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arXiv:cond-mat/9606118 · Physical Review Letters

Common Trends in the Critical Behavior of the Ising and Directed Walk Models

Ferenc Iglói, Loïc Turban

Published Aug 12, 1996• 35 citations
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Abstract

We consider layered two-dimensional Ising and directed walk models and show that the two problems are inherently related. The information about the zero-field thermodynamical properties of the Ising model is contained in the transfer matrix of the directed walk. For several hierarchical and aperiodic distributions of the couplings, critical exponents for the two problems are obtained exactly through renormalization.

DOI
10.1103/physrevlett.77.1206

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We consider layered two-dimensional Ising and directed walk models and show that the two problems are inherently related. The information about the zero-field thermodynamical properties of the Ising model is contained in the transfer matrix of the directed walk. For several hierarchical and aperiodic distributions of the couplings, critical exponents for the two problems are obtained exactly through renormalization.
We consider the Ising model and the directed walk on two-dimensional layered lattices and show that the two problems are inherently related: The zero-field thermodynamical properties of the Ising model are contained in the spectrum of the transfer matrix of the directed walk. The critical properties of the two models are connected to the scaling behavior of the eigenvalue spectrum of the transfer matrix which is studied exactly through renormalization for different self-similar distributions of the couplings. The models show very rich bulk and surface critical behaviors with nonuniversal critical exponents, coupling-dependent anisotropic scaling, first-order surface transition, and stretched exponential critical correlations. It is shown that all the nonuniversal critical exponents obtained for the aperiodic Ising models satisfy scaling relations and can be expressed as functions of varying surface magnetic exponents.
We consider the critical behavior of two-dimensional layered Ising models where the exchange couplings between neighboring layers follow hierarchical sequences. The perturbation caused by the nonperiodicity could be irrelevant, relevant, or marginal. For marginal sequences we have performed a detailed study, which involved analytical and numerical calculations of different surface and bulk critical quantities in the two-dimensional classical as well as in the one-dimensional quantum version of the model. The critical exponents are found to vary continuously with the strength of the modulation, while close to the critical point the system is essentially anisotropic: the correlation length is diverging with different exponents along and perpendicular to the layers.
The equivalence of the critical behavior of the ddd-dimensional Ising model at a finite temperature to that of a ddd- 1 dimensional system at zero temperature as a function of a transverse magnetic field is used to calculate the critical exponents for the three-dimensional model by applying the Niemeijer-Van Leeuwen renormalization-group method to the two-dimensional problem.
The author considers layered systems with aperiodically modulated couplings and study the effect of inhomogeneity on the critical behaviour. Using scaling arguments a relevance/irrelevance-type criterion is formulated and for relevant inhomogeneities the scaling form of singular quantities is determined. The author has performed exact calculations for the surface magnetization of aperiodic quantum Ising chains and for a (1+1)-dimensional directed polymer in a longitudinal aperiodic environment. The results for both systems are in accord with scaling considerations. For a marginal aperiodicity the critical exponents are found to be non-universal.
The ddd-dimensional Ising transfer matrix is expressed exactly in terms of Fermi-type operators. In two dimensions this leads to the standard solution of the model. In three or more dimensions the transfer matrix is no longer a quadratic form in the Fermi variables, but rather contains quartic and higher-order terms. However, it can be expressed as a well-defined expansion about a quadratic form in Fermi operators. The zeroth-order transfer-matrix-called herein the "fermion model"---can be exactly diagonalized, and is well approximated in the critical region by ${V}^{(0)}\ensuremath{\approx}{[2sinh(2K)]}^{\frac{N}{2}}\mathrm{exp}\left[\ensuremath{-}N\ensuremath{\int}d\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}{\left({\ensuremath{\kappa}}^{2}+\ensuremath{\Sigma}{n=1}^{d\ensuremath{-}1}\ensuremath{\Sigma}{m=1}^{d\ensuremath{-}1}{q}_{n}{q}_{m}\right)}^{\frac{1}{2}}t(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}})\right]$where NNN is the number of sites per layer, $\ensuremath{\mu}$ is the inverse correlation length, $t(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}})$ takes values \ifmmode\pm\else\textpm\fi{}1/2, KKK is the reduced inverse temperature, and the integral extends over a Brillouin zone of the ($d\ensuremath{-}1$)-dimensional layer. The critical point of the model is located by the equation ${x}_{c}^{d}+{x}_{c}^{d\ensuremath{-}1}+{x}_{c}=1$, where ${x}_{c}={e}^{\ensuremath{-}2{K}_{c}}$. The critical exponent $\ensuremath{\nu}$ describing the divergence of the correlation length at Kc{K}_{c}Kc​ is found to take on its two-dimensional value, $\ensuremath{\nu}=1$, in all dimensions ddd. The possibility of using the fermion model as the basis for an expansion of the Ising critical exponents as a function of $\ensuremath{\delta}(=d\ensuremath{-}2)$ is pointed out.