Extended nonergodic regime and spin subdiffusion in disordered SU(2)-symmetric Floquet systems
AAnomalous thermalization and spin subdiffusion in disordered SU(2)-symmetricFloquet systems
Zhi-Cheng Yang,
1, 2, ∗ Stuart Nicholls, and Meng Cheng † Joint Quantum Institute, University of Maryland, College Park, MD 20742, USA Joint Center for Quantum Information and Computer Science,University of Maryland, College Park, MD 20742, USA Department of Physics, Yale University, New Haven, CT 06520-8120, USA (Dated: September 2, 2020)We explore thermalization and quantum dynamics in a one-dimensional disordered SU(2)-symmetric Floquet model, where a many-body localized phase is prohibited by the non-abeliansymmetry. Despite the absence of localization, we find an extended non-ergodic regime at strongdisorder where the system reaches thermal equilibrium in the thermodynamic limit in an anomalousmanner. In the strong disorder regime, the level spacing statistics exhibit neither a Wigner-Dysonnor a Poisson distribution, and the spectral form factor does not show a linear-in-time growth atearly times characteristic of random matrix theory. The average entanglement entropy of the Flo-quet eigenstates is subthermal, although violating an area-law scaling with system sizes. We furthercompute the expectation value of local observables and find strong deviations from the eigenstatethermalization hypothesis. The infinite temperature spin autocorrelation function decays at longtimes as t − β with β < .
5, indicating subdiffusive transport at strong disorders.
I. INTRODUCTION
The presence of strong quenched disorders in non-integrable quantum systems often impedes thermaliza-tion. A prototypical example in one dimension is many-body localization (MBL) [1–3], where highly-excitedeigenstates of the Hamiltonian violate the eigenstatethermalization hypothesis (ETH) [4, 5] and the systemfails to reach thermal equilibrium starting from genericinitial states having a finite energy density. However, dis-orders have a dramatic effect on the quantum dynamicseven in the ergodic phase of systems exhibiting a MBLphase transition. It has been shown that thermalizationin this regime is anomalous, featuring subdiffusive trans-port as well as slow relaxation of local observables totheir thermal expectation values [6–17].On the other hand, a true MBL phase is incompatiblewith non-abelian global symmetries, such as an SU(2)spin rotation symmetry [18–23]. The quasi-local integralsof motion, a defining feature of MBL, form exactly de-generate multiplets under a non-abelian symmetry group,which are unstable against any infinitesimal interactionsbetween them, and hence must break down. One maythus naively expect that systems with non-abelian sym-metries are trivially thermal even at strong disorders.However, Ref. [21] recently studied an SU(2)-symmetricrandom Heisenberg chain using a real-space renormaliza-tion group approach and identified a broad regime in sys-tem sizes where the system appears nonergodic. Withinthis regime, the eigenstates are well-approximated by treetensor networks with faster than area-law but stronglysubthermal entanglement entropy scaling, and expecta-tion values of local observables exhibit deviations from ∗ [email protected] † [email protected] generic thermalizing systems. The extremely long lengthscale beyond which the eventual thermalization sets inmakes this intermediate nonergodic regime directly rele-vant in typical experiments with moderate system sizes.While the approach taken in Ref. [21] depends cruciallyon energetics in a Hamiltonian system, it is natural to askwhat will happen in a strongly disordered periodically-driven Floquet system with SU(2) symmetry, where en-ergy conservation is absent and a tree tensor networkstructure for the eigenstates does not seem to hold. Thelack of energy conservation tends to allow Floquet sys-tems to thermalize faster and more completely to infinitetemperature than Hamiltonian systems [24–29]. Previousstudies suggest that thermalization and transport in Flo-quet systems near a MBL transition are anomalous [30–34]. However, it is unclear whether such anomalous ther-malization regime becomes more fragile once the nearbyMBL phase is absent. Moreover, the transport propertyin disordered systems with SU(2) symmetry remains anopen question.In this work, we address the above questions by study-ing a one-dimensional disordered SU(2)-symmetric Flo-quet model. The key properties of a time periodic Hamil-tonian H ( t + T ) = H ( t ) are encoded in the eigenstates ofthe Floquet operator U F = T e − i (cid:82) T dtH ( t ) , which gener-ates time evolution over integer multiples of periods. Wefirst look at the level spacing statistics of the eigenenergyspectrum of U F using exact diagonalization and find notransition into a MBL phase, which is consistent with theSU(2) symmetry. However, at strong disorder, the levelspacing statistics exhibit neither a Wigner-Dyson nor aPoisson distribution, and the drift towards a Wigner-Dyson distribution upon increasing the system size isvery slow. To further probe the long-range spectral cor-relations beyond nearest-neighboring levels, we calculatethe disorder averaged spectral form factor. We find that a r X i v : . [ c ond - m a t . d i s - nn ] S e p the spectral form factor also deviates from random ma-trix behaviors. In particular, within system sizes accessi-ble numerically, the linear-in-time growth at early timesis absent, and the curves coincide with random matrixtheory predictions only at timescales comparable to theinverse level spacings. The average entanglement entropyof the Floquet eigenstates is subthermal, although ex-hibiting a faster than area-law scaling with system sizes.The spectral properties of the Floquet model suggestan intermediate regime that is neither MBL nor quan-tum chaotic in the usual sense. To directly test ETHin our model, we calculate the distribution of the expec-tation values of local observables under Floquet eigen-states, and find that it strongly deviates from the Floquetversion of ETH, which predicts a Gaussian distributioncentered around the infinite temperature average value.This suggests the existence of an extended delocalizedand yet non-ergodic regime [35, 36] where the systemreaches thermal equilibrium in the thermodynamic limitin an anomalous manner. Finally, we study transportproperties in the strong disorder regime by computing theinfinite temperature spin autocorrelation function, whichdecays at long times as t − β with β < .
