Many-body delocalization via symmetry emergence
MMany-body delocalization via symmetry emergence
N. S. Srivatsa, Roderich Moessner, and Anne E. B. Nielsen ∗ Max-Planck-Institut f¨ur Physik komplexer Systeme, N¨othnitzer Str. 38, D-01187 Dresden, Germany
Many-body localization (MBL) provides a mechanism to avoid thermalization in many-body quan-tum systems. Here, we show that an emergent symmetry can protect a state from MBL. Specifically,we propose a Z symmetric model with nonlocal interactions, which has an analytically known,SU(2) invariant, critical ground state. At large disorder strength all states at finite energy densityare in a glassy MBL phase, while the lowest energy states are not. These do, however, localizewhen a perturbation destroys the emergent SU(2) symmetry. The model also provides an exampleof MBL in the presence of nonlocal, disordered interactions that are more structured than a powerlaw. The presented ideas raise the possibility of an ‘inverted quantum scar’, in which a state thatdoes not exhibit area law entanglement is embedded in an MBL spectrum, which does. The eigenstate thermalization hypothesis suggests thatclean quantum systems typically thermalize [1, 2]. Al-ternatively, disorder may prevent thermalization and in-stead give rise to MBL [3]. This opens the door for gen-erating interesting phases, in which e.g. excited statesexhibit entanglement properties similar to ground states[4, 5]. Examples of MBL have been found numerically insimple spin models [3, 4] and studied in experiment [8],and many aspects of MBL are currently under investiga-tion.One area of intense interest is the interplay betweensymmetry and MBL. For instance, while it is known thatMBL can be present in systems where the Hamiltonianhas Z symmetry [6], it has been argued that short-rangeHamiltonians with SU(2) symmetry cannot support anMBL phase, because the eigenstates of such Hamiltoniansdo not have area law entanglement [9]. Also, it was notedthat symmetry-constrained dynamics can yield a many-body mobility edge [10].Another area of intense interest is the presence or ab-sence of MBL in models with nonlocal interactions. Re-cent studies, which have focused on power law interac-tions and/or hopping in a random potential or randommagnetic field, suggest that MBL can occur in long-rangemodels [11–14]. MBL has also been studied in modelswith power law interactions with random strengths orsigns combined with a random magnetic field [15, 16].Here, we introduce a new type of model which exhibitsMBL with a number of novel properties. The main ideais to use emergent symmetry of a single state in the spec-trum to protect it from MBL, while not preventing therest of the spectrum from localizing. Specifically, we in-vestigate a Hamiltonian which has only a Z symmetry,but nevertheless has a critical ground state with SU(2)symmetry. We introduce disorder into this model via ran-dom positions of the spins, and we show that all statesat finite energy density form a glassy phase [5, 6, 17] atan appropriate disorder strength, while the ground stateremains critical.The model exhibits several interesting features. First,it shows that an emergent symmetry can have interestingand nontrivial effects with respect to MBL. Specifically, for a wide range of disorder strengths all the states at fi-nite energy density form a MBL glass, while a few statesat the bottom of the spectrum do not. Crucially, thesestates also become glassy as the emergent SU(2) symme-try vanishes upon perturbing the Hamiltonian. Second,it gives an example of MBL in a system with nonlocal,disordered interactions that have more structure than apower law. Third, the ground state can be found analyt-ically, which allows for a detailed study of its propertieseven for large system sizes. Fourth, the model showsas yet unexplained gaps in the disorder averaged energyspectrum. Model — We study a system of N spin-1 / H = (cid:88) i (cid:54) = j F Aij ( S xi S xj + S yi S yj ) + (cid:88) i (cid:54) = j F Bij S zi S zj + F C (1)in terms of the spin operators S ai = σ ai / a ∈ { x, y, z } ,where σ ai are the Pauli matrices acting on the i th spin.The coupling strengths F Aij = − w ij , w jk = − i/ tan[( φ j − φ k ) / , (2) F Bij = − w ij + 2 w ij (cid:0) (cid:88) k ( (cid:54) = i ) w ik − (cid:88) k ( (cid:54) = j ) w jk (cid:1) ,F C = ( N − N − − N ) / − (cid:88) i (cid:54) = j w ij + ( N − N − (cid:88) i S zi , depend on the positions e iφ j of the spins on the unitcircle. We introduce disorder into the model by choosing φ f ( j ) = 2 π ( j + α j ) /N, j = 1 , , . . . , N, (3)where α j is a random number chosen with constant prob-ability density in the interval [ − δ , δ ] and δ ∈ [0 , N ] isthe disorder strength. We choose the indices f ( j ) ∈{ , , . . . , N } such that the spins are always numberedin ascending order when going around the circle. For theclean case, δ = 0, the spins are uniformly distributed onthe circle, and for maximal disorder, δ = N , all the spins a r X i v : . [ c ond - m a t . d i s - nn ] F e b may be anywhere on the circle. For δ ≥
