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arXiv:1804.01956 · Journal of High Energy Physics

Entanglement entropy, dualities, and deconfinement in gauge theories

Mohamed M. Anber, Benjamin J. Kolligs

Published Aug 1, 2018• 12 citations
arXivHTMLPDF

Abstract

Abstract Computing the entanglement entropy in confining gauge theories is often accompanied by puzzles and ambiguities. In this work we show that compactifying the theory on a small circle $$ {\mathbb{S}}_L^1 $$ S L 1 evades these difficulties. In particular, we study Yang-Mills theory on $$ {\mathrm{\mathbb{R}}}^3\times {\mathbb{S}}_L^1 $$ ℝ 3 × S L 1 with double-trace deformations or adjoint fermions and hold it at temperatures near the deconfinement transition. This theory is dual to a multi-component (electric-magnetic) Coulomb gas that can be mapped either to an XY-spin model with ℤ p symmetry-preserving perturbations or dual Sine-Gordon model. The entanglement entropy of the dual Sine-Gordon model exhibits an extremum at the critical temperature/crossover. We also compute Rényi mutual information (RMI) of the XY-spin model by means of the replica trick and Monte Carlo simulations. These are expensive calculations, since one in general needs to suppress lower winding vortices that do not correspond to physical excitations of the system. We use a T-duality that maps the original XY model to its mirror image, making the extraction of RMI a much efficient process. Our simulations indicate that RMI follows the area law scaling, with subleading corrections, and this quantity can be used as a genuine probe to detect deconfinement transitions. We also discuss the effect of fundamental matter on RMI and the implications of our findings in gauge theories and beyond.

