Regulated Entropy and Replica Methods
Entanglement entropy in QFT is usually computed through a regulated subsystem. Correlation matrices and replicas offer complementary methods, but their continuum interpretation depends on ultraviolet subtraction, infrared control, and analytic continuation.
Enter this chapter
Section titled “Enter this chapter”Begin with Entropy of a Regulated Subregion to define the regulated region and its reduced-state entropy.
Helpful background. Restricted states and factorization failure distinguish a continuum restriction from a regulated density matrix; Euclidean correlators and Gaussian fields supply the path-integral and free-field inputs.
Topics
Section titled “Topics”- Entropy of a Regulated Subregion — Specifies the density matrix and cutoff before computing entropy.
- Replica Trick and Branched Geometries — Represents integer moments by sewing replicated geometries.
- Rényi Entropies and Replica Analytic Continuation — Separates spectral continuation from knowing integer replica values.
- Twist Operators, Replica Defects, and Sewing Data — Encodes replica permutations as localized defects.
- Analytic Continuation: Uniqueness and Failure Modes — Tests nonuniqueness, singular limits, and branch choices.
- Gaussian States and Correlation-Matrix Entropy — Computes Gaussian entropy from covariance spectra.
- Entanglement Entropy of Free Fields — Develops free-field benchmarks for the different methods.
- Zero Modes, Boundaries, and Infrared Sensitivity — Tracks zero modes, boundaries, and infrared sensitivity.
- Ultraviolet Divergences and the Area Law — Identifies local geometric ultraviolet divergences.
- Entropy Counterterms and Renormalization Ambiguities — Classifies entropy counterterms and finite ambiguities.
- Numerical Replica and Thermodynamic-Integration Estimators — Constructs numerical estimators for replica quantities.
- From Lattice Entropy to a Continuum Claim — Controls the limiting procedure from a lattice calculation.
- Checking Replica and Entropy Calculations — Checks normalization, spectra, and independent entropy calculations.
How the ideas fit together
Section titled “How the ideas fit together”For a regulated Gaussian state, the covariance spectrum determines the entropy, giving a direct check of replica calculations Peschel 2003, pp. L205–L208. Replicas instead begin with integer moments of the same density matrix and need a justified continuation to recover the von Neumann entropy.
Removing the cutoff introduces another limit. Geometric ultraviolet terms may be subtracted or combined into finite quantities, while zero modes can cause independent infrared problems. Agreement at one cutoff therefore does not establish a unique continuation or a regulator-independent entropy.
The comparison below separates the assumptions and tests needed for these different conclusions.
Methods and supported conclusions
Section titled “Methods and supported conclusions”| Method | Required input | Direct output | Main validity check |
|---|---|---|---|
| Density-matrix or exact diagonalization | Finite factorization, normalized state, and declared subsystem | Spectrum, Sn, and S at fixed regulator | Trace one, positivity, pure-state complement spectrum, and exact limits |
| Gaussian correlation matrix | Gaussian state and canonically normalized restricted covariance | Entropy from bosonic symplectic or fermionic occupation eigenvalues | Uncertainty cone or 0 ≤ CA ≤ 1, plus a non-Gaussianity check |
| Replica path integral | Integer n, Euclidean state preparation, cyclic sewing, and Z1n normalization | Tr ρAn and integer Rényi entropy | Correct n = 1 limit, complement symmetry, and a spectral cross-check |
| Analytic continuation | Analytic domain, branch, growth or spectral control, and integer data covariance | Noninteger Rényi data or the derivative at n = 1 | Alternative admissible continuations and nearby-singularity sensitivity |
| Continuum extrapolation | Fixed physical geometry, multiple cutoffs, infrared control, and a finite target | Universal coefficient, difference, derivative, or continuum combination | Fit-window, discretization, orientation, and finite-volume stability |
Where to go next
Section titled “Where to go next”Continue to Relative Entropy, Distinguishability, and Recovery to define relative entropy on a chosen algebra and reference state.