Calabrese-Cardy Formula
Overview
The Calabrese-Cardy formula gives the bipartite entanglement entropy of a 1D critical system as a function of the subsystem size, the total system size, and the central charge $c$ of the underlying conformal field theory.
This formula is the primary tool for extracting the central charge from a ground-state wavefunction, and is used throughout QAtlas's entanglement verification.
Statement
Periodic boundary conditions (PBC)
For a system of $N$ sites on a ring, with subsystem $A$ consisting of $l$ consecutive sites:
\[S(l) = \frac{c}{3}\ln\!\left[\frac{N}{\pi a}\sin\!\left(\frac{\pi l}{N}\right)\right] + s_1\]
The prefactor $c/3$ arises because PBC creates two entanglement cuts (between sites $l, l+1$ and sites $N, 1$).
Open boundary conditions (OBC)
For a system of $N$ sites on an open chain, with subsystem $A$ being sites $1, \ldots, l$:
\[S(l) = \frac{c}{6}\ln\!\left[\frac{2N}{\pi a}\sin\!\left(\frac{\pi l}{N}\right)\right] + s_1'\]
The prefactor is $c/6$ because OBC creates only one entanglement cut (between sites $l$ and $l+1$).
Using $c/3$ (PBC) for an OBC system halves the extracted central charge. This is a common mistake — always match the prefactor to the boundary conditions of the system being studied.
Parameters
\[c\]
: central charge of the CFT (universal)\[a\]
: UV cutoff (lattice spacing, typically $a = 1$)\[s_1, s_1'\]
: non-universal additive constants
Physical Origin
In a 1D critical system, the ground state is described by a conformal field theory. The entanglement entropy of a subsystem of size $l$ in a total system of size $N$ is determined by the two-point function of twist operators in the orbifold CFT $\mathcal{C}^n / \mathbb{Z}_n$ (replica trick):
\[S = -\lim_{n \to 1} \frac{\partial}{\partial n} \mathrm{Tr}(\rho_A^n)\]
The $\sin(\pi l / N)$ factor is the chord length in the conformal mapping from the infinite plane to the cylinder (PBC) or strip (OBC).
Central Charge Extraction
To extract $c$ from numerical entropy data $\{S(l)\}$:
- Compute $\xi(l) = \ln[(2N/\pi)\sin(\pi l/N)]$ (OBC conformal coordinate)
- Perform linear regression: $S = (\text{slope}) \cdot \xi + \text{const}$
\[c = 6 \times \text{slope}\]
(OBC) or $c = 3 \times \text{slope}$ (PBC)
Practical tips:
- Skip boundary-adjacent points ($l = 1, 2$ and $l = N-1, N-2$) to reduce lattice artifacts.
- For the Heisenberg chain, use only even $l$ values to suppress the SU(2) alternating correction $(-1)^l f(l)$.
Known Central Charges
| System | $c$ | QAtlas verification |
|---|---|---|
| TFIM at $h = J$ | $1/2$ | → |
| Heisenberg chain | $1$ | → |
| Free boson (Luttinger liquid) | $1$ | — |
| Free Dirac fermion | $1/2$ | — |
Full derivation
Step-by-step: replica trick, branch-point twist operators with scaling dimension $\Delta_n = (c/24)(n - 1/n)$, the cylinder → plane conformal map for PBC, the strip → upper half-plane map with method of images for OBC, explicit $n \to 1$ derivative to extract $S$, and the factor-of-2 origin tabulated: Calabrese–Cardy entanglement entropy: OBC vs PBC prefactor .
References
- P. Calabrese, J. Cardy, "Entanglement entropy and quantum field theory", J. Stat. Mech. 0406, P06002 (2004) — original derivation. Eq. (7) for PBC, Eq. (19) for OBC.
- P. Calabrese, J. Cardy, "Entanglement entropy and conformal field theory", J. Phys. A 42, 504005 (2009) — review with extensions.
- C. Holzhey, F. Larsen, F. Wilczek, "Geometric and renormalized entropy in conformal field theory", Nucl. Phys. B 424, 443 (1994) — earlier result for the $c/3$ formula.