Notes on Entanglement in Abelian Gauge Theories

TL;DR

This work reframes entanglement entropy in Abelian gauge theories around buffer zones that separate spatial regions, showing that entanglement can be computed without enlarging the Hilbert space and that different boundary-algebra choices are merely basis changes for buffer-zone degrees of freedom. The authors develop explicit constructions for both electric and magnetic boundary conditions, extend the framework to $U(1)$ and matter, and demonstrate a unified, algebra-independent approach. The results recover known topological entropy pieces, illuminate continuum-limit divergences, and provide a versatile toolkit for analyzing gauge entanglement in lattice and continuum settings. The work lays groundwork for extensions to non-Abelian theories and potential links to holography and gravity.

Abstract

We streamline and generalize the recent progress in understanding entanglement between spatial regions in Abelian gauge theories. We provide an unambiguous and explicit prescription for calculating entanglement entropy in a $\mathbb Z_N$ lattice gauge theory. The main idea is that the lattice should be split into two disjoint regions of links separated by a buffer zone of plaquettes. We show that the previous calculations of the entanglement entropy can be realized as special cases of our setup, and we argue that the ambiguities reported in the previous work can be understood as basis choices for gauge-invariant operators living in the buffer zone. The proposed procedure applies to Abelian theories with matter and with continuous symmetry groups, both on the lattice and in the continuum.

Figures (4)

• Figure 1: (color online) Three examples of boundary electric operators defined in eq. \ref{['bdry el ops']}. Full lines denote links in $V$ and dotted lines denote links in $\bar{V}$. Gray dots denote elements of $\partial V$. The boundary electric operators $E_i$, defined on each boundary site, are products of electric generators on all links that enter that site and belong to $V$ (thick red lines). Similarly, boundary electric operators $\bar{E}_i$ are products of electric generators on all links entering site $i$ and belonging to $\bar{V}$ (wavy blue lines). The pictured choice of $(V, \bar{V})$ has 16 boundary lattice sites (i.e. elements of $\partial V$) and equally many pairs $(E_i, \bar{E}_i)$.
• Figure 2: (color online) A $d = 2$ example of one insertion of a magnetic boundary operator at link $(i, j)$ into a partition with purely electric boundary conditions. To the left: as before, solid black lines are links in $V$, dashed black lines are links in $\bar{V}$, and gray circles are elements of $\partial V$. The product of electric generators on the red solid links gives the boundary electric operator $E_i$, and corresponding products on green solid links gives $E_j$. The inverses of these operators are respectively equal to $\bar{E}_i$, the product of electric generators on blue wavy links, and to $\bar{E}_j$, the product of electric generators on teal wavy links. To the right: Solid black lines are links in $V'$, dashed black lines are links in $\bar{V}'$, the shaded green plaquette is the newly inserted buffer zone, which (together with the gray circles) forms the new boundary $\partial V'$. Electric generators on the solid red links multiply to give $E_{ij} = E_i E_j$, the new boundary electric operator that measures the electric flux flowing into the buffer zone. Its inverse is equal to $\bar{E}_{ij} = \bar{E}_i \bar{E}_j$, which is in turn given by the product of electric generators on the blue wavy links. The magnetic flux through the buffer zone placed at link $(i, j)$ is measured by $W_{ij}$, the magnetic generator around the green shaded buffer plaquette.
• Figure 3: (color online) A $d = 2$ example of a partition implementing purely magnetic boundary conditions. The "outer links" of $V$ in original, purely electric boundary conditions --- the ones that corresponded to the partition $(V, \bar{V})$ on Fig. \ref{['fig el bdry ops']} --- are shown by a black dashed line, and as before black solid lines correspond to links in $V'$ while black dotted lines correspond to links in $\bar{V}'$. The boundary $\partial V'$ now consists solely of 16 buffer plaquettes shaded in green. The magnetic generators around these plaquettes generate the magnetic center. Both the product of electric generators on links leading into the buffer zone (solid red lines) and out of it (wavy blue lines) are identically unity, due to the global Gauss law constraint. At each prior step that lead from the purely electric to the purely magnetic boundary conditions, the number of generators of the center was kept constant.
• Figure 4: The simplest lattice which supports magnetic boundary conditions. The solid line is the single link in $V$, and the dotted lines are the three links in $\bar{V}$. The entire plaquette makes up the buffer zone $\partial V'$ (shaded region). The set of $V$ links not in the buffer zone ($V'$) and the set of $\bar{V}$ links not in the buffer zone ($\bar{V}'$) are both empty. The eigenvalues of the position operators $U_\ell$ on the four links $\ell \in \{\mathrm{U}, \mathrm{D}, \mathrm{L}, \mathrm{R}\}$ are $n_\ell$.