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arXiv:2404.01987 · Journal of High Energy Physics

Duality transformations and the entanglement entropy of gauge theories

Andrea Bulgarelli, Marco Panero C

Published Jun 7, 2024• 12 citations
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Abstract

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.

DOI
10.1007/jhep06(2024)041

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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 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^1 SL1​ S L 1 evades these difficulties. In particular, we study Yang-Mills theory on R3×SL1 {\mathrm{\mathbb{R}}}^3\times {\mathbb{S}}_L^1 R3×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.
We investigate the quantum entanglement entropy for the four-dimensional Euclidean SU(3) gauge theory. We present the first non-perturbative calculation of the entropic |ccc|-function (⁠|C(l)C(l)C(l)|⁠) of SU(3) gauge theory in lattice Monte Carlo simulation using the replica method. For |0⩽l⩽0.70 \leqslant l \leqslant 0.70⩽l⩽0.7| fm, where |lll| is the length of the subspace, the entropic |ccc|-function is almost constant, indicating conformally invariant dynamics. The value of the constant agrees with that perturbatively obtained from free gluons, with 20% discrepancy. When |lll| is close to the |ΛQCD−1(∼Tc−1)\Lambda _{{\rm QCD}}^{-1} \big (\sim T_c^{-1}\big )ΛQCD−1​(∼Tc−1​)| scale, the entropic |ccc|-function decreases smoothly, and it is consistent with zero within error bars at |l≳0.9l \gtrsim 0.9l≳0.9| fm.
We investigate the quantum entanglement entropy for the four-dimensional Euclidean SU(3) gauge theory. We present the first non-perturbative calculation of the entropic |ccc|-function (⁠|C(l)C(l)C(l)|⁠) of SU(3) gauge theory in lattice Monte Carlo simulation using the replica method. For |0⩽l⩽0.70 \leqslant l \leqslant 0.70⩽l⩽0.7| fm, where |lll| is the length of the subspace, the entropic |ccc|-function is almost constant, indicating conformally invariant dynamics. The value of the constant agrees with that perturbatively obtained from free gluons, with 20% discrepancy. When |lll| is close to the |ΛQCD−1(∼Tc−1)\Lambda _{{\rm QCD}}^{-1} \big (\sim T_c^{-1}\big )ΛQCD−1​(∼Tc−1​)| scale, the entropic |ccc|-function decreases smoothly, and it is consistent with zero within error bars at |l≳0.9l \gtrsim 0.9l≳0.9| fm.
The determination of entanglement measures in SU(N) gauge theories is a non-trivial task. With the so-called "replica trick", a family of entanglement measures, known as "Rényi entropies", can be determined with lattice Monte Carlo. Unfortunately, the standard implementation of the replica method for SU(N) lattice gauge theories suffers from a severe signal-to-noise ratio problem, rendering high-precision studies of Rényi entropies prohibitively expensive. In this work, we propose a method to overcome the signal-to-noise ratio problem and show some first results for SU(N) in 4 dimensions.
The determination of entanglement measures in SU(N) gauge theories is a non-trivial task. With the so-called "replica trick", a family of entanglement measures, known as "Rényi entropies", can be determined with lattice Monte Carlo. Unfortunately, the standard implementation of the replica method for SU(N) lattice gauge theories suffers from a severe signal-to-noise ratio problem, rendering high-precision studies of Rényi entropies prohibitively expensive. In this work, we propose a method to overcome the signal-to-noise ratio problem and show some first results for SU(N) in 4 dimensions.