Distinguishing localization from chaos: Challenges in finite-size systems

Abanin, Dmitry A.; Bardarson, Jens H.; Tomasi, Giuseppe De; Gopalakrishnan, Sarang; Khemani, Vedika; Parameswaran, S. A.; Pollmann, Frank; Potter, Andrew C.; Serbyn, Maksym; Vasseur, Romain

doi:10.1016/j.aop.2021.168415

Cited by 198 publications

(100 citation statements)

References 79 publications

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“…In this Article we deal with this issue by investigating one of the most striking exceptions to thermal behavior in manybody quantum systems: the transition to many-body localization (MBL) [4,. The MBL phase is an insulating quantum phase of matter that emerges in some disordered interacting many-body systems, like the paradigmatic onedimensional spin chain, when the disorder is large enough [4,[13][14][15]. Several experiments in one-dimensional lattice fermions and bosons [16,17], two-dimensional interacting bosons [18], trapped ultracold ions [19], and superconducting qubits [20,21] have found its signatures.…”

Section: Introductionmentioning

confidence: 99%

Thouless energy challenges thermalization on the ergodic side of the many-body localization transition

2020

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We study the ergodic side of the many-body localization transition in its standard model, the disordered Heisenberg quantum spin chain. We show that the Thouless energy, extracted from long-range spectral statistics and the power-spectrum of the full momentum distribution fluctuations, is not large enough to guarantee thermalization. We find that both estimates coincide and behave non-monotonically, exhibiting a strong peak at an intermediate value of the disorder. Furthermore, we show that non-thermalizing initial conditions occur well within the ergodic phase with larger probability than expected. Finally, we propose a mechanism, driven by the Thouless energy and the presence of anomalous events, for the transition to the localized phase.

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Section: Introductionmentioning

confidence: 99%

Thouless energy challenges thermalization on the ergodic side of the many-body localization transition

2020

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“…Whether the MBL phase is reached in the thermodynamic limit from the ergodic phase in the form of an actual phase transition (i.e., by means of a critical point), or as a dynamical crossover, exhibiting a finite transition region (dubbed bad metal) with surviving non-ergodic but extended states is an open question [39,[64][65][66]]. The answer seems to be forever eluding the community due to the (strictly speaking) impossibility to access the thermodynamic limit and the importance of finite size effects [44,45], which are strong in these chains.…”

Section: Model: the Disordered J 1 -J Chain And Many-body Localizationmentioning

confidence: 99%

“…A variety of ergodicity breaking indicators [15,23,24,28,[36][37][38][39][40][41][42][43] have been used in the past to try and identify a hypothetical value of disorder strength that, in the thermodynamic limit, completely separates the ergodic and the nonergodic phases. The search for such a critical point implicitly assumes a nature of real phase transition in MBL systems, although this has not been proved and some apparent contradictions seem to exist which have led to much debate in the community [28,[44][45][46][47]. To obtain this value, different kinds of involved finite size scalings of such indicators have been employed.…”

Section: Introductionmentioning

confidence: 99%

“…To obtain this value, different kinds of involved finite size scalings of such indicators have been employed. However, the problem is not simple at all and a strong dependence on the particular indicator and the number of sites means that some of the previously obtained values for the critical point are often incompatible with each other; today much uncertainty remains about the properties and the phenomenology of the MBL transition [19,23,28,30,38,39,44,45,[47][48][49][50][51].…”

Section: Introductionmentioning

confidence: 99%

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Signatures of a critical point in the many-body localization transition

2021

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Disordered interacting spin chains that undergo a many-body localization transition are characterized by two limiting behaviors where the dynamics are chaotic and integrable. However, the transition region between them is not fully understood yet. We propose here a possible finite-size precursor of a critical point that shows a typical finite-size scaling and distinguishes between two different dynamical phases. The kurtosis excess of the diagonal fluctuations of the full one-dimensional momentum distribution from its microcanonical average is maximum at this singular point in the paradigmatic disordered J_1J1-J_2J2 model. For system sizes accessible to exact diagonalization, both the position and the size of this maximum scale linearly with the system size. Furthermore, we show that this singular point is found at the same disorder strength at which the Thouless and the Heisenberg energies coincide. Below this point, the spectral statistics follow the universal random matrix behavior up to the Thouless energy. Above it, no traces of chaotic behavior remain, and the spectral statistics are well described by a generalized semi-Poissonian model, eventually leading to the integrable Poissonian behavior. We provide, thus, an integrated scenario for the many-body localization transition, conjecturing that the critical point in the thermodynamic limit, if it exists, should be given by this value of disorder strength.

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“…This computational advantage makes extensive finite size scaling studies of Floquet MBL systems possible, and can help to make progress on the recent debate on the stability of manybody localization in the thermodynamic limit [42][43][44][45][46][47][48][49], which highlights the importance of finite size effects.…”

Section: Introductionmentioning

confidence: 99%

Polynomial filter diagonalization of large Floquet unitary operators

Luitz

2021

SciPost Phys.

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Periodically driven quantum many-body systems play a central role for our understanding of nonequilibrium phenomena. For studies of quantum chaos, thermalization, many-body localization and time crystals, the properties of eigenvectors and eigenvalues of the unitary evolution operator, and their scaling with physical system size LL are of interest. While for static systems, powerful methods for the partial diagonalization of the Hamiltonian were developed, the unitary eigenproblem remains daunting. % In this paper, we introduce a Krylov space diagonalization method to obtain exact eigenpairs of the unitary Floquet operator with eigenvalue closest to a target on the unit circle. Our method is based on a complex polynomial spectral transformation given by the geometric sum, leading to rapid convergence of the Arnoldi algorithm. We demonstrate that our method is much more efficient than the shift invert method in terms of both runtime and memory requirements, pushing the accessible system sizes to the realm of 20 qubits, with Hilbert space dimensions \geq 10^6≥106.

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Distinguishing localization from chaos: Challenges in finite-size systems

Cited by 198 publications

References 79 publications

Thouless energy challenges thermalization on the ergodic side of the many-body localization transition

Thouless energy challenges thermalization on the ergodic side of the many-body localization transition

Signatures of a critical point in the many-body localization transition

Polynomial filter diagonalization of large Floquet unitary operators

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