For pt.I see ibid., vol.14, p.241-57 (1981). Two efficient methods for finding the low-lying states of Hamiltonians on finite lattices are described. The first involves constructing a finite representation of the Hamiltonian using strong-coupling eigenstates, while the second is based on the Lanczos recursion method. The methods are used to determine the mass gap of the O(2) and O(3) Heisenberg Hamiltonians in (1+1) dimensions for a sequence of finite chains. The critical behaviour of the infinite chain is then analysed by extrapolating the finite-lattice estimates using finite-size scaling. A remarkably sensitive test is developed for the presence of a phase transition. For the O(2) model data this test yields strong evidence for a phase transition with the weak-coupling phase massless, while, in the O(3) case the test supports, although more weakly, the absence of any transition.
The critical behaviour of the spin- Ising model with a transverse field at zero temperature in triangular and square lattices is investigated through a finite-size rescaling transformation. This technique enables the critical properties of infinite systems to be calculated from the scaling properties of systems of finite size, in a generalisation of the so-called phenomenological renormalisation group. The critical field and correlation length exponent calculated using systems of fairly small sizes have reasonable accuracy, but calculations on larger clusters are limited by computer time and storage.
Abstract A number of numerical calculations, like Monte Carlo or transfer matrix calculations, are performed with systems in which the size in several or all dimensions is finite. To extrapolate the results to the infinite system, it is thus necessary to understand how the infinite size limit is reached. In particular in a system in which the forces are short range, no phase transition can occur in a finite volume, or in a geometry in which the size is infinite only in one dimension. This indicates that the infinite-size extrapolation is somewhat non-trivial. In this chapter, the problem is analysed in the case of second-order phase transitions, in the framework of the N-vector model. The existence of a finite-size scaling is established, extending renormalization group (RG) arguments to this new situation. Then, finite volume geometry and cylindrical geometry, in which the size is finite in all dimensions except one, are distinguished. It is explained how to adapt the methods used in the case of infinite systems to calculate the new universal quantities appearing in finite-size effects, for example, in d = 4−ϵ or d = 2+ϵ dimensions. Special properties of the commonly used periodic boundary conditions are emphasized.
In this paper we propose an efficient routine to carry out finite size scaling studies of one-dimensional massive lattice models. Unlike massless systems, for which the lattice and continuum models yield the same results for low-lying excitations due to the infinite correlation length, the massive continuum models can at best approximate their lattice counterparts because of the intrinsic length scale $\ensuremath{\xi}\ensuremath{\sim}{\ensuremath{\Delta}}^{\ensuremath{-}1},$ where $\ensuremath{\Delta}$ is the mass gap. On examples of antiferromagnetic gapped spin-1/2
XXZ chains we show explicitly that several relations between physical quantities like the mass gap, spin-wave velocity
v, and the correlation length \ensuremath{\xi}, derived from the continuum models deviate significantly from the numerical results obtained for finite chains, when $\ensuremath{\xi}$ is comparable with the lattice spacing
a0. On the other hand, we find if the dispersion for elementary excitations is modified from the ``Lorentz invariant'' form $\sqrt{{\ensuremath{\Delta}}^{2}{+v}^{2}{p}^{2}}$ to $\sqrt{{\ensuremath{\Delta}}^{2}{+v}^{2}{\mathrm{sin}}^{2}p},$ the spin-wave velocity
v becomes almost size independent, and the appropriately defined correlation length using that dispersion agrees very well with numerical results. In fact, this follows from the Bethe ansatz solution for the spin-1/2
XXZ chains. Based on our previous experience and these considerations we propose scaling equations to obtain the ground-state energy, mass gap, correlation length, spin-wave velocity, and scattering length between the massive elementary excitations. Only three low energy levels are needed in this method. Although the method is illustrated on the gapped
XXZ quantum spins, it is applicable to all one-dimensional lattice models with massive relativistic low energy dispersions. To substantiate our statement we show explicitly that the numerical data for the Ising model in a transverse magnetic field fully agree with the finite-size scaling based on the correlation length newly defined from the exact analytic solution.