On the universality of entanglement entropy in quantum field theory

Technical Summary: On the Universality of Entanglement Entropy in Quantum Field Theory

Problem Statement
The standard definition of entanglement entropy in quantum field theory (QFT) faces a fundamental ambiguity: the Hilbert space of a QFT does not generally admit a tensor-product decomposition into spatial subregions. To compute entanglement entropy, one must regularize the entangling surface (e.g., by excising a tubular neighborhood of width $\epsilon$) and impose boundary conditions to factorize the Hilbert space. This procedure raises two critical, open questions:

1. Existence of Unitary Maps: Under what conditions do the chosen boundary conditions yield a unitary map between the original Hilbert space and the factorized one? Non-unitary maps fail to capture the entanglement entropy of the original state.
2. Universality: When multiple valid (unitary) boundary conditions exist, do they all yield the same entanglement entropy, specifically regarding the "universal" terms (those invariant under redefinitions of the regulator $\epsilon$)?

Current literature often assumes specific boundary conditions implicitly, leading to discrepancies in computed universal terms (e.g., in four-dimensional Maxwell theory). This paper investigates whether the choice of shrinkable boundary conditions—those that reduce to the identity defect in the $\epsilon \to 0$ limit—ensures the universality of the entropy.

Methodology
The paper employs a combination of path integral techniques, replica trick formalism, and topological field theory arguments:

• Regularization and Factorization: The entangling surface is regularized by removing a small neighborhood, turning the surface into a codimension-one boundary $S^{d-2} \times S^1_\epsilon$.
• Shrinkability Criterion: A boundary condition is defined as "shrinkable" if, in the limit $\epsilon \to 0$, the boundary reduces to the identity defect (possibly with a line counterterm). This ensures the map between the original and factorized states is unitary.
• Replica Trick: Entanglement entropy is computed via $S = \lim_{n \to 1} (1 - \partial_n) \log Z_n$, where $Z_n$ is the partition function on an $n$-fold cover (replica manifold) of the original space.
• Case Studies:
  • Topological Theories: The author analyzes 3D $U(1)_k$ Chern-Simons theory. The bulk is topological, allowing metric deformations that smooth out conical singularities in the replica manifold without changing the partition function.
  • Conformal Field Theories (CFTs): The argument is extended to 2D CFTs using local Weyl rescalings to flatten the geometry near the boundary while preserving the induced metric. The applicability to higher-dimensional CFTs is examined via the behavior of the metric under similar transformations.

Key Contributions and Results

1. Distinction Between Shrinkable and Non-Shrinkable Boundaries:
   The paper demonstrates that not all boundary conditions lead to a well-defined entanglement entropy.

   • Non-shrinkable boundaries (e.g., the topological boundary $A=B$ in $U(1)_k \times U(1)_{-k}$ theory) correspond to non-unitary maps. They project the state onto a specific charge sector, potentially mapping a non-zero norm state to a null state, thereby altering or destroying the entanglement entropy.
   • Shrinkable boundaries (e.g., holomorphic boundary conditions $A_{\bar{z}}=0$ in Chern-Simons theory) ensure the boundary reduces to the identity, preserving the state's norm and unitarity.

2. Universality in Topological Field Theories:
   For topological theories like Chern-Simons, the paper proves that the topological entanglement entropy (the constant term in the entropy expansion) is universal among all shrinkable boundary conditions.

   • Different shrinkable boundaries may introduce different counterterms (e.g., cosmological constant terms depending on the central charge of a decoupled 2D CFT attached to the boundary).
   • These counterterms affect only the divergent terms (proportional to $1/\epsilon$) or non-universal additive constants.
   • The constant term, $-\frac{1}{2}\log(k)$ for $U(1)_k$, remains invariant regardless of the specific shrinkable boundary chosen, provided the boundary remains shrinkable.

3. Universality in 2D CFTs:
   Using a local Weyl rescaling argument, the paper shows that for 2D CFTs, different local shrinkable boundary conditions yield the same universal logarithmic term, $S_{univ} = \frac{c}{3} \log(l/\epsilon)$.

   • While different boundary conditions may alter non-universal additive constants, the coefficient of the logarithmic divergence is fixed by the central charge $c$ and is independent of the boundary choice.
   • This relies on the assumption that the leading small-$\epsilon$ behavior of the replica partition function can be evaluated before taking the $n \to 1$ derivative without altering the universal term.

4. Obstacles in Higher Dimensions:
   The paper notes that the argument for universality does not directly extend to higher-dimensional CFTs ($d > 2$).

   • In $d > 2$, a Weyl rescaling that flattens the transverse metric introduces curvature in the longitudinal directions (the entangling surface), generating a divergent Ricci scalar as $\epsilon \to 0$.
   • Consequently, ordinary shrinkability on smooth backgrounds is insufficient to guarantee universality in the replica construction for $d > 2$.
   • The paper formulates a necessary condition for universality in general theories: the small-$\epsilon$ behavior of the expectation value of the modular Hamiltonian (or stress tensor) in the presence of the boundary must be consistent across different shrinkable boundaries.

Significance and Claims
The paper claims to establish a rigorous framework for understanding the "universality" of entanglement entropy. Its primary significance lies in:

• Clarifying the Role of Boundary Conditions: It identifies "shrinkability" as the necessary condition for a boundary to define a valid entanglement entropy.
• Resolving Ambiguities in Topological Theories: It proves that for topological theories, the topological entanglement entropy is a robust, universal quantity independent of the specific choice of shrinkable boundary condition.
• Defining Limits of Universality: It explicitly delineates where the universality argument holds (topological theories, 2D CFTs) and where it requires further investigation (higher-dimensional CFTs), suggesting that discrepancies in literature (e.g., regarding 4D Maxwell theory) may stem from the use of non-shrinkable boundaries or a lack of understanding of the required boundary conditions in replica geometries.

The author concludes that while universality is guaranteed for shrinkable boundaries in topological and 2D conformal settings, a general criterion for universality in arbitrary QFTs remains an open problem, to be addressed in future work.