Entanglement Entropy and Central Charge
Overview
The entanglement entropy of a 1D critical ground state encodes the central charge $c$ of the underlying conformal field theory via the Calabrese-Cardy formula. QAtlas extracts $c$ from exact-diagonalization ground states and compares against the known CFT values, providing a verification that connects quantum information (entanglement) to conformal field theory (central charge).
Method
Calabrese-Cardy Formula (OBC)
For an open chain of $N$ sites with subsystem $A = \{1, \ldots, l\}$:
\[S(l) = \frac{c}{6}\ln\!\left[\frac{2N}{\pi}\sin\!\left(\frac{\pi l}{N}\right)\right] + s_1'\]
The prefactor is $c/6$ (not $c/3$) because OBC creates only one entanglement cut. See calc/calabrese-cardy-obc-vs-pbc.md for the derivation and physical reason for the factor-of-2 difference.
Extraction Procedure
- Compute the ground-state wavefunction $|\psi_0\rangle$ via ED.
- For each bipartition $l = 1, \ldots, N-1$, compute the reduced density matrix $\rho_A = \mathrm{tr}_B |\psi_0\rangle\langle\psi_0|$ and the von Neumann entropy $S(l) = -\mathrm{tr}(\rho_A \ln \rho_A)$.
- Define the conformal coordinate $\xi(l) = \ln[(2N/\pi)\sin(\pi l/N)]$.
- Perform linear regression: $S = \text{slope} \cdot \xi + \text{const}$.
- Extract $c = 6 \times \text{slope}$.
Results
TFIM at $h = J$ (Ising CFT, $c = 1/2$)
| Parameter | Value |
|---|---|
| System size | $N = 14$, OBC |
| Boundary conditions | Open |
| Expected $c$ | $1/2$ |
| Extracted $c$ | $\approx 0.5$ (within 10%) |
| Dominant error source | Finite-size corrections at small $N$ |
At the TFIM critical point $h = J$, the system is in the Ising CFT universality class with $c = 1/2$. The entanglement entropy $S(l)$ shows the characteristic logarithmic growth, and the extracted central charge agrees with $1/2$ to approximately 10% at $N = 14$.
Heisenberg Chain (SU(2)$_1$ WZW, $c = 1$)
| Parameter | Value |
|---|---|
| System size | $N = 12$, OBC |
| Boundary conditions | Open |
| Expected $c$ | $1$ |
| Extracted $c$ | $\approx 1.0$ (within 20%, even-$l$ fit) |
| Dominant error source | SU(2) alternating correction $(-1)^l f(l)$ |
The spin-1/2 Heisenberg chain is a $c = 1$ Luttinger liquid (SU(2)$_1$ WZW model). A crucial practical detail: the SU(2) symmetry produces an alternating correction $(-1)^l$ to $S(l)$ that contaminates the fit if all $l$ values are used. Restricting the fit to even $l$ only suppresses this correction and yields $c \approx 1$ within 20%.
Gapped Phases: Area Law
Away from criticality, the entanglement entropy saturates to a constant as $l$ increases (area law). QAtlas verifies this for:
- TFIM $h \gg J$: paramagnetic phase, product-state ground state, $S \to 0$.
- TFIM $h \ll J$: ferromagnetic phase, $S$ saturates to a small constant (boundary contribution).
The absence of logarithmic growth in the gapped phase confirms that the Calabrese-Cardy formula applies specifically at criticality.
Limitations
Peschel Method and Kramers-Wannier Duality
For the TFIM, an alternative route to entanglement entropy uses the Peschel method: compute the correlation matrix of Jordan-Wigner fermions and extract the entropy from its eigenvalues. This avoids full ED and scales as $O(N^3)$.
However, the Peschel method requires the ground state to be expressible as a free-fermion Slater determinant. The Kramers-Wannier (KW) duality maps the TFIM to a dual model where the boundary conditions may differ. Under OBC, the KW duality introduces a subtlety at the boundary that can shift the effective system size by one site. QAtlas uses the full ED approach to avoid this issue.
Finite-Size Accuracy
At the system sizes accessible to ED ($N \leq 16$), the extracted $c$ has $\sim 10$–$20\%$ error. This is expected: the Calabrese-Cardy formula is asymptotically exact as $N \to \infty$, and subleading corrections are significant at small $N$. The test asserts qualitative agreement (correct order of magnitude and sign) rather than high precision.
QAtlas Test
# test/verification/test_entanglement_central_charge.jl
# TFIM: c ≈ 1/2
c_tfim = extract_central_charge(tfim_ground_state, N=14, bc=:OBC)
@test abs(c_tfim - 0.5) / 0.5 < 0.10
# Heisenberg: c ≈ 1 (even-l fit)
c_heis = extract_central_charge(heis_ground_state, N=12, bc=:OBC, even_only=true)
@test abs(c_heis - 1.0) / 1.0 < 0.20Connections
- Calabrese-Cardy formula: method page, OBC vs PBC derivation
- Ising universality: $c = 1/2$ is the central charge of the minimal model $\mathcal{M}(3,4)$; see Ising
- Cross-check #3: TFIM $c = 1/2$ appears in the cross-verification table
- Disordered systems: entanglement in random chains; see disordered
References
- P. Calabrese, J. Cardy, J. Stat. Mech. 0406, P06002 (2004) –- original Calabrese-Cardy formula.
- I. Peschel, J. Phys. A 36, L205 (2003) –- free-fermion entanglement from correlation matrix.
- G. Vidal, J. I. Latorre, E. Rico, A. Kitaev, Phys. Rev. Lett. 90, 227902 (2003) –- entanglement entropy in spin chains.