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arXiv:2012.10264 · Physical review. E

Generative neural samplers for the quantum Heisenberg chain

Johanna Vielhaben, Nils Strodthoff

Published Jun 7, 2021• 6 citations
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Abstract

Generative neural samplers offer a complementary approach to Monte Carlo methods for problems in statistical physics and quantum field theory. This paper tests the ability of generative neural samplers to estimate observables for real-world low-dimensional spin systems. It maps out how autoregressive models can sample configurations of a quantum Heisenberg chain via a classical approximation based on the Suzuki-Trotter transformation. We present results for energy, specific heat, and susceptibility for the isotropic XXX and the anisotropic XY chain are in good agreement with Monte Carlo results within the same approximation scheme.

DOI
10.1103/physreve.103.063304

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Generative neural samplers offer a complementary approach to Monte Carlo methods for problems in statistical physics and quantum field theory. This paper tests the ability of generative neural samplers to estimate observables for real-world low-dimensional spin systems. It maps out how autoregressive models can sample configurations of a quantum Heisenberg chain via a classical approximation based on the Suzuki-Trotter transformation. We present results for energy, specific heat, and susceptibility for the isotropic XXX and the anisotropic XY chain are in good agreement with Monte Carlo results within the same approximation scheme.
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We design generative neural networks that generate Monte Carlo configurations with complete absence of autocorrelation and from which direct measurements of physical observables can be employed, irrespective of the system locating at the classical critical point, fermionic Mott insulator, Dirac semimetal and quantum critical point. We further propose a generic parallel-chain Monte Carlo scheme based on such neural networks, which provides independent samplings and accelerates the Monte Carlo simulations by reducing the thermalization process. We demonstrate the performance of our approach on the two-dimensional Ising and fermion Hubbard models.
Entanglement entropy is an essential metric for characterizing quantum many-body systems, but its numerical evaluation for neural network representations of quantum states has so far been inefficient and demonstrated only for the restricted Boltzmann machine architecture. Here we estimate generalized R\'enyi entropies of autoregressive neural quantum states with up to N=256N=256N=256 spins using quantum Monte Carlo methods. A naive ``direct sampling'' approach performs well for low-order R\'enyi entropies but fails for larger orders when benchmarked on a one-dimensional Heisenberg model. We therefore propose an improved ``conditional sampling'' method exploiting the autoregressive structure of the network ansatz, which outperforms direct sampling and facilitates calculations of higher-order R\'enyi entropies in both one- and two-dimensional Heisenberg models. Access to higher-order R\'enyi entropies allows for an approximation of the von Neumann entropy as well as extraction of the single-copy entanglement. Both methods elucidate the potential of neural network quantum states in quantum Monte Carlo studies of entanglement entropy for many-body systems.
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