Stability of 2D quantum many-body scar states against random disorder
SStability of 2D quantum many-body scar states against random disorder
Ke Huang, Yu Wang, and Xiao Li
2, 3, ∗ School of Physics and Technology, Wuhan University, Wuhan 430072, China Department of Physics, City University of Hong Kong, Kowloon, Hong Kong SAR City University of Hong Kong Shenzhen Research Institute, Shenzhen 518057, Guangdong, China (Dated: February 17, 2021)Recently a class of quantum systems exhibiting weak ergodicity breaking has attracted muchattention. These systems feature a special band of eigenstates called quantum many-body scarstates in the energy spectrum. In this work we study the fate of quantum many-body scar states ina two-dimensional lattice against random disorders. We show that in both the square lattice and thehoneycomb lattice the scar states can persist up to a finite disorder strength, before eventually beingkilled by the disorder. We further study the localization properties of the system in the presence ofeven stronger disorders and show that whether a full localization transition occurs depends on thetype of disorder we introduce. Our study thus reveals the fascinating interplay between disorderand quantum many-body scarring in a two-dimensional system.
Introduction .— Recently rapid experimental progressin creating and manipulating isolated quantum sys-tems [1, 2] has enabled the simulation of novel quantumphases of matter in out-of-equilibrium systems. In par-ticular, whether an isolated quantum system can reachthermal equilibrium under its own dynamics is currentlyone of the central research topics in condensed matterphysics. The answer to this question is crucial becauseit is closely related to the ergodicity hypothesis, whichis one of the fundamental hypotheses of statistical me-chanics. To date the eigenstate thermalization hypothe-sis (ETH) [3–5] provides the most plausible mechanismfor thermalization in an isolated quantum system. In par-ticular, it states that thermalization occurs at the eigen-state level: the expectation value of local observables ineach eigenstate is thermal.Meanwhile, several systems that exhibit ergodicitybreaking have been identified, including noninteractingsystems and certain fine-tuned integrable models [6].However, they are unstable against perturbations, andhence do not constitute a phase of matter. Recently,however, it has been realized that strong disorders canprovide a generic and robust mechanism for ergodicitybreaking. In particular, the localization can persist evenwhen finite interactions are present, leading to the so-called many-body localization (MBL) phase [7–10]. Onehallmark of the MBL phase is that ergodicity breakingoccurs for all typical eigenstates. In particular, it is be-lieved that in an MBL system quench dynamics from ageneric initial nonequilibrium state is nonergodic, andslow relaxations can be expected.In the past few years, yet another type of ETH break-ing has been identified, and rapid progress has been madeto understand this phenomenon. In particular, persistentrevivals from a specific initial state—the | Z (cid:105) state—havebeen observed in an array of strongly interacting Ryd-berg atoms [11], whereas much faster relaxation is found ∗ [email protected] in the quench dynamics from most other initial states.The fact that relaxation this system has a strong depen-dence on the initial conditions indicates that this systemis qualitatively different from an MBL system. It turnsout that the persistent revival can be attributed to a spe-cial band of eigenstates in the energy spectrum that arenon-ergodic (characterized by their low entanglement en-tropies). which are now dubbed as quantum many-bodyscar states [12–29]. Several interesting properties of thescar states have been uncovered. For example, it wasshown that the scar states are associated with a hiddenunstable periodic orbit [17, 18], in analogy to a classicallychaotic system [30]. In addition, an approximate su (2)-algebra has been