1, neighboringspins can be arbitrarily close.The Hamiltonian (11) is fully connected, meaning thatevery spin interacts with every other spin. If we interpreta spin up as a particle and a spin down as an empty site,the first term in the Hamiltonian is a hopping term. Thecoefficient F Aij decreases rapidly with distance betweenthe spins, following a one-over-distance-squared behaviorfor short distances. The coefficient F Bij of the interactionterm has a more complicated behavior. The maximuminteraction strength can occur at different distances, andFig. 1(a) shows the probability that a given spin interactsmost strongly with the m th neighbor on either the leftor the right side. In the rest of this article, we restrictourselves to the zero magnetization sector (cid:80) i S zi = 0.It was shown analytically in [18] that the state | ψ (cid:105) = (cid:88) s ,...,s N δ s (cid:89) k χ k (cid:89) i
1. We first investi-gate the entanglement entropy S = − Tr[ ρ ln( ρ )] averagedover disorder realizations for states in the middle of thespectrum [6], where ρ is the reduced density operator forhalf of the chain. The mean entanglement entropy andthe variance of the distribution as a function of the disor-der parameter δ (see Fig. 1(c-d)) show a phase transitionat δ ≈
1. The variance close to the phase transitionpoint is large, which suggests that the transition is con-tinuous. On the left hand side of the transition, the meanentanglement entropy scales with the system size N andis bounded by the thermodynamic entropy, while on theright hand side of the transition it displays area law be-haviour indicating MBL (see the inset in Fig. 1(c)).The level spacing statistics is another diagnostic ofMBL. We see that the energy spectrum at strong dis-order ( δ = N ) has pairs of eigenvalues that are almost S N=6N=8N=10N=12N=14 S -3 N=6N=8N=10N=12N=14 N r =N=0.3 m P r ob a b ili t y N < S > =0.5=N (b)(a)(c)(d) N =16 =16
FIG. 1. (a) Probability that a given spin in the chain inter-acts most strongly with the m th nearest neighbor. (b) Theadjacent gap ratio (averaged over 10 disorder realizationsand shown as a function of system size) is close to the Gaus-sian orthogonal ensemble for weak disorder and close to thePoisson distribution, indicating MBL, for strong disorder. (c)The transition to the MBL phase is also seen in the entan-glement entropy S of half of the chain for the state closest tothe middle of the spectrum averaged over 10 disorder real-izations as a function of the disorder strength δ for differentsystem sizes N . The blue dashed lines indicate the thermalvalue [ N ln(2) − / σ of theentanglement entropy computed from the same set of datashows a peak at the transition point. degenerate and have opposite parity with respect to theglobal Z symmetry of the Hamiltonian. We hence com-pute the adjacent gap ratio [23] for different system sizesby restricting ourselves to one of the Z sectors. Figure1(b) shows that the disorder averaged gap ratio r con-verges towards the Poisson distribution ( r ≈ . r ≈ .
53) for weak disorder. This confirms thatthe system is indeed MBL for strong disorder.The fact that pairs of almost degenerate states with op-posite parity appear in the spectrum suggests that thereis spin glass order in the excited states [6, 17]. In aneigenstate | ψ n (cid:105) this can be identified by the divergence / h @ S G i N=6N=8N=10N=12N=14 ln(E/E max ) -5 -4 -3 -2 -1 0 l n ( G ( E )) -10-9-8-7-6-5-4-3-2-1 / =16 / =10 / =1.5 / =1.2 / =1.1 / =1 / =0.9 / =0.8 / =0.7 / =0.6 / =0.2 / =0 / h @ S G i N=6N=8N=10N=12N=14Ground State(N=20)
Eigenstates h @ S G i / =0.1 / =12x L -100 0 100 y L n E n / E m a x (d) a = 0 : = 0 : / c = 0 : (c) (b)(a) FIG. 2. (a) Disorder averaged spin glass order parameter (cid:104) χ SG (cid:105) for a state in the middle of the spectrum ( E/E max =0 .
5, circles), a low energy state (
E/E max = 0 .
