Spectral flow between conformal field theories in 1 + 1 dimensions

Abstract

Consider the φ 1,3-perturbed minimal unitary conformal field theories M p (with diagonal modular invariant partition function) of central charge c p = 1 − ( 6 p(p + 1) ), p = 3, 4,… It is believed that for one sign of the perturbing parameter they are massive scattering theories of kinks, whereas for the other they are massless but not scale-invariant quantum field theories interpolating between M p in the UV and M p−1 in the IR limit. We propose integral equations for the exact finite-volume energies of the first and kth excited states in φ 1,3- perturbed M 2k + 2 on the cylinder (of circumference R). These integral equations are similar to the thermodynamic Bethe Ansatz equations recently proposed by Al. Zamolodchikov for the ground-state energy in these theories. The behaviour of the conjectured expressions for the finite-volume energies at small and large R, which we investigate analytically and numerically, is in excellent agreement with predictions of conformal perturbation theory around the UV and IR fixed point, respectively, providing strong support for our proposal. In particular, we study in some detail the flow of the spin field ( dimension d = 3 40 ) of the tricritical Ising model M 4 to the spin field (d = 1 8 ) of the critical Ising model M 3 . For φ 1,3- perturbed M 4 we also compare the small- R behaviour of the conjectured first energy-gap with results from the “truncated conformal-space approach”, and discuss the limitations of the latter method when conformal perturbation theory has UV divergencies.