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Physical review. B, Solid state

Kac-Baker model of partially finite spin systems

G. Costache

Published Dec 1, 1976• 1 citation
Publisher

Abstract

The free energy of the Kac-Baker model for spin systems with free surfaces is exactly calculated. The critical temperature is obtained and its shift index $\ensuremath{\lambda}$ evaluated. It is shown that for Ising-like interaction $\ensuremath{\lambda}=\frac{1}{\ensuremath{\nu}}$ ($\ensuremath{\nu}$ is the correlation-length critical index of the bulk), while for the spherical model with overall constraint $\ensuremath{\lambda}=1$; it is argued that the value $\ensuremath{\lambda}=1$ is determined by the imposed constraint. As a byproduct an exact analytical expression for the magnetization of an open Ising chain with random fields is obtained.

DOI
10.1103/physrevb.14.4983

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arXiv:cond-mat/9807025•Physical review. B, Condensed matter•Dec 1, 1998•16 citations
The free energy of the Kac-Baker model for spin systems with free surfaces is exactly calculated. The critical temperature is obtained and its shift index \ensuremathλ\ensuremath{\lambda}\ensuremathλ evaluated. It is shown that for Ising-like interaction \ensuremathλ=1\ensuremathν\ensuremath{\lambda}=\frac{1}{\ensuremath{\nu}}\ensuremathλ=\ensuremathν1​ (\ensuremathν\ensuremath{\nu}\ensuremathν is the correlation-length critical index of the bulk), while for the spherical model with overall constraint \ensuremathλ=1\ensuremath{\lambda}=1\ensuremathλ=1; it is argued that the value \ensuremathλ=1\ensuremath{\lambda}=1\ensuremathλ=1 is determined by the imposed constraint. As a byproduct an exact analytical expression for the magnetization of an open Ising chain with random fields is obtained.
This thesis is devoted to establishing rigorous derivations of exact solutions of the 2D classical Ising model and 1D quantum spin systems via the Kac-Ward method More precisely, we give a rigorous derivation of the free energy of (i) the classical Ising model on the triangular lattice with translation-invariant coupling constants, and (ii) the one-dimensional quantum Ising model. We use the method of the Kac and Ward. The novel aspect is that the coupling constants may have negative signs. We describe the logarithmic singularity of the specific heat of the classical model and the validity of the Cimasoni–Duminil-Copin–Li formula for the critical temperature. We also discuss the quantum phase transition of the quantum model. Furthermore, we explore the possibility of extending the Kac-Ward method to 1D quantum spin systems. These systems enjoy stochastic representations given by random loop models. Specifically, we aim at obtaining exact formulas for their free energy and correlation functions.
Phase transitions in Ising models with tree surfaces are studied from various points of view, including a phenomenological Landau theory, high-temperature series expansions, and a scaling theory for thermodynamic quantities and correlation functions. In the presence of a surface a number of new critical exponents must be defined. These arise because of the existence of "surface" terms in the thermodynamic functions, and because of the anisotropy of space and lack of translational symmetry introduced by the surface. The need for these new critical exponents already appears in the phenomenological theory, which is discussed in detail and related to the microscopic mean-field approximation. The essential new parameter appearing in this theory is an extrapolation length $\ensuremath{\lambda}$ which enters the boundary condition on the magnetization at the surface. For magnetic systems this length is of the order of the interaction range, in contrast to superconductors, where it is usually much larger. In order to go beyond the mean-field theory, high-temperature series expansions are carried out for the Ising half-space, to tenth order in two dimensions and to eighth order in three dimensions. A scaling theory is developed both for thermodynamic functions and for spin correlations near the surface, and relations are found among the exponents of the half-space. Both the scaling theory and the numerical calculations are compared with the exact solution of the Ising half-plane (two dimensions) by McCoy and Wu, and agreement is found wherever the theory is applicable. In analogy to the bulk situation, the scaling theory is found to agree with mean-field theory in four dimensions. The prediction of the present work which is most easily accessible to experiment is the temperature dependence of the magnetization at the surface, with critical exponent estimated to be ${\ensuremath{\beta}}_{1}=\frac{2}{3}$. The mean-field result, ${\ensuremath{\beta}}_{1}=1$, seems to agree more closely with presently available experiment, and more work is needed to clarify the situation.