01, triangles),and the ground state ( E = 0, dashed line) as a function ofthe disorder strength δ . The excited states show glassinessfor strong disorder, and the ground state is not glassy. Afinite size scaling collapse (inset) for the state in the middleof the spectrum gives the phase transition point δ c ≈ . δ ≈
1. (c) (cid:104) χ SG (cid:105) for alleigenstates for a system with 12 spins and weak ( δ = 0 . δ = 12) disorder. The states close to the groundstate have different values of (cid:104) χ SG (cid:105) compared to the statesin the middle of the spectrum. The inset shows that thedisorder averaged energy spectrum (normalised) at δ = 12 hasgaps, and these coincide with the jumps in (cid:104) χ SG (cid:105) . (d) Whenwe destroy the emergent SU(2) symmetry by perturbing theHamiltonian (we modify F Aij to − . w ij ), the ground statebecomes glassy for δ > ∼
1. In all cases, the number of disorderrealizations is 10 . of an Edwards-Anderson order parameter [6, 24] χ SG = 1 N N (cid:88) i (cid:54) = j (cid:104) ψ n | σ zi σ zj | ψ n (cid:105) . (5)For eigenstates in the middle of the spectrum, we find(see Fig. 2(a)) that there is glassiness for strong disor- der ( (cid:104) χ SG (cid:105) increases with system size), but not for weakdisorder ( (cid:104) χ SG (cid:105) approaches zero with increasing systemsize). We perform a finite size scaling analysis to getan estimate of the critical disorder strength. The scal-ing parameters are given in the inset where we define thescaling function as x L = ( δ − δ c ) N ν and y L = (cid:104) χ SG (cid:105) /N a .Glassiness sets in at around the same disorder strength( δ c ≈
1) as MBL.The transition around δ = 1 is also visible in the dis-order averaged integral of the density of states G ( E ) = (cid:82) E ρ ( E ) dE/d H plotted as a function of energy E/E max in Fig. 2(b), where d H is the total number of states inthe Hilbert space and E max is the highest energy in thespectrum. Ground state —The low energy physics of the Haldane-Shastry model is described by Luttinger liquid theory.For strong disorder, we show that various properties ofthe ground state remain the same rather than reflectinga phase transition to, e.g., a random singlet phase or aglassy phase.The disordered Haldane-Shastry state has been studiedpreviously for weak disorder [20, 25]. In [20], the Renyientropy was investigated. For critical systems it is knownthat the Renyi entropy of order two shows a universalbehavior given by [26–28] S L = C ln [sin( πL/N )] + α, (6)where L is the number of spins in the considered sub-system, and C is a universal constant that takes thevalue 1 / / δ = 0 . δ = 0 .
5, and δ = 0 .
75 in [20] showed that C might be closer to ln(2) / / δ = 0 . δ = 0 .
75. We have redone the computation for δ = 0 . C might rathergo to 1 / L valuesclose to N/ C due to, e.g., finite size ef-fects and ambiguity in the fitting procedure is not smallcompared to the difference between 1 / /
3. Theconclusions may not be reliable, and one should also con-sider other diagnostics.In [25], the second cumulant C ( N/
2) = (cid:104) M (cid:105) − (cid:104) M (cid:105) , M ≡ N/ (cid:88) i =1 S zi , (7)of the total magnetization M of half of the systemwas observed to show a Luttinger liquid behavior at δ = 1. Specifically, the second cumulant is known todiverge logarithmically with system size, C ( N/ ∼ ξ ln( N/
2) + constant for large N , with different coef-ficients for the Luttinger liquid ( ξ = 1 / (2 π )) and for N
16 32 64 128 256 [ C ( N ) ! C ( N = ) ] = l n disorderedRSPLuttingerclean N h C i SlopeRSPLuttinger ln[sin( : L=N )] -2 -1 0 h S L i h S i Fit ln(2)3
Fit : (a) (b) h C i =0.2443 , = 1 :
79 12 : FIG. 3. (a) The Renyi entropy of the ground state (plottedfor N = 600 and δ = 0 .
75 in the inset) follows the logarithmicrelation (6). The main plot shows the coefficient C as a func-tion of the system size for δ = 0 .
