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arXiv:1910.00859 · Physical review. D/Physical review. D.

Lattice QCD study of static quark and antiquark correlations via entanglement entropies

TORU T. TAKAHASHI, Yoshiko Kanada-En’yo

Published Dec 4, 2019• 8 citations
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Abstract

We study the color correlation between the static quark and antiquark ($q\overline{q}$) in the confined phase via reduced density matrices $\ensuremath{\rho}$ defined in color space. We adopt the standard Wilson gauge action and perform quenched calculations with the Coulomb gauge condition for reduced density matrices. The spatial volumes are ${L}^{3}={8}^{3}$, $1{6}^{3}$, $3{2}^{3}$, and $4{8}^{3}$, with the gauge couplings $\ensuremath{\beta}=5.7$, 5.8, and 6.0. Each element of the reduced density matrix in the subspace of quarks' color degrees of freedom of the $q\overline{q}$ pair is calculated from staples defined by link variables. As a result, we find that $\ensuremath{\rho}$ is well written by a linear combination of the strongly correlated $q\overline{q}$ pair state with the color-singlet component and the uncorrelated $q\overline{q}$ pair state with random color configurations. We compute the Renyi entropies ${S}^{\text{Renyi}}$ from $\ensuremath{\rho}$ to investigate the $q\overline{q}$ distance dependence of the color correlation of the $q\overline{q}$ pair and find that the color correlation is quenched as the distance increases.

