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arXiv:1104.1011 · International Symposium on Lattice Field Theory (LATTICE)

Quantum entanglement in SU(3) lattice Yang-Mills theory at zero and finite temperatures

Yoshiyuki Nakagawa

Published Jun 6, 2011• 15 citations
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Abstract

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.

DOI
10.22323/1.105.0281

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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.
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 propose a novel prescription for calculating the entanglement entropy of the S U ( N ) Yang-Mills gauge theories on the lattice under the strong coupling expansion in powers of β = 2 N / g 2 , where g is the coupling constant. Using the replica method, our Lagrangian formalism maintains gauge invariance on the lattice. At O ( β 2 ) and O ( β 3 ) , the entanglement entropy is solely contributed by the central plaquettes enclosing the conical singularity of the n -sheeted Riemann surface. The area law emerges naturally to the highest order O ( β 3 ) of our calculation. The leading O ( β ) term is negative, which could in principle be canceled by taking into account the “cosmological constant” living in interface of the two entangled subregions. This unknown cosmological constant resembles the ambiguity of edge modes in the Hamiltonian formalism. We further speculate this unknown cosmological constant can show up in the entanglement entropy of scalar and spinor field theories as well. Furthermore, it could play the role of a counterterm to absorb the ultraviolet divergence of entanglement entropy and make entanglement entropy a finite physical quantity.
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.