-
and quenched randomness when using replicated vari- dent susceptibility χ have multiplicity m . Parame-
l l
ables. In both cases we average a replicated partition ters m must be divisors of ν and obey a sum rule
l
5
Z = cosh(cid:104)β(cid:16)h+η√q+(cid:80)K λ √∆χ (cid:17)(cid:105). Free energy
0 χ χ χ χ χ χ χ  0 l=1 l l
1 2 2 3 3 3 3
χ1 0 χ2 χ2 χ3 χ3 χ3 χ3 fK is an analytic function of geometric parameters mi
χ χ 0 χ χ χ χ χ  thatcannowbearbitrarypositivenumbers. Equilibrium
 2 2 1 3 3 3 3
χ χ χ 0 χ χ χ χ  values of order parameters χ ,m in representation (14a)
 2 2 1 3 3 3 3 l l
χ3 χ3 χ3 χ3 0 χ1 χ2 χ2 or ∆χl,ml from (14b) are determined from extremal
χ3 χ3 χ3 χ3 χ1 0 χ2 χ2 points of the free energy functional.
χ3 χ3 χ3 χ3 χ2 χ2 0 χ1 It is clear that complexity of the solution increases
χ χ χ χ χ χ χ 0
3 3 3 3 2 2 1 rapidly with the increasing number of different values
∆χ , that is, with number K of replica hierarchies. We
i
FIG. 1: Matrix of overlap susceptibilities χ for ν = 8 and give here an example of the lowest replica-symmetry
ab
with three levels (hierarchies) of symmetry breaking K = 3 breaking free energy (K =1)
exemplifying the structure allowing for analytic continuation
to arbitrary positive ν. β
f (q;χ ,m )=− (1−q−χ )2
1 1 1 4 1
β 1 (cid:90) ∞ (cid:90) ∞
+ m χ (2q+χ )− Dηln Dλ
ν =1+(cid:80)Ki=1mi. We generally denote K the number of 4 1 1 1 βm√1 −∞ √ −∞ 1
independent values of overlap susceptibility χab. An ex- {2cosh[β(h+η q+λ1 χ1)]}m1 . (15)
ample of such a matrix for ν =8, K =3 and m =2l−1
l
is illustrated in Fig. 1. It is now straightforward to cal- It has three parameters, q,χ1,m1 to be determined from
culate free energy for matrices χ fulfilling criteria (13). stationarity of the free-energy functional from Eq. (15).
ab
We obtain It represents a free energy with the first level of ergodic-
itybreakingorreplica-symmetrybreaking(1RSB).Gen-
β β (cid:88)K erally, free energy fK stands for ergodicity breaking on
f (q,{χ};{m})=− (1−q)2+ (m −m )χ K levels, K generations of replicas (KRSB).
K 4 4 l l−1 l
Free energy f (q;∆χ ,...,∆χ ,m ,...,m ) con-
l=1 K 1 K 1 K
×(2q+χl)+ β2χ1− βm1K (cid:90)−∞∞ √d2ηπe−η2/2ln(cid:20)(cid:90)−∞∞ 1ftra,ei2en,se.n.2.eKKrgy+thw1aittvhaarrrieeastpdioeencttaerltmopisanmreaadmllefflrtoeumrcst,usqat,at∆itoionχnsi,aomrfittiyhfeoosrfeitpha=e-
dλ (cid:26) (cid:90) ∞ dλ
√ Ke−λ2K/2 ... √ 1 e−λ21/2{2cosh[β(h rameters. Thereplicaconstruction introducedanew pa-
2π −∞ 2π rameterK thatisnot`aprioridetermined. Itcanassume
K (cid:33)(cid:35)(cid:41)m1 (cid:41)mK/mK−1 anyintegervalueinthetrueequilibrium. Thenumberof
√ (cid:88) (cid:112) replicahierarchiesisinthisconstructiondeterminedfrom
+η q+ λl χl−χl+1 ... 
stabilityconditionsthatrestrictadmissiblesolutions,sta-
l=1
tionaritypoints. AsolutionwithK levelsislocallystable
(14a)
ifitdoesnotdecayintoasolutionwithK+1hierarchies.
