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Journal of Physics C Solid State Physics

Correlation function and transition temperature bounds for bond-disordered Ising systems

Harold Falk, G A Gehring

Published Jul 7, 1975• 20 citations
Publisher

Abstract

For spin 1 / 2 Ising models with ferromagnetic pair interactions J ij which are taken to be statistically independent, non-negative, random variables, the authors show that the averaged two-spin correlation function ( sigma p sigma q ) is less than or equal to the value ( sigma p sigma q ) J for the corresponding non-random system in which each J ij is replaced by its average value J ij . The inequality implies for sufficiently high temperatures that the averaged isothermal susceptibility chi <or= chi j . With additional assumptions about the existence and nature of the phase transitions for the systems, one concludes that the critical temperature tau c for the averaged system is not greater than the corresponding critical temperature tau (J) for the nonrandom system. In the process of obtaining the results it is noticed that Griffiths' inequalities ( sigma p sigma q )>or=0 and delta ( sigma p sigma q )/ delta J rs >or=0 imply that delta 2 ( sigma p sigma q )/ delta J rs 2 <or=0.

DOI
10.1088/0022-3719/8/13/005

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For spin 1 / 2 Ising models with ferromagnetic pair interactions J ij which are taken to be statistically independent, non-negative, random variables, the authors show that the averaged two-spin correlation function ( sigma p sigma q ) is less than or equal to the value ( sigma p sigma q ) J for the corresponding non-random system in which each J ij is replaced by its average value J ij . The inequality implies for sufficiently high temperatures that the averaged isothermal susceptibility chi <or= chi j . With additional assumptions about the existence and nature of the phase transitions for the systems, one concludes that the critical temperature tau c for the averaged system is not greater than the corresponding critical temperature tau (J) for the nonrandom system. In the process of obtaining the results it is noticed that Griffiths' inequalities ( sigma p sigma q )>or=0 and delta ( sigma p sigma q )/ delta J rs >or=0 imply that delta 2 ( sigma p sigma q )/ delta J rs 2 <or=0.
Studies a correlation function which is given by the canonical average of a product of one or more spin variables, for the random-bond Ising model, in which the exchange integrals are +J(J>O) and -J with probabilities p and 1-p, respectively. The authors show that an upper bound to the configurational average of the correlation functions, calculated in the thermodynamic limit in the zero external field limit, is the product of the same correlation function for the corresponding ferromagnetic Ising model at the temperature under consideration and the same quantity at the temperature T 1 which is determined by the condition T 1 =2J/ mod k B 1n(p/(1-p)) mod , where k B is the Boltzmann constant. The results are given for the random-bond Ising model of an arbitrary spin S, and also for the diluted random-bond Ising models of an arbitrary spin S, with the pair interaction and with general interaction, and for the diluted and undiluted random-bond n-vector model.
It is shown that for any dynamical quantity phi associated with an impurity system the variance, var ( phi ) p calculated after averaging the partition function, is greater than or equal to the corresponding variance var ( phi ) q calculated according to the usual quenched prescription in which the free energy is averaged. In particular, if chi p , chi q denote the respective isothermal susceptibilities subject to the same conditions, then chi p >or= chi q . This is a general result valid for classical and quantum systems in which the susceptibility is proportional to a variance, and it implies that if a quenched spin system displays a phase transition at a temperature of T c q at which a certain variance, say the susceptibility chi q diverges, then chi p must also be infinite at the same temperature. It is also shown that for a large class of spin S ferromagnetic Ising systems which satisfy the conditions of Griffiths' correlation inequalities and other specified conditions, the covariance between any two spins s i and s j is similarly ordered, i.e. cov (s i ,s j ) p >or=cov (s i ,s j ) q .
