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arXiv:0809.4502 · International Symposium on Lattice Field Theory (LATTICE)

Entanglement entropy in SU(N) gauge theory

Alexander Velytsky

Published Jun 30, 2009• 6 citations
arXivHTMLPDF

Abstract

The entanglement entropy of SU(N) lattice gauge theory is studied exactly in 1+1 space-time dimensions and in Migdal-Kadanoff approximation in higher dimensional space. The existence of a non-analytical behavior reminiscent of a phase transition for a characteristic size of the entangled region is demonstrated for higher dimensional theories.

DOI
10.22323/1.066.0256

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The entanglement entropy of SU(N) lattice gauge theory is studied exactly in 1+1 space-time dimensions and in Migdal-Kadanoff approximation in higher dimensional space. The existence of a non-analytical behavior reminiscent of a phase transition for a characteristic size of the entangled region is demonstrated for higher dimensional theories.
We consider the entanglement entropy for a subsystem in d+1d+1d+1 dimensional SU(N)SU(N)SU(N) lattice gauge theory. The 1+11+11+1 gauge theory is treated exactly and shows trivial behavior. Gauge theories in higher dimensions are treated within Migdal-Kadanoff approximation. We consider the gauge theory in the confinement phase. We demonstrate the existence of a nonanalytical change from the short distance to long distance form in the entanglement entropy in such systems (d>2d>2d>2) reminiscent of phase transition. The transition is manifested in nontrivial change in the renormalization group flow of character expansion coefficients defining the partition function.
We report on the recent progress in theoretical and numerical studies of entanglement entropy in lattice gauge theories. It is shown that the concept of quantum entanglement between gauge fields in two complementary regions of space can only be introduced if the Hilbert space of physical states is extended in a certain way. In the extended Hilbert space, the entanglement entropy can be partially interpreted as the classical Shannon entropy of the flux of the gauge fields through the boundary between the two regions. Such an extension leads to a reduction procedure which can easily be implemented in lattice simulations by constructing lattices with special topology. This enables us to measure the entanglement entropy in lattice Monte-Carlo simulations. On the simplest example of Z 2 lattice gauge theory in dimensions we demonstrate the relation between the entanglement entropy and the classical entropy of the field flux. For lattice gauge theory in four dimensions, we find a signature of non-analytic dependence of the entanglement entropy on the size of the region. We also comment on the holographic interpretation of the entanglement entropy.
We report on the recent progress in theoretical and numerical studies of entanglement entropy in lattice gauge theories.It is shown that the concept of quantum entanglement between gauge fields in two complementary regions of space can only be introduced if the Hilbert space of physical states is extended in a certain way.In the extended Hilbert space, the entanglement entropy can be partially interpreted as the classical Shannon entropy of the flux of the gauge fields through the boundary between the two regions.Such an extension leads to a reduction procedure which can be easily implemented in lattice simulations by constructing lattices with special topology.This enables us to measure the entanglement entropy in lattice Monte-Carlo simulations.On the simplest example of Z 2 lattice gauge theory in (2 + 1) dimensions we demonstrate the relation between entanglement entropy and the classical entropy of the field flux.For SU (2) lattice gauge theory in four dimensions, we find a signature of non-analytic dependence of the entanglement entropy on the size of the region.We also comment on the holographic interpretation of the entanglement entropy.
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 entanglement entropy in 1+11+11+1-dimensional SU(N)SU(N)SU(N) gauge theories with various matter fields using the lattice regularization. Here we use extended Hilbert space definition for entanglement entropy, which contains three contributions; (1) classical Shannon entropy associated with superselection sector distribution, where sectors are labeled by irreducible representations of boundary penetrating fluxes, (2) logarithm of the dimensions of their representations, which is associated with ``color entanglement,'' and (3) EPR Bell pairs, which give ``genuine'' entanglement. We explicitly show that entanglement entropies (1) and (2) above indeed appear for various multiple ``meson'' states in gauge theories with matter fields. Furthermore, we employ transfer matrix formalism for gauge theory with fundamental matter field and analyze its ground state using hopping parameter expansion (HPE), where the hopping parameter KKK is roughly the inverse square of the mass for the matter. We evaluate the entanglement entropy for the ground state and show that all (1), (2), (3) above appear in the HPE, though the Bell pair part (3) appears in higher order than (1) and (2) do. With these results, we discuss how the ground state entanglement entropy in the continuum limit can be understood from the lattice ground state obtained in the HPE.
We focus on the issue of proper definition of entanglement entropy in lattice gauge theories, and examine a naive definition where gauge invariant states are viewed as elements of an extended Hilbert space which contains gauge non-invariant states as well. Working in the extended Hilbert space, we can define entanglement entropy associated with an arbitrary subset of links, not only for abelian but also for non-abelian theories. We then derive the associated replica formula. We also discuss the issue of gauge invariance of the entanglement entropy. In the Z N gauge theories in arbitrary space dimensions, we show that all the standard properties of the entanglement entropy, e.g. the strong subadditivity, hold in our definition. We study the entanglement entropy for special states, including the topological states for the Z N gauge theories in arbitrary dimensions. We discuss relations of our definition to other proposals.