Aneworderparameterinthenextreplicageneration∆χ
with χ = 0 and m = 1. It may appear convenient
K+1 0
may emerge so that ∆χ > ∆χ > ∆χ for arbitrary
to rewrite the free-energy density to another equivalent l l+1
l. That is, the new order parameter may peel off from
form
∆χ and shifts the numeration of the order parameters
l
β for i > l in the existing K-level solution. To guaran-
fK(q;∆χ1,...,∆χK,m1,...,mK)=−4 (1−q− tee that this does not happen and that the averaged free
energy depends on no more geometric parameters than
K (cid:33)2 K (cid:34) (cid:32) K (cid:33)
(cid:88) 1 β (cid:88) (cid:88) m ,...,m we have to fulfill a set of K +1 generalized
∆χ − ln2+ m ∆χ 2 q+ ∆χ 1 K
l β 4 l l i stability criteria that for our hierarchical solution read
l=1 l=1 i=l for l=0,1,...,K
1 (cid:90) ∞
−∆χ ]− Dη ln Z (14b)
l β −∞ K ΛKl =1−β2(cid:10)(cid:10)(cid:10)1−t2+
where we ordered the parameters so that ∆χ = χ − l (cid:43)2(cid:43) (cid:43)
χl+1 ≥ ∆χl+1 ≥ 0. We further used a short-halnd nolta- (cid:88)mi(cid:0)(cid:104)t(cid:105)2i−1−(cid:104)t(cid:105)2i(cid:1) ≥0 (16)
tion for iterative partition functions i=0 l K η
(cid:20)(cid:90) ∞ (cid:21)1/ml with m0 = 0 and formally (cid:104)t(cid:105)−1 =
Z = Dλ Zml 0. We introduced short-hand notations
l −∞ l l−1 t ≡ tanh(cid:104)β(cid:16)h+η√q+(cid:80)K λ √∆χ (cid:17)(cid:105) and
l=1 l l
wenittihal aDnλ ab≡brevidaλtione−λf2o/r2/√a2π.GaussTiahne diniffiteiar-l (cid:104)(cid:104)tX(cid:105)l((ληl;)λ(cid:105)λKl,.=..,(cid:82)λ−∞l∞+1D)λl X=(λl)(cid:104)aρnld...ρ(cid:104)lρ1=t(cid:105)λZ1lm.−.l1.(cid:105)/λ(cid:104)lZlm−lw1(cid:105)iλtlh.
partition function for the Ising spin glass is ThelowestK forwhichallstabilityconditions, Eq.(16),
6
are fulfilled is an allowed equilibrium state. It need not, It is an implicit but closed functional equation for the
however,bethetrueequilibriumstate,sincethestability derivative g(cid:48)(x;h) on interval [0,X] for a given function
conditions test only local stability and cannot decide m(x). Wehavetoknowthefulldependenceofthisfunc-
whichofseveralextremalpointsisthetruegroundstate. tion on parameter x to evaluate the free energy with
The stability conditions, Eq. (16), are necessary for the continuous replica-symmetry breaking. It is important
system to be thermodynamically homogeneous. They to note that free energy f(q,X;m(x)) defines a ther-
are, however, not sufficient to guarantee global thermo- modynamic theory independently of the replica method
dynamic homogeneity. Note that stability conditions within which it was derived. It means that we can look
from Eq. (16) guarantee only local homogeneity, since for equilibrium states of spin-glass models without the
they hold only for the optimal geometric parameters necessity to go through instabilities of the discrete hier-
m determined by stationarity equations. The global archical replica-symmetry breaking solutions.
l
thermodynamic homogeneity, Eq. (4), would demand Analogously we can perform the limit K → ∞ with
ΛK ≥0 for arbitrary positive m . m −m =∆m/K →dm in representation Eq. (14a).
