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arXiv:1502.01722 · Physical Review B

Detecting Goldstone modes with entanglement entropy

Bohdan Kulchytskyy, C. M. Herdman, Stephen Inglis, Roger G. Melko

Published Sep 25, 2015• 40 citations
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Abstract

In the face of mounting numerical evidence, Metlitski and Grover (arXiv:1112.5166) have given compelling analytical arguments that systems with spontaneous broken continuous symmetry contain a subleading contribution to the entanglement entropy that diverges logarithmically with system size. They predict that the coefficient of this log is a universal quantity that depends on the number of Goldstone modes. In this paper, we confirm the presence of this log term through quantum Monte Carlo calculations of the second R\'enyi entropy on the spin-1/2 $XY$ model. Devising an algorithm to facilitate convergence of entropy data at extremely low temperatures, we demonstrate that the single Goldstone mode in the ground state can be identified through the coefficient of the log term. Furthermore, our simulation accuracy allows us to obtain an additional geometric constant additive to the R\'enyi entropy, that matches a predicted fully universal form obtained from a free bosonic field theory with no adjustable parameters.

DOI
10.1103/physrevb.92.115146

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In the face of mounting numerical evidence, Metlitski and Grover (arXiv:1112.5166) have given compelling analytical arguments that systems with spontaneous broken continuous symmetry contain a subleading contribution to the entanglement entropy that diverges logarithmically with system size. They predict that the coefficient of this log is a universal quantity that depends on the number of Goldstone modes. In this paper, we confirm the presence of this log term through quantum Monte Carlo calculations of the second R'enyi entropy on the spin-1/2 XYXYXY model. Devising an algorithm to facilitate convergence of entropy data at extremely low temperatures, we demonstrate that the single Goldstone mode in the ground state can be identified through the coefficient of the log term. Furthermore, our simulation accuracy allows us to obtain an additional geometric constant additive to the R'enyi entropy, that matches a predicted fully universal form obtained from a free bosonic field theory with no adjustable parameters.
We study entanglement properties of systems with spontaneously broken continuous symmetry. We find that in addition to the expected area law behavior, the entanglement entropy contains a subleading contribution which diverges logarithmically with the subsystem size in agreement with the Monte Carlo simulations of A. Kallin et. al. (Phys. Rev. B 84, 165134 (2011)). The coefficient of the logarithm is a universal number given simply by NG(d−1)/2N_G (d-1)/2NG​(d−1)/2, where NGN_GNG​ is the number of Goldstone modes and ddd is the spatial dimension. This term is present even when the subsystem boundary is straight and contains no corners, and its origin lies in the interplay of Goldstone modes and restoration of symmetry in a finite volume. We also compute the "low-energy" part of the entanglement spectrum and show that it has the same characteristic "tower of states" form as the physical low-energy spectrum obtained when a system with spontaneously broken continuous symmetry is placed in a finite volume.
We explore the R\'enyi entanglement entropies of a one-dimensional (line) subsystem of length LLL embedded in a two-dimensional $L\ifmmode\times\else\texttimes\fi{}L$ square lattice for quantum spin models whose ground state breaks a continuous symmetry in the thermodynamic limit. We first study the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model (with antiferromagnetic nearest-neighbor J1>0{J}_{1}>0J1​>0 and ferromagnetic second-neighbor couplings ${J}_{2}\ensuremath{\le}0)$, breaking SU(2) symmetry in the ground state, using quantum Monte Carlo simulations. The signature of SU(2)-symmetry breaking on finite-size systems, ranging from L=4L=4L=4 up to L=40L=40L=40, clearly appears as a universal additive logarithmic correction to the R\'enyi entanglement entropies: lqlnL{l}_{q}lnLlq​lnL with ${l}_{q}\ensuremath{\simeq}1$, independent of the R\'enyi index and values of J2{J}_{2}J2​. We then confirm this result using a spin-wave analysis (with restored spin rotational symmetry) on the same ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ model, as well as on the XY model, with a U(1)U(1)U(1) broken symmetry in the ground state. The high-precision spin-wave calculations are performed on finite lattices up to ${10}^{5}\ifmmode\times\else\texttimes\fi{}{10}^{5}$ sites, allowing us to explore further nonuniversal finite-size corrections. Our results fully agree with the prediction lq=nG/2{l}_{q}={n}_{G}/2lq​=nG​/2, where nG{n}_{G}nG​ is the number of Goldstone modes, by Metlitski and Grover (arXiv:1112.5166).