We consider long strips of finite width L\ensuremath{\le}13 sites of ferromagnetic Ising spins with random couplings distributed according to the binary distribution P(Jij{\mathit{J}}_{\mathit{ij}}Jij​)=1/2[\ensuremath{\delta}(Jij{\mathit{J}}_{\mathit{ij}}Jij​-J0{\mathit{J}}_{0}J0​)+\ensuremath{\delta}(Jij{\mathit{J}}_{\mathit{ij}}Jij​- rJ0{\mathit{rJ}}_{0}rJ0​)], 0r1. Spin-spin correlation functions 〈${\mathrm{\ensuremath{\sigma}}}_{0}$${\mathrm{\ensuremath{\sigma}}}_{\mathit{R}}$〉 along the ``infinite'' direction are computed by transfer-matrix methods, at the critical temperature of the corresponding two-dimensional system, and their probability distribution is investigated. We show that, although in-sample fluctuations do not die out as strip length is increased, averaged values converge satisfactorily. These latter are very close to the critical correlation functions of the pure Ising model, in agreement with recent Monte Carlo simulations. A scaling approach is formulated, which provides the essential aspects of the R and L dependence of the probability distribution of ln〈${\mathrm{\ensuremath{\sigma}}}_{0}$${\mathrm{\ensuremath{\sigma}}}_{\mathit{R}}$〉, including the result that the appropriate scaling variable is R/L. Predictions from scaling theory are borne out by numerical data, which show the probability distribution of ln〈${\mathrm{\ensuremath{\sigma}}}_{0}$${\mathrm{\ensuremath{\sigma}}}_{\mathit{R}}$〉 to be remarkably skewed at short distances, approaching a Gaussian only as R/L\ensuremath{\gg}1. \textcopyright{} 1996 The American Physical Society.
In a previous paper of this series, we studied $\mathfrak{P}(\overline{\ensuremath{\sigma}})$, the probability that the average boundary spin for a half-plane of Ising spins is $\overline{\ensuremath{\sigma}}$. We extend our study of such functions by generalizing the previous derivation to any magnetic system and using them to relate the magnetic behavior at the critical isotherm to the spin-spin correlations at the critical temperature. We also clarify the meaning of the secondary maxima previously found in $\mathfrak{P}(\overline{\ensuremath{\sigma}})$ by a more accurate calculation and by examining the magnetization on interior rows of the half-plane. These considerations show that $\mathfrak{P}(\overline{\ensuremath{\sigma}})$ is more properly to be considered as the sum of two separate spin probability functions. The various ways in which the thermodynamic limit may be taken and the effects that these different limiting procedures have on $\mathfrak{P}(\overline{\ensuremath{\sigma}})$ are discussed in detail. The influence of boundary conditions on spin probability functions is studied by reversing the sign of one column of horizontal bonds. We compute the additional free energy resulting from the misfit bonds and use this to show that $\mathfrak{P}(\overline{\ensuremath{\sigma}})$ is no longer bimodal but is rectangular for $|\overline{\ensuremath{\sigma}}|$ less than the boundary spontaneous magnetization. Finally, we present in graphical form numerical integrations of the boundary magnetization and susceptibility.
Using conformal field theory techniques, we compute the disorder-averaged p -th power of the spin-spin correlation function (overline⟨σ(0)σ( R )⟩ p , p ∊ Z ) for the ferromagnetic random bond Potts model. We thus generalize the calculations of Dotsenko, Dotsenko and Picco ( hep-th/9709136 , 1997), where the case p = 2 was considered, and of Ludwig ( Nucl. Phys. B , 85 (1987) 97, 330 (1990) 639), where first-order computations where made for general p . Perturbative calculations are made up to second order in ( being proportional to the central charge deviation of the pure model from the Ising model value). The explicit dependence of the correlation function on p gives an upper bound for the validity of the -expansion, which seems to be valid, in the three-states case, only if p ⩽ 4.
A method is described for obtaining perturbation expansions of the pair correlation function about the ferromagnetically ordered state of the S=1/2 Ising model. The expansions take the form of a double power series in temperature and field variables and may be used in any region of the T-H plane of the ferromagnetic and in the disordered region of the antiferromagnet. Coefficients are calculated for the face-centred cubic lattice which allow the low-temperature expansions in zero field to be obtained through u31 and the high-field expansions along the critical isotherm to be obtained through mu 6. Agreement is obtained with existing high-temperature expansions.