I present reaults of simulations of the q=10 and q=20 2-d Potts models in the transition region. The asymptotic finite size behavior sets in only for extremely large lattices. We learn from this simulation that finite size scaling cannot be used to decide that a transition is first order.
In this paper, I give an overview over a recently developed rigorous theory of finite-size scaling near first order transitions. Leaving out details of the mathematical proofs, the main emphasis is put on the underlying physical ideas and the discussion of the validity of the results for regions which are, in the sense of mathematical rigour, not covered by the original papers. I present both the finite-size scaling for cubic systems and for long cylinders, discussing also a recent controversy which stems from our work on the finite-size scaling of the mass gap in long cylinders.
We develop a detailed finite-size-scaling theory at a general, asymmetric, temperature-driven, strongly first-order phase transition in a system with periodic boundary conditions. We compute scaling functions for various cumulants of energy in the form U(L,t)=U0(tLd)+${\mathit{L}}^{\mathrm{\ensuremath{-}}\mathit{d}}$U1(tLd) with t=1-Tc/T. In particular, we consider the specific heat and Binder's fourth cumulant and show this has a minimum value of 2/3-(e1/e2-e2/e1)2/12+O(${\mathit{L}}^{\mathrm{\ensuremath{-}}\mathit{d}}$) at a temperature Tc(2)(L)-Tc=O(${\mathit{L}}^{\mathrm{\ensuremath{-}}\mathit{d}}$). Various other pseudocritical temperatures corresponding to extrema of other cumulants are evaluated. We compare these theoretical predictions with extensive Monte Carlo simulations of the nominally strong first-order transitions in the eight- and ten-state Potts models in two dimensions for system sizes L\ensuremath{\le}50. The ten-state simulations agree with theory in all details in contrast to the eight-state data, and we give estimates for the bulk specific heats at Tc using all exactly known analytic results. A criterion is developed to estimate numerically whether or not system sizes used in a simulation of a first-order transition are in the finite-size-scaling regime.
We investigate finite-size effects at first-order quantum transitions. For this purpose we consider the one-dimensional q-state quantum Potts chain, in particular with q=10, which undergoes a first-order transition, separating the quantum disordered and ordered phases with a discontinuity in the energy density of the ground state. In agreement with the general theory, around the transition the low-energy properties show finite-size scaling with respect to appropriate scaling variables. Their size dependence is particularly sensitive to boundary conditions, which is a specific feature of first-order quantum transitions. Finally, we also discuss the finite-size behavior of the q-state Potts model (q≥2) at the first-order transitions driven by a parallel magnetic field, occurring in the ferromagnetic phase.
We present a large- q expansion of the 2d q -states Potts model free energies up to order 9 in 1/√ q . Its analysis leads us to an ansatz which, in the first-order region, incorporates properties inferred from the known critical regime at q = 4, and predicts, for q > 4, the n -th energy cumulant scales as the power (3 n /2 - 2) of the correlation length. The parameter-free energy distributions reproduce accurately, without reference to any interface effect, the numerical data obtained in a simulation for q = 10 with lattices of linear dimensions up to L = 50. The pure-phase specific heats are predicted to be much larger, at q ⩽ 10, than the values extracted from the current finite-size scaling analysis of the extrema. Implications for safe numerical determinations of interface tensions are discussed.