DOI
10.1007/jhep08(2018)175

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Abstract Computing the entanglement entropy in confining gauge theories is often accompanied by puzzles and ambiguities. In this work we show that compactifying the theory on a small circle SL1{\mathbb{S}}_L^1SL1​ S L 1 evades these difficulties. In particular, we study Yang-Mills theory on R3×SL1{\mathrm{\mathbb{R}}}^3\times {\mathbb{S}}_L^1R3×SL1​ ℝ 3 × S L 1 with double-trace deformations or adjoint fermions and hold it at temperatures near the deconfinement transition. This theory is dual to a multi-component (electric-magnetic) Coulomb gas that can be mapped either to an XY-spin model with ℤ p symmetry-preserving perturbations or dual Sine-Gordon model. The entanglement entropy of the dual Sine-Gordon model exhibits an extremum at the critical temperature/crossover. We also compute Rényi mutual information (RMI) of the XY-spin model by means of the replica trick and Monte Carlo simulations. These are expensive calculations, since one in general needs to suppress lower winding vortices that do not correspond to physical excitations of the system. We use a T-duality that maps the original XY model to its mirror image, making the extraction of RMI a much efficient process. Our simulations indicate that RMI follows the area law scaling, with subleading corrections, and this quantity can be used as a genuine probe to detect deconfinement transitions. We also discuss the effect of fundamental matter on RMI and the implications of our findings in gauge theories and beyond.
A bstract The study of entanglement in gauge theories is expected to provide insights into many fundamental phenomena, including confinement. However, calculations of quantities related to entanglement in gauge theories are limited by ambiguities that stem from the non-factorizability of the Hilbert space. In this work we study lattice gauge theories that admit a dual description in terms of spin models, for which the replica trick and Rényi entropies are well defined. In the first part of this work, we explicitly perform the duality transformation in a replica geometry, deriving the structure of a replica space for a gauge theory. Then, in the second part, we calculate, by means of Monte Carlo simulations, the entropic c-function of the ℤ 2 gauge theory in three spacetime dimensions, exploiting its dual description in terms of the three-dimensional Ising model.
The study of entanglement in quantum field theories provides insight into universal properties which are typically challenging to extract by means of local observables. However, calculations of quantities related to entanglement in gauge theories are limited by ambiguities that stem from the non-factorizability of the Hilbert space. On the other hand, (2+1)-dimensional lattice gauge theories are known to admit a dual description in terms of spin models, for which the replica trick and Rényi entropies are well defined. In this contribution, we exploit Kramers-Wannier duality to perform a numerical study of the entropic c-function of the (2+1)-dimensional Z2\mathbb{Z}_2Z2​ gauge theory in the continuum limit. Our results are compared with analytical predictions from holographic models.
Abstract Two-dimensional Yang-Mills theory is a useful model of an exactly solvable gauge theory with a string theory dual at large N. We calculate entanglement entropy in the 1/N expansion by mapping the theory to a system of N fermions interacting via a repulsive entropic force. The entropy is a sum of two terms: the “Boltzmann entropy,” log dim(R) per point of the entangling surface, which counts the number of distinct microstates, and the “Shannon entropy,” − Σ p R log p R , which captures fluctuations of the macroscopic state. We find that the entropy scales as N 2 in the large N limit, and that at this order only the Boltzmann entropy contributes. We further show that the Shannon entropy scales linearly with N, and confirm this behaviour with numerical simulations. While the term of order N is surprising from the point of view of the string dual — in which only even powers of N appear in the partition function — we trace it to a breakdown of large N counting caused by the replica trick. This mechanism could lead to corrections to holographic entanglement entropy larger than expected from semiclassical field theory.
We develop a nonequilibrium increment method to compute the Rényi entanglement entropy and investigate its scaling behavior at the deconfined critical (DQC) point via large-scale quantum Monte Carlo simulations. To benchmark the method, we first show that, at a conformally invariant critical point of O(3) transition, the entanglement entropy exhibits universal scaling behavior of area law with logarithmic corner corrections, and the obtained correction exponent represents the current central charge of the critical theory. Then we move on to the deconfined quantum critical point, where we still observe similar scaling behavior, but with a very different exponent. Namely, the corner correction exponent is found to be negative. Such a negative exponent is in sharp contrast with the positivity condition of the Rényi entanglement entropy, which holds for unitary conformal field theories (CFTs). Our results unambiguously reveal fundamental differences between DQC and quantum critical points described by unitary CFTs.
Although entanglement is a well studied property in the context of quantum systems, the ability to measure it in Monte Carlo methods is relatively new. \nThrough measures of the Renyi entanglement entropy and mutual information one is able to examine and characterize criticality, pinpoint phase transitions, and probe universality. \nWe describe the most basic algorithms for calculating these quantities in straightforward Monte Carlo methods and state of the art techniques used in high performance computing. \nThis description emphasizes the core principal of these measurements and allows one to both build an intuition for these quantities and how they are useful in numerical studies. \n \nUsing the Renyi entanglement entropy we demonstrate the ability to detect thermal phase transitions in the Ising model and XY model without use of an order parameter. \nThe scaling near the critical point also shows signatures identifying the universality class of the model. \nImproved methods are explored using extended ensemble techniques that can increase calculation efficiency, and show good agreement with the standard approach. \nWe explore the "ratio trick" at finite temperature and use it to explore the quantum critical fan of the one dimensional transverse field Ising model, showing agreement with finite temperature and finite size scaling from field theory. \nThis same technique is used at zero temperature to explore the geometric dependence of the entanglement entropy and examine the universal scaling functions in the two dimensional transverse field Ising model. \nAll of this shows the multitude of ways in which the study of the Renyi entanglement entropy can be efficiently and practically used in conventional and exotic condensed matter systems, and should serve as a reference for those wishing to use it as a tool.
A bstract We develop a transfer operator approach for the calculation of Rényi entanglement entropies in arbitrary (i.e. Abelian and non-Abelian) pure lattice gauge theory projected entangled pair states in 2+1 dimensions. It is explicitly shown how the long-range behavior of these quantities gives rise to an entanglement area law in both the thermodynamic limit and in the continuum. We numerically demonstrate the applicability of our method to the ℤ 2 lattice gauge theory and relate some entanglement properties to the confinement-deconfinement transition therein. We provide evidence that Rényi entanglement entropies in certain cases do not provide a complete probe of (de)confinement properties compared to Wilson loop expectation values as other genuine (nonlocal) observables.