5, indicating thatspin transport is subdiffusive at strong disorders.
II. THE MODEL
We consider a spin-1/2 system with Heisenberg inter-actions that respect the global SU(2) spin rotation sym-metry. Time evolutions are generated by switching be-tween two alternating Hamiltonians: H = (cid:88) i (cid:112) − g J i S i · S i +1 H = (cid:88) i g J S i · S i +2 , (1)with the Floquet operator given by: U F = exp ( − iH T /
2) exp ( − iH T /
2) (2)with period T . J i ’s are Gaussian distributed randomnearest-neighbor couplings with zero mean and vari-ance unity, and J is uniform across all next-nearest-neighboring spins. We included the J term to makethe model more generic while respecting the symmetry.The parameter 0 < g < g varies. The limit when g = 1 is inte-grable, corresponding to two copies of the clean Heisen-berg model on odd and even sites respectively. However,taking g to be strictly less than one makes the modelnonintegrable. A disorder-free Hamiltonian system corre-sponding to H + H has been shown to thermalize, withthe equilibrium distribution described by a non-abelian FIG. 1. The average level spacing ratio as a function of thedisorder strength g for different system sizes. Each data pointis averaged over 500 disorder realizations for L = 12, 250realizations for L = 14, and 50 realizations for L = 16. thermal ensemble [37]. We hereafter choose J = 1 and T = 4 for numerical simulations.As a result of the SU(2) symmetry, model (1) hastwo commuting conserved quantities: ( S , S z tot ) with S tot = (cid:80) i S i being the total spin and S z tot being its z -projection. Therefore, the Hilbert space of our modelfalls into distinct blocks labeled by two quantum num-bers S z tot and S = S ( S + 1). For simplicity, we shallfocus on the sector with S z tot = 0 and total spin S = 0. III. SPECTRAL STATISTICS
We shall now present our numerical results on the spec-tral statistics of the quasi-energy spectrum of the Flo-quet operator U F , and demonstrate that the spectrumat strong disorder shows deviations from random matrixtheory predictions. A. Level spacing statistics
The analog of eigenenergies in a Floquet system isgiven by the eigenvalues of the Floquet operator, whichare unimodular and can be denoted as { e iθ n } . Thequasi-energies { θ n } are 2 π periodic, hence we take themto be within the principal zone [ − π, π ). Let { θ n } berank-ordered descendingly, such that θ n > θ n +1 , anddefine the gap between adjacent quasi-energy levels as∆ θ n = θ n − − θ n >
0. The level spacing distribution canbe captured by the ratio between adjacent gaps: r n = min(∆ θ n , ∆ θ n +1 )max(∆ θ n , ∆ θ n +1 ) . (3)The average value of r n over different levels is able tocapture the distributions of level spacings, and since Flo-quet quasi-energies have a uniform spectral density, we (a)(b) FIG. 2. The spectral form factor K ( t ) for (a) g = 0 . g = 0 . L = 12 ,
14, and 500 realizations for L = 16. take the average over the entire spectrum. This quantityserves as the canonical diagnostic for the phase transitionbetween a thermalizing phase and MBL phase [1, 2]. Inthe localized phase with a Poisson distributed spectrum, (cid:104) r (cid:105) ≈ .
