Remarks on entanglement entropy for gauge fields

TL;DR

The paper analyzes how entanglement entropy for gauge fields on a lattice is affected by gauge constraints that introduce centers in the local operator algebras, preventing a straightforward tensor-product decomposition. It shows that different natural algebra choices (electric, magnetic, or boundary-fixed via a maximal tree) yield different 'entanglement' entropies, though relative entropy and mutual information remain finite and gauge-independent in the continuum. The extended lattice construction and boundary-gauge-fixing perspectives connect algebraic centers to gauge choices, revealing that many entropy ambiguities are boundary-local and vanish in universal continuum quantities. The authors discuss scalar and Z2 models to illustrate the scope and limitations of these ambiguities and argue that in the continuum the universal content is captured by boundary-regularized, gauge-invariant information, with potential extensions to non-Abelian gauge theories.

Abstract

In gauge theories the presence of constraints can obstruct expressing the global Hilbert space as a tensor product of the Hilbert spaces corresponding to degrees of freedom localized in complementary regions. In algebraic terms, this is due to the presence of a center --- a set of operators which commute with all others --- in the gauge invariant operator algebra corresponding to finite region. A unique entropy can be assigned to algebras with center, giving place to a local entropy in lattice gauge theories. However, ambiguities arise on the correspondence between algebras and regions. In particular, it is always possible to choose (in many different ways) local algebras with trivial center, and hence a genuine entanglement entropy, for any region. These choices are in correspondence with maximal trees of links on the boundary, which can be interpreted as partial gauge fixings. This interpretation entails a gauge fixing dependence of the entanglement entropy. In the continuum limit however, ambiguities in the entropy are given by terms local on the boundary of the region, in such a way relative entropy and mutual information are finite, universal, and gauge independent quantities.

Figures (7)

• Figure 1: The product of three link operators on the square $\hat{L}_g^{(12)} \hat{L}_g^{(13)} \hat{L}_g^{(14)}$ is equal to a link operator outside the square, $\hat{L}_g^{(51)}$, and hence it commutes with the rest of the operators on the square. The same occurs for the product $\hat{L}_g^{(67)} \hat{L}_g^{(68)}=\hat{L}_g^{(96)} \hat{L}_g^{(10\,6)}$ on the corner.
• Figure 2: (a) The algebra of the square with an electric center choice. The center is formed by the link operators shown with dashed lines. The commutant is represented by the shaded region, having the same center. (b) The square with a magnetic center choice. The center is formed by a single loop at the boundary in this two-dimensional example.
• Figure 3: The links cut by the boundary $\partial V$ (dashed line) are duplicated. The crosses are new vertices of the lattice, but no gauge invariance is required for them.
• Figure 4: A surface $\Sigma$ intersecting the region $V$. If the link operators on $\partial V$ coming out of $\Sigma$ are in the algebra ${\cal A}_V$, this algebra has a non trivial center as a result of the Gauss law constraint.
• Figure 5: Left panel: Choice of an algebra with trivial center for $d=2$. A maximal tree on the boundary is shown with a dashed line. Only one link operator on the boundary of $V$ is included in ${\cal A}_V$. Two links on the boundary lead to a non trivial center through the electric Gauss law. In this example the total number of electric and magnetic degree of freedom equals $9$: $9$ plaquettes, and $13$ link operators with $4$ constraint equations. Right panel: tree of links on the surface (dashed line). The number of electric and magnetic degrees of freedom equals $5$ in this example (there are $6$ plaquettes but the product of all plaquettes is $1$).