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arXiv:1506.06654 · Physical Review B

Hybrid Monte Carlo approach to the entanglement entropy of interacting fermions

Joaquín E. Drut, William J. Porter

Published Sep 14, 2015• 31 citations
arXivHTMLPDF

Abstract

The Monte Carlo calculation of R\'enyi entanglement entropies ${S}_{n}^{}$ of interacting fermions suffers from a well-known signal-to-noise problem, even for a large number of situations in which the infamous sign problem is absent. A few methods have been proposed to overcome this issue, such as ensemble switching and the use of auxiliary partition-function ratios. Here, we present an approach that builds on the recently proposed free-fermion decomposition method; it incorporates entanglement in the probability measure in a natural way; it takes advantage of the hybrid Monte Carlo algorithm (an essential tool in lattice quantum chromodynamics and other gauge theories with dynamical fermions); and it does not suffer from noise problems. This method displays no sign problem for the same cases as other approaches and is therefore useful for a wide variety of systems. As a proof of principle, we calculate ${S}_{2}^{}$ for the one-dimensional, half-filled Hubbard model and compare with results from exact diagonalization and the free-fermion decomposition method.

DOI
10.1103/physrevb.92.125126

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The Monte Carlo calculation of R'enyi entanglement entropies Sn{S}_{n}^{}Sn​ of interacting fermions suffers from a well-known signal-to-noise problem, even for a large number of situations in which the infamous sign problem is absent. A few methods have been proposed to overcome this issue, such as ensemble switching and the use of auxiliary partition-function ratios. Here, we present an approach that builds on the recently proposed free-fermion decomposition method; it incorporates entanglement in the probability measure in a natural way; it takes advantage of the hybrid Monte Carlo algorithm (an essential tool in lattice quantum chromodynamics and other gauge theories with dynamical fermions); and it does not suffer from noise problems. This method displays no sign problem for the same cases as other approaches and is therefore useful for a wide variety of systems. As a proof of principle, we calculate S2{S}_{2}^{}S2​ for the one-dimensional, half-filled Hubbard model and compare with results from exact diagonalization and the free-fermion decomposition method.
We put forward a simpler and improved variation of a recently proposed method to overcome the signal-to-noise problem found in Monte Carlo calculations of the entanglement entropy of interacting fermions. The present method takes advantage of the approximate log-normal distributions that characterize the signal-to-noise properties of other approaches. In addition, we show that a simple rewriting of the formalism allows circumvention of the inversion of the restricted one-body density matrix in the calculation of the nth Rényi entanglement entropy for n>2. We test our technique by implementing it in combination with the hybrid Monte Carlo algorithm and calculating the n=2,3,4,⋯,10 Rényi entropies of the one-dimensional attractive Hubbard model. We use that data to extrapolate to the von Neumann (n=1) and n→∞ cases.
A, by now, well-known signal-to-noise problem plagues Monte Carlo calculations of quantum-information-theoretic observables in systems of interacting fermions, particularly the Renyi entanglement entropies Sn, even in many cases where the infamous sign problem does not appear. Several methods have been put forward to circumvent this affliction including ensemble-switching techniques using auxiliary partition-function ratios. This dissertation presents an algorithm that modifies the recently proposed free-fermion decomposition in an essential way: we incorporate the entanglement-sensitive correlations directly into the probability measure in a natural way. Implementing this algorithm, we demonstrate that it is compatible with the hybrid Monte Carlo algorithm, the workhorse of the lattice quantum chromodynamics community and an essential tool for studying gauge theories that contain dynamical fermions. By studying a simple one-dimensional Hubbard model, we demonstrate that our method does not exhibit the same debilitating numerical difficulties that naive attempts to study entanglement often encounter. Following that, we illustrate some key probabilistic insights, using intuition derived from the previous method and its successes to construct a simpler, better behaved, and more elegant algorithm. Using this method, in combination with new identities which allow us to avoid seemingly necessary numerical difficulties, the inversion of the restricted one-body density matrices, we compute high order Renyi entropies and perform a thorough comparison to this new algorithm's predecessor using the Hubbard model mentioned before. Finally, we characterize non-perturbatively the Renyi entropies of degree n=2,3,4, and 5 of three-dimensional, strongly coupled many-fermion systems in the scale-invariant regime of short interaction range and large scattering length, i.e. in the unitary limit using the algorithms detailed herein. We also detail an exact, few-body projective method which we use to characterize the entanglement properties of the two-body sector across a broad range of attractive couplings. In the many-body case, we determine universal scaling properties of this system, and for the two-body case, we compute the entanglement spectrum exactly, successfully characterizing a vast range of entanglement behavior across the BCS-BEC crossover.
Quantum Monte Carlo simulations of fermions are hampered by the notorious sign problem whose most striking manifestation is an exponential growth of sampling errors with the number of particles. With the sign problem known to be an NP-hard problem and any generic solution thus highly elusive, the Monte Carlo sampling of interacting many-fermion systems is commonly thought to be restricted to a small class of model systems for which a sign-free basis has been identified. Here we demonstrate that entanglement measures, in particular the so-called R\'enyi entropies, can intrinsically exhibit a certain robustness against the sign problem in auxiliary-field quantum Monte Carlo approaches and possibly allow for the identification of global ground-state properties via their scaling behavior even in the presence of a strong sign problem. We corroborate these findings via numerical simulations of fermionic quantum phase transitions of spinless fermions on the honeycomb lattice at and below half filling.
Entanglement measures such as the entanglement entropy have become an indispensable tool to identify the fundamental character of ground states of interacting quantum many-body systems. For systems of interacting spin or bosonic degrees of freedom much recent progress has been made not only in the analytical description of their respective entanglement entropies but also in their numerical classification. Systems of interacting fermionic degrees of freedom, however, have proved to be more difficult to control, in particular with regard to the numerical understanding of their entanglement properties. Here we report a generalization of the replica technique for the calculation of Rényi entropies to the framework of determinantal Quantum Monte Carlo simulations—the numerical method of choice for unbiased, large-scale simulations of interacting fermionic systems. We demonstrate the strength of this approach over a recent alternative proposal based on a decomposition in free fermion Green's functions by studying the entanglement entropy of one-dimensional Hubbard systems both at zero and finite temperatures.
In the absence of a fermion sign problem, auxiliary-field (or determinantal) quantum Monte Carlo (DQMC) approaches have long been the numerical method of choice for unbiased, large-scale simulations of interacting many-fermion systems. More recently, the conceptual scope of this approach has been expanded by introducing ingenious schemes to compute entanglement entropies within its framework. On a practical level, these approaches, however, suffer from a variety of numerical instabilities that have largely impeded their applicability. Here we report on a number of algorithmic advances to overcome many of these numerical instabilities and significantly improve the calculation of entanglement measures in the zero-temperature projective DQMC approach, ultimately allowing us to reach similar system sizes as for the computation of conventional observables. We demonstrate the applicability of this improved DQMC approach by providing an entanglement perspective on the quantum phase transition from a magnetically ordered Mott insulator to a band insulator in the bilayer square lattice Hubbard model at half filling.
The determination of entanglement measures in SU(N) gauge theories is a non-trivial task. With the so-called "replica trick", a family of entanglement measures, known as "Rényi entropies", can be determined with lattice Monte Carlo. Unfortunately, the standard implementation of the replica method for SU(N) lattice gauge theories suffers from a severe signal-to-noise ratio problem, rendering high-precision studies of Rényi entropies prohibitively expensive. In this work, we propose a method to overcome the signal-to-noise ratio problem and show some first results for SU(N) in 4 dimensions.