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arXiv:1207.4188 · Physical Review B

Renyi entropy of the interacting Fermi liquid

Jeremy McMinis, Norm M. Tubman

Published Feb 28, 2013• 67 citations
arXivHTMLPDF

Abstract

Entanglement properties, including the Renyi $\ensuremath{\alpha}$ entropies and scaling laws, are becoming increasingly important in condensed-matter physics because they can be used to determine physical properties and the ``fingerprint'' of quantum phases. In this work we use variational quantum Monte Carlo to compute the Renyi $\ensuremath{\alpha}$ entropies, their scaling laws, and the relationship between different $\ensuremath{\alpha}$ entropies for one of the most important phases in condensed matter, the interacting Fermi liquid. We also investigate the relationship between the scaling laws and the discontinuity in the momentum distribution at the Fermi surface. Contrary to recent theoretical predictions, we find that interactions increase the prefactor for the $\ensuremath{\alpha}$-entropy scaling laws for all particle interaction strengths and forms. We also show that a theory of these scaling laws for the interacting systems may be developed by extending the free theory to incorporate properties of the momentum distribution.

DOI
10.1103/physrevb.87.081108

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Entanglement properties, including the Renyi \ensuremathα\ensuremath{\alpha}\ensuremathα entropies and scaling laws, are becoming increasingly important in condensed-matter physics because they can be used to determine physical properties and the ``fingerprint'' of quantum phases. In this work we use variational quantum Monte Carlo to compute the Renyi \ensuremathα\ensuremath{\alpha}\ensuremathα entropies, their scaling laws, and the relationship between different \ensuremathα\ensuremath{\alpha}\ensuremathα entropies for one of the most important phases in condensed matter, the interacting Fermi liquid. We also investigate the relationship between the scaling laws and the discontinuity in the momentum distribution at the Fermi surface. Contrary to recent theoretical predictions, we find that interactions increase the prefactor for the \ensuremathα\ensuremath{\alpha}\ensuremathα-entropy scaling laws for all particle interaction strengths and forms. We also show that a theory of these scaling laws for the interacting systems may be developed by extending the free theory to incorporate properties of the momentum distribution.
In this work we compute subleading oscillating terms in the Renyi entropy of Fermi gases and liquids corresponding to 2kF2{k}_{F}2kF​-like oscillations. Our theoretical tools are the one-dimensional formulation of Fermi liquid entanglement familiar from discussions of the logarithmic violation of the area law and quantum Monte Carlo calculations. The main result is a formula for the oscillating term for any region geometry and a spherical Fermi surface in any dimension. Specializing to two dimensions, we compare this term to numerical calculations of Renyi entropies using the correlation function method and find excellent agreement. We also compare with quantum Monte Carlo data on interacting Fermi liquids where we also find agreement up to moderate interaction strengths.
Momentum space entanglement entropy probes quantum correlations in interacting fermionic phases. It is very sensitive to interactions, obeying volume-law scaling in general, while vanishing in the Fermi gas. We show that the R\'enyi entropy in momentum space has a systematic expansion in terms of the phase space volume of the partition, which holds at all orders in perturbation theory. This permits, for example, the controlled computation of the entropy of thin shells near the Fermi wave vector in isotropic Fermi liquids and BCS superconductors. In the Fermi liquid, the thin-shell entropy is a universal function of the quasiparticle residue. In the superconductor, it reflects the formation of Cooper pairs. Momentum space R\'enyi entropies are accessible in cold atomic and molecular gas experiments through a time-of-flight generalization of previously implemented measurement protocols.
Quantum phases characterized by surfaces of gapless excitations are known to violate the otherwise ubiquitous boundary law of entanglement entropy in the form of a multiplicative log correction: $S\ensuremath{\sim}{L}^{d\ensuremath{-}1}logL$. Using variational Monte Carlo, we calculate the second R\'enyi entropy for a model wave function of the $\ensuremath{\nu}=1/2$ composite Fermi liquid (CFL) state defined on the two-dimensional triangular lattice. By carefully studying the scaling of the total R\'enyi entropy and, crucially, its contributions from the modulus and sign of the wave function on various finite-size geometries, we argue that the prefactor of the leading LlogLL\phantom{\rule{0.16em}{0ex}}logLLlogL term is equivalent to that in the analogous free fermion wave function. In contrast to the recent results of Shao et al. [Phys. Rev. Lett. 114, 206402 (2015)], we thus conclude that the ``Widom formula'' holds even in this non-Fermi liquid CFL state. More generally, our results further elucidate---and place on a more quantitative footing---the relationship between nontrivial wave function sign structure and $S\ensuremath{\sim}L\phantom{\rule{0.16em}{0ex}}logL$ entanglement scaling in such highly entangled gapless phases.
Quantum spin liquids are phases of matter whose internal structure is not captured by a local order parameter. Particularly intriguing are critical spin liquids, where strongly interacting excitations control low energy properties. Here we calculate their bipartite entanglement entropy that characterizes their quantum structure. In particular we calculate the Renyi entropy S(2) on model wave functions obtained by Gutzwiller projection of a Fermi sea. Although the wave functions are not sign positive, S(2) can be calculated on relatively large systems (>324 spins) using the variational Monte Carlo technique. On the triangular lattice we find that entanglement entropy of the projected Fermi sea state violates the boundary law, with S(2) enhanced by a logarithmic factor. This is an unusual result for a bosonic wave function reflecting the presence of emergent fermions. These techniques can be extended to study a wide class of other phases.
We compute the leading contribution to the ground-state R\'enyi entropy ${S}_{\ensuremath{\alpha}}$ for a region of linear size LLL in a Fermi liquid in ddd dimensions. The result contains a universal boundary law violating term simply related to the more familiar entanglement entropy. We also obtain a universal crossover function that smoothly interpolates between the zero-temperature result and the ordinary thermal R\'enyi entropy of a Fermi liquid. Formulas for the entanglement entropy of more complicated regions, including nonconvex and disconnected regions, are obtained from the conformal field theory formulation of Fermi surface dynamics. These results permit an evaluation of the quantum mutual information between different regions in a Fermi liquid. We also study the number fluctuations in a Fermi liquid. Taken together, these results give a reasonably complete characterization of the low-energy quantum information content of Fermi liquids.
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.