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arXiv:2112.11457 · Physical review. B./Physical review. B

Simulation of many-body localization and time crystals in two dimensions with the neighborhood tensor update

Jacek Dziarmaga

Published Feb 14, 2022• 14 citations
arXivHTMLPDF

Abstract

The Heisenberg antiferromagnet with discrete disorder on an infinite square lattice is evolved in time from an initial N\'eel state. The simulation is performed with the neighborhood tensor update algorithm for an infinite projected entangled pair state [Dziarmaga, Phys. Rev. B 104, 094411 (2021)]. Ancillary spins are used to average over two or five discrete values of disorder. With a bond dimension up to 20, evolution times are long enough to identify a many-body localized regime for a strong enough disorder. Furthermore, the same Hamiltonian is subject to periodic spin flips. Simulations of the Floquet dynamics show that it can sustain a time crystalline stage for a strong enough disorder.

DOI
10.1103/physrevb.105.054203

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The Heisenberg antiferromagnet with discrete disorder on an infinite square lattice is evolved in time from an initial N'eel state. The simulation is performed with the neighborhood tensor update algorithm for an infinite projected entangled pair state [Dziarmaga, Phys. Rev. B 104, 094411 (2021)]. Ancillary spins are used to average over two or five discrete values of disorder. With a bond dimension up to 20, evolution times are long enough to identify a many-body localized regime for a strong enough disorder. Furthermore, the same Hamiltonian is subject to periodic spin flips. Simulations of the Floquet dynamics show that it can sustain a time crystalline stage for a strong enough disorder.
Many-body localization is a striking mechanism that prevents interacting quantum systems from thermalizing. The absence of thermalization behavior manifests itself, for example, in a remanence of local particle number configurations, a quantity that is robust over a parameter range. Local particle numbers are directly accessible in programmable quantum simulators, in systems of cold atoms, even in two spatial dimensions. Yet, the classical simulation aimed at building trust in quantum simulations is highly challenging. In this work, we present a comprehensive tensor network simulation of a many-body localized systems in two spatial dimensions using a variant of an infinite projected entangled pair states algorithm. The required translational invariance can be restored by implementing the disorder into an auxiliary spin system, providing an exact disorder average under dynamics. We can quantitatively assess signatures of many-body localization for the infinite system: Our methods are powerful enough to provide crude dynamical estimates for the transition between localized and ergodic phases. Interestingly, in this setting of finitely many disorder values, which we also compare with simulations involving noninteracting fermions and for which we discuss the emergent physics, localization emerges in the interacting regime, for which we provide an intuitive argument, while Anderson localization is absent.
A discrete time crystal is an out-of-equilibrium phase of matter characterized by the spontaneous breaking of discrete time-translation symmetry. Using extensive numerical simulations based on matrix-product-state methods, we provide evidence for discrete time-crystalline behavior in strongly disordered spin chains with Heisenberg interactions, including the isotropic point, subject to periodic driving. Starting from a many-body localized regime, we observe that rotations induced by delta kicks produce a pronounced subharmonic response at half the drive frequency in spin observables. We investigate the stability of this response against rotation-angle errors through entanglement entropy, quantum Fisher information, short-range spin correlations, and restricted-control ergotropy. Increasing the rotation error reveals an intermediate dynamical regime separating the time-crystalline and Floquet-localized responses. In this regime, most observables exhibit signatures of weakly correlated dynamics reminiscent of Anderson localization.
Time evolution of an infinite two-dimensional (2D) many body quantum lattice system can be described by the Suzuki-Trotter decomposition applied to the infinite projected entangled pair state (iPEPS). Each Trotter gate increases the bond dimension of the tensor network, DDD, that has to be truncated back in a way that minimizes a suitable error measure. This paper goes beyond simplified error measures--like the one used in the full update, the simple update, and their intermediate neighborhood tensor update--and directly maximizes an overlap between the exact iPEPS with the increased bond dimension and the new iPEPS with the truncated one. The optimization is performed in a tangent space of the iPEPS variational manifold. This gradient tensor update is benchmarked by a simulation of a sudden quench of a transverse field in the 2D quantum Ising model and the quantum Kibble-Zurek mechanism in the same 2D system.
The recent progress in the optimization of two-dimensional tensor networks [H.-J. Liao, J.-G. Liu, L. Wang, and T. Xiang, Phys. Rev. X 9, 031041 (2019)] based on automatic differentiation opened the way towards precise and fast optimization of such states and, in particular, infinite projected entangled-pair states (iPEPS) that constitute a generic-purpose Ansatz for lattice problems governed by local Hamiltonians. In this work, we perform an extensive study of a paradigmatic model of frustrated magnetism, the J_1-J_2 J 1 − J 2 Heisenberg antiferromagnet on the square lattice. By using advances in both optimization and subsequent data analysis, through finite correlation-length scaling, we report accurate estimations of the magnetization curve in the N'eel phase for J_2/J_1 \le 0.45 J 2 / J 1 ≤ 0.45 . The unrestricted iPEPS simulations reveal an U(1) U ( 1 ) symmetric structure, which we identify and impose on tensors, resulting in a clean and consistent picture of antiferromagnetic order vanishing at the phase transition with a quantum paramagnet at J_2/J_1 \approx 0.46(1) J 2 / J 1 ≈ 0.46 ( 1 ) . The present methodology can be extended beyond this model to study generic order-to-disorder transitions in magnetic systems.
Infinite projected entangled pair states (iPEPS), the tensor network ansatz for two-dimensional systems in the thermodynamic limit, already provide excellent results on ground-state quantities using either imaginary-time evolution or variational optimisation. Here, we show (i) the feasibility of real-time evolution in iPEPS to simulate the dynamics of an infinite system after a global quench and (ii) the application of disorder-averaging to obtain translationally invariant systems in the presence of disorder. To illustrate the approach, we study the short-time dynamics of the square lattice Heisenberg model in the presence of a bi-valued disorder field.
We demonstrate that Monte Carlo sampling can be used to efficiently extract the expectation value of projected entangled pair states with a large virtual bond dimension. We use the simple update rule introduced by H. C. Jiang et al. [Phys. Rev. Lett 101, 090603 (2008)] to obtain the tensors describing the ground state wave function of the antiferromagnetic Heisenberg model and evaluate the finite size energy and staggered magnetization for square lattices with periodic boundary conditions of linear sizes up to L=16L=16L=16 and virtual bond dimensions up to D=16D=16D=16. The finite size magnetization errors are 0.003(2)0.003(2)0.003(2) and 0.013(2)0.013(2)0.013(2) at D=16D=16D=16 for a system of size L=8,16L=8,16L=8,16, respectively. Finite DDD extrapolation provides exact finite size magnetization for L=8L=8L=8, and reduces the magnetization error to 0.005(3)0.005(3)0.005(3) for L=16L=16L=16, significantly improving the previous state-of-the-art results.