Entanglement Entropy of U(1) Quantum Spin Liquids

TL;DR

This work analyzes the entanglement structure of a (3+1)D U(1) quantum spin liquid in its deconfined phase, where a gapless photon coexists with gapped matter. By leveraging the Bisognano-Wichmann theorem and a local-thermal approximation, the authors separate entanglement into a boundary particle contribution arising from the closed-loop constraint and a bulk photon contribution from photon fluctuations, obtaining two universal logarithmic corrections to the entanglement entropy. They show that the particle contribution generalizes topological entanglement entropy to this gapless phase, while the photon contribution is essentially local and geometry-dependent; together these yield S ∼ α(L/a)^2 − [γ_top + γ_ph] log L, with γ_top = 1 and γ_ph = 1/45 for a sphere. A 3D generalization of the Kitaev-Preskill construction, Grover-Turner-Vishwanath, isolates the topological piece S_top = −log L, enabling a clean separation of universal terms, and the framework is extended to higher dimensions with a conjectured generalized topological log term. The findings offer a robust entanglement-based characterization of gapless U(1) spin liquids and motivate entanglement-spectrum diagnostics for symmetry-enriched variants and other gapless phases.

Abstract

We here investigate the entanglement structure of the ground state of a (3+1)-dimensional U(1) quantum spin liquid, which is described by the deconfined phase of a compact U(1) gauge theory. A gapless photon is the only low-energy excitation, with matter existing as deconfined but gapped excitations of the system. It is found that, for a given bipartition of the system, the elements of the entanglement spectrum can be grouped according to the electric flux between the two regions, leading to a useful interpretation of the entanglement spectrum in terms of electric charges living on the boundary. The entanglement spectrum is also given additional structure due to the presence of the gapless photon. Making use of the Bisognano-Wichmann theorem and a local thermal approximation, these two contributions to the entanglement (particle and photon) are recast in terms of boundary and bulk contributions, respectively. Both pieces of the entanglement structure give rise to universal subleading terms (relative to the area law) in the entanglement entropy, which are logarithmic in the system size (log L), as opposed to the subleading constant term in gapped topologically ordered systems. The photon subleading logarithm arises from the low-energy conformal field theory and is essentially local in character. The particle subleading logarithm arises due to the constraint of closed electric loops in the wavefunction and is shown to be the natural generalization of topological entanglement entropy to the U(1) spin liquid. This contribution to the entanglement entropy can be isolated by means of the Grover-Turner-Vishwanath construction (which generalizes the Kitaev-Preskill scheme to three dimensions).

Figures (7)

• Figure 1: A typical closed-loop configuration. The ground state of the $U(1)$ spin liquid will be a superposition of such configurations.
• Figure 2: A typical configuration with open strings. The endpoints of the strings represent matter fields. States composed of open strings are taken to be gapped in the $U(1)$ spin liquid phase.
• Figure 3: When we partition the system into two regions, each configuration has a corresponding configuration of flux points between the two regions. The closed loop constraint ensures that the boundary conditions must match on the two sides. We define a partition of the system as a partition of the links, and the links on the partitioning line above have been taken as part of the left subsystem to avoid ambiguity in defining the location of the flux points.
• Figure 4: In a one-dimensional chain, the only states consistent with Gauss's law have uniform electric flux throughout the whole chain. These states can be labeled by integers, representing the quantized value of the electric flux. Since the flux is uniform throughout the whole system, in particular it represents the flux across the boundary between any two partitions of the system.
• Figure 5: The first figure displays the function $\tilde{f}(b) = 1+2\cos b$ appropriate to the hard-core example in blue, along with its Gaussian approximation in yellow. The second figure shows both functions to the fifth power, and the last figure shows them to the tenth, at which point the functions are essentially indistinguishable, with the error becoming negligible as a fraction of the maximum.