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arXiv:1706.06062 · Physical review. B./Physical review. B

Eigenstate phases with finite on-site non-Abelian symmetry

Abhishodh Prakash, Sriram Ganeshan, Lukasz Fidkowski, Tzu-Chieh Wei

Published Oct 18, 2017• 18 citations
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Abstract

We study the eigenstate phases of disordered spin chains with on-site finite non-Abelian symmetry. We develop a general formalism based on standard group theory to construct local spin Hamiltonians invariant under any on-site symmetry. We then specialize to the case of the simplest non-Abelian group, ${S}_{3}$, and numerically study a particular two-parameter spin-1 Hamiltonian. We observe a thermal phase and a many-body localized phase with a spontaneous symmetry breaking (SSB) from ${S}_{3}$ to ${\mathbb{Z}}_{3}$ in our model Hamiltonian. We diagnose these phases using full entanglement distributions and level statistics. We also use a spin-glass diagnostic specialized to detect spontaneous breaking of the ${S}_{3}$ symmetry down to ${\mathbb{Z}}_{3}$. Our observed phases are consistent with the possibilities outlined by A. C. Potter and R. Vasseur [Phys. Rev. B 94, 224206 (2016)], namely thermal/ergodic and many-body-localized spin-glass (MBL-SG) phases. We also speculate on the nature of an intermediate region between the thermal and MBL + SSB regions where full ${S}_{3}$ symmetry exists.

