Calabrese–Cardy Entanglement (Universality-Level API)

Overview

For any 1+1D conformal field theory of central charge $c$, the Calabrese–Cardy formulae give a closed-form universal value for the von Neumann (and Rényi) entanglement entropy of a contiguous block of length $\ell$ on a chain of length $L$. QAtlas exposes these as universality-level fetch methods so they can be evaluated without ever instantiating a specific lattice model — the only physics input is the universality class itself, from which the central charge is read out via fetch(Universality{C}(), CentralCharge()).

This page documents the universality dispatch added in #149. The rigorous twist-operator derivation of the $c/3$ vs $c/6$ prefactors lives separately in docs/src/calc/calabrese-cardy-obc-vs-pbc.md.

Closed forms

Let $c$ be the central charge of the universality class. Drop the non-universal additive constant $c_{1}^{\prime}$ (UV cutoff) and the Affleck–Ludwig boundary entropy $\log g$ (boundary-state-dependent), both of which require model-specific input not available at the universality level. What remains is the universal log-prefactor piece:

\[S_{\rm PBC}(\ell, L) = \frac{c}{3}\,\log\!\left[\frac{L}{\pi}\sin\!\left(\frac{\pi\ell}{L}\right)\right],\]

\[S_{\rm OBC}(\ell, L) = \frac{c}{6}\,\log\!\left[\frac{2L}{\pi}\sin\!\left(\frac{\pi\ell}{L}\right)\right],\]

\[S_{\rm \infty}(\ell) = \frac{c}{3}\,\log\ell.\]

The Rényi-$\alpha$ extension uses the substitution

\[c \;\to\; c\,\frac{1 + 1/\alpha}{2}\]

in the same closed form, which reduces to $c$ at $\alpha = 1$ (von Neumann).

API

fetch(::Universality{C}, ::VonNeumannEntropy, ::PBC; ℓ::Real, L::Real, kwargs...)
fetch(::Universality{C}, ::VonNeumannEntropy, ::OBC; ℓ::Real, L::Real, kwargs...)
fetch(::Universality{C}, ::VonNeumannEntropy, ::Infinite; ℓ::Real, kwargs...)

fetch(::Universality{C}, ::RenyiEntropy, ::PBC; ℓ::Real, L::Real, kwargs...)
fetch(::Universality{C}, ::RenyiEntropy, ::OBC; ℓ::Real, L::Real, kwargs...)
fetch(::Universality{C}, ::RenyiEntropy, ::Infinite; ℓ::Real, kwargs...)

The central charge is fetched internally via fetch(Universality{C}(), CentralCharge(); kwargs...). Universality classes that do not have a 1+1D-CFT central charge defined (e.g. KPZ, Percolation, the 3D O($n$) classes) raise ErrorException with a clear message.

Supported universality classes

Class	$d$	$c$	Reference
`Universality(:Ising)`	2	$1/2$	M(3,4) — Belavin–Polyakov–Zamolodchikov, Nucl. Phys. B 241, 333 (1984)
`Universality(:Potts3)`	2	$4/5$	M(5,6) — Dotsenko, Nucl. Phys. B 235, 54 (1984)
`Universality(:Potts4)`	2	$1$	Compact boson at marginal point — di Francesco–Mathieu–Sénéchal §12.3
`Universality(:XY)`	2	$1$	BKT free boson — Kosterlitz J. Phys. C 7, 1046 (1974)

For all other classes — Universality(:KPZ), Universality(:Percolation), Universality(:Heisenberg) (3D O(3)), and any future class — the fetch(::Universality{C}, ::CentralCharge; ...) method is intentionally not defined, so any Cardy entanglement call routes through the helper _cardy_central_charge and raises ErrorException with the message:

Universality{:<C>}: Calabrese-Cardy entanglement requires a 1+1D CFT central
charge, which is not defined for this universality class. Define
fetch(::Universality{:<C>}, ::CentralCharge; ...) first, or use a class that
lives in a 1+1D CFT (e.g. :Ising d=2, :Potts3 d=2, :Potts4 d=2, :XY d=2).

Examples

Ising critical chain, finite size

using QAtlas

# c = 1/2; (1/6) log[(8/π) sin(π·4/8)] = (1/6) log(8/π) ≈ 0.1558
S_pbc = QAtlas.fetch(Universality(:Ising), VonNeumannEntropy(), PBC(); ℓ=4.0, L=8.0)
S_obc = QAtlas.fetch(Universality(:Ising), VonNeumannEntropy(), OBC(); ℓ=4.0, L=8.0)

Thermodynamic limit

# (1/2)/3 · log(10) = (1/6) log 10
S_inf = QAtlas.fetch(Universality(:Ising), VonNeumannEntropy(), Infinite(); ℓ=10.0)

Rényi at α = 2

The substitution $c \to c \cdot (1 + 1/\alpha)/2 = (3/4)c$ at $\alpha = 2$ makes the Rényi-2 entropy exactly $3/4$ of the von Neumann value with the same log argument:

S_vn = QAtlas.fetch(Universality(:Ising), VonNeumannEntropy(), PBC(); ℓ=4.0, L=8.0)
S_r2 = QAtlas.fetch(Universality(:Ising), RenyiEntropy(2.0), PBC(); ℓ=4.0, L=8.0)
@assert S_r2 ≈ (3//4) * S_vn

Caveats

Non-universal constants dropped. The UV cutoff term $c_{1}^{\prime}$ and the Affleck–Ludwig boundary entropy $\log g$ are not included in the returned value. Tests that fit lattice entanglement-entropy data against this reference must include an additive offset.
Endpoints. ℓ = 0 and ℓ = L (the formula limit where the block contains zero or all sites) return -Inf rather than NaN or a finite floating-point artefact of sin(π). This is the correct continuum limit — the entropy diverges as the cut encloses arbitrarily fewer sites and is regularised by the UV cutoff that we have dropped.
No off-critical extension. These formulae assume the universality class is realised at criticality; gapped universality classes do not have a CFT central charge. Off-critical entanglement (mass crossover) is the responsibility of model-level fetch methods (e.g. TFIM in src/models/quantum/TFIM/TFIM_cft_entanglement.jl).

References

P. Calabrese, J. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech. P06002 (2004).
P. Calabrese, J. Cardy, Entanglement entropy and conformal field theory, J. Phys. A 42, 504005 (2009).
I. Affleck, A. W. W. Ludwig, Universal noninteger "ground-state degeneracy" in critical quantum systems, Phys. Rev. Lett. 67, 161 (1991) — Affleck–Ludwig $\log g$ term.