DOI
10.1103/physrevd.100.114502

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We study the color correlation between the static quark and antiquark (qq‾q\overline{q}qq​) in the confined phase via reduced density matrices \ensuremathρ\ensuremath{\rho}\ensuremathρ defined in color space. We adopt the standard Wilson gauge action and perform quenched calculations with the Coulomb gauge condition for reduced density matrices. The spatial volumes are L3=83{L}^{3}={8}^{3}L3=83, 1631{6}^{3}163, 3233{2}^{3}323, and 4834{8}^{3}483, with the gauge couplings \ensuremathβ=5.7\ensuremath{\beta}=5.7\ensuremathβ=5.7, 5.8, and 6.0. Each element of the reduced density matrix in the subspace of quarks' color degrees of freedom of the qq‾q\overline{q}qq​ pair is calculated from staples defined by link variables. As a result, we find that \ensuremathρ\ensuremath{\rho}\ensuremathρ is well written by a linear combination of the strongly correlated qq‾q\overline{q}qq​ pair state with the color-singlet component and the uncorrelated qq‾q\overline{q}qq​ pair state with random color configurations. We compute the Renyi entropies SRenyi{S}^{\text{Renyi}}SRenyi from \ensuremathρ\ensuremath{\rho}\ensuremathρ to investigate the qq‾q\overline{q}qq​ distance dependence of the color correlation of the qq‾q\overline{q}qq​ pair and find that the color correlation is quenched as the distance increases.
With the aim of clarification of color correlations among quarks, we investigate the color correlation between a static quark and an antiquark (static qq‾q\overline{q}qq​) below and above the phase transition temperature Tc{T}_{c}Tc​ through the entanglement entropy (EE). By a quenched lattice QCD calculation on an anisotropic lattice adopting the standard Wilson gauge action in the Coulomb gauge, we compute a reduced density matrix $\ensuremath{\rho}$ defined in the color space and the entanglement entropy SEE{S}_{\mathrm{EE}}SEE​ constructed from $\ensuremath{\rho}$. The spatial volume is L3=243{L}^{3}=2{4}^{3}L3=243, and the temporal extents are NT=12{N}_{T}=12NT​=12, 13, 14, 15, 16, 18, 20 and 24, with a gauge coupling $\ensuremath{\beta}=5.75$ and a renormalized anisotropy $\ensuremath{\xi}=4.0$, which corresponds to temperatures between 180 and 370 MeV. From an analysis of $\ensuremath{\rho}$ and SEE{S}_{\mathrm{EE}}SEE​, the color correlation between qq‾q\overline{q}qq​ pairs is obtained as a function of the interquark distance RRR and the temperature TTT. Below Tc{T}_{c}Tc​, the RRR-dependence of the color correlation resembles that at T=0T=0T=0: the quark's color correlation gradually decreases as RRR increases due to the color screening by in-between gluons. Above Tc{T}_{c}Tc​, due to the deconfinement phase transition, the color correlation among quarks is found to quickly disappear. Further, we investigate the color screening effect via the screening mass defined by $\ensuremath{\rho}$ and discuss the differences in the screening properties between the small and large R regions.
We study the entanglement of gluon fields in presence of a static QQˉQ\bar QQQˉ​ pair in quenched QCD. Using the replica method, we investigate the q=2q=2q=2 Renyi entropy of the entanglement of gluon fields inside and in the vicinity of the confining QCD string between the quark and the antiquark. We find that there is excess entropy of gluon entanglement compared to vacuum fluctuations. This excess of entanglement entropy is associated with the gluon flux tube, and we find that it has a finite non-zero value in the continuum. We investigate the dependence of gluon entanglement on the geometry of longitudinal and transverse partitioning of the flux tube. Our preliminary results suggest scaling of the entanglement entropy with the area of the boundary overlapping with the flux tube.
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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
Recent lattice QCD studies have revealed an excess entanglement entropy (flux tube entanglement entropy, FTE 2 ) in the color flux tube between static quark–antiquark pairs, which grows linearly with the quark separation. In this paper, we illustrate a possible analogy in a simplified string-net model based on the Fibonacci fusion category, where the non-trivial object [Formula: see text] (analogous to color charge) cannot exist in isolation due to the fusion rule [Formula: see text], naturally exhibiting “confinement”. We compute the entanglement entropy of the flux tube connecting two [Formula: see text] objects and find an entropy density of [Formula: see text] ([Formula: see text] is the golden ratio), which quantitatively agrees with the lattice QCD measured FTE 2 density (approximately 0.3–0.5). Minimizing the free energy [Formula: see text] at fixed temperature and volume, the competition between the energy term [Formula: see text] and the entropy term [Formula: see text] yields a confinement-deconfinement phase transition at critical temperature [Formula: see text]. This model provides a heuristic demonstration platform as a minimal model for understanding the entropic properties of the QCD flux tube.
We define a finite-volume lattice-QCD observable for the vacuum-subtracted spatial Rényi response of a source--sink-prepared, momentum-projected hadron. At fixed regulator, integer replica index n>1n>1n>1, spatial region and gauge-theory cut prescription, double-sided state projection expresses this response as a ratio of replicated and ordinary hadron correlators. The first numerical target is the n=2n=2n=2 ratio; its proposed L−3L^{-3}L−3 scaling at fixed physical region is to be tested with matched volumes. We implement the measurement in quenched SU(3) for a charged pseudoscalar meson with heavy degenerate Wilson valence quarks, ameff≃2.07am_{\rm eff}\simeq2.07ameff​≃2.07, on two 163×3216^3\times32163×32 sheets with a central-plaquette cut. Five finite balls and empty/full controls are measured at nine source/sink placements. Their finite-ball means increase with radius at every placement. At the reference placement the three largest responses are 1.6--2.5 times an ideal single-quasiparticle occupancy reference. A full-covariance fit Δcp=κoccS2occΔ_{\rm cp}=κ_{\rm occ}S_2^{\rm occ}Δcp​=κocc​S2occ​ gives κocc=1.68±0.12κ_{\rm occ}=1.68\pm0.12κocc​=1.68±0.12; radius-window and full-control comparisons give values spanning 1.491.491.49--1.731.731.73. An exact pure-state quark--antiquark pair model interpolates, in the dilute limit, between the unit-occupancy reference (κocc=1κ_{\rm occ}=1κocc​=1) for an unresolved meson and the double-occupancy reference (κocc=2κ_{\rm occ}=2κocc​=2) for a separately resolved quark and antiquark, and yields a boundary-area correction governed by their separation.