identified in the PXP model [12–15],which represents a good approximation for the scar statesin the original Rydberg Hamiltonian. Subsequently, sim-ilar “spectrum generating algebras” [29] have been iden-tified in a variety of other models, such as the Afflect-Kennedy-Lieb-Tasaki (AKLT) model [16], and severalother noninteracting spin models [31–35].One of the important open questions in this field isthe stability of these scar states against various pertur-bations [13, 27, 28]. In particular, the question of whetherscar states are stable against random external disordershas not been discussed much in the literature. Recently,the stability of scar states in the one-dimensional PXPmodel against random disorders has been carefully an-alyzed [28], and it was shown that the scar states arestable against finite disorders. However, the stability oftwo-dimensional scar states [14, 15] in the presence ofrandom disorders has not been addressed. This ques-tion is crucial because the effect of random disorders in aone-dimensional system are often qualitatively differentfrom that in higher dimensional systems. For example,it is generally believed that MBL is a stable phase in aone-dimensional disordered system [36], whereas its exis-tence in a two-dimensional system is still an open ques-tion. Therefore, it is important to analyze this questionthoroughly.In this Letter, we study the robustness of revivals froma | Z (cid:105) state in a two-dimensional PXP model against ran- a r X i v : . [ c ond - m a t . d i s - nn ] F e b dom disorders. In particular, we consider both the squarelattices and the honeycomb lattices. We find that scarstates in the two-dimensional PXP model are genericallystable against finite random disorders before it is even-tually killed at stronger disorders. We also analyze thefate of the PXP model itself against very strong disor-ders and find that whether the system eventually entersa many-body localized phase depends on the type of dis-order present in the system. The PXP model .— To begin with, we introduce theHamiltonian for Rydberg atoms in a square lattice of N sites, which can be written as [12, 13] H R = (cid:88) r (cid:20) Ω2 σ x r + J R P + r (cid:88) (cid:104) r (cid:48) , r (cid:105) P + r (cid:48) (cid:21) , (1)where r = ( i, j ) goes over all lattice sites and (cid:104) r (cid:48) , r (cid:105) de-notes summation over nearest neighbors of r . In addi-tion, σ x r = |•(cid:105)(cid:104)◦| + |◦(cid:105)(cid:104)•| represents a Pauli matrix de-scribing the transition between the ground state |◦(cid:105) andthe excited state |•(cid:105) at site r . The projection operators P + r (cid:48) = |•(cid:105)(cid:104)•| and P − r (cid:48) = |◦(cid:105)(cid:104)◦| project states into theirexcited or ground parts respectively. Finally, the Rabifrequency Ω represents the rapidity of the oscillationsbetween the excited state and the ground state, and J R exerts an extra energy if two adjacent sites are both ex-cited. Throughout this work we will maintain periodicboundary conditions in both directions.In the limit J R (cid:29) Ω, the probability that two nearestneighbors are both excited is strongly suppressed due tothe overwhelming energy cost. Hence, in this strong in-teraction limit, the Hilbert space is almost limited to itssubspace spanned by states having no two nearest neigh-bors in the exited state. As a result, the system can becaptured by the so-called PXP model [12–15] H PXP = P H R P , (2)where P = (cid:81) (cid:104) r , r (cid:48) (cid:105) ( I − P + r P + r (cid:48) ). Equivalently, the Hamil-tonian can be expressed as H PXP = Ω2 (cid:88) r σ x r (cid:89) (cid:104) r , r (cid:48) (cid:105) P − r (cid:48) . (3)The above Hamiltonian possesses a separated band of N + 1 eigenstates known as quantum many-body scarstates, and they have a high overlap with | Z (cid:105) , whichis the maximum excited states in this subspace. There-fore, the evolution of | Z (cid:105) is mostly controlled by these N + 1 scar states. Furthermore, because of