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arXiv:1407.0707 · Physical Review Letters

Renyi Entanglement Entropy of Interacting Fermions Calculated Using the Continuous-Time Quantum Monte Carlo Method

Lei Wang, Matthias Troyer

Published Sep 10, 2014• 51 citations
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Abstract

We present a new algorithm for calculating the Renyi entanglement entropy of interacting fermions using the continuous-time quantum Monte Carlo method. The algorithm only samples the interaction correction of the entanglement entropy, which by design ensures the efficient calculation of weakly interacting systems. Combined with Monte Carlo reweighting, the algorithm also performs well for systems with strong interactions. We demonstrate the potential of this method by studying the quantum entanglement signatures of the charge-density-wave transition of interacting fermions on a square lattice.

DOI
10.1103/physrevlett.113.110401

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We present a new algorithm for calculating the Renyi entanglement entropy of interacting fermions using the continuous-time quantum Monte Carlo method. The algorithm only samples the interaction correction of the entanglement entropy, which by design ensures the efficient calculation of weakly interacting systems. Combined with Monte Carlo reweighting, the algorithm also performs well for systems with strong interactions. We demonstrate the potential of this method by studying the quantum entanglement signatures of the charge-density-wave transition of interacting fermions on a square lattice.
The entanglement entropy probing novel phases and phase transitions numerically via quantum Monte Carlo has made great achievements in large-scale interacting spin/boson systems. In contrast, the numerical exploration in interacting fermion systems is rare, even though fermion systems attract more attentions in condensed matter. The fundamental restrictions is that the computational cost of fermion quantum Monte Carlo (∼βN3\sim βN^3∼βN3) is much higher than that of spin/boson (∼βN\sim βN∼βN). Here, NNN is the total number of sites and βββ is the inverse temperature or projection length. To tackle this problem, we propose a fermionic quantum Monte Carlo algorithm based on the incremental technique along physical parameters, which greatly improves the efficiency of extracting entanglement entropy. We benchmark the developed algorithm by calculating the scaling behavior of the entanglement entropy in a two-dimensional square lattice Hubbard model. The obtained phase diagram including Fermi surface and Goldstone modes validates the correctness of the algorithm. Remarkably, our method shows the high-efficiency with respect to the existing algorithms, while keeping the high computation precision. We proceed to apply this algorithm to explore the scaling behavior of the entanglement entropy and particularly its derivative at Gross-Neveu criticality. Our results elucidate that such critical behavior can be quantified by the correlation length exponent.
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We show that the two recently proposed methods to compute Renyi entanglement entropies in the realm of determinant quantum Monte Carlo methods for fermions are in principle equivalent, but differ in sampling strategies. The analogy allows us to formulate a numerically stable calculation of the entanglement spectrum at strong coupling. We demonstrate the approach by studying static and dynamical properties of the entanglement Hamiltonian across the interaction driven quantum phase transition between a topological insulator and quantum antiferromagnet in the Kane-Mele Hubbard model. The formulation is not limited to fermion systems and can readily be adapted to world-line-based simulations of bosonic systems.
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.
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