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arXiv:1512.01334 · Progress of Theoretical and Experimental Physics

Entanglement in four-dimensional SU(3) gauge theory

Etsuko Itou, Keitaro Nagata, Yoshiyuki Nakagawa, Atsushi Nakamura, V. I. Zakharov

Published Jun 1, 2016• 38 citations
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Abstract

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 |$c$|-function (⁠|$C(l)$|⁠) of SU(3) gauge theory in lattice Monte Carlo simulation using the replica method. For |$0 \leqslant l \leqslant 0.7$| fm, where |$l$| is the length of the subspace, the entropic |$c$|-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 |$l$| is close to the |$\Lambda _{{\rm QCD}}^{-1} \big (\sim T_c^{-1}\big )$| scale, the entropic |$c$|-function decreases smoothly, and it is consistent with zero within error bars at |$l \gtrsim 0.9$| fm.

DOI
10.1093/ptep/ptw050

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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.
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.
We examine the entanglement properties of the Yang-Mills theory by calculating α entanglement entropy with α = 2 using a SU(3) quenched lattice gauge simulation both in the confinement and the deconfinement phases.In the confinement phase, the derivative of the α entropy with respect to the size l of the subregion, whose entanglement properties are interested in, scales as 1/l 3 , and a clear discontinuity cannot be found within our statistical errors.The α entropy in the deconfinement phase saturates at large l.The saturation value is comparable with the thermal entropy of the pure Yang-Mills theory, indicating that the α entropy obeys the volume law at large l in the deconfinement phase.
Entanglement entropy encodes important features of strongly interacting quantum many-body systems and gauge theories, but its analytical study is still limited to systems with high level of symmetry. This motivates the search for efficient techniques to investigate this quantity numerically, through Monte Carlo calculations on the lattice. In this contribution, we discuss the computation of the entropic c-function by means of an algorithm based on Jarzynski's equality, which is an exact theorem from non-equilibrium statistical mechanics. After presenting benchmark results for the Ising model in two dimensions, where our algorithm successfully reproduces the analytical predictions from conformal field theory, we discuss its generalization to the three-dimensional Ising model, for which we were able to extract universal terms beyond the area law. Finally we point out some future generalizations of this calculation.
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.