B. Georgeot and D. L. Shepelyansky- Integrability and Quantum Chaos in Spin Glass Shards

8/3/2019 B. Georgeot and D. L. Shepelyansky- Integrability and Quantum Chaos in Spin Glass Shards

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VOLUME 81, NUMBER 23 P H Y S I C A L R E V I E W L E T T E R S 7 DECEMBER 1998

Integrability and Quantum Chaos in Spin Glass Shards

B. Georgeot and D. L. Shepelyansky*

Laboratoire de Physique Quantique, UMR 5626 du CNRS, Universit Paul Sabatier, F-31062 Toulouse Cedex 4, France(Received 8 July 1998)

We study spin glass clusters (shards) in a random transverse magnetic field, and determinethe regime where quantum chaos and random matrix level statistics emerge from the integrable

limits of weak and strong fields. Relations with quantum phase transition are also discussed.[S0031-9007(98)07880-6]

PACS numbers: 05.45.+b, 75.10.Jm, 75.10.Nr

Quantum manifestations of classical chaos and integra-bility have been intensively investigated in the past decade[1]. It has been realized that the statistical properties ofthe quantum energy spectrum are strongly influenced bythe underlying classical dynamics. Indeed, classical in-tegrability generally implies the absence of level repul-sion and a Poissonian statistics of energy level spacings.On the contrary, chaotic dynamics leads to level repulsion

with the level spacing statistics Ps being the same as inthe random matrix theory (RMT), i.e., the Wigner-Dyson(WD) distribution [2]. These two distributions of Psalso characterize, respectively, the localized and metallicphases in the Anderson model of disordered systems. Atthe critical point between these two phases an intermedi-ate level spacing statistics occurs [3].

While for one-particle systems the statistical propertiesof spectra are well understood, the same problem in many-body systems has been addressed only recently. The firstresults demonstrated that many-body integrable systemsare still characterized by Poisson statistics PPs, whereasin the absence of integrals of motion the Wigner-Dysonstatistics PWDs has been found [4]. More recently, in-vestigations of finite Fermi systems such as the Ce atom[5] and the 28Si nucleus [6] put forward the question of thestatistical description and thermalization induced by inter-action. A quantum chaos criterion for emergence of RMTstatistics and dynamical thermalization induced by inter-action was established in Ref. [7]. Namely, a crossoverfrom Poisson statistics to the WD distribution takes placewhen the coupling matrix elements become comparableto the level spacing between directly coupled states. Fortwo-body interaction, the critical interaction strength be-comes exponentially larger than the multiparticle level

spacing. Indeed, the latter decreases exponentially withthe number of particles, while the former decreases typi-cally only quadratically. This result was corroborated inRef. [8], and should apply to various physical systems suchas complex nuclei, atoms, clusters, and quantum dots.

In this Letter, we develop and apply these concepts todisordered spin systems. These systems are of great ex-perimental and theoretical interest [9]; in particular, quan-tum spin glasses have recently attracted a great deal ofattention, and various approaches have been developed to

investigate their properties [10]. Such important problemsas zero temperature quantum phase transition, structure ofthe phase diagram, correlations, and susceptibility prop-erties have been intensively investigated both analyticallyand numerically [11 16]. However, to the best of ourknowledge, the level spacing statistics has not been stud-ied in disordered spin systems. Here we present theoreti-cal estimates for spin glass shards which determine the

crossover from integrability to quantum chaos in a wayanalogous to the case of finite interacting fermionic sys-tems. Numerical investigations of Ps statistics give apowerful test of this crossover both at high temperatureT and at T 0 where a quantum phase transition is be-lieved to take place.

The spin glass shards we study are described by theHamiltonian,

HX

i,j

Jijsxi s

xj 1

X

i

Giszi , (1)

where the si are the Pauli matrices for the spin i, andthe first sum runs over all spin pairs. The local randommagnetic field is represented by Gi uniformly distributedin the interval 0,G. The exchange interactions Jij aredistributed in the same way in 2J

pn, J

pn , where

n is the total number of spins. For G 0, this systemis the classical Sherrington-Kirkpatrick spin glass model[9]. The

pn factor in the definition of Jij ensures

a proper thermodynamic limit at n ! `. In the limitJG ! 0, one obtains a paramagnetic phase with all spinsin the field direction. The opposite limit (JG ! `)corresponds to a spin glass phase at low temperature [9].