K K l l+1
We recall that m < m . We further on assume that
l+1 l
χ −χ >0isnotinfinitesimallysmall. Thelimitingfree
1 2
C. Continuous limit energy can then be represented as
Iffreeenergyf (q;∆χ ,...,∆χ ,m ,...,m )isun- β
K 1 K 1 K f(q,χ ,m ,m ;x(m))=− (1−q−χ )2
stable for all finite K’s one has to perform the limit 1 1 0 4 1
K → ∞. Parisi derived a continuous version of the β (cid:104) (cid:105) β (cid:90) m1
+ m (q+χ )2−m q2 − dm[q+X (m)]2
infinitely times replicated system by assuming ∆χl = 4 1 1 0 4 0
∆χ/K → dx, and neglecting second and higher powers m0
1 √
of ∆χ with the fixed index l. This ansatz was based on − (cid:104)g (m ,h+η q)(cid:105) , (19)
l β 1 0 η
the analysis of the first few hierarchical solutions of the
Isingspinglass.12–14 WhenperformingthelimitK →∞ where we denoted X (m)=(cid:82)m dm(cid:48)x(m(cid:48)),
inrepresentationEq.(14b)thefree-energyfunctionalcan 0 m0
then be represented as15
g (m ,h)=T exp(cid:26)1(cid:90) m1dmx(m)(cid:2)∂2
β 1 β (cid:90) X 1 0 m 2 h¯
f(q,X;m(x))=− (1−q−X)2− ln2+ dx m0
4 β 2 (cid:12)
1 √ 0 +mg1(cid:48)(m;h+h¯)∂h¯(cid:3)(cid:9)g1(h+h¯)(cid:12)(cid:12)(cid:12) . (20a)
m(x)[q+X−x]− (cid:104)g(X,h+η q)(cid:105) (17) h¯=0
β η
andT isnowmincreasingordering. Theinputfunction
(cid:82)∞ m
where (cid:104)X(η)(cid:105)η = −∞DηX(η). This free energy is is
onlyimplicitsinceitsinteractingpartg(X,h)canbeex-
pressedonlyviaanintegralrepresentationcontainingthe 1 (cid:90) ∞ dφ
g (h)= ln √ e−φ2/2[2cosh(β(h
solution itself 1 m1 −∞ 2π
(cid:40) (cid:112) (cid:17)(cid:17)(cid:105)m1
g(X,h)=T exp 1(cid:90) Xdx(cid:2)∂2 +φ χ1−X0(m1) . (20b)
x 2 h¯
0
This free energy better suits the case when the solution
(cid:12)
+m(x)g(cid:48)(x;h+h¯)∂h¯(cid:3)(cid:9)g(h+h¯)(cid:12)(cid:12)(cid:12)h¯=0 , (18a) wleivtehlaRcSoBntoinrutohuestRwSoBsoplueetilosnosffcforoexmisat.soTluhteiosnpawciethofotnhee-
order parameters is restricted to an interval 0≤m ≤1
1
with g(h) = ln[coshβh]. The ”time-ordering” operator
and 0≤X (m)≤χ ≤1.
T ordersproductsofx-dependentnon-commutingoper- 0 1
x Iffreeenergyf isnotlocallystableandatleastoneof
K
ators from left to right in a x-decreasing succession. The
stabilityconditions,Eq.(16),isbrokenforallK’s,itisa
exponent of the ordered exponential contains function
question whether the continuous limit K →∞ is locally
g(cid:48)(x;h) = ∂g(x;h)/∂h for x ∈ [0,X] and is not known
stable. It can be shown that the continuous free energy
when g(x;h) is not know on the whole definition inter-
f(q,X;m(x)) is marginally stable, fulfills the continuous
val. This derivative can also be expressed via an ordered version of stability conditions with equality.15 It means
exponential
thatthecontinuousfreeenergydoesnotbreakergodicity,
isalwaysmarginallyergodicinthewholespin-glassphase
(cid:40)(cid:90) X (cid:20)1 as the ferromagnetic model is only at the critical point.
g(cid:48)(X,h)=T exp dx ∂2
x 2 h¯ Both continuous free energies, Eq. (17) and (19), were
0
(cid:12) derived as the limit of the number of replica hierarchies
+m(x)g(cid:48)(x;h+h¯)∂h¯(cid:3)(cid:9)g(cid:48)(h+h¯)(cid:12)(cid:12)(cid:12)h¯=0 . (18b) Kera→rch∞ieswisheinrefinthiteesdimistaal,nctehabtetiswe∆eχnlt∝heKne−ig1,hbaosrwinegllhais-
7
χ ∝ K−1, and ∆χ /∆m < ∞ for each l ≤ K. Repre- leading order of the order parameters
K l l
sentation (19) is, however, more general, since it allows
. 2
for a mixture of a discrete and continuous order param- ∆χK = θ , (21a)
l 2K+1
eters.