39; in the thermalized phase, the quasi-energyspectrum of our model follows a circular orthogonal en-semble (COE) with (cid:104) r (cid:105) ≈ .
53, since both H and H aretime reversal symmetric [24, 38].In Fig. 1, we show the average level spacing ratio (cid:104) r (cid:105) as a function of the disorder strength g for different sys-tem sizes. First of all, we find that there is no transi-tion into a MBL phase, as expected for SU(2)-symmetricsystems in general. Second, the flow of (cid:104) r (cid:105) upon increas-ing system sizes is always monotonic, as opposed to theHamiltonian system studied in Ref. [21]. This is due tothe removal of energy conservation as well as the uniformspectral density in Floquet systems, which allows one totake all quasi-energies into the average on equal footing.At weak disorders, (cid:104) r (cid:105) approaches the COE value, indi-cating that the system is nonintegrable and thermalizing.
10 12 14 162345
FIG. 3. Average entanglement entropy as a function of systemsizes, for different disorder strengths. The disorder is strongfor small g and weak for large g . At strong disorders, theentanglement entropy scales faster than area-law, but has asubthermal value. At weak disorders, a linear fitting yieldsan entropy density S/L ≈ . S th /L = ln2 /
2. The numbers of realizationsin obtaining each data point are the same as in Fig. 1.
On the other hand, at strong disorder, (cid:104) r (cid:105) approaches avalue that is intermediate between a Poisson and COEdistribution, with short-range level repulsion. While oneexpects that the system will eventually thermalize in thethermodynamic limit, for system sizes accessible in ournumerics, the flow towards COE with increasing systemsizes is extremely slow. It is therefore possible that thereis a regime where thermalization is anomalous at strongdisorders, similar to the strong disorder regime on theergodic side of a MBL transition [11–17]. B. Spectral form factor
While the level spacing statistics capture the repulsionbetween nearest-neighboring quasi-energy levels, we nowconsider a complementary spectral measure that is ca-pable of describing correlations beyond nearest-neighborlevels. The spectral form factor of the quasi-energy spec-trum is defined as: K ( t ) = (cid:42)(cid:88) i,j e i ( θ i − θ j ) t (cid:43) , (4)where the average is taken over different disorder realiza-tions. This quantity is intimately related to the temporaltwo-point correlation functions of local observables, andhas been playing a central role in characterizing quantumchaos [39–41]. Since the definition (4) involves all pairsof quasi-energy levels, it is able to capture spectral corre-lations beyond the scale of level spacing. For orthogonal (a) (b)(c) (d) FIG. 4. Probability distributions of the local observable S i · S i +1 over all eigenstates for strong disorders g = 0 . g = 0 . | J i | are picked; in (c), the weakestbonds with the smallest | J i | are picked; in (b) the bonds are picked randomly. The distributions are obtained for system size L = 14 and over 250 disorder realizations. ensembles, K ( t ) takes the form [42]: K ( t ) = N [2 τ − τ ln(1 + 2 τ )] ( τ ≤ N (cid:104) − τ ln (cid:16) τ +12 τ − (cid:17)(cid:105) ( τ > , (5)where N is the Hilbert-space dimension, and τ = t/ N .Notice that the behavior of K ( t ) in Eq. (5) is differentfrom that in the unitary ensembles, where K ( t ) is simplya linear ramp for τ < τ > τ yields K ( t ) ≈ t . Thus at early times, K ( t ) growslinearly in time with a different slope from the unitaryensembles.In Fig. 2, we plot the spectral form factor of our modelat weak and strong disorders. One can see that at weakdisorders, the spectral form factor agrees very well withrandom matrix theory predictions. On the other hand, atstrong disorders, K ( t ) strongly deviates from Eq. (5). In particular, the linear-in- t growth at early times is absent,and the curves only agree with the random matrix the-ory behavior at late times comparable to the Heisenbergtimescale ∼ N . This implies that the long range spectralcorrelations in the quasi-energy spectrum does not fol-low the random matrix theory behavior. For the systemsizes accessible in our numerics, level repulsion betweenquasi-energy levels exists only within the order of a fewlevel spacings, as indicated by the non-Poissonian levelspacing ratio. IV. ENTANGLEMENT ENTROPY AND LOCALOBSERVABLES
We next turn to the entanglement entropy scaling andlocal observables of our model. In Fig. 3, we plot theentanglement entropy averaged over all eigenstates as afunction of system sizes. We take an equi-bipartitioningof the system in the middle, and compute the von-Neumann entropy: S = − Tr ( ρ A ln ρ A ) , (6)where ρ A = Tr ¯ A | ψ (cid:105)(cid:104) ψ | is the reduced density matrix ofsubsystem A . We find that the entanglement entropy atstrong disorders scales faster than area-law, which is aconstant in one dimension. This is again consistent withthe general expectation that the system is not many-body localized at strong disorders. However, the valuesof the entanglement entropy for small g are well below the(infinite temperature) thermal values for the given size ofHilbert space. This indicates that, although the systemdoes not localize at strong disorders, it also does notheat up to infinite temperature, as would be the case forgeneric thermalizing Floquet systems. Thermalization inthe strong disordered regime is thus different from theusual scenario.To further characterize the anomalous thermalizationat strong disorders, we study expectation values of localobservables