Let us first discuss the properties of the model forhighly excited states near the center of the energy band.

At J

0, the system is integrable, since there are n in-tegrals of motion, and Ps should follow the Poissondistribution. When JG increases, we expect integra-bility to be destroyed and therefore a crossover towardsPWDs statistics typical of RMT should take place. AtG 0, the model again becomes integrable since thereare n operators (sxi ) commuting with the Hamiltonian;hence, Ps PPs. The latter may seem surprising, es-pecially in light of recent discussions on chaos in classicalspin glass [17]. However, in spite of a possible complex

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thermodynamic behavior (Monte Carlo dynamics), theHamiltonian dynamics of a classical spin glass is inte-grable. As a result, we expect another crossover from aWD statistics back to a Poissonian one for large JG.This picture is confirmed by the numerical results forPs displayed in Fig. 1. These Ps distributions wereobtained for the states of the same symmetry class (S2),namely, the number of spins up is always an even num-

ber (the interaction does not mix states with even and oddnumbers of spins up). We will call the other symmetryclass with an odd numbers of spins up S1.

To analyze the evolution of the Ps distributionswith respect to J, it is convenient to use the parameterh

Rs00 Ps 2 PWDsds

Rs00 PPs 2 PWDsds,

where s0 0.4729... is the intersection point of PPsand PWDs [7]. In this way h 1 corresponds to thePoissonian case, and h 0 to the WD distribution. Thedata of Fig. 1 show that the distributions with the samevalue of h are very close, even if JG varies morethan 10 times. It is convenient to determine a criticalcoupling strength Jc at which the crossover from PPs

to PWDs takes place by the condition hJcG 0.3.The variation of h with respect to J is shown in Fig. 2for different n. The global behavior ofh is in agreementwith the above picture of transition between integrabilityand quantum chaos. Indeed, when JG increases, hdrops to zero and then starts to grow again back to onewhen the spin glass term in (1) dominates. The fact thata WD statistics sets in is not trivial. Indeed, in dimensiond 1 the model (1) with nearest-neighbor coupling onlyand Jii11 drawn randomly from 0, J, studied in [10,11]

0.0 1.0 2.0 3.0 4.0s

0.0

0.2

0.4

0.6

0.8

1.0

P(s)

0 1 2 3 4s0

1P(s)

FIG. 1. Crossover from Poisson to WD statistics in the model(1) for the states in the middle of the energy band ( 612.5%around the center) for n 12: J 0, h 0.984 (+); JG JcpG 0.38, h 0.3 (3); JG 0.866, h 0.027 ();JG JcsG 6.15, h 0.3 ( ); G 0, h 0.99 ( ).Full curves show the Poisson and WD distributions. Totalstatistics (NS) is more than 3 3 104; s is in units of meanlevel spacing. Inset shows Ps for the first excitation from theground state in the chaotic regime for n 15, JG 0.465,h0 0.018 (NS 3000) (3); the full line shows PWDs.

the statistics remains Poissonian (h 1) at arbitraryJG, as is shown in Fig. 2. The critical point JG 1[11] does not manifest itself. Indeed, this d 1 modelcan be mapped into a model of noninteracting fermions(see, e.g., [11]). Therefore the total energy is the sumof one-particle energies that generically leads to PPs.As a result, RMT can never be applied to this modeland dynamical (interaction induced) thermalization never

takes place. Thermalization can appear only througha coupling to a thermal bath that was implicitly usedin Refs. [16,18]. On the contrary, in the system (1)there are two transitions from integrability to chaoswhich determine two critical couplings Jcp , from theparamagnetic side, and Jcs from the spin glass side[hJcs hJcp 0.3; Jcs . Jcp].