. 4(K−l+1)
mK = θ , (21b)
ThecontinuousfreeenergieswerederivedfortheIsing l 2K+1
spin glass but they can be straightforwardly generalized . 1
to other spin-glass models. The symmetry of the order qK = θ (21c)
2K+1
parameters has to be adapted and the input single-site
free energy g or g is to be appropriately modified.16,17 that clearly indicate that the limit K → ∞ leads to the
1
Parisi continuous replica-symmetry breaking. Since each
solution with a finite number of replica generations is
unstable
4 θ2
ΛK =− <0 , (21d)
l 3 (2K+1)2
V. EQUILIBRIUM STATES IN SPIN-GLASS
MODELS: ASYMPTOTIC SOLUTIONS itisexplicitlyprovedthattheParisisolutionistheequi-
librium state in the glassy phase of the Sherrington-
Kirkpatrick model. Other physical quantities in the
Spin-glass models experience a nontrivial ergodicity asymptotic limit are the Edwards-Anderson parameter
breaking that can be studied with hierarchical replica- defined as QK =q+(cid:80)K ∆χK
l=1 l
tions. Thereplicamethoddoesnottellus,however,how
the true equilibrium looks like. Whether finite number QK =. θ+ 12K(K+1)+1 θ2 , (22a)
of replica hierarchies are enough to restore ergodicity or 3(2K+1)2
it is the marginally ergodic continuous limit that repre-
local spin susceptibility
sentstheequilibriumstate. Theproblemwiththemean-
field theories of spin glasses is that we are unable to find (cid:32) (cid:88)K (cid:33) . θ2
χ =β 1−QK + m ∆χ =1− .
full solutions in most models beyond the first level of T l l 3(2K+1)2
ergodicity breaking, 1RSB free energy, Eq. (15). The l=1
(22b)
only way to resolve the stationarity equations of the full
andthe free-energy differenceto the paramagnetic state,
mean-field models is to use an asymptotic expansion in
small order parameters below the transition point to the . (cid:18)1 7 29 (cid:19) 1 (cid:18) 1 (cid:19)4
glassy phase, if they exist. That is, when the transition ∆f = θ3+ θ4+ θ5 − θ5 . (22c)
6 24 120 360 K
iscontinuous. Wenowdiscusthethreespin-glassmodels
defined in Sec. III. Each of the models shows a different Differences between different levels of RSB manifest
behavior in the glassy phase. themselves in free energy first in fifth order.
B. Potts glass
The Potts model with p states reduces to Ising for
A. Ising spin glass p = 2, but differs from it for p > 2 in that it breaks
the spin-reflection symmetry. This property was used to
arguethattheParisischemefails.22 Ithadbeenlongbe-
Mean-field Ising spin glass is paradigm for the the-
lieved that it is the one-level replica-symmetry breaking
ory of spin glasses. It is called Sherrington-Kirkpatrick
that determines the equilibrium state below the transi-
model.18 It is this model for which Parisi derived tiontemperature.23 ThePottsglassdisplaysadiscontin-
a free energy with a continuous replica-symmetry
uous transition into the replica-symmetry broken state
breaking.12–14 It was also later proved that the hierar- for p>4.24 Discontinuous transitions do not allow us to
chical scheme of replica-symmetry breaking covers the
use an asymptotic expansion in a small parameter below
exact equilibrium state.19,20 The rigorous proof does
the transition temperature. It is, nevertheless, possible
not, however, tell us whether the equilibrium state
totesttheorderedphaseofthePottsglassfor2<p<4.
is described only by a finite number of replica hi-
WediditinRefs.16,25andfoundanunexpectedbehav-
erarchies or a continuous limit is needed. Only a
ior.
few years ago we resolved the hierarchical free energy
Studying the discrete replica-symmetry breaking we
f (q;∆χ ,...,∆χ ,m ,...,m ) for arbitrary K via
K 1 K 1 K found two 1RSB solutions with the same geometric pa-
the asymptotic expansion below the transition temper-
rameter
ature in a small parameter θ = 1−T/T .21 Only this
c . p−2 36−12p+p2
asymptotic solution was able to resolve the question of m= + θ . (23)
the structure of the equilibrium state. We found the 2 8(4−p)
8
One non-trivial 1RSB solution then leads to order pa-
rameters
.
