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International Symposium on Lattice Field Theory (LATTICE)

Entanglement entropy of SU(3) and SU(2) Yang-Mills theories at finite temperature

Atsushi Nakamura, Yoshiyuki Nakagawa, V.I. Zakharov, Shinji Motoki

Published Jun 23, 2010• 16 citations
Publisher

Abstract

We calculate the entanglement entropy using a SU(3) quenched lattice gauge simulation.We find that the entanglement entropy scales as 1/l 2 at small l as in the conformal field theory.Here l is the size of the system, whose degrees of freedom is left after the other part are traced out.The derivative of the entanglement entropy with respect to l hits zero at about l * = 0.6 ∼ 0.7 [fm] and vanishes above the length.It may imply that the Yang-Mills theory has the mass gap of the order of 1/l * .Within our statistical errors, no discontinuous change can be seen in the entanglement entropy.We discuss also a subtle point appearing in gauge systems when we divide a system with cuts.

DOI
10.22323/1.091.0188

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We calculate the entanglement entropy using a SU(3) quenched lattice gauge simulation.We find that the entanglement entropy scales as 1/l 2 at small l as in the conformal field theory.Here l is the size of the system, whose degrees of freedom is left after the other part are traced out.The derivative of the entanglement entropy with respect to l hits zero at about l * = 0.6 ∼ 0.7 [fm] and vanishes above the length.It may imply that the Yang-Mills theory has the mass gap of the order of 1/l * .Within our statistical errors, no discontinuous change can be seen in the entanglement entropy.We discuss also a subtle point appearing in gauge systems when we divide a system with cuts.
We calculate the entanglement entropy using a SU(3) quenched lattice gauge simulation. We find that the entanglement entropy scales as 1/l21/l^21/l2 at small lll as in the conformal field theory. Here lll is the size of the system, whose degrees of freedom is left after the other part are traced out. The derivative of the entanglement entropy with respect to lll hits zero at about l∗=0.6∼0.7l^{\ast} = 0.6 \sim 0.7l∗=0.6∼0.7 [fm] and vanishes above the length. It may imply that the Yang-Mills theory has the mass gap of the order of 1/l∗1/l^{\ast}1/l∗. Within our statistical errors, no discontinuous change can be seen in the entanglement entropy. We discuss also a subtle point appearing in gauge systems when we divide a system with cuts.
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 in SU(2)SU(2)SU(2) lattice gauge theory
The entropy of entanglement between a three-dimensional slab of thickness l and its complement is studied numerically for four-dimensional SU (2) lattice gauge theory. We find a signature of a nonanalytic behavior of the entanglement entropy, which was predicted recently for large Nc confining gauge theories in the framework of AdS/CFT correspondence. The derivative of the entanglement entropy over l is likely to have a discontinuity at some l = lc ≈ 0.5 fm. The application of the concepts of quantum information theory to quantum field theories in continuous space-times and on the lattices has led recently to many important advances in our understanding of their quantum behavior [1–3]. One such concept is the quantum entanglement of states of systems with many degrees of freedom, which turned out to be a very useful modelindependent characteristic of the structure of the ground state of quantum fields. In particular, for quantum fields on the lattice in the vicinity of a quantum phase transition (i.e. a phase transition which occurs at zero temperature when some parameters of the system, such as the coupling constants, are varied) the ground state is a strongly entangled superposition of the states of all elementary lattice degrees of freedom (such as spins in Heisenberg model, or link variables in lattice gauge theory), and different phases of lattice theories can be characterized by different patterns of entanglement [1–3]. Quantum entanglement thus appears to be an adequate concept for the description of the emergence of collective degrees of freedom in quantum field theories [1–3]. A commonly used measure of quantum entanglement of the ground state of quantum fields in (D − 1) + 1-dimensional space-time is the entropy of entanglement S [A] between some (D − 1)-dimensional region A and its (D − 1)-dimensional complement B, which characterizes the amount of information shared between A and B [1, 2, 4]. Entanglement entropy is defined as the usual von Neumann entropy for the reduced density matrix ρA associated with the region A: S [A] = −Tr A (ρA ln ρA) (1) The reduced density matrix is obtained from the density matrix of the ground state of the theory, ρAB = |0〉〈0| , by tracing over all degrees of freedom which are localized outside of A, i.e. within B [2, 4]: ρA = Tr B ρAB = Tr B |0〉〈0| (2) This density matrix describes the state of quantum fields as seen by an observer who can only perform measurements within A. ∗ e-mail: buividovich@tut.by † e-mail: polykarp@itep.ru
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.
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.