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arXiv:1704.01180 · Physical review. B./Physical review. B

Entanglement entropy and computational complexity of the Anderson impurity model out of equilibrium: Quench dynamics

Zhuoran He, Andrew J. Millis

Published Aug 7, 2017• 20 citations
arXivHTMLPDF

Abstract

We study the growth of entanglement entropy in density-matrix renormalization-group calculations of the real-time quench dynamics of the Anderson impurity model. We find that with an appropriate choice of basis, the entropy growth is logarithmic in both the interacting and noninteracting single-impurity models. The logarithmic entropy growth is understood from a noninteracting chain model as a critical behavior separating regimes of linear growth and saturation of entropy, which correspond respectively to overlapping and gapped energy spectra of the set of bath states. We find that a logarithmic entropy growth is the generic behavior of quenched impurity models when the bath orbitals in the matrix product state are ordered in energy. A noninteracting calculation of the double-impurity Anderson model supports the conclusion in the multi-impurity case. The logarithmic growth of entanglement entropy enables studies of quench dynamics to very long times.

DOI
10.1103/physrevb.96.085107

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We study the growth of entanglement entropy in density-matrix renormalization-group calculations of the real-time quench dynamics of the Anderson impurity model. We find that with an appropriate choice of basis, the entropy growth is logarithmic in both the interacting and noninteracting single-impurity models. The logarithmic entropy growth is understood from a noninteracting chain model as a critical behavior separating regimes of linear growth and saturation of entropy, which correspond respectively to overlapping and gapped energy spectra of the set of bath states. We find that a logarithmic entropy growth is the generic behavior of quenched impurity models when the bath orbitals in the matrix product state are ordered in energy. A noninteracting calculation of the double-impurity Anderson model supports the conclusion in the multi-impurity case. The logarithmic growth of entanglement entropy enables studies of quench dynamics to very long times.
We study the growth of entanglement entropy and bond dimension with time in density matrix renormalization group simulations of the periodically driven single-impurity Anderson model. The growth of entanglement entropy is found to be related to the ordering of the bath orbitals and to the relation of the driving period TTT to the convergence radius of the Floquet-Magnus expansion. Ordering the bath orbitals by their Floquet quasienergy is found to reduce the exponential growth rate of the computation time at intermediate driving periods, suggesting new ways to optimize matrix product state calculations of driven systems.
We numerically investigate the growth of the entanglement entropy Sent(t){S}_{\mathrm{ent}}(t)Sent​(t) in time ttt, after a global quench from a product state, in quantum chains with various kinds of disorder. The main focus is, in particular, on fermionic chains with bond disorder. In the noninteracting case at criticality we numerically test recent predictions by the real-space renormalization group for the entanglement growth in time, the maximal entanglement as a function of block size, and the decay of a density-wave order parameter. We show that multiprecision calculations are required to reach the scaling regime and perform such calculations for specific cases. For interacting models with binary bond disorder we present data based on infinite-size density matrix renormalization group calculations and exact diagonalizations. We obtain numerical evidence of a many-body localized phase in bond-disordered systems where ${S}_{\mathrm{ent}}(t)\ensuremath{\sim}lnt$ seems to hold. Our results for bond disorder are contrasted with the well-studied case of potential disorder.
The authors study the metal-insulator transition in the three-dimensional Anderson model from the entanglement perspective. They show that at the transition point the entanglement entropy after a quantum quench grows logarithmically in time, while the number entropy shows a double logarithmic growth. This behavior is consistent with exact bounds. It is also the same scaling recently found in many-body localized (MBL) phases, suggesting that the MBL phase might be more akin to an extended critical regime, rather than a fully localized phase.
Quantum quench is a non-equilibrium process where the Hamiltonian is suddenly changed during the quantum evolution. The change can be made by spatially local perturbations (local quench) or globally switching to a completely different Hamiltonian (global quench). This thesis investigates the post-quench non-equilibrium dynamics with an emphasis on the time dependence of the quantum entanglement. We inspect the scaling of entanglement entropy (EE) to learn how correlation and entanglement built up in a quench. We begin with two local quench examples. In Chap. 2, we apply a local operator to the groundstate of the quantum Lifshitz model and monitor the change of the EE. We find that the entanglement grows according to the dynamical exponent z = 2 and then saturates to the scaling dimension of the perturbing operator -- a value representing its strength. In Chap. 3, we study the evolution after connecting two different one-dimensional critical chains at their ends. The Loschmidt echo which measures the similarity between the evolved state and the initial one decays with a power law, whose exponent is the scaling dimension of the defect (junction). Among other conclusions, we see that the local quench dynamics contain universal information of the (critical) theory. In the global quench scenario, the change of the Hamiltonian affects all parts of the system. In this thesis, we focus on the global chaotic quench driven by generic non-integrable Hamiltonians. In Chap. 4, we propose to use the operator entanglement entropy of the unitary operator as a probe. Its fast linear entanglement production is sharply contrasted to the slow logarithmic spreading of the many-body localized system. The entanglement saturation suggests that the evolution operator in the long time can be modeled by a random unitary matrix. In Chap. 5, we construct a random tensor network which consists of random unitary matrices connected locally to model chaotic evolution with local interactions. We find that the entanglement dynamics is mapped to the statistical mechanics of interacting random walks. This appealing emergent picture allows us to understand the universal linear growth as well as the fluctuations of entanglement in a chaotic quench.
Abstract We study the dynamics of the quenched Anderson model at finite temperature using matrix product states (MPSs). Exploiting a chain mapping for the electron bath, we investigate the entanglement structure in the MPS for various orderings of the two chains, which emerge from the thermofield transformation employed to deal with nonzero temperature. We show that merging both chains can significantly lower the entanglement at finite temperatures as compared to an intuitive nearest-neighbor implementation of the Hamiltonian. Analyzing the population of the free bath modes—possible when simulating the full dynamics of impurity plus bath—we find clear signatures of the Kondo effect in the quench dynamics.
The equilibrium physics of quantum impurities frequently involves a universal crossover from weak to strong reservoir-impurity coupling, characterized by single-parameter scaling and an energy scale TK{T}_{K}TK​ (Kondo temperature) that breaks scale invariance. For the noninteracting resonant level model, the nonequilibrium time evolution of the Loschmidt echo after a local quantum quench was recently computed explicitly [R. Vasseur, K. Trinh, S. Haas, and H. Saleur, Phys. Rev. Lett. 110, 240601 (2013)]. It shows single-parameter scaling with variable TKt{T}_{K}tTK​t. Here, we scrutinize whether similar universal dynamics can be observed in various interacting quantum impurity systems. Using density matrix and functional renormalization group approaches, we analyze the time evolution resulting from abruptly coupling two noninteracting Fermi or interacting Luttinger liquid leads via a quantum dot or a direct link. We also consider the case of a single Luttinger liquid lead suddenly coupled to a quantum dot. We investigate whether the field-theory predictions for the universal scaling as well as for the large-time behavior successfully describe the time evolution of the Loschmidt echo and the entanglement entropy of microscopic models. Our study shows that for the considered local quench protocols the above quantum impurity models fall into a class of problems for which the nonequilibrium dynamics can largely be understood based on the knowledge of the corresponding equilibrium physics.