q(1) =0 , (24a) 0.4 0.8
. 2 228−96p+p2
∆χ(1) = θ+ θ2 (24b)
4−p 6(4−p)3 0.2 0.4
S Λ
while the second one has both parameters nonzero
0 0
. −12+24p−7p2
q(2) = θ2 , (25a)
3(4−p)2(p−2) -0.2 -0.4
. 2 360−204p−6p2+13p3
∆χ(2) = θ− θ2 . (25b) 0.2 0.4 0.6 0.8 1.0
4−p 6(4−p)3(p−2)
T/TC
We can see that the asymptotic expansion with small
parameters q and ∆χ breaks down already at p = 4 FIG.2: EntropyS (leftscale,dashedline)andlocalstability
above which we expect a discontinuous transition from Λ(rightscale,solidline)ofthe1RSBsolutionfromEq.(24)of
the3-statePottsglass. Thesolutionbecomeslocallyunstable
the paramagnetic to a 1RSB state at T > T = 1.
0 c
at ≈0.33T and entropy negative at T ≈0.16T .
Note that a transition to the replica-symmetric solution c c
q >0,∆χ=0 is continuous up to p=6.
The 1RSB solution has a higher free energy than the
replica-symmetric one. The difference is of order θ3,
stable. Negativity of q means that local magnetizations
. (p−2)2(p−1)θ3 are imaginary and the solution is unphysical. This de-
f1RSB −fRS = 3(4−p)(6−p)2 . (26) ficiency, however, decreases with the increasing number
of spin hierarchies and disappears in the limit K → ∞.
Thetwostationarystatesofthe1RSBfreeenergybehave It means that the resulting solution with a continuous
differently as a function of parameter p. The former so- replica-symmetry breaking shows no unphysical behav-
lution is physical for all values of p unlike the latter that ior. It is analogous to negativity of entropy in the low-
becomes unphysical for p > p∗ ≈ 2.82 where q(2) from temperature solutions of KRSB approximations of the
Eq. (25a) turns negative. It is also the region of the pa- Sherrington-Kirkpatrick model.
rameter p where the first solution is locally stable as can Potts glass hence shows a degeneracy for p∗ < p < 4
be seen from the stability function with a marginally stable solution continuously break-
ing the replica symmetry and a locally stable one-level
Λ(0) =. θ2(p−1)(cid:0)7p2−24p+12(cid:1)>0 . (27) replica-symmetry breaking. To decide which one is the
1 6(4−p)2
true equilibrium state one has to compare free energies.
The difference of the continuous free energy f and that
in this region. That is why the solution with q = 0 was c
of the KRSB solution is
assumedtobethetrueequilibriumandasolutionwitha
continuous replica-symmetry breaking had not been ex-
pectedtoexist. We,however,foundthatthereisaParisi- . (p−1)(p(7p−24)+12)2θ5
β(f −f )= (29a)
likesolutionevenintheregionofstabilityofthesolution c KRSB 720K4(4−p)5
from Eq. (24). The second 1RSB solution is unstable
and decays to solutions with higher numbers of replica and and that of the replica-symmetric one reads
hierarchies as
qK =. − 1 12−24p+7p2θ2 , (28a) β(fc−fRS)=. (3p(4−−1)p()p(6−−2)p2)θ23 . (29b)
3K2(4−p)2(p−2)
. 1 2
∆χK = θ , (28b) WeseethatthesolutionwiththecontinuousRSBhasthe
l K (4−p)
highest free energy as the true equilibrium state should
(cid:20)
mK =. p−2 + 2 3+ 3p−p2 have for geometric factors m < 1. In this situation en-
l 2 4−p 2 tropyreachesminimumandfreeenergymaximuminthe
(cid:18) 7 (cid:19)2l−1(cid:21) phase space of the order parameters. The locally stable
+ 3−6p+ p2 θ . (28c) 1RSB solution becomes unstable at lower temperatures
4 2K
and entropy turns negative at very low temperatures as
We can see that the KRSB solution behaves unphysi- demonstrated on the 3-state model in Fig. 2. This leads
cally in the same way as the second 1RSB solution does. us to the conclusion that the Parisi solution with a con-
The averaged square of the local magnetization is nega- tinuous replica-symmetry breaking represents the equi-
tive for p > p∗ where the first 1RSB solution is locally librium state for the Potts glass with p<4.