as a direct test for ETH. For Hamiltoniansystems, ETH suggests that the diagonal matrix elementof local observables is a continuous function of the en-ergy, and centered around its microcanonical ensembleaverage value. The Floquet version of ETH thus impliesthat the expectation value of local observables under Flo-quet eigenstates should be narrowly peaked around theirinfinite temperature average value, with fluctuations de-caying exponentially with increasing system sizes. Wechoose the observable S i · S i +1 associated with a partic-ular bond between site i and i + 1. The eigenvalue of S i · S i +1 equals to − for singlet bonds, and for tripletbonds.In Fig. 4(a)-(c), we plot the probability distributionsof (cid:104) S i · S i +1 (cid:105) at strong disorders across all eigenstates fordifferent choices of bonds. In Fig. 4(a), we choose thestrongest bonds with the largest | J i | for each disorderrealization. We find that the distribution is nearly bi-modal, with dominating weights centered around anda weaker peak around − . This clearly shows that thepair of spins coupled via the strongest bond almost forma triplet or singlet instead of thermalizing with the rest ofthe system. One expects that the probability of findinga triplet or a singlet bond is proportional to the num-ber of multiplets (2 S + 1), which is apparently bigger fortriplets with S = 1. This explains the peak around in Fig. 4(a), which corresponds to triplet bonds. In con-trast, for weak disorders, the probability distribution foreven the strongest bond is a Gaussian centered aroundthe infinite temperature average value with a narrowwidth, as shown in Fig. 4(d), in agreement with ETH.We further show the probability distributions for theweakest bond with the smallest | J i | [Fig. 4(c)], and a ran-domly chosen bond [Fig. 4(b)]. In Fig. 4(b), we find thatfor a randomly chosen bond, the probability distributionalso deviates from ETH behavior, featuring a broad non-Gaussian distribution within its domain. To quantify theultimate approach to ETH in the thermodynamic limit,
10 11 12 13 14 15 1610 -4 -3 -2 -1 FIG. 5. Variance of the expectation values of S i · S i +1 onrandomly chosen bonds for strong and weak disorders. Theensemble includes both different Floquet eigenstates and dis-order realizations. we compute the variance of local observable S i · S i +1 fora randomly chosen bond as a function of system sizes, asshown in Fig. 5. For weak disorders g = 0 .
9, the fluctu-ations decay exponentially with increasing system sizes,consistent with ETH. On the other hand, the fluctuationsat strong disorders are significantly larger than the weakdisordered case. Our numerics indicate that the fluctua-tions at strong disorders also decay as the system size in-creases, although much more slowly than the weak disor-dered case. Due to this extremely slow decay, we cannottell from numerics on small system sizes whether or notthe asymptotic decay is exponential in system size. Fi-nally, for the weakest bond for each disorder realization,the expectation value is no longer peaked around thatof a triplet. Instead, it is now centered near zero, withonly small weights around the singlet and triplet values.This implies that pairs of spins that are weakly coupleddo not form singlets or triplets between themselves andtend to thermalize with the rest of the system. However,comparing with Fig. 4(d), the distribution still shows de-viations from ETH predictions, namely, the distributionin Fig. 4(c) has heavy tails away from its peak.Therefore, we conclude that the diagonal matrix el-ement of S i · S i +1 exhibits deviations from ETH pre-dictions. Its expectation value shows a broad non-Gaussian distribution, with fluctuations decaying muchmore slowly than predicted by ETH. Spins that arestrongly coupled tend to form triplets, and even in thepresence of driving, such strongly coupled pairs have ahard time absorbing energy from the drive and hence arenearly decoupled from the rest of the system. Expecta-tion values associated with a typical bond also show abroad distribution within their domain. V. SPIN AUTOCORRELATION FUNCTION
In this section, we study the transport property of ourmodel. Since the total z -magnetization is conserved, onecan thus focus on transport of local magnetizations. Fordisordered non-integrable systems that are thermalizing,one usually expects that the transport of the conservedcharge should be diffusive. However, several studies havefound anomalous subdiffusion behavior in the ergodicregime of systems exhibiting a MBL phase transition [6–17, 30–32]. Here we present numerical evidence of a simi-lar subdiffusive regime in our model, despite the absenceof a true MBL phase.We consider the infinite temperature spin autocorrela-tion function: C zz ( t ) = 1 N Tr [ S zi ( t ) S zi (0)] , (7)where the trace is taken within a fixed total magneti-zation sector. Physically, this autocorrelation functionprobes the probability of finding an initially localizedcharge at the same position at time t . At late times, C zz ( t ) decays as a power law: C zz ( t ) ∼ t − β , where β = 0 . < β < . β ≈ . β from C zz ( t ) is typically not extremely accurate, due tothe oscillations on top of the power-law decay as well asthe arbitrariness in the choice of time window. Nonethe-less, different choices of time windows in our fitting con-sistently yield a value for β that is smaller than 0.5. Wethus conclude that spin transport at strong disorder isindeed subdiffusive.As the disorder strength is decreased, the power β in-creases continuously. For example, β ≈ .
