Manifest Modular Invariance in the Near-Critical Ising Model

Abstract

Using recent results in mathematics, I point out that free energies and scale-dependent central charges away from criticality can be represented in compact form where modular invariance is manifest. The main example is the near-critical Ising model on a thermal torus, but the methods are not restricted to modular symmetry, and apply to automorphic symmetries more generally. One application is finite-size effects.

Figures (8)

• Figure 1: A point $w=\alpha \omega_2+\beta\omega_1$ in terms of $\tau=\omega_2/\omega_1$ is $z=\alpha\tau+\beta$. Switching the lattice basis vectors as $\tilde{\omega}_1=\omega_2$ and $\tilde{\omega}_2=-\omega_1$ (S modular transformation) makes $\tilde{\tau}=-1/\tau$ and $\tilde{z}=z/\tau$. Here $\sim \tilde{z}$ means shifted by a lattice vector from $\tilde{z}$.
• Figure 2: Running central charges for two fermion quasiperiodicities as function of the mass parameter.
• Figure 3: The Mellin-dual energy $\widetilde{E}(s)$ in the complex $s$ plane. The three arrows give an idea of the integration contour, which closes at infinity. The argument ${\rm arg}(\widetilde{E}(s))$ sets the color.
• Figure 4: Black: Bessel sum representation of the zero-point energy $E(\mu)$ in \ref{['restot']}. Blue dashed: truncation of eq. \ref{['close']} at order 4,8,12 in $\mu$. For large mass, $E(\mu)$ becomes negligible, as expected.
• Figure 5: Black: the free energy building block ${\mathcal{E}}_{1,\mu}$ in \ref{['FEB']}. Blue dashed: truncation of eq. \ref{['muexp']} at order 2,3 in $\mu$. The $E_0'$ value was fixed to $-0.1$ to match the exact result (cf. eq. \ref{['E0vals']}).