In analogy with finite interacting fermionic systems [7],we expect that quantum chaos sets in when the couplingstrength U becomes comparable to the spacing betweendirectly coupled states Dc (U Dc). For small J inthe middle of the spectrum, one has U J

pn and

Dc 16Gn2 since each state is coupled to nn 2 12

states in an energy band of 8G. This gives the quantumchaos border from the paramagnetic side:

Jcp CpGn32, (2)

where Cp is some numerical constant. We note that Jcp isexponentially larger than the multiparticle spacing Dn 2nG2n. For large J, the transitions between unperturbedstates at J G are determined by the G term in (1), andhave typical value G. The number of such transitions, ina typical energy interval J, is n; hence, Dc Jn. Thisgives the quantum chaos border from the spin glass side:

0 10 20 300.0

0.2

0.4

0.6

0.8

1.0

0.5 1.00

1

J/Jcp

J0 1 2 / 3

FIG. 2. Dependence of h on the rescaled coupling strengthJJcp for the states in the middle of the energy band for n 7(), 8 (), 9 ( ), 10 (), 11 (+), 12 (), 13 (3), 14 (),and symmetry S2; NS varies between 12 500 and 160 000. Thefull line for n 12 shows the global behavior. Data for thed 1 model (see text) are given for n 12 () as a functionof JG (upper scale); NS 30000. The inset magnifies theregion near JJcp 1.

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Jcs CsGn , (3)

where, again, Cs is some constant.These theoretical predictions are confirmed by the

numerical results presented in Figs. 2 and 3 which giveCp 16 and Cs 0.5. We attribute the deviation from(2) in Fig. 3 seen for small n to the fact that for thesevalues Jcp is rather large; this can slightly modify theunperturbed spectrum and the estimates for Dc. Also,for small n the proximity of the second transition at Jcscan affect the actual value ofJcp . The data shown in theinset of Fig. 2 indicate that h depends only on the ratioJJcp ; the global scaling behavior is less visible than inthe fermionic model [7], apparently due to the reasonsabove.

The analysis at the band center corresponds to a veryhigh energy or temperature. Therefore the properties of(1) near the ground state energy Eg should be studiedseparately. To do that, we investigated Ps for the firstexcitations from Eg within the same symmetry class. Forthe class S1, the distribution Ps was computed for the

first four level spacings starting from Eg by averagingover different disorder realizations. The first excitationand its h value (h0) was analyzed separately while theother three were used to generate one cumulative Pswith an h noted by h1 [19]. There are several physicalreasons for this separation: in the presence of a gap, h0 isrelated to the gap fluctuations while h1 characterizes thequasiparticle excitations of higher energy, above the gap;in a metallic quantum dot with noninteracting electrons,h0 reflects the chaotic character of the dot giving h0 0while the spectrum of higher excitations becomes close toPPs with h1 1.

0.8 0.9 1.0 1.1 1.2 1.3-

1.0

-0.5

0.0

0.5

1.0

Log(n)

Log(Jc/

)

FIG. 3. Critical coupling strength Jc as a func-tion of n: Jcp at band center for symmetry S2 (),and near the ground state ( : S1; : S2); Jcs atband center for S2 ( ), and near the ground state(: S1; : S2). Full lines show the theory [Eqs. (2) and(3)] for the band center; the dashed line indicates the quantumcritical point at T 0, JG 0.5. Logarithms are decimal.

The global behavior of h0 and h1 as a function ofJ JJ 1 G is shown in Fig. 4. This parametrizationnaturally represents the two integrable limits J 0 andG 0 in a symmetric way. In both extreme cases (J0; 1), h0 and h1 approach the Poissonian value 1. Inbetween there is a pronounced minimum with h0 0and h1 0.1. When the number of spins in the shardincreases, the values of h to the right from the minimum

go up markedly, whereas the minimal h value itselfchanges only weakly. These h data suggest the existenceof a crossing point around J Jq 0.5G. For J , Jq,the difference between curves for different n is small,whereas it is much bigger for J . Jq. The data in Fig. 3show the variation of Jcp , Jcs (for which h1 0.3) withn; the behavior of Jcs is consistent with Jcs ! Jq forn ! ` for both symmetry classes. Also, according tothese data Jcp ~ n