9
C. p-spin glass solvable. It coincides with the random energy model of
Derrida.8,9 For this reason the p-spin glass was also in-
The spin model generalized to random interactions tended to be used to study and understand the genesis
connectingpspins,p-spinglasshasbeenusedtosimulate of the Parisi free energy when studying the asymptotic
the dynamical transition in real glasses.26,27 This model, limit p→∞.
analogously to the Potts glass, generalizes the Ising spin Tocoverboththeboundarysolutionsp=2andp=∞
glass to p > 2 and allows one to study the behavior of we have to mix up the one-level RSB scheme and the
the equilibrium state as a function of parameter p. In Parisi continuous RSB. Such free energy density of the
particular, the limit p → ∞ is accessible7 and is exactly mean-field p-spin glass reads
β (cid:104) (cid:105)
f(p)(q,χ ,µ ,µ ;x(µ))=− 1−p(q+χ )+(p−1)(q+χ )p/(p−1)
T 1 1 0 4 1 1
+ p−1(cid:104)µ (q+χ ))p/(p−1)−µ qp/(p−1)(cid:105)− p−1(cid:90) µ1dµ[q+X (µ)]p/(p−1)−(cid:68)g (cid:16)µ ,h+η(cid:112)pq/2(cid:17)(cid:69) , (30)
4 1 1 0 4 µ0 0 1 0 η
with temperature entropy in the 1RSB solution. We obtain
g1(µ1,h)=Tµexp(cid:26)p4(cid:90)µµ01dµ x(µ)(cid:2)∂h¯2 S0(h)∝−p(p8−1)(cid:20)exp{√−πµp21pχχ11/4} exp2{C−Hhµ2/(hp)χ1}(cid:21)2 ,
+ µg1(cid:48)(µ,h+h¯)∂h¯(cid:3)(cid:27)g1(h+h¯)(cid:12)(cid:12)(cid:12)(cid:12)h¯=0 , (31a) (32)
where we used the following notation
where X (µ)=(cid:82)µ dµ(cid:48)x(µ(cid:48)). The generating free energy
is 0 µ0 2CHµ(h)=eµ1hEµ(p)(−h)+e−µ1hEµ(p)(h) , (33a)
(cid:90) ∞ dφ (cid:16) √ (cid:17)2
g1(h)= µ1 ln(cid:90) ∞ √d2φπe−φ2/2[2cosh(β(h Eµ(p)(h)= h/√pχ1/2 √2πe− φ−µ1 pχ1/2 /2 . (33b)
1 −∞
(cid:112) (cid:17)(cid:17)(cid:105)µ1/β Negativityofthelow-temperatureentropyindicatesthat
+φ p(χ −X (µ )/2 . (31b)
1 0 1 1RSBcannotproduceastablegroundstateforarbitrary
p < ∞. Negativity of entropy decreases with increasing
We rescaled the function m→µ=βm. If M =µ0, free p, see Fig. 3, but only if a condition µ2pχ = ∞ is ful-
energy f(p) reduces to the 1RSB approximation. On the filled the 1RSB solution (µ > 0) leads1to1zero entropy
T 1
otherhandifχ =0,orµ =β freeenergyf(p) coincides at zero temperature. Nonnegative entropy is a necessary
1 1
withthatoftheParisisolutionwithacontinuousreplica- conditionforphysicalconsistencyofthelow-temperature
symmetry breaking. solution. Itthenmeansthatthelow-temperatureequilib-
The p-spin glass can not only be used to investigate rium state for p<∞ must contain the Parisi continuous
analytically the p → ∞ limit but also the T → 0 limit. order-parameter function x(µ) with β ≥ µ ≥ M > µ .