DOI
10.1103/physrevb.96.165136

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We study the eigenstate phases of disordered spin chains with on-site finite non-Abelian symmetry. We develop a general formalism based on standard group theory to construct local spin Hamiltonians invariant under any on-site symmetry. We then specialize to the case of the simplest non-Abelian group, S3{S}_{3}S3​, and numerically study a particular two-parameter spin-1 Hamiltonian. We observe a thermal phase and a many-body localized phase with a spontaneous symmetry breaking (SSB) from S3{S}_{3}S3​ to Z3{\mathbb{Z}}_{3}Z3​ in our model Hamiltonian. We diagnose these phases using full entanglement distributions and level statistics. We also use a spin-glass diagnostic specialized to detect spontaneous breaking of the S3{S}_{3}S3​ symmetry down to Z3{\mathbb{Z}}_{3}Z3​. Our observed phases are consistent with the possibilities outlined by A. C. Potter and R. Vasseur [Phys. Rev. B 94, 224206 (2016)], namely thermal/ergodic and many-body-localized spin-glass (MBL-SG) phases. We also speculate on the nature of an intermediate region between the thermal and MBL + SSB regions where full S3{S}_{3}S3​ symmetry exists.
An important challenge in the field of many-body quantum dynamics is to identify nonergodic states of matter beyond many-body localization (MBL). Strongly disordered spin chains with non-Abelian symmetry and chains of non-Abelian anyons are natural candidates, as they are incompatible with standard MBL. In such chains, real space renormalization group methods predict a partially localized, nonergodic regime known as a quantum critical glass (a critical variant of MBL). This regime features a treelike hierarchy of integrals of motion and symmetric eigenstates with entanglement entropy that scales as a logarithmically enhanced area law. We argue that such tentative nonergodic states are perturbatively unstable using an analytic computation of the scaling of off-diagonal matrix elements and accessible level spacing of local perturbations. Our results indicate that strongly disordered chains with non-Abelian symmetry display either spontaneous symmetry breaking or ergodic thermal behavior at long times. We identify the relevant length and timescales for thermalization: Even if such chains eventually thermalize, they can exhibit nonergodic dynamics up to parametrically long timescales with a nonanalytic dependence on disorder strength.
We study the nonequilibrium phase structure of the three-state random quantum Potts model in one dimension. This spin chain is characterized by a non-Abelian D3{D}_{3}D3​ symmetry recently argued to be incompatible with the existence of a symmetry-preserving many-body localized (MBL) phase. Using exact diagonalization and a finite-size scaling analysis, we find that the model supports two distinct broken-symmetry MBL phases at strong disorder that either break the Z3{\mathbb{Z}}_{3}Z3​ clock symmetry or a Z2{\mathbb{Z}}_{2}Z2​ chiral symmetry. In a dual formulation, our results indicate the existence of a stable finite-temperature topological phase with MBL-protected parafermionic end zero modes. While we find a thermal symmetry-preserving regime for weak disorder, scaling analysis at strong disorder points to an infinite-randomness critical point between two distinct broken-symmetry MBL phases.
SU(N)\mathrm{SU}(N)SU(N) symmetry is incompatible with the many-body localized (MBL) phase, even when strong disorder is present. However, recent studies have shown that finite-size SU(2)\mathrm{SU}(2)SU(2) systems exhibit nonergodic, subthermal behavior, characterized by the breakdown of the eigenstate thermalization hypothesis, and by the excited eigenstates entanglement entropy that is intermediate between area and volume law. In this paper, we extend previous studies of the SU(2)\mathrm{SU}(2)SU(2)-symmetric disordered Heisenberg model to larger systems, using the time-dependent density matrix renormalization group (tDMRG) method. We simulate quench dynamics from weakly entangled initial states up to long times, finding robust subthermal behavior at stronger disorder. Although we find an increased tendency towards thermalization at larger system sizes, the subthermal regime persists at intermediate time scales, nevertheless, and therefore should be accessible experimentally. At weaker disorder, we observe signatures of thermalization; however, entanglement entropy exhibits slow sublinear growth, in contrast to conventional thermalizing systems. Furthermore, we study dynamics of the SU(3)\mathrm{SU}(3)SU(3)-symmetric disordered Heisenberg model. Similarly, strong disorder drives the system into subthermal regime, albeit thermalizing phase is broader compared to the SU(2)\mathrm{SU}(2)SU(2) case. Our findings demonstrate the robustness of the subthermal regime in spin chains with non-Abelian continuous symmetry, and are consistent with eventual thermalization at large system sizes and long time scales, suggested by previous studies.
Many-body localization (MBL) enables new nonergodic dynamical phases of matter that avoid the fate of thermalization. Do other ergodicity-breaking phases beyond the MBL paradigm exist? Here, the authors introduce a method for constructing and probing the stability of candidate nonergodic phases. As an example, they consider disordered spin systems, in which a non-Abelian SU(2) symmetry precludes the conventional MBL phase that is characterized by quasilocal integrals of motion and area-law entanglement entropy of eigenstates. Instead, the authors construct a candidate nonergodic phase characterized by eigenstates having logarithmic scaling of entanglement entropy. They find that the phase is only marginally unstable and, therefore, nonthermal yet non-MBL dynamics can be observed in such disordered systems.
Non-abelian symmetries are thought to be incompatible with many-body localization, but have been argued to produce in certain disordered systems a broad non-ergodic regime distinct from many-body localization. In this context, we present a numerical study of properties of highly-excited eigenstates of disordered chains with SU(3) symmetry. We find that while weakly disordered systems rapidly thermalize, strongly-disordered systems indeed exhibit non-thermal signatures over a large range of system sizes, similar to the one found in previously studied SU(2) systems. Our analysis is based on the spectral, entanglement, and thermalization properties of eigenstates obtained through large-scale exact diagonalization exploiting the full SU(3) symmetry.
The two primary categories for eigenstate phases of matter at a finite temperature are many-body localization (MBL) and the eigenstate thermalization hypothesis (ETH). We show that, in the paradigmatic quantum p-spin models of the spin-glass theory, eigenstates violate the ETH yet are not MBL either. A mobility edge, which we locate using the forward-scattering approximation and replica techniques, separates the nonergodic phase at a small transverse field from an ergodic phase at a large transverse field. The nonergodic phase is also bounded from above in temperature, by a transition in configuration-space statistics reminiscent of the clustering transition in the spin-glass theory. We show that the nonergodic eigenstates are organized in clusters which exhibit distinct magnetization patterns, as characterized by an eigenstate variant of the Edwards-Anderson order parameter.