2a with 0 # a # 0.2. This value ofa is smaller than the one expected from the quantumchaos criterion, used to derive (2). Indeed, for small Jnear the ground state Dc Gn whereas the coupling isstill J

pn, and therefore one expects a 0.5. For large

n, this would imply that h drops quickly to zero with Jand then increases again after the crossing point Jq. Insuch a scenario, in the thermodynamic limit (n ! `) weexpect Jcp ! 0 and Jcs ! Jq, so that h changes fromzero (J , Jq) to one (J . Jq) with some intermediatestatistics at the critical point Jq. The determination of theh value at that point for n ! ` requires larger systemsizes. We expect that this critical point corresponds tothe quantum phase transition between paramagnetic andspin glass phases discussed in Refs. [10,12,15] for a casewhen all Gi G. The above scenario is similar to thesituation for the Anderson transition in d 3 [3]: asfor the Anderson insulator, the lack of space ergodicity

in the spin glass phase implies h 1. In this phase

0.0 0.5 1.0 1.50.0

0.5

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0

J/(J+)

J/

FIG. 4. Dependence of h0 on J JJ 1 G: n 7 (),n 9 ( ), n 11 (1), n 13 (3), n 15 (), n 17(), n 19 (); 2000 # NS # 30000. Full curves connectdata for n 7,17. Inset shows h1 in the region near JG JqG 0.5 in more detail; 6000 # NS # 90000. Data arefor S1 symmetry.

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0 2 4 6 80.0

0.2

0.4

0.6

0.80 4 8 12 16

E/

E/

FIG. 5. Dependence of h on dEG for n 12 (1000realizations of disorder): JG 0.52 () (lower scale); JG 3.46 (+) (upper scale). Data are for S1 symmetry.

J . Jq the eigenstates are expected to be localized in

different parts of the phase space with small overlapbetween them. This would lead to uncorrelated levels(Poisson distributed) as in integrable systems. Even if inthis regime the levels can be rather sensitive to a smallparameter variation [17], the eigenstates are not ergodicand the usual properties of quantum chaos [1,2] areabsent. The situation in the paramagnetic phase J , Jqmay be more complicated. Indeed, the criterion U Dcwhich gives a 0.5 assumes that the couplings remainsmall and do not strongly modify the unperturbed Dc.But near Eg the typical mean field acting on a spin isof the order J and for J G

pn is much larger than the

local fields Gi Gn. The strong mean field near Eg can

change the effective Dc. Such a fact was seen for the Ceatom [5]. This may lead to another scenario in which inthe thermodynamic limit Jcp tends to some nonzero valuesmaller than Jq. Such a behavior would be in agreementwith the small variation ofJcp with n in Fig. 3.

The discussions above dealt with the two extreme casesof energy values. The variation of h between these twolimits is shown in Fig. 5 near the critical point and in thespin glass phase as a function of the excitation energydE E 2 Eg. Both cases show an unusual behaviorwhen h initially grows with energy and starts to decreaseonly later. This tendency is more pronounced near thecritical point. A possible reason for this behavior isthat the relative influence of the mean field becomesweaker as dEG increases. It is also possible that thesituation is similar to the case of a chaotic quantumdot with noninteracting electrons discussed above: theground state is chaotic but the interaction between the firstquasiparticle excitations is weak and so initially h growswith dE, approaching the Poissonian value. As a result,it is only at higher energy when the interaction betweenquasiparticles becomes stronger that h starts to decrease

towards the WD value. Larger system sizes are requiredto clarify this issue.

We thank F. Mila and E. Srensen for stimulatingdiscussions.

*Also Budker Institute of Nuclear Physics, 630090

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(1997).[19] In the S2 symmetry class, h0 0.4 at J 0; this

complicates the analysis of h0, but the results for h1 inFig. 3 are similar to those of S1.

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B. Georgeot and D. L. Shepelyansky- Integrability and Quantum Chaos in Spin Glass Shards