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Physical Review Letters

Nonuniversal and Anomalous Decay of Boundary Spin Correlations in Inhomogeneous Ising Systems

H. J. Hilhorst, J. M. J. van Leeuwen

Published Oct 26, 1981• 80 citations
Publisher

Abstract

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 $m$ from the boundary. By means of a repeated star-triangle transformation the boundary spin-spin correlation $g(\mathcal{r})$ is studied for critical bulk coupling. For $y=1$ 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 $A$. For $0<y<1$ 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 $A$ and $y$ are given.

DOI
10.1103/physrevlett.47.1188

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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−Amy\ensuremath{-}\frac{A}{{m}^{y}}\ensuremath−myA​ 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(r)\ensuremath∼1r\ensuremathηg(\mathcal{r})\ensuremath{\sim}\frac{1}{{\mathcal{r}}^{\ensuremath{\eta}}}g(r)\ensuremath∼r\ensuremathη1​ as r\ensuremath→\ensuremath∞r\ensuremath{\rightarrow}\ensuremath{\infty}r\ensuremath→\ensuremath∞, with a nonuniversal exponent \ensuremathη\ensuremath{\eta}\ensuremathη depending on AAA. For 0<y<10<y<10<y<1 anomalous decay g(r)\ensuremath∼exp[\ensuremath−(r\ensuremathξ)1\ensuremath−y]g(\mathcal{r})\ensuremath{\sim}\mathrm{exp}[\ensuremath{-}{(\frac{r}{\ensuremath{\xi}})}^{1\ensuremath{-}y}]g(r)\ensuremath∼exp[\ensuremath−(\ensuremathξr​)1\ensuremath−y] is found. Expressions for \ensuremathη\ensuremath{\eta}\ensuremathη and \ensuremathξ\ensuremath{\xi}\ensuremathξ in terms of AAA and yyy are given.
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 ${\mathrm{Am}}^{\ensuremath{-}y}$ 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 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.
We consider the critical and off-critical properties at the boundary of the random transverse-field Ising spin chain when the distribution of the couplings and/or transverse fields, at a distance l from the surface, deviates from its uniform bulk value by terms of order ${l}^{\ensuremath{-}\ensuremath{\kappa}}$ with an amplitude A. Exact results are obtained using a correspondence between the surface magnetization of the model and the surviving probability of a random walk with time-dependent absorbing boundary conditions. For slow enough decay, $\ensuremath{\kappa}<1/2,$ the inhomogeneity is relevant: Either the surface stays ordered at the bulk critical point or the average surface magnetization displays an essential singularity, depending on the sign of A. In the marginal situation $\ensuremath{\kappa}=1/2,$ the average surface magnetization decays as a power law with a continuously varying, A-dependent, critical exponent which is obtained analytically. The behavior of the critical and off-critical autocorrelation functions as well as the scaling form of the probability distributions for the surface magnetization and the first gaps are determined through a phenomenological scaling theory. In the Griffiths phase, the properties of the Griffiths-McCoy singularities are not affected by the inhomogeneity. The various results are checked using numerical methods based on a mapping to free fermions.
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