Universal term in the free energy at a critical point and the conformal anomaly

Abstract

We show that the leading finite-size correction to lnZ for a two-dimensional system at a conformally invariant critical point on a strip of length L width \ensuremath{\beta}(\ensuremath{\beta}\ensuremath{\ll}L) is (\ensuremath{\pi}/6)c(L/\ensuremath{\beta}), where c is the conformal anomaly. Equivalently, the leading low-temperature correction to the free energy of a one-dimensional quantum system is -(\ensuremath{\pi}/6)cL(kT${)}^{2}$\ensuremath{\Elzxh}v, where v is the effective ``velocity of light.'' The latter formula is used to check recently derived critical theories of spin-s quantum chains against Bethe-Ansatz solutions.