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Physical review. B, Condensed matter

Role of boundary conditions in the finite-size Ising model

G. G. Cabrera, Rémi Jullien

Published May 1, 1987• 85 citations
Publisher

Abstract

Boundary conditions monitor the finite-size dependence of scaling functions for the Ising model. We study the low-temperature phase for the extremely anisotropic limit, or quantum version of the 2D classical Ising model, by means of combined exact results and large-size numerical calculations. The mass gap (inverse of correlation length) is the suitable order parameter for the finite system, and its finite-size behavior is studied as a function of variable boundary conditions. We find that the well-known exponential convergence to zero of the mass gap is only valid in a limited range of parameters; it strikingly changes into a power law for antiperiodic boundary conditions. We suggest that this puzzling phenomenon is associated with topological excitations.

DOI
10.1103/physrevb.35.7062

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Boundary conditions monitor the finite-size dependence of scaling functions for the Ising model. We study the low-temperature phase for the extremely anisotropic limit, or quantum version of the 2D classical Ising model, by means of combined exact results and large-size numerical calculations. The mass gap (inverse of correlation length) is the suitable order parameter for the finite system, and its finite-size behavior is studied as a function of variable boundary conditions. We find that the well-known exponential convergence to zero of the mass gap is only valid in a limited range of parameters; it strikingly changes into a power law for antiperiodic boundary conditions. We suggest that this puzzling phenomenon is associated with topological excitations.
Finite-size scaling functions are investigated both for the mean-square magnetization fluctuations and for the probability distribution of the magnetization in the one-dimensional Ising model. The scaling functions are evaluated in the limit of the temperature going to zero (T → 0), the size of the system going to infinity (N → ∞) while N[1 − tanh(J/kBT)] is kept finite (J being the nearest neighbour coupling). Exact calculations using various boundary conditions (periodic, antiperiodic, free, block) demonstrate explicitly how the scaling functions depend on the boundary conditions. We also show that the block (small part of a large system) magnetization distribution results are identical to those obtained for free boundary conditions.
First order phase transitions are rounded in solids of finite size. It is shown here that the above rounding is monitored by the correlation length ξL of the finite system, or equivalently by the so-called mass gap for the quantum Hamiltonian version of the model. Scaling with size is studied as a function of variable boundary conditions for the cylinder geometry (infinite strips with finite width), and a striking crossover is found in the mass gap behavior when the coupling g along the boundary becomes anti-periodic. For g>0, the rounding is exponential with size and an accurate determination of the spontaneous magnetization (order parameter) of the infinite system is obtained from numerical extrapolations.
Finite-size scaling functions are investigated both for the mean-square magnetization fluctuations and for the probability distribution of the magnetization in the one-dimensional Ising model. The scaling functions are evaluated in the limit of the temperature going to zero (T -> 0), the size of the system going to infinity (N -> oo) while N[1-tanh(J/k_BT)] is kept finite (J being the nearest neighbor coupling). Exact calculations using various boundary conditions (periodic, antiperiodic, free, block) demonstrate explicitly how the scaling functions depend on the boundary conditions. We also show that the block (small part of a large system) magnetization distribution results are identical to those obtained for free boundary conditions.
I study the universal finite-size scaling function for the lowest gap of the quantum Ising chain with a one-parameter family of ``defect'' boundary conditions, which includes periodic, open, and antiperiodic boundary conditions as special cases. The universal behavior can be described by the Majorana fermion field theory in 1+11+11+1 dimensions, with the mass proportional to the deviation from the critical point. Although the field theory appears to be symmetric with respect to the inversion of the mass (Kramers-Wannier duality), the actual gap is asymmetric, reflecting the spontaneous symmetry breaking in the ordered phase which leads to the two-fold ground-state degeneracy in the thermodynamic limit. The asymptotic ground-state degeneracy in the ordered phase is realized by (i) formation of a bound state at the defect (except for the periodic/antiperiodic boundary condition) and (ii) effective reversal of the fermion number parity in one of the sectors (except for the open boundary condition), resulting in a rather nontrivial crossover ``phase'' diagram in the space of the boundary condition (defect strength) and mass.
We present detailed calculations for the partition function and the free energy of the finite two-dimensional square lattice Ising model with periodic and antiperiodic boundary conditions, variable aspect ratio, and anisotropic couplings, as well as for the corresponding universal free energy finite-size scaling functions. Therefore, we review the dimer mapping, as well as the interplay between its topology and the different types of boundary conditions. As a central result, we show how both the finite system as well as the scaling form decay into contributions for the bulk, a characteristic finite-size part, and – if present – the surface tension, which emerges due to at least one antiperiodic boundary in the system. For the scaling limit we extend the proper finite-size scaling theory to the anisotropic case and show how this anisotropy can be absorbed into suitable scaling variables.