47 for g = 0 . β appears to exceed 0.5 which implies su-perdiffusion. However, this conclusion is false. As thedisorder strength decreases, the mean free path of thesystem l mfp increases. At some point, the mean free pathat that disorder strength becomes the order of the sys-tem size used in our simulation l mfp ∼ L = 20, and thussimulations on small system sizes yield superdiffusive be-haviors. Beyond that point, the system size accessiblein our numerics is insufficient to draw any conclusion onthe nature of transport in the thermodynamic limit andhence can no longer be trusted. On the other hand, atstrong disorders l mfp is typically much smaller than thesystem size, and hence simulations on moderate systemsizes are good enough for inferring transport propertiesin the thermodynamics limit.Although our numerics are inconclusive for weak dis-orders, the g = 1 limit is well understood. At g = 1, thesystem becomes two decoupled clean Heisenberg chainson the even and odd sites, respectively. This limit is -2 -1 FIG. 6. Spin autocorrelation function C zz ( t ) for different dis-order strengths with L = 20 and averaged over 500 disor-der realizations. For strong disorders ( g = 0 . C zz ( t ) ∼ t − . . For weak disorders( g = 0 . β > . integrable, and spin transport is superdiffusive with anexponent β = [43, 44]. One may thus conjecture thatupon adding disorder, transport becomes diffusive simi-lar to the random field XXZ chain with a MBL phasetransition [13], although one needs a much larger systemsize to see diffusion.
VI. SUMMARY AND OUTLOOK
In this work, we study themalization and spin trans-port in a disordered Floquet model with SU(2) symmetry.This model can be viewed as an extension of disorderedHeisenberg Hamiltonians when energy conservation is re-moved, or an extension of Floquet-MBL models when anadditional SU(2) symmetry is imposed. We find that,despite the absence of a true MBL phase, thermaliza-tion in this model is anomalous, which is characterizedby both the spectral statistics and a direct comparisonwith ETH using expectation values of local observables.Moreover, we provide numerical evidence from the spinautocorrelation function indicating that spin diffusion atstrong disorders is also anomalous.Our result raises several interesting questions for fu-ture study. First, in Hamiltonian systems, a lengthscale beyond which resonances proliferate and the systemeventually thermalizes can be extracted, using a real-space strong disorder renormalization group approach.While such a procedure is not directly applicable to Flo-quet systems where energy cannot be defined, is there asimilar length scale controlling the ultimate thermaliza-tion in the strong disorder regime? Second, the eigen-states in strongly disordered Heisenberg chain can bewell-approximated by tree tensor networks. What is thestructure of the eigenstates in a Floquet system? Fi-nally, the transport properties of the strongly disorderedHeisenberg chain has remained unexplored. It will beinteresting to see if there is a subdiffusive regime thereas well. We focus on C zz ( t ) in this work, but one canalso look at other quantities such as the ac conductiv-ity σ ( ω ), whose scaling exponent at low frequencies is infact related to β . Furthermore, it is desirable to iden-tify the crossover from subdiffusion to diffusion by usingdifferent numerical methods that are amenable for muchbigger system sizes, e.g. probing the steady-state currentby coupling the system to leads [13]. ACKNOWLEDGMENTS
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