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Journal of Physics C Solid State Physics

Upper bounds on susceptibilities and correlations of quenched impurity systems

Harold Falk

Published Jul 7, 1975• 4 citations
Publisher

Abstract

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 .

DOI
10.1088/0022-3719/8/13/004

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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 .
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.
The Griffiths phase in systems with quenched disorder occurs below the ordering transition of the pure system down to the ordering transition of the actual disordered system. While it does not exhibit long-range order, large fluctuations in the disorder degrees of freedom result in exponentially rare, long-range ordered states and hence the occurrence of broad distributions in response functions. Inside the Griffiths phase of the two-dimensional bond-diluted Ising model the distribution of the magnetic susceptibility is expected to have such a broad, exponential tail. A large-deviation Monte Carlo algorithm is used to sample this distribution and the exponential tail is extracted over a wide range of the support down to very small probabilities of the order of 10^{-300}. We study the behavior of the susceptibility distribution across the full phase diagram, from the paramagnetic state through the Griffiths phase to the ferromagnetically ordered system and down to the zero-temperature point. We extract the rate function of large-deviation theory as well as its finite-size scaling behavior and we reveal interesting differences and similarities between the cases. A connection between the fraction of ferromagnetic bonds in a given disorder sample and the size of the magnetic susceptibility is demonstrated numerically.
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.
In the asymptotic critical region of a ferromagnetic spin system, the magnetic susceptibility χ and the correlation length ξ are related by the power law: χ∼ξ2−η, where η is the correlation function exponent. Somewhat farther from the critical point, but still within the critical region, we expect the behavior to be mean-field in character: χ∼ξ2. We give an exact relationship between χ and ξ in terms of a differential renormalization group approach. The leading behavior in the disordered phase is explicitly calculated for the n-component Wilson-Fisher model to O(ε2), ε≡4−d. The result includes both limiting power laws in a smooth crossover function. For the Ising case, n=1, the relationship is extended to the entire critical region, including the ordered phase. In this case, even in the asymptotic limit, the susceptibility differs from ξ2−η by a factor which depends on the magnetization M. This function is calculated to leading order and correctly gives the associated amplitude ratios to O(ε). Applications to the general p-point vertex function are discussed.
Fisher's rigorous bound chi < chi saw , where chi saw is the susceptibility calculated from the self-avoiding walk (SAW) generating function, is applied to (i) the ferromagnetic Ising alloy, (ii) the dilute Ising ferromagnet, (iii) the layer susceptibility of a free surface, (iv) the Ising ferromagnet in a transverse field, and (v) the dilute Ising ferromagnet in a transverse field. The author obtains an upper bound T UB for the temperature T 0 below which a spontaneous magnetization occurs. The use of a microscopically varying molecule field is analysed. A heuristic argument is given to show that, for the dilute Ising ferromagnet in a transverse field, the critical transverse field at zero temperature jumps from zero to a value at least equal to that of the one-dimensional ground state as x passes through the critical percolation concentration.
Expressions for general correlation functions in Ising ferromagnets are derived in terms of the Lee-Yang theorem. By the use of these expressions, the asymptotic forms of correlation functions in the presence of magnetic field are analyzed, assuming those in the absence of the external field. In particular, the critical indices of spin-spin, energy-density-energy-density, energy-density-spin correlation functions are related with the critical indices of thermal susceptiblity χT0, specific heat, susceptibility χH0, and magnetization. These results agree in form with those predicted by the scaling law approach.