CFT Finite-Size Casimir Correction
Overview
For a 1+1D conformal field theory at a critical point with central charge $c$ and CFT velocity $v$, the ground-state energy on a finite system of size $L$ admits a universal $1/L$ correction (Cardy 1986; Blöte–Cardy–Nightingale 1986; Affleck 1986):
\[E_0^{\mathrm{PBC}}(L) = L\,\varepsilon_\infty - \frac{\pi c v}{6 L} + O(L^{-2})\]
\[E_0^{\mathrm{OBC}}(L) = L\,\varepsilon_\infty + \varepsilon_{\mathrm{surf}} - \frac{\pi c v}{24 L} + O(L^{-2})\]
The PBC term comes from quantising the CFT on a cylinder of circumference $L$; the OBC term comes from the corresponding strip. Their ratio is exactly 4, a kinematic consequence of the conformal map.
QAtlas exposes only the universal $1/L$ correction term — the CasimirEnergyCorrection quantity — via Universality{C} dispatch. The extensive piece $L\,\varepsilon_\infty$ and the OBC surface term $\varepsilon_{\mathrm{surf}}$ are model-specific and live on the model side (e.g. TFIM ground-state energy at the critical field).
API
using QAtlas
# 1+1D Ising at criticality, periodic chain of size L = 16, v = 2J = 2
QAtlas.fetch(Universality(:Ising), CasimirEnergyCorrection(), PBC(); L=16.0, v=2.0)
# -> -π/96 ≈ -0.0327249...
# Same Ising, OBC
QAtlas.fetch(Universality(:Ising), CasimirEnergyCorrection(), OBC(); L=16.0, v=2.0)
# -> -π/384 ≈ -0.0081812...
# Heisenberg chain (SU(2)_1 WZW; v = (π/2) J at the AFM point)
QAtlas.fetch(Universality(:Heisenberg), CasimirEnergyCorrection(), PBC(); L=16.0, v=π/2)The CFT velocity v is model-dependent and supplied by the caller. Each row below quotes the velocity in the spin convention used by the named model in QAtlas, so that passing model.J (and model.h) through this column gives the correct numerical v:
| Model | Spin convention | Velocity $v$ at criticality |
|---|---|---|
TFIM, $h = J$ | Pauli, $H = -J\sum\sigma^z\sigma^z - h\sum\sigma^x$ | $v = 2J$ |
| AFM Heisenberg chain | $\mathbf{S} = oldsymbol{\sigma}/2$, $H = J\sum \mathbf{S}\cdot\mathbf{S}$ | $v = (\pi/2)\,J$ |
| XXZ Luttinger liquid | $\mathbf{S} = oldsymbol{\sigma}/2$, $H = J\sum(S^xS^x+S^yS^y+\Delta S^zS^z)$ | $v = \pi J\,\sin(\gamma)/\gamma$ (Bethe ansatz, $\Delta=\cos\gamma$) |
If a downstream code uses Pauli normalisation ($\mathbf{S}=oldsymbol{\sigma}$ without the $1/2$ factor) for a spin-$ frac12$ chain, all three velocities pick up a factor of $4$ relative to this table.
QAtlas already exposes LuttingerVelocity / FermiVelocity / SpinWaveVelocity for the relevant models — read those, then pass the value through as the v kwarg.
Supported Universality Classes
C | $c$ | 1+1D CFT origin |
|---|---|---|
:Ising | $1/2$ | Virasoro minimal model $\mathcal{M}(3,4)$ |
:Potts3 | $4/5$ | Virasoro minimal model $\mathcal{M}(5,6)$ |
:Potts4 | $1$ | Free-boson radius limit (marginal) |
:XY | $1$ | Compact free boson / 1+1D Luttinger liquid |
:Heisenberg | $1$ | $\mathrm{SU}(2)_1$ Wess–Zumino–Witten model |
Other classes (:KPZ, :Percolation, :MeanField, …) raise ErrorException:
- KPZ is non-equilibrium and has no CFT central charge in the Cardy sense.
- Percolation has $c = 0$ but the underlying CFT is logarithmic (non-unitary); the simple Cardy formula does not apply in the same form.
- Mean-field lives above the upper critical dimension, with no 1+1D CFT representation.
Verification properties
PBC : OBC ratio = 4 (kinematic, class-independent):
\[\frac{E_{0,\mathrm{Casimir}}^{\mathrm{PBC}}}{E_{0,\mathrm{Casimir}}^{\mathrm{OBC}}} = \frac{-\pi c v / (6 L)}{-\pi c v / (24 L)} = 4.\]
$L \to \infty$ $\Rightarrow$ value $\to 0$ as $1/L$.
Sign: always negative (the conformal cylinder lowers the ground-state energy below the bulk value).
These properties are exercised in test/standalone/test_universality_cft_casimir.jl.
Phase 2 (TODO)
The conformal tower of states —
\[E_n - E_0 = \frac{2\pi v}{L}\,(h_n + \bar h_n) + O(L^{-2})\]
with primary scaling dimensions $(h, \bar h)$ — is tracked separately as future work (issue #150 Phase 2) and will be exposed via a ConformalTower quantity once implemented. For 2D Ising the primary spectrum is
\[\{(0,0),\ (1/16, 1/16),\ (1/2, 1/2)\}\]
(identity, spin field $\sigma$, energy field $\varepsilon$); see Ising universality for the corresponding scaling dimensions.
References
- J. Cardy, "Operator content of two-dimensional conformally invariant theories", Nucl. Phys. B 270, 186 (1986).
- H. W. J. Blöte, J. L. Cardy, M. P. Nightingale, "Conformal invariance, the central charge, and universal finite-size amplitudes at criticality", Phys. Rev. Lett. 56, 742 (1986).
- I. Affleck, "Universal term in the free energy at a critical point and the conformal anomaly", Phys. Rev. Lett. 56, 746 (1986).