75. The results might suggesta Luttinger liquid, but the method is not accurate enough tomake clear conclusions. Averaging is done over 10 disorderrealizations, and the error occurring from the Monte Carlosimulations and disorder averaging is of order 10 − . (b) Thecoefficient ξ in the second cumulant C computed for differentsystems divided into two halves is close to the value for theLuttinger liquid both for the clean ( δ = 0) and the disordered( δ = 4) Haldane-Shastry state. Each data point is averagedover 10 disorder realizations. the random singlet phase ( ξ = 1 / C ( N ) − C ( N/ / ln(2) ≈ ξ approaches the valuefor the Luttinger liquid for large system sizes. Figure3 shows that the same is true for δ = 4. This showsthat the ground state retains its Luttinger liquid behav-ior also for a disorder strength for which the highly ex-cited states are MBL. Note that relatively large systemsizes can be reached in these computations because thetwo point correlations can be obtained by solving a setof linear equations derived in [32].Finally, we consider the spin glass order parametershown in Fig. 2(a). Here, the ground state yet againstands out, with no indication of a phase transition. Infact, for the ground state, χ SG = 1 is constant, irrespec-tive of disorder strength and system size. Low-lying excited states —The observation that thehighly excited states undergo a transition to MBL, whilethe ground state does not undergo a transition, naturallyraises the question, whether it is only the ground statethat is special, or the ground state properties are to someextent inherited to the low-lying excitations. By study-ing the spin glass order parameter, we find that a smallnumber of low-lying excitations behave differently, butas soon as we consider a finite energy density, the statesappear to show glassiness for strong disorder.In Fig. 2(a), we show data for (cid:104) χ SG (cid:105) for the case where we choose the state in the spectrum that is closest to E/E max = 0 .
01 in every disorder realization. Even forthis low value of the energy density, glassiness is stillobserved for strong disorder. A more detailed view for12 spins is given in Fig. 2(c), where we plot (cid:104) χ SG (cid:105) forall states in the spectrum for weak and strong disor-der. For all the highly excited states, there is a largeincrease in (cid:104) χ SG (cid:105) when going from weak to strong dis-order, which shows the transition into the glassy phase.For the ground state, there is no change as χ SG = 1. Afew states close to the ground state show an intermedi-ate behavior and have particularly low values of (cid:104) χ SG (cid:105) forstrong disorder. The inset shows the disorder averagedspectrum, and we note that the sudden jumps observedin (cid:104) χ SG (cid:105) coincide with gaps in the spectrum. Symmetry —Finally, we show that the special behaviorof the ground state disappears together with the emer-gent SU(2) symmetry. To do so, we slightly modify thehopping strengths F Aij to − . w ij . This preserves the Z symmetry of the Hamiltonian, but not the emergentSU(2) symmetry of the ground state. Figure 2(d) showsthat the ground state is now glassy for strong disorder.If instead we add a small amount of the Haldane-ShastryHamiltonian, the ground state is unaltered, and the Z symmetry of the Hamiltonian is preserved. In this case,the spin glass order parameter behaves similarly to theresults in Fig. 2(a). Conclusions —We have constructed a new type of MBLmodel, in which an emergent symmetry protects theground state from MBL. While states at a finite energydensity show MBL for sufficiently strong disorder, theground state remains critical. It seems likely that the ob-served behavior is a general mechanism to protect statesfrom MBL, and it would be interesting to search for asimilar behavior in other models.The model has the unexpected property that the disor-der averaged energy spectrum has gaps. The backgroundfor this is not understood and would be interesting to in-vestigate further.While we do find glassiness to be present in all ex-cited states already at very low energy densities, we finda different behavior for the states adjacent to the groundstate. The possibility of a ‘critical regime’ with a diverg-ing localization length warrants further study here.If the delocalized state should turn out to be genuinelyisolated in a sea of localized ones, this would have theflavour of an ‘inverted scar state’, i.e. a state with above-area-law entanglement in a sea of area law entangledstates; it is the converse of what is found in the cele-brated many-body scars [33].At any rate, it would be interesting to consider thescope of constructing models with multiple states in thespectrum – including at finite energy density – being pro-tected from localization by an emergent symmetry.Another interesting direction for further investigationsis to study the transport properties of our model, in par-ticular in a regime where the ground state and some ofthe lower lying excited states are populated.
