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arXiv:2302.10940 · Physical review. B./Physical review. B

Dynamical mean-field theory for Rényi entanglement entropy and mutual information in the Hubbard model

Surajit Bera, Arijit Haldar, Sumilan Banerjee

Published Jan 24, 2024• 5 citations
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Abstract

Quantum entanglement, lacking any classical counterpart, provides a fundamental new route to characterize the quantum nature of many-body states. In this work, we discuss an implementation of a new path integral method [Phys. Rev. Res. 2, 033505 (2020)] for fermions to compute entanglement for extended subsystems in the Hubbard model within dynamical mean-field theory (DMFT) in one and two dimensions. The new path integral formulation measures entanglement by applying a ``kick'' to the underlying interacting fermions. We show that the R\'enyi entanglement entropy can be extracted efficiently within the DMFT framework by integrating over the strength of the kick term. Using this method, we compute the second R\'enyi entropy as a function of subsystem size for metallic and Mott insulating phases of the Hubbard model. We explore the thermal entropy to entanglement crossover in the subsystem R\'enyi entropy in the correlated metallic phase. We show that the subsystem-size scaling of the second R\'enyi entropy is well described by the crossover formula which interpolates between the volume-law thermal R\'enyi entropy and the universal boundary-law R\'enyi entanglement entropy with logarithmic violation, as predicted by conformal field theory. We also study the mutual information across the Mott metal-insulator transition.

DOI
10.1103/physrevb.109.035156

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Quantum entanglement, lacking any classical counterpart, provides a fundamental new route to characterize the quantum nature of many-body states. In this work, we discuss an implementation of a new path integral method [Phys. Rev. Res. 2, 033505 (2020)] for fermions to compute entanglement for extended subsystems in the Hubbard model within dynamical mean-field theory (DMFT) in one and two dimensions. The new path integral formulation measures entanglement by applying a ``kick'' to the underlying interacting fermions. We show that the R'enyi entanglement entropy can be extracted efficiently within the DMFT framework by integrating over the strength of the kick term. Using this method, we compute the second R'enyi entropy as a function of subsystem size for metallic and Mott insulating phases of the Hubbard model. We explore the thermal entropy to entanglement crossover in the subsystem R'enyi entropy in the correlated metallic phase. We show that the subsystem-size scaling of the second R'enyi entropy is well described by the crossover formula which interpolates between the volume-law thermal R'enyi entropy and the universal boundary-law R'enyi entanglement entropy with logarithmic violation, as predicted by conformal field theory. We also study the mutual information across the Mott metal-insulator transition.
The subsystem-size dependence of ground-state entanglement entropy and its crossover to thermal entropy as a function of temperature are well understood for one-dimensional (1D) gapless systems described by conformal field theory (CFT), and for free fermions with a Fermi surface in any dimension. However, little is known about the entanglement entropy for gapless fermionic systems without quasi-particles, such as a strange metal. Here we study the entanglement entropy of fermions in a solvable large-NNN 1D lattice model akin to the Yukawa-Sachdev-Ye-Kitaev (Yukawa SYK) model. In this model, two Fermi points are coupled to scalar bosons via spatially random Yukawa interactions, providing a solvable model of a strange metal when the bosons become critical at a quantum critical point. We exactly compute the second Rényi entropy of fermions in a spatial subregion in this model. Our results unravel crucial role of intra-subregion entanglement between fermionic and bosonic degrees of freedom along with the inter-subregion entanglement in understanding the ground states of such strongly coupled fermion-boson systems. We show that the crossover from thermal entropy to entanglement entropy, is captured by a single scaling ansatz, that collapses the second Rényi entropy of fermions for different subregion sizes and temperatures into a single universal curve. We further show that the universal scaling curve for the critical strange metal is well described by the standard CFT formula, albeit with an effective central charge substantially larger than the non-interacting value.
The entanglement entropy is a unique probe to reveal universal features of strongly interacting many-body systems. In two or more dimensions these features are subtle, and detecting them numerically requires extreme precision, a notoriously difficult task. This is especially challenging in models of interacting fermions, where many such universal features have yet to be observed. In this Letter we tackle this challenge by introducing a new method to compute the Rényi entanglement entropy in auxiliary-field quantum Monte Carlo simulations, where we treat the entangling region itself as a stochastic variable. We demonstrate the efficiency of this method by extracting, for the first time, universal subleading logarithmic terms in a two-dimensional model of interacting fermions, focusing on the half-filled honeycomb Hubbard model at T=0. We detect the universal corner contribution due to gapless fermions throughout the Dirac semi-metal phase and at the Gross-Neveu-Yukawa critical point, where the latter shows a pronounced enhancement depending on the type of entangling cut. Finally, we observe the universal Goldstone mode contribution in the antiferromagnetic Mott insulating phase.
Entanglement and information are powerful lenses to probe phases transitions in many-body systems. Motivated by recent cold atom experiments, which are now able to measure the corresponding information-theoretic quantities, we study the Mott transition in the half-filled two-dimensional Hubbard model using cellular dynamical mean-field theory, and focus on two key measures of quantum correlations: entanglement entropy and a measure of total mutual information. We show that they detect the first-order nature of the transition, the universality class of the end point, and the crossover emanating from the end point.
Using dynamical mean field theory (DMFT) in combination with a continuous-time quantum Monte Carlo (DMFT+CTQMC) tool, we studied the von Neumann entanglement entropy (S) in a bilayer Hubbard model on the Bethe lattice. Jump behaviors in S are observed at the critical points among the Fermi liquid, the Mott insulator, and the band insulator at a low temperature, similar to the results in a double occupancy which the DMFT+CTQMC method can easily calculate. As novel results that the double occupancy cannot account for, the behaviors of S in the weak region with metallic states can supply information on the coherence temperature and the Fermi temperature without the need to analyze the transport properties through analytical continuation of the self-energy or differentiation of the energy. In addition, when the strength of the interlayer hopping is tuned in the strong interaction region, S increases in the Mott phase owing to the short-range spatial fluctuations between impurities of bilayers while it decreases in the band phase. These behaviors mean that S might provide a criterion for separation between the Mott insulator and the band insulator as an order parameter in the strong interaction region.
A new numerical approach to entanglement entropies of the Rényi type is proposed for one-dimensional quantum field theories. The method extends the truncated conformal spectrum approach and we will demonstrate that it is especially suited to study the crossover from massless to massive behavior when the subsystem size is comparable to the correlation length. We apply it to different deformations of massless free fermions, corresponding to the scaling limit of the Ising model in transverse and longitudinal fields. For massive free fermions the exactly known crossover function is reproduced already in very small system sizes. The new method treats ground states and excited states on the same footing, and the applicability for excited states is illustrated by reproducing Rényi entropies of low-lying states in the transverse field Ising model.