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Entanglement entropy in lattice gauge theories

P. V. Buividovich, M. I. Polikarpov

TL;DR

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, which leads to a reduction procedure which can be implemented in lattice simulations by constructing lattices with special topology.

Abstract

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 Z2 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 lattice gauge theories

TL;DR

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, which leads to a reduction procedure which can be implemented in lattice simulations by constructing lattices with special topology.

Abstract

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 Z2 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.

Paper Structure

This paper contains 6 equations, 3 figures.

Figures (3)

  • Figure 1: Pair production for a confining gauge theory in the vicinity of a black hole.
  • Figure 2: Lattice derivatives of the entanglement entropy and the entropy of intersection points over the size of the region for $16^{3}$ lattice. On the left: as a function of $l$ at $\beta_{g} = 0.788$, on the right: as a function of $\beta_{g}$ at $l = 8$.
  • Figure 3: The dependence of the derivative of the entanglement entropy $\frac{1}{|\partial A|} \, \frac{S_{f} \left( l \right) }{\partial l}$ on $l$. Solid line is the fit of the data by the function $C \, l^{-3}$.