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Physical review. B, Condensed matter

Boundary magnetization and spin correlations in inhomogeneous two-dimensional Ising systems

Theodore W. Burkhardt, Ihnsouk Guim, H. J. Hilhorst, J. M. J. van Leeuwen

Published Aug 1, 1984• 38 citations
Publisher

Abstract

We consider a system of Ising spins on a semi-infinite triangular lattice with nearest-neighbor coupling constants that depend on the distance $m$ from the boundary. An exact technique for calculating the boundary magnetization and the boundary pair correlation function is described. It relies on repeated application of a mapping based on the star-triangle transformation. The case of coupling constants that differ from the bulk coupling by an amount ${\mathrm{Am}}^{\ensuremath{-}y}$ for large $m$ is examined in detail. For $y<1$ the inhomogeneity of the couplings leads to an interesting variety of modifications in the boundary critical behavior. For $y=1$, $A>{A}_{c}$ (${A}_{c}$ being a positive critical value), and for $y<1$, $A>0$, there is a spontaneous surface magnetization at the bulk critical temperature.

DOI
10.1103/physrevb.30.1486

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We consider a system of Ising spins on a semi-infinite triangular lattice with nearest-neighbor coupling constants that depend on the distance mmm from the boundary. An exact technique for calculating the boundary magnetization and the boundary pair correlation function is described. It relies on repeated application of a mapping based on the star-triangle transformation. The case of coupling constants that differ from the bulk coupling by an amount Am\ensuremath−y{\mathrm{Am}}^{\ensuremath{-}y}Am\ensuremath−y for large mmm is examined in detail. For y<1y<1y<1 the inhomogeneity of the couplings leads to an interesting variety of modifications in the boundary critical behavior. For y=1y=1y=1, A>AcA>{A}_{c}A>Ac​ (Ac{A}_{c}Ac​ being a positive critical value), and for y<1y<1y<1, A>0A>0A>0, there is a spontaneous surface magnetization at the bulk critical temperature.
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 mmm, mmm being the distance from the edge. The case A<0A<0A<0 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 A>0A>0A>0. 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.
A triangular, half-infinite, nearest-neighbor Ising system is studied whose coupling strengths close to the boundary are weaker than those in the bulk. The deviations decay as $\ensuremath{-}\frac{A}{{m}^{y}}$ with the distance mmm from the boundary. By means of a repeated star-triangle transformation the boundary spin-spin correlation g(r)g(\mathcal{r})g(r) is studied for critical bulk coupling. For y=1y=1y=1 the behavior is $g(\mathcal{r})\ensuremath{\sim}\frac{1}{{\mathcal{r}}^{\ensuremath{\eta}}}$ as $r\ensuremath{\rightarrow}\ensuremath{\infty}$, with a nonuniversal exponent $\ensuremath{\eta}$ depending on AAA. For 0<y<10<y<10<y<1 anomalous decay $g(\mathcal{r})\ensuremath{\sim}\mathrm{exp}[\ensuremath{-}{(\frac{r}{\ensuremath{\xi}})}^{1\ensuremath{-}y}]$ is found. Expressions for $\ensuremath{\eta}$ and $\ensuremath{\xi}$ in terms of AAA and yyy are given.
A semi-infinite nearest-neighbor square Ising system whose couplings at a distance lll from the boundary differ from homogeneity by an amount $\ensuremath{\delta}K\ensuremath{\sim}\ensuremath{-}\frac{A}{l}$ is investigated. On the basis of the Pfaffian method we obtain the critical behavior at the surface of this system. The exponents ${\ensuremath{\eta}}_{\ensuremath{\parallel}}$, ${\ensuremath{\nu}}_{\ensuremath{\parallel}}$, ${\ensuremath{\beta}}_{1}$, ${\ensuremath{\gamma}}_{11}$, and ${\ensuremath{\delta}}_{11}$ display rich nonuniversal behavior as a function of the amplitude AAA. For AAA below a critical value, there is exponent asymmetry and a spontaneous surface magnetization when the bulk ($l=\ensuremath{\infty}$) is critical.
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 .
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.