We study the scaling of the (basis dependent) Shannon entropy for two-dimensional quantum antiferromagnets with N\'eel long-range order. We use a massless free-field description of the gapless spin wave modes and phase space arguments to treat the fact that the finite-size ground state is rotationally symmetric, while there are degenerate physical ground states which break the symmetry. Our results show that the Shannon entropy (and its R\'enyi generalizations) possesses some universal logarithmic term proportional to the number NNGN_\text{NG}NNG​ of Nambu-Goldstone modes. In the case of a torus, we show that Sn>1≃O(N)+NNG4nn−1log⁡NS_{n>1} \simeq \mathcal{O}(N)+ \frac{N_\text{NG}}{4}\frac{n}{n-1} \log{N}Sn>1​≃O(N)+4NNG​​n−1n​logN and S1≃O(N)−NNG4log⁡NS_1 \simeq \mathcal{O}(N) - \frac{N_\text{NG}}{4} \log{N}S1​≃O(N)−4NNG​​logN, where NNN is the total number of sites and nnn the R\'enyi index. The result for n>1n>1n>1 is in reasonable agreement with the quantum Monte Carlo results of Luitz et al. [Phys. Rev. Lett. 112, 057203 (2014)], and qualitatively similar to those obtained previously for the entanglement entropy. The Shannon entropy of a line subsystem (embedded in the two-dimensional system) is also considered. Finally, we present some density-matrix renormalization group (DMRG) calculations for a spin12\frac{1}{2}21​ XY model on the square lattice in a cylinder geometry. These numerical data confirm our findings for logarithmic terms in the n=∞n=\inftyn=∞ R\'enyi entropy (also called −log⁡pmax-\log{p_{\rm max}}−logpmax​). They also reveal some universal dependence on the cylinder aspect ratio, in good agreement with the fact that, in that case, pmaxp_{\rm max}pmax​ is related to a non-compact free-boson partition function in dimension 1+1.
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.
Our understanding of physical phenomena hinges on finding universal core mechanisms that unite them. The concept of universality is deeply ingrained in the study of quantum many-body systems. At zero temperature, microscopically different systems with long-range order collapse into their universal state described by a handful of universal parameters. Establishing those parameters implies identification of the universal theory that is effectively describing the system and ultimately providing the desired understanding. We take up this task with the help of Renyi entanglement entropy. Defined with respect to a system bipartition, this measure quantifies information shared between the subsystems. Understanding what insights into universality are encoded within this information-theoretic quantity as well as developing numerical tools to efficiently estimate the Renyi entanglement entropy are the subjects of this thesis. \n \nOn the computational end of this far-reaching goal, we develop a novel theoretical framework for constructing improved Renyi entanglement entropy estimators in the context of d+1 quantum Monte Carlo methods. The discovery of a connection of this methodology to the well-established Kandel-Domany formalism provides a clear path towards generalization. Additionally, we embrace a data-driven approach towards learning the ground state wavefunction. We demonstrate how a restricted Boltzmann machine can be used to reconstruct the Renyi entanglement entropy from projective measurements of a quantum ground state. Furthermore, we extend this classical generative architecture to a quantum analogue that we call the quantum Boltzmann machine. \n \nOn the theoretical side of this endeavour, we study the Renyi entanglement entropy scaling terms for two quantum lattice models embedded in two dimensional space via extensive quantum Monte Carlo simulations. For the ground state of the XY model, we provide conclusive numerical evidence for a logarithmic contribution that uniquely characterizes the continuous symmetry of the emerging order parameter. Moreover, we confirm the form of the subleading universal geometric contribution arising due to the bosonic nature of low-energy degrees of freedom in this model. For the critical ground state of the transverse field Ising model, we develop a novel scaling procedure to extract a universal number κ revealed via a cylindrical entangling bipartition in the thin-slice limit. The combined product of our work sheds new light on the entanglement-based classification of universality and brings a suite of new powerful numerical tools to continue illuminating this theoretical program in the future.
In this paper, we derive corrections to the subleading logarithmic term of the entanglement entropy in systems with spontaneous broken continuous symmetry. Using quantum Monte Carlo simulations, we show that the improved scaling formula leads to much better estimations of the number of Goldstone modes in the two-dimensional square lattice spin-1/2 Heisenberg model and bilayer spin-1/2 Heisenberg model in systems of rather small sizes, compared with previous results. In addition, the universal geometry-dependent finite constant in the entanglement entropy scaling is also obtained in good agreement with the theoretical value.