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arXiv:1505.01675 · Journal of Statistical Mechanics Theory and Experiment

How to calculate quantum quench distributions with a weighted Wang–Landau Monte Carlo

Simone Ziraldo, Giuseppe E. Santoro

Published Jun 3, 2015• 0 citations
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Abstract

We present here an extension of the Wang–Landau Monte Carlo method which allows us to get very accurate estimates of the full probability distributions of several observables after a quantum quench for large systems, whenever the relevant matrix elements are calculable, but the full exponential complexity of the Hilbert space would make an exhaustive enumeration impossible beyond very limited sizes. We apply this method to quenches of free-fermion models with disorder, further corroborating the fact that a generalized Gibbs ensemble fails to capture the long-time average of many-body operators when disorder is present.

DOI
10.1088/1742-5468/2015/06/p06009

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We present here an extension of the Wang–Landau Monte Carlo method which allows us to get very accurate estimates of the full probability distributions of several observables after a quantum quench for large systems, whenever the relevant matrix elements are calculable, but the full exponential complexity of the Hilbert space would make an exhaustive enumeration impossible beyond very limited sizes. We apply this method to quenches of free-fermion models with disorder, further corroborating the fact that a generalized Gibbs ensemble fails to capture the long-time average of many-body operators when disorder is present.
In the present thesis we study the unitary dynamics and the thermalization properties of free-fermion-like Hamiltonians after a sudden quantum quench in presence of disorder. With analytical and numerical arguments, we show that the existence of a stationary state and its description with a generalized Gibbs ensemble (GGE) depend crucially on the observable considered (local versus extensive, one-body versus many-body) and on the localization properties of the final Hamiltonian. We then show an extension of the Wang-Landau algorithm which allows the computation of weighted distributions associated to quantum quenches, like the diagonal and the GGE ensemble expectation-value distributions. We present results on three one-dimensional models, the Anderson model, a disordered one-dimensional fermionic chain with long-range hopping, and the disordered Ising/XY spin chain.
We study the unitary dynamics and the thermalization properties of free-fermion-like Hamiltonians after a sudden quantum quench, extending the results of S. Ziraldo et al. [Phys. Rev. Lett. 109, 247205 (2012)]. With analytical and numerical arguments, we show that the existence of a stationary state and its description with a generalized Gibbs ensemble (GGE) depend crucially on the observable considered (local versus extensive) and on the localization properties of the final Hamiltonian. We present results on two one-dimensional (1D) models, the disordered 1D fermionic chain with long-range hopping and the disordered Ising/XYXYXY spin chain. We analytically prove that, while time averages of one-body operators are perfectly reproduced by GGE (even for finite-size systems, if time integrals are extended beyond revivals), time averages of many-body operators might show clear deviations from the GGE prediction when disorder-induced localization of the eigenstates is at play.
The sign problem remains a major obstacle to extending the success of Monte Carlo methods to generic quantum many-body systems. A powerful alternative relies on cluster expansions, which provide convergent representations of free energies and related quantities, with applications ranging from tensor network contractions to quantum dynamics and circuits. Within their convergence regime, these expansions often yield deterministic polynomial-time algorithms. However, such methods require exhaustive enumeration of connected clusters, leading to a computational overhead of (Nε−1)O(log⁡Δ)(N\varepsilon^{-1})^{\mathcal{O}(\logΔ)}(Nε−1)O(logΔ), where NNN is the system size, ε\varepsilonε the desired precision, and ΔΔΔ the interaction-graph degree. This becomes particularly severe in long-range systems, where Δ≡NΔ\equiv NΔ≡N, resulting in quasi-polynomial runtimes. Here, we introduce a Markov chain algorithm that samples cluster expansions directly in the space of connected clusters, replacing deterministic enumeration by Monte Carlo sampling. Because sampling occurs in an abstract cluster space rather than a physical configuration space, the conventional exponential sign problem is absent within the convergence regime. For general polymer models, we establish an FPRAS with runtime O~(N4ε−3)\tilde{\mathcal{O}}(N^4\varepsilon^{-3})O~(N4ε−3). For quantum partition functions at temperatures β≤βsampling≡O(1)β\leqβ_{\mathrm{sampling}}\equiv\mathcal{O}(1)β≤βsampling​≡O(1), this improves to O~(N3+1/Dε−3)\tilde{\mathcal{O}}(N^{3+1/D}\varepsilon^{-3})O~(N3+1/Dε−3), where DDD is the lattice dimension. The framework directly applies to long-range quantum systems with interactions ∝1/rα\propto1/r^α∝1/rα, including 222-local models with α>2Dα>2Dα>2D.
By combining classical Monte Carlo and Bethe ansatz techniques we devise a numerical method to construct the Truncated Generalized Gibbs Ensemble (TGGE) for the spin-1/2 isotropic Heisenberg (XXXXXXXXX) chain. The key idea is to sample the Hilbert space of the model with the appropriate GGE probability measure. The method can be extended to other integrable systems, such as the Lieb-Liniger model. We benchmark the approach focusing on GGE expectation values of several local observables. As finite-size effects decay exponentially with system size, moderately large chains are sufficient to extract thermodynamic quantities. The Monte Carlo results are in agreement with both the Thermodynamic Bethe Ansatz (TBA) and the Quantum Transfer Matrix approach (QTM). Remarkably, it is possible to extract in a simple way the steady-state Bethe-Gaudin-Takahashi (BGT) roots distributions, which encode complete information about the GGE expectation values in the thermodynamic limit. Finally, it is straightforward to simulate extensions of the GGE, in which, besides the local integral of motion (local charges), one includes arbitrary functions of the BGT roots. As an example, we include in the GGE the first non-trivial quasi-local integral of motion.
Determining the dynamics of interacting integrable many-particle quantum systems at finite times after homogeneous quantum quenches is a long-standing challenge. We present a Monte Carlo sampling scheme that numerically evaluates the Lehmann representation for time-dependent expectation values of local operators, allowing us to access system sizes and times significantly beyond the reach of existing methods. The approach accommodates both the full Lehmann sum and the Quench Action formalism. We benchmark against exact results for non-interacting lattice and continuum models and short-time results at weak interactions, finding excellent agreement. We apply the method to quantum quenches from a Bose-Einstein condensate in the repulsive Lieb-Liniger model and determine the time evolution of the order parameter for a wide range of interaction strengths. We discuss the emergence of a "sign problem" for more general dynamical correlators and setups.