an su (2)-likealgebraic structure in H PXP [20, 21, 25], the energies ofthe scar states are separated almost evenly, resulting ina strong revival of | Z (cid:105) .However, the su (2) algebra in the pure PXP model isonly approximate and the energies of scar states are notperfectly aligned, especially in a two-dimensional lattice.It has been pointed out that one can enhance its su (2)-like structure and stabilize the revival of | Z (cid:105) in a square -10 -5 0 5 10-12-10-8-6-4-20 -10 -5 0 5 10-12-10-8-6-4-20 (c) -10 -5 0 5 1000.20.40.60.81 -10 -5 0 5 1000.20.40.60.81 (b)(d)(a) FIG. 1. (a) The overlap between the eigenstates of thePXP model and the | Z (cid:105) state in a 4 × W = 0 . ω scar . Blue dots are obtained byexact diagonalization methods and red circles in (a), (b) areresults from the FSA approximation. lattice by adding some weak perturbations [15] δH = (cid:88) r σ x r (cid:2) aP l r + 2 aP d r + cP r (cid:3) (cid:89) (cid:104) r , r (cid:48) (cid:105) P − r (cid:48) , (4)where a ≈ . b ≈ . P li,j = P − i +1 ,j + · · · , P di,j = P − i +1 ,j +1 + · · · , (5) P i,j = P − i +1 ,j +1 P − i +1 ,j − P − i +2 ,j + · · · . In order to study the stability of the scar states againstrandom disorders we introduce a specific type of disorderperturbation to the Rydberg Hamiltonian H R , H w = (cid:88) r h a ( r ) σ a r , (6)where a = x, y, z and all components of h a ( r ) are uni-formly distributed in [ − W/ , W/ H = H PXP + δH + P H w P . (7) Properties of eigenstates .— We first verify the exis-tence of quantum many-body scar states in a square lat-tice and demonstrate that the scars in square lattice sharesimilar properties to those in a 1D chain. One hallmarkof scar states in a one-dimensional chain is that | Z (cid:105) approximately resides in the subspace spanned by thescar states. In Fig. 1(a), we show that in a clean 6 × | Z (cid:105) . -2 -1 0 1 2024681012 -2 -1 0 1 2024681012 -2 -1 0 1 2024681012 (a)(e) (b) (c)(f) (g) (h)(d) FIG. 2. (a)-(d) show systems with a fixed disorder realization and disorder strength W = 0, 0 . ω scar , 0 . ω scar , and ω scar ,respectively. (e)-(g) in the low panel show the Fourier transformation of the corresponding upper panels. The blue linesrepresent ˆ M ( k , t ) or ˜ M ( k , ν ) at Γ and the orange lines represent ˆ M ( k , t ) or ˜ M ( k , ν ) at M . More importantly, these scar states are spaced evenlyby a frequency ω scar = 2 πν scar = 0 . × L ( i ≤
3) and a right half R ( i ≥ S ( | φ (cid:105) ) = − Tr { ρ L ( | φ (cid:105) ) ln ρ L ( | φ (cid:105) ) } , where ρ L ( | φ (cid:105) ) = Tr R ( | φ (cid:105)(cid:104) φ | ).In Fig. 1(b), the EE of these scar states are clearly muchsmaller than that of the rest of the eigenstates. Fur-ther, we find that the forward scattering approximation(FSA) [12, 13, 15, 20] remains a good approximation forboth the overlap and EE in this square lattice.We now study the stability of these scar states againstrandom disorders. In Fig. 1(c) and 1(d), we apply a fixeddisorder realization with W = 0 . ω scar and find that | Z (cid:105) no longer resides in the subspace spanned by a small setof eigenstates. Instead, the dynamics of | Z (cid:105) are now con-trolled by all states in the Hilbert space, implying that | Z (cid:105) may lose its revival if the disorder strength is strongenough, as we verify in the following. Additionally, themajority of the eigenstates follows the volume law, ap-proaching the Page value S T = ln(181) − / × | Z (cid:105) . Quench dynamics from the Z state .— We now discussthe quench dynamics from | Z (cid:105) . A useful quantity tocharacterize the dynamics of | Z (cid:105) is the magnetization M ( r , t ) = (cid:104) Z ( t ) | σ z r | Z ( t ) (cid:105) [28]. In particular, it is helpfulto consider this quantity in its momentum space ˆ M ( k , t )and the corresponding Fourier transformation ˜ M ( k , ν )ˆ M ( k , t ) = (cid:88) r e i k · r M ( r , t ) , (8) (a) (b)(c) (d) FIG. 3. (a)-(d) shows | ˜ M ( k , ω ) | in systems with W/ω scar =0, 0 .
1, 0 .
5, 1 .
