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Physical review. B, Condensed matter

Surface behavior of the fully frustrated two-dimensional Ising model

Ihnsouk Guim, Theodore W. Burkhardt, Tianyou Xue

Published Dec 1, 1990• 2 citations
Publisher

Abstract

We consider the fully frustrated Villain or ``odd'' Ising model on a semi-infinite square lattice with surface couplings ${\mathit{J}}_{\mathit{S}}$ and bulk couplings \ifmmode\pm\else\textpm\fi{}J. The zero-temperature properties of this model have an interpretation in terms of dimers. From previous work it is known that at T=0 there is no bulk long-range order and the bulk spin-spin correlation function decays algebraically with critical exponent \ensuremath{\eta}=1/2. We analyze the surface behavior in the T=0 transition. For ${\mathit{J}}_{\mathit{S}}$\ensuremath{\ne}1/2J the surface spins do order at T=0, and the connected pair-correlation function decays as ${\mathit{r}}^{\mathrm{\ensuremath{-}}2}$ parallel to the surface. For ${\mathit{J}}_{\mathit{S}}$=1/2J there is no long-range surface order at T=0, and the pair-correlation function of the surface spins decays as ${\mathit{r}}^{\mathrm{\ensuremath{-}}1}$ and ${\mathit{r}}^{\mathrm{\ensuremath{-}}3}$ for r even and odd, respectively. We also calculate the free energy, the correlation length, and the profile of the energy density in strips of finite width and compare the results with predictions of conformal invariance for ordinary critical systems.

DOI
10.1103/physrevb.42.10298

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We consider the fully frustrated Villain or ``odd'' Ising model on a semi-infinite square lattice with surface couplings JS{\mathit{J}}_{\mathit{S}}JS​ and bulk couplings \ifmmode\pm\else\textpm\fi{}J. The zero-temperature properties of this model have an interpretation in terms of dimers. From previous work it is known that at T=0 there is no bulk long-range order and the bulk spin-spin correlation function decays algebraically with critical exponent \ensuremath{\eta}=1/2. We analyze the surface behavior in the T=0 transition. For JS{\mathit{J}}_{\mathit{S}}JS​\ensuremath{\ne}1/2J the surface spins do order at T=0, and the connected pair-correlation function decays as r\ensuremath−2{\mathit{r}}^{\mathrm{\ensuremath{-}}2}r\ensuremath−2 parallel to the surface. For JS{\mathit{J}}_{\mathit{S}}JS​=1/2J there is no long-range surface order at T=0, and the pair-correlation function of the surface spins decays as r\ensuremath−1{\mathit{r}}^{\mathrm{\ensuremath{-}}1}r\ensuremath−1 and r\ensuremath−3{\mathit{r}}^{\mathrm{\ensuremath{-}}3}r\ensuremath−3 for r even and odd, respectively. We also calculate the free energy, the correlation length, and the profile of the energy density in strips of finite width and compare the results with predictions of conformal invariance for ordinary critical systems.
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.
The spin-spin correlation function in the ground state of some two-dimensional fully frustrated spin models are investigated. The correlations are long ranged and decay as $\frac{1}{{r}^{\ensuremath{\eta}}}$, where rrr is the distance between the two spins and $\ensuremath{\eta}=\frac{1}{2}$. This result is true for both the square and triangular fully frustrated lattices, thus suggesting that these models form a universality class. It is shown that the fully frustrated Ising model on a square lattice can be mapped exactly into a special Baxter model. The spin-spin correlation function of this Baxter model can be calculated exactly. The ground state of the fully frustrated square Ising model corresponds to the free fermion point of the FFF model and therefore, through duality, to the decoupling point of the critical Baxter line. The relationship between the FFF model above the critical temperature and the Gaussian model above the multicritical point is discussed.
We consider long, finite-width strips of Ising spins with randomly distributed couplings. Frustration is introduced by allowing both ferromagnetic and antiferromagnetic interactions. Free energy and spin-spin correlation functions are calculated by transfer-matrix methods. Numerical derivatives and finite-size scaling concepts allow estimates of the usual critical exponents $\ensuremath{\gamma}/\ensuremath{\nu},$ $\ensuremath{\alpha}/\ensuremath{\nu},$ and $\ensuremath{\nu}$ to be obtained, whenever a second-order transition is present. Low-temperature ordering persists for suitably small concentrations of frustrated bonds, with a transition governed by pure-Ising exponents. Contrary to the unfrustrated case, subdominant terms do not fit a simple, logarithmic-enhancement form. Our analysis also suggests a vertical critical line at and below the Nishimori point. Approaching this point along either the temperature axis or the Nishimori line, one finds nondiverging specific heats. A percolationlike ratio $\ensuremath{\gamma}/\ensuremath{\nu}$ is found upon analysis of the uniform susceptibility at the Nishimori point. Our data are also consistent with frustration inducing a breakdown of the relationship between correlation-length amplitude and critical exponents, predicted by conformal invariance for pure systems.
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.
The authors present a simple argument showing that the surface energy density of a semi-infinite d-dimensional spin system has, in general, a leading thermal singularity of the same form, mod T-T c mod 2- alpha , as the bulk free energy. At bulk criticality energy-energy correlations decay as r -2d parallel to the surface. These results hold for both free and fixed boundary spins, i.e. the 'ordinary' and 'extraordinary' transitions. The extraordinary critical behaviour of the magnetisation is the same as that of the energy density. They also confirm and generalise these results in two dimensions with an independent approach based on conformal invariance.
The fully frustrated XY model with Villain interaction on a square lattice is studied by means of Monte Carlo simulations. On the basis of the universal jump condition it is argued that there are two distinct transitions in the model, corresponding to the loss of XY order and Z2{\mathrm{Z}}_{2}Z2​ order, respectively. The Kosterlitz-Thouless (KT) transition is analyzed by finite-size scaling of the helicity modulus at lattices of size L=32 through 128, giving TKT{\mathrm{T}}_{\mathrm{KT}}TKT​\ensuremath{\approx}0.8108(1). The vorticity-vorticity correlation function is used to determine two different characteristic lengths, the Z2{\mathrm{Z}}_{2}Z2​ correlation length \ensuremath{\xi}, and the screening length \ensuremath{\lambda}, associated with the KT transition and free vortices. The temperature dependence of \ensuremath{\xi} is examined in order to determine Tc{\mathrm{T}}_{\mathrm{c}}Tc​ and the correlation length exponent, \ensuremath{\nu}. The exponent is found to be consistent with the two-dimensional (2D) Ising value, \ensuremath{\nu}=1, and the obtained critical temperature is Tc{\mathrm{T}}_{\mathrm{c}}Tc​=0.8225(5). The determinations of both \ensuremath{\xi} and \ensuremath{\nu} are done carefully, first applying the techniques to the 2D Ising model, which serves as a convenient testing ground.