Acknowledgements —We thank Giuseppe De Tomasiand Ivan Khaymovich for discussions. This work wasin part supported by the Deutsche Forschungsgemein-schaft under grants SFB 1143 (project-id 247310070) andthe cluster of excellence ct.qmat (EXC 2147, project-id39085490). ∗ On leave from Department of Physics and Astronomy,Aarhus University, DK-8000 Aarhus C, Denmark.[1] J. M. Deutsch, Phys. Rev. A , 2046 (1991).[2] M. Srednicki, Phys. Rev. E , 888 (1994).[3] D. Basko, I. Aleiner, and B. Altshuler, Annals of Physics , 1126 (2006).[4] R. Nandkishore and D. A. Huse, Annual Review of Con-densed Matter Physics , 15 (2015).[5] S. A. Parameswaran and R. Vasseur, Reports on Progressin Physics , 082501 (2018).[6] J. A. Kj¨all, J. H. Bardarson, and F. Pollmann, Phys.Rev. Lett. , 107204 (2014).[7] D. J. Luitz, N. Laflorencie, and F. Alet, Phys. Rev. B , 081103 (2015).[8] M. Schreiber, S. S. Hodgman, P. Bordia, H. P. L¨uschen,M. H. Fischer, R. Vosk, E. Altman, U. Schneider, andI. Bloch, Science , 842 (2015).[9] I. V. Protopopov, W. W. Ho, and D. A. Abanin, Phys.Rev. B , 041122 (2017).[10] I. Mondragon-Shem, A. Pal, T. L. Hughes, and C. R.Laumann, Phys. Rev. B , 064203 (2015).[11] N. Y. Yao, C. R. Laumann, S. Gopalakrishnan, M. Knap,M. M¨uller, E. A. Demler, and M. D. Lukin, Phys. Rev.Lett. , 243002 (2014).[12] A. L. Burin, Phys. Rev. B , 104428 (2015). [13] R. M. Nandkishore and S. L. Sondhi, Phys. Rev. X ,041021 (2017).[14] S. Nag and A. Garg, Phys. Rev. B , 224203 (2019).[15] K. S. Tikhonov and A. D. Mirlin, Phys. Rev. B ,214205 (2018).[16] T. Botzung, D. Vodola, P. Naldesi, M. M¨uller, E. Erco-lessi, and G. Pupillo, Phys. Rev. B , 155136 (2019).[17] D. A. Huse, R. Nandkishore, V. Oganesyan, A. Pal, andS. L. Sondhi, Phys. Rev. B , 014206 (2013).[18] H.-H. Tu, A. E. B. Nielsen, J. I. Cirac, and G. Sierra,New Journal of Physics , 033025 (2014).[19] See the supplemental material.[20] J. I. Cirac and G. Sierra, Phys. Rev. B , 104431 (2010).[21] F. D. M. Haldane, Phys. Rev. Lett. , 635 (1988).[22] B. S. Shastry, Phys. Rev. Lett. , 639 (1988).[23] V. Oganesyan and D. A. Huse, Phys. Rev. B , 155111(2007).[24] S. F. Edwards and P. W. Anderson, Journal of PhysicsF: Metal Physics , 965 (1975).[25] J.-M. St´ephan and F. Pollmann, Phys. Rev. B , 035119(2017).[26] C. Holzhey, F. Larsen, and F. Wilczek, Nuclear PhysicsB , 443 (1994).[27] G. Vidal, J. I. Latorre, E. Rico, and A. Kitaev, Phys.Rev. Lett. , 227902 (2003).[28] P. Calabrese and J. Cardy, Journal of Statistical Mechan-ics: Theory and Experiment , P06002 (2004).[29] D. S. Fisher, Phys. Rev. B , 3799 (1994).[30] N. Laflorencie, Phys. Rev. B , 140408 (2005).[31] Y.-R. Shu, D.-X. Yao, C.-W. Ke, Y.-C. Lin, and A. W.Sandvik, Phys. Rev. B , 174442 (2016).[32] A. E. B. Nielsen, J. I. Cirac, and G. Sierra, Journalof Statistical Mechanics: Theory and Experiment ,P11014 (2011).[33] N. Shiraishi and T. Mori, Phys. Rev. Lett. , 030601(2017). Supplemental material —Our starting point is theHamiltonian H = (cid:88) i Λ † i Λ i − (cid:88) i Γ † i Γ i (8)for hardcore bosons on a lattice introduced in [18]. Here,Λ i = (cid:88) j ( (cid:54) = i ) w ij [ c j − c i (2 n j − , (9)Γ i = (cid:88) j ( (cid:54) = i ) w ij c i c j , (10)where c j is the operator that annihilates a hardcore bosonon site j , and n j = c † j c j . It can be shown [18] that bothΛ i and Γ i annihilate the state in Eq. (4) in the main text.Inserting (9) and (10) into (8), we obtain H = (cid:88) i (cid:54) = j ( F Aij c † i c j + F Bij n i n j ) + (cid:88) i F Ci n i + F D , (11) where the coupling coefficients are given by F Aij = − w ij ,F Bij = 2 w ij + 4 (cid:88) l ( (cid:54) = j (cid:54) = i ) w ij w il ,F Ci = − (cid:88) j ( (cid:54) = i ) w ij − (cid:88) k,l ( (cid:54) = i ) w ik w il ,F D = − N ( N − N − . (12)One arrives at the spin version of the Hamiltonian de-scribed in Eq. (1) of the main text by introducing thetransformation S + i = c † i S − i = c i S zi = c † i c i − / , (13)where S ± i = S xi ± iS yiyi