0, respectively. All results are averaged over 20disorder realizations. The radius of the spheres represents thestrength of | ˜ M ( k , ω ) | at each k point. ˜ M ( k , ω ) = (cid:90) + ∞−∞ M ( k , t ) e − iωt d t. (9)We start with the properties of ˆ M ( k , t ) at two specialmomentum points Γ = (0 ,
0) and M = ( π, π ). As shownin Fig. 2(a), the magnetization ˆ M ( k , t ) at Γ and M in aclean system oscillates at a frequency of 2 ν scar and ν scar ,respectively, and the oscillations show no sign of decay.This result is further confirmed by its Fourier transfor-mation in which there are peaks at ± ν scar and ± ν scar asshown in Fig. 2(b). When a weak disorder ( W = 0 . ω ) isturned on [Fig. 2(c)], ˆ M ( k , t ) still oscillates at the same -2 -2 (a) (b) FIG. 4. (a) and (b) plot the EE and level statistics respec-tively. The blue lines represent the system with h x , h y , h z distributed uniformly in [ − W/ , W/
2] and the orange linesrepresent the results with h x = h y = 0 and h z distributeduniformly in [ − W/ , W/ frequency initially, but eventually decays, correspondingto the widening of the peaks in Fig. 2(d). Moreover,we notice that the magnetization at M is more resistantto disorder than its counterpart at Γ , as evidenced byits slower decay and stronger peaks of its Fourier trans-formation. If the disorder is strong enough [ W = ω scar ,Fig. 2(e)] the magnetization loses all its non-ergodic prop-erties, and quickly reaches equilibrium. Correspondingly,˜ M ( k , ν ) peaks at ν = 0 with a strength equal to theirequilibrium values, as shown in Fig. 2(f).We further consider the behavior of ˜ M ( k , ω ) for k atgeneric points in the momentum space in order to betterunderstand the stability of the system and the effects ofdisorder. In Fig. 3 we calculated ˜ M ( k , ω ) for four differ-ent disorder strengths W = 0 , . ω scar , . ω scar , ω scar .The results are all averaged over 100 realizations when W (cid:54) = 0. The W = 0 limit is shown in Fig. 3(a). Due totranslational symmetry of Z and the Hamiltonian, onlymagnetization peaks at k = (0 , , ( π, π ) are allowed.When weak disorder is turned on [Fig.3(b)], translationalsymmetry of the system is broken, and anticipatory peaksappear at all k points, though peaks at Γ and M stronglyobscure the others. Moreover, in the vicinity of Γ and M ,peaks only appear with a frequency of ω = ± ω scar or ω = ± ω scar respectively. In contrast, for k points notin the vicinity of Γ and M , the magnetization M ( k , ω )peaks at five frequencies ω = 0 , ± ω scar , ± ω scar . Asthe disorder strength further increases [ ω = 0 . ω scar ,Fig. 3 (c)], peaks away from Γ and M are almost com-pletely killed by the disorder, and only peaks close to M are left with finite residues. Finally, if the disorderis strong enough ( W = ω scar ), all peaks now appear at ω = 0 only, as shown in Fig. 3(d). Possible localization transitions .— Having shown thatsubstantial disorder ( W ∼ ω scar ) can destroy the weaknon-ergodicity of the PXP model, we now study itsfate at even stronger disorders. In particular, a natu-ral question to ask is whether the PXP model can be-come many-body localized as the disorder becomes evenstronger. We will use both the level statistics and EEto study this problem. We start from the level spacingratios r n = min( δE n , δE n +1 ) / max( δE n , δE n +1 ), where δE n = E n +1 − E n is the energy gap between two adja-cent energy eigenvalues E n and E n +1 . In the thermalphase, because the disorder Hamiltonian Eq. (6) pos-sesses no symmetry, the average of level spacing ratios r approaches r = 0 . r obeys the Poisson distribution and ap-proaches r = 0 .