A model Hamiltonian H=−JΣ(ij)Si·Sj is considered, where Si are isotropically interacting D-dimensional classical spins. Thus the S=½ Ising, classical planar, and classical Heisenberg models are obtained if D=1, 2, and 3, respectively; moreover, as D→∞, the free energy corresponding to H approaches the free energy of the exactly-soluble spherical model of Berlin and Kac. H can also be solved exactly providing the lattice dimensionality d=1. For lattices of higher dimensionality, we must resort to approximation techniques; hence we calculate high-temperature expansions for arbitrary D and d of the susceptibility, internal energy (specific heat), and the second moment. (With a few exceptions, such expansions have been available in the past only for D=1, 3 and d=1–3.) Somewhat smoother series to facilitate more reliable extrapolation are then obtained by re-expanding all of the series in terms of the nearest-neighbor spin correlation function for a one-dimensional lattice. We then study the dependence on spin and lattice dimensionality of the critical temperatures Tc(D, d) and the critical exponents γ(D, d) [(χ∼(T-Tc)−γ], α(D, d) [C∼(T-Tc)−α], ν(D, d) [inverse correlation range κ∼(T-Tc)−v]. These ``critical properties'' are all found to vary monotonically (and in most cases, smoothly) with D and d. The assumptions behind making critical phenomena predictions from a dozen or so terms (of an infinite series!) are given increased plausibility by the fact that we have extended the various expansions for the spherical model to 80–100 terms and find that none of the predictions based upon ∼10 terms is changed [e.g., Tc(∞, 2)=0, γ(∞, 3)=2, etc.].
Dynamical correlations of the spin and the energy density are investigated in the critical region of the random transverse-field Ising chain by numerically exact calculations in large finite systems ( L ⩽ 128). The spin-spin autocorrelation function is found to decay proportionally to (log t ) −2 x m and (log t ) −2 x m s in the bulk and on the surface, respectively, with x m and x m s the bulk and surface magnetization exponents, respectively. On the other hand, the critical energy-energy autocorrelation functions have a power law decay, which are characterized by novel critical exponents η e ≈ 2.2 in the bulk and η e s ≈ 2.5 at the surface, respectively. The numerical results are compared with the predictions of a scaling theory.
Let fN{f}_{N}fN​ and ${\ensuremath{\xi}}_{N}^{\ensuremath{-}1}$ represent, respectively, the free energy per spin and the inverse spin-spin correlation length of the critical Ising model on a $N\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}$ lattice, with ${f}_{N}\ensuremath{\rightarrow}{f}_{\ensuremath{\infty}}$ as $N\ensuremath{\rightarrow}\ensuremath{\infty}$. We obtain analytic expressions for ak{a}_{k}ak​ and bk{b}_{k}bk​ in the expansions $N({f}_{N}{\ensuremath{-}f}_{\ensuremath{\infty}})\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{\Sigma}{k\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1}^{\ensuremath{\infty}}{a}_{k}/{N}^{2k\ensuremath{-}1}$ and ${\ensuremath{\xi}}_{N}^{\ensuremath{-}1}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{\Sigma}{k\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1}^{\ensuremath{\infty}}{b}_{k}/{N}^{2k\ensuremath{-}1}$ for square, honeycomb, and plane-triangular lattices, and find that ${b}_{k}/{a}_{k}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}({2}^{2k}\ensuremath{-}1)/({2}^{2k\ensuremath{-}1}\ensuremath{-}1)$ for all of these lattices, i.e., the amplitude ratio bk/ak{b}_{k}/{a}_{k}bk​/ak​ is universal. We also obtain similar results for a critical quantum spin chain and find that such results could be understood from a perturbated conformal field theory.
We study the dynamical properties of the random transverse-field Ising chain at criticality using a mapping to free fermions with which we can obtain numerically exact results for system sizes L as large as 256. The probability distribution of the local imaginary time correlation function $S(\ensuremath{\tau})$ is investigated and found to be simply a function of $\ensuremath{\alpha}\ensuremath{\equiv}\ensuremath{-}\mathrm{ln}S(\ensuremath{\tau})/\mathrm{ln}\ensuremath{\tau}.$ This scaling behavior implies that the typical correlation function decays algebraically, ${S}_{\mathrm{typ}}(\ensuremath{\tau})\ensuremath{\sim}{\ensuremath{\tau}}^{\ensuremath{-}{\ensuremath{\alpha}}_{\mathrm{typ}}},$ where the exponent ${\ensuremath{\alpha}}_{\mathrm{typ}}$ is determined from $P(\ensuremath{\alpha}),$ the distribution of $\ensuremath{\alpha}.$ The precise value for ${\ensuremath{\alpha}}_{\mathrm{typ}}$ depends on exactly how the ``typical'' correlation function is defined. The form of $P(\ensuremath{\alpha})$ for small $\ensuremath{\alpha}$ gives a contribution to the average correlation function ${S}_{\mathrm{av}}(\ensuremath{\tau})$ namely, ${S}_{\mathrm{av}}(\ensuremath{\tau})\ensuremath{\sim}(\mathrm{ln}\ensuremath{\tau}{)}^{\ensuremath{-}{2x}_{m}},$ where xm{x}_{m}xm​ is the bulk magnetization exponent, which was reported recently in Europhys. Lett. 39, 135 (1997). These results represent a type of ``multiscaling'' different from the well-known ``multifractal'' behavior.