Semi-infinite Two-dimensional Ising Model Research Articles

Marginally inhomogeneous semiinfinite two-dimensional Ising models

The scaling dimensions of the semiinfinite two-dimensional Ising model with smoothly inhomogeneous couplings are determined, in general, by the eigenvalues of a Schrodinger-type equation. The author obtains exact results for a marginal, two-parameter inhomogeneity that corresponds to a supersymmetric quantum mechanical potential. The results are compared with predictions of conformal invariance.

Semi-infinite two-dimensional Ising model with marginally inhomogeneous couplings and conformal invariance

The authors consider a semi-infinite two-dimensional Ising model with nearest-neighbour couplings that deviate from the bulk critical couplings by Al-2, where l is the distance from the surface. The surface critical exponents of this system are non-universal. Under a conformal mapping onto a strip of width L, the Al-1 inhomogeneity transforms into A((L/ pi )sin( pi L/L))-1. For the square lattice the spectrum of the transfer matrix in the strip geometry is calculated exactly in the extreme anisotropic limit. The analytical results and numerical results for the triangular lattice are compared with predictions of conformal invariance.

Correlation functions of semi-infinite two-dimensional ising model. III. influence of a ?fixed? boundary

Correlation functions of semi-infinite two-dimensional ising model. III. influence of a ?fixed? boundary

Bulk, surface, and interface properties of the Ising model and conformal invariance.

By solving partial differential equations for the correlation functions implied by conformal invariance, we obtain exact results at the critical point for the bulk six-point function 〈\ensuremath{\sigma}(1)...\ensuremath{\sigma}(6)〉 of the two-dimensional Ising model and the three-point functions 〈\ensuremath{\sigma}(1)\ensuremath{\sigma}(2)\ensuremath{\sigma}(3)〉, 〈\ensuremath{\sigma}(1)\ensuremath{\mu}(2)\ensuremath{\mu}(3)〉 of the semi-infinite two-dimensional Ising model with fixed boundary spins. Here \ensuremath{\sigma} and \ensuremath{\mu} denote order (spin) and disorder variables, respectively. We also derive the explicit form of the magnetization profile at the bulk critical temperature in an Ising strip of infinite length and finite width, with spin-up boundary conditions on one edge and spin-down boundary conditions on the other.

Surface critical behavior of the smoothly inhomogeneous Ising model

We consider a semi-infinite two-dimensional Ising model with nearest-neighbor coupling constants that deviate from the bulk coupling by $A{m}^{\ensuremath{-}y}$ for large $m$, $m$ being the distance from the edge. The case $Al0$ of couplings which are weaker near the surface has been discussed by Hilhorst and van Leeuwen. We report exact results for the boundary magnetization and boundary pair-correlation function when $Ag0$. At the bulk critical temperature there is a rich variety of critical behavior in the $A \ensuremath{-}y$ plane with both paramagnetic and ferromagnetic surface phases. Some of our results can be derived and generalized with simple scaling arguments.

Real-space renormalisation of the smoothly inhomogeneous semi-infinite Ising model

Using the Migdal-Kadanoff renormalisation method, the authors study the semi-infinite two-dimensional Ising model with inhomogeneous nearest-neighbour coupling constants, that deviate from the bulk coupling by Am-y for large m, m being the distance from the edge. The approximation correctly predicts irrelevance of the inhomogeneity for y>v-1, a surface magnetic phase at the bulk critical temperature for y<v-1, A>0, and a non-universal A-dependent surface magnetic index in the marginal case y=v-1.

Correlation functions of the semi-infinite two-dimensional ising model. II. Two-point correlation functions

Correlation functions of the semi-infinite two-dimensional ising model. II. Two-point correlation functions

Correlation functions of the semi-infinite two-dimensional ising model

Correlation functions of the semi-infinite two-dimensional ising model

Thermodynamics of an anisotropic boundary of a two-dimensional Ising model

We consider a semi-infinite two-dimensional Ising model with its spins on the boundary row having a different interaction energy E′1 from the ferromagnetic bulk. We find that the boundary specific heat has two divergent terms: one of which diverges linearly at the bulk critical temperature Tc, and the other, logaritmically. The linearly divergent term is independent of E′1, and the coefficient of the logaritmically divergent term is a decreasing function of E′1. There is a boundary latent heat at Tc, which is identical to McCoy and Wu's result. The boundary spins, which can be either ferromagnetic or antiferromagnetic, are aligned for temperature lower than Tc. The boundary spontaneous magnetization approaches zero in the form of A(E′1)(1−T/Tc)1/2, and the boundary zero field magnetic susceptibility diverges at Tc in the form −B (E′1)ln|1−T/Tc|, where A(E′1) and B(E′) are increasing functions of E′1.