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Finite-Size Scaling in the transverse Ising Model on a Square Lattice

C. J. Hamer

Published Jan 1, 2000• 16 citations
OpenAlex

Abstract

Energy eigenvalues and order parameters are calculated by exact diagonalization for the transverse Ising model on square lattices of up to 6x6 sites. Finite-size scaling is used to estimate the critical parameters of the model, confirming universality with the three-dimensional classical Ising model. Critical amplitudes are also estimated for both the energy gap and the ground-state energy.

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Finite-size scaling in the three-dimensional Ising model

Michael N. Barber, Robert B. Pearson, D. Toussaint +1 more

We have performed extensive studies of finite-size scaling in the three-dimensional Ising model, using a specially designed Ising processor. We show results for the susceptibility, the energy and specific heat, and the connected part of the fourth moment of the magnetization. Corrections to finite-size scaling are found to be small only for lattices of linear dimension greater than about L=24. This effect severely limits the accuracy to which critical parameters can be estimated. We estimate ${K}_{c}$=0.221 650\ifmmode\pm\else\textpm\fi{}0.000 005 for the critical coupling, while exponent estimates are consistent with those obtained by other methods.

Physical review. B, Condensed matter•Aug 1, 1985•134 citations
Energy eigenvalues and order parameters are calculated by exact diagonalization for the transverse Ising model on square lattices of up to 6x6 sites. Finite-size scaling is used to estimate the critical parameters of the model, confirming universality with the three-dimensional classical Ising model. Critical amplitudes are also estimated for both the energy gap and the ground-state energy.
We investigate the critical parameters of an order-disorder quantum phase transition in the spin-½ J - J ' Heisenberg and XY antiferromagnets on a square lattice. Based on the excitation gaps calculated by the exact diagonalization technique for systems up to 32 spins and finite-size scaling analysis we estimate the critical couplings and exponents of the correlation length for both models. Our analysis confirms the universal critical behaviour of these quantum phase transitions: they belong to 3D O(3) and 3D O(2) universality classes, respectively.
We have adjusted the density-matrix renormalization method to handle two-dimensional systems of limited width. The key ingredient for this extension is the incorporation of symmetries in the method. The advantage of our approach is that we can force certain symmetry properties to the resulting ground-state wave function. Combining the results obtained for system sizes up to $30\ifmmode\times\else\texttimes\fi{}6$ and finite-size scaling, we derive the phase-transition point and the critical exponent for the gap in the Ising model in a transverse field on a two-dimensional square lattice.
We have performed extensive studies of finite-size scaling in the three-dimensional Ising model, using a specially designed Ising processor. We show results for the susceptibility, the energy and specific heat, and the connected part of the fourth moment of the magnetization. Corrections to finite-size scaling are found to be small only for lattices of linear dimension greater than about L=24. This effect severely limits the accuracy to which critical parameters can be estimated. We estimate Kc{K}_{c}Kc​=0.221 650\ifmmode\pm\else\textpm\fi{}0.000 005 for the critical coupling, while exponent estimates are consistent with those obtained by other methods.