1 0
In this limit simple solutions of mean-field models lead It can also be seen from the asymptotic free energy for
to negative entropy. It is easy to calculate the zero- p→∞ that reads
1 1
f(p→∞)(q,χ ,µ )=− [1−(q+χ )(1−ln(q+χ ))]− ln[2cosh(µ h)]
T 1 1 4T 1 1 µ 1
1
− µ1 [χ −(q+χ )ln(q+χ )]− µ1q (cid:2)lnq+p(cid:0)1−tanh2(µ h)(cid:1)(cid:3) , (34)
4 1 1 1 4 1
giving the leading-order solution for the variational pa- deronewhilethelatterisexponentiallysmallforlargep,
rametersχ ,µ andq. Thefirsttwoparametersareofor-
1 1
10
bath. Weintroducedasmallinter-replicainteractionand
lookedatthelinearresponsetoit. Thisschemewashier-
χ1 =1−q , (35a) archically used to test and restore at least local thermo-
q =exp{−p(1−tanh2(µ h))} , (35b) dynamic homogeneity. The principal step in this proce-
1
durewastoselectanadequatesymmetry-breakingofthe
(cid:112)
µ1 =2 ln[2cosh(µ1h)]−htanh(µ1h) . (35c) replicated variables so that to make thermodynamic po-
tentials analytic functions of the originally integer repli-
The above nontrivial solution holds only if
cation index. Only then it is possible to test and restore
(cid:112)
β > 2 ln[2cosh(βh)]−htanh(βh) (low-temperature
thermodynamic homogeneity and ergodicity.
phase), otherwise µ = β and χ + q = 0 (high-
1 1
temperature phase). To derive an equation for the
order-parameter function x(µ) one needs to include the
next-to-leading order contributions. To go beyond the
leading asymptotic order one can use the Landau-type Weexplicitlydemonstratedthisconstructiononmean-
theory for the order-parameter function developed in field models of spin glasses. Randomness in the spin-
Ref.17. NotethattheasymptoticsolutionfromEq.(35) exchange in these models makes them frustrated and no
with µ(x) = 0 suffers from a negative entropy as can be regularlong-rangeorderisestablishedinequilibrium. Er-
seen from Eq. (32) and is plotted in Fig. 3. Note that godicity is broken in the whole low-temperature, spin-
the transition to the ordered phase in the p-spin glass glassphaseandthethermodynamiclimitisnotuniquely
is discontinuous and hence, an asymptotic expansion defined. Replica method allows one to restore ergodicity
belowthetransitiontemperatureisnotapplicable. Only in hierarchical steps by breaking successively the replica
an asymptotic expansion p→∞ makes sense. symmetry or independence of the replicated spaces.
We applied the real replicas on mean-field Ising, p-
state Potts and p-spin glass models and calculated their
VI. CONCLUSIONS asymptoticsolutionsbelowthetransitiontemperatureto
the glassy phase. While the Ising spin glass is known to
We studied ergodicity breaking that is not accompa- have continuously broken replica symmetry in the equi-
nied by any broken symmetry of the Hamiltonian. Equi- librium state, the Potts and p-spin glasses allow for lo-
librium thermodynamics and the equilibrium cannot be cally stable solutions with a one-level discrete replica-
found then by standard methods, since ergodicity can- symmetry breaking. The continuous RSB and 1RSB co-
notbestraightforwardlyrestored. Whennosymmetryof exist in the p-state Potts model, while the p-spin glass
the Hamiltonian is broken we do not have direct means, reduces to a 1RSB solution in the limit p→∞. In both
physicalmacroscopicallycontrollableexternalfields,with cases for p < ∞ the 1RSB state leads to negative en-
which we could remove dependence of the thermody- tropy at very low temperatures and it seems that the
namic limit on properties of the thermal bath. Broken ultimate equilibrium state for the mean-field spin-glass
ergodicity impedes the existence of a uniquely defined models breaks the replica symmetry in a continuos form
thermodynamic limit that depends on the behavior of as suggested by Parisi in the Ising model. Our analy-
the thermal bath. The system is not thermodynamically sis indicates that a continuous RSB is indispensable to
homogeneous. prevententropytobecomenegativeatzerotemperature.
We used replications of the phase space of the dy- Spinreflectionsymmetryishencenotsubstantialforthe
namical variables to simulate the impact of the thermal existence of a solution with a continuous RSB.
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