39. Though there is a small dip around W = 2 ω scar in Fig. 4(a), r shows no sign of approaching0 .
39 for all values of W in the figure. This result im-plies that the system does not actually enter MBL phase,which is also testified by the EE of the energy eigenstates.In Fig. 4(b), we plot the mean EE of the middle-quartereigenstates in the spectrum. We note that EE shouldapproach S T in the thermal phase, and approach zero asthe system transits into MBL phase. However, the EE ofour model seems to follow volume law, but never couldreach S T . This indicates that for the PXP model, thesystem never really resides in either the thermal phase orthe MBL phase.We note that the reason why the current model doesnot approach a localization phase even at strong disor-ders may be attributed to the specific type of disorderwe introduce in Eq. (6), which belongs to the GUE. Incontrast, if we turn off h x and h y , keeping only h z , thedisorder potential then belongs to the Gaussian orthog-onal ensemble (GOE), similar to the disordered Heisen-berg XXZ chain [39]. In such a case the resulting phasediagram becomes very different. As shown in Fig. 4(a)and 4(b), the PXP model now seems to approach an MBLphase at a disorder strength of W = 2 ω scar . Summary and outlook .— In this Letter we studied thefate of two-dimensional quantum many-body scar statesagainst random disorders. In particular, we showed thatthe scar states in a two-dimensional square lattice arestable against random disorders up to W ∼ . ω scar , be-fore eventually being killed. We demonstrated this phasetransition by studying the magnetization of the system M ( r , t ). We also studied whether the two-dimensionalPXP model enters a many-body localized phase at strongdisorders. We found that the behavior of the PXP modelstrongly depends on the type of disorder present in thesystem. Meanwhile, we have also studied the results in atwo-dimensional honeycomb lattice [40], and found qual-itatively similar results. This Letter raises several inter-esting questions for future studies. For example, it isinteresting to understand the stability of scar states inthe presence of quasiperiodic potentials instead of ran-dom disorders. In addition, it is not clear whether theintermediate phase found in Fig. 4 has some similarity tothe intermediate phase identified during the transitionbetween a thermal and an MBL phase in a quasiperiodicsystem [41, 42]. Future experiments can help verify theresults in our work and also help us gain insights intothese interesting open questions. Acknowledgements .— X.L. thanks Dong E. Liu forfruitful discussions. X.L. acknowledges support fromCity University of Hong Kong (Project No. 9610428), theNational Natural Science Foundation of China (GrantNo. 11904305), as well as the Research Grants Councilof Hong Kong (Grant No. CityU 21304720). Y.W. ac- knowledges the support from the National Natural Sci-ence Foundation of China under Grants No. 11874292,No. 11729402, and No. 11574238. [1] I. Bloch, J. Dalibard, and W. Zwerger, Rev. Mod. 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In the Appendix, we study the stability of quantummany-body scars against random disorders in a two-dimensional honeycomb lattice. In particular, the PXPHamiltonian H PXP and the disorder potential defined inthe main text are still applicable to the honeycomb lat-tice, which we quote below: H PXP = Ω2 (cid:88) r σ x r (cid:89) (cid:104) r , r (cid:48) (cid:105) P − r (cid:48) , H w = (cid:88) r h a ( r ) σ a r . (S1)However, the stabilizer δH defined in the main text nolonger applies to the honeycomb lattice. Instead, thereexists a different perturbation δH to enhance the weaknon-ergodicity in the honeycomb lattice [15], δH = (cid:88) r σ x r (cid:2) aP l r + bP r (cid:3) (cid:89) (cid:104) r , r (cid:48) (cid:105) P − r (cid:48) , (S2)where P l r = (cid:88) (cid:104)(cid:104) r (cid:48) , r (cid:105)(cid:105) P − r (cid:48) , P r = (cid:88) (cid:104) r (cid:48) , r (cid:105) P − r (cid:48) (cid:20) (cid:89) (cid:104) r (cid:48)(cid:48) , r (cid:48) (cid:105) P − r (cid:48)(cid:48) (cid:21) . (S3)In the above equation, (cid:104)(cid:104) r (cid:48) , r (cid:105)(cid:105) indicates summing overall next nearest neighbors of site r . In addition, a and b are optimized according to the revival of the fidelity |(cid:104) Z | Z ( t ) (cid:105)| [15]. Note that there are two sublattices inthe honeycomb lattice, which we name as the A and B sublattice, respectively. Hence | Z (cid:105) in the honeycomblattice is defined as the state with all atoms on one sub-lattice getting excited while those on the other sublat-tice remaining at the ground state. In this Appendix wespecifically choose | Z (cid:105) to be the state with all atoms onthe A sublattice being excited. In a 3 × a = 0 . b = 0 . × × | Z (cid:105) and EE. In particu-lar, in Fig. S1 we observed a similar structure for the scarstates in a clean system and verified that it is eventuallykilled by strong disorders.We now discuss the dynamics of the magnetization M ( r , t ) = (cid:104) Z ( t ) | σ z r | Z ( t ) (cid:105) in this 3 × M j ( k , t ) = (cid:88) r ∈ j e i k · r M ( r , t ) , ˜ M j ( k , ω ) = (cid:90) + ∞−∞ M j ( k , t ) e − iωt d t. (S4) In the above equation j = A , B, k = k x b + k y b and b , b are the bases of the reciprocal space. Further, weintroduce the following combinationsˆ M ± ( k , t ) = ˆ M A ( k , t ) ± ˆ M B ( k , t ) , ˜ M ± ( k , ω ) = ˜ M A ( k , ω ) ± ˜ M B ( k , ω ) . (S5)It turns out that ˆ M ± ( , t ) corresponds to ˆ M ( k , t ) with k = (0 ,
0) and k = ( π, π ) in the square lattice, respec-tively. We find from Fig. S2 and Fig. S3 that as the dis-order strength increases, there is also a transition fromweak ergodicity breaking to a thermal phase. In par-ticular, the strongest peak only appears at k = , and˜ M ± ( , ω ) peaks at ω = ± ω scar or ω = ± ω scar initially,while as disorder turned on, all peaks move to ω = 0.Finally, we discuss the behavior of the system at verystrong disorders. In Fig. S4, we plot the EE and levelstatistics in a 3 × × × D L = 260 and D R = 18 respec-tively, and therefore the EE in thermal phase is given by S T = ln D L − D R / (2 D L ) [37]. It turns out that the re-sult again depends on the type of disorder we introduce.In particular, if h x , h y , and h z are all present, a strongdisorder still cannot localize the system. In contrast, if h x = h y = 0 and only h z is present, the small systemseems to approach an MBL phase quickly. -10 -5 0 5 10-12-10-8-6-4-20 -10 -5 0 5 1000.20.40.60.81-10 -5 0 5 10-12-10-8-6-4-20 (c) -10 -5 0 5 1000.20.40.60.81 (b)(d)(a) FIG. S1. (a) The overlap between the eigenstates of thePXP model and the | Z (cid:105) state in a 3 × W = 0 . ω scar . Blue dots are obtained byexact diagonalization methods and red circles in (a), (b) areresults from the FSA approximation. -2 -1 0 1 2024681012 -2 -1 0 1 2024681012 -2 -1 0 1 2024681012 -2 -1 0 1 2024681012 (d)(c)(b)(a) (h)(g)(f)(e) FIG. S2. (a)-(d) are the results in a 3 × W = 0 , . ω scar , . ω scar , ω scar , respectively. The lower panels show the Fourier transformation of the correspondingupper panels. The blue lines represent ˆ M + or ˜ M + , and the orange lines represent ˆ M − or ˜ M − , respectively. (d)(c)(b)(a) (g)(f)(e) (h) FIG. S3. (a)-(d) plot ˜ M + in a 3 × W = 0 , . ω scar , . ω scar , ω scar respectively. All results areaveraged over 20 disorder realizations. The radii of the spheres represent the strength of | ˜ M ( k , ω ) | . The lower panels displaythe corresponding ˜ M − for each upper panel. -2 -2 (b)(a) FIG. S4. (a) and (b) plot the EE and level statistics ina 3 × h x , h y , h z distributed uniformly in [ − W/ , W/ h x = h y = 0 and h z distributed uniformly in [ − W/ , W/, W/