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Progress of Theoretical Physics

Concentration Expansion for Random Ising Systems. II

Tsuguhiro Tamaribuchi

Published Nov 1, 1981• 6 citations
Publisher

Abstract

A new method for the concentration expansion of the free energy of the two-dimensional Ising system with binary bond mixture is given. This method is simpler and more systematic compared with that used in a previous paper by the author and Takano. By using this new method, the expansion is carried out up to the order of c3, where c is the concentration of impurity bonds. The result is shown to be consistent with the assumption on functional form of the internal energy proposed in the previous paper. This assumption leads us to a conclusion that the specific heat anomaly in the quenched system is a cusp as in the annealed system.

DOI
10.1143/ptp.66.1574

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Concentration Expansion for Random Ising Systems

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Journal of Physics C Solid State Physics•Dec 14, 1977•6 citations
A new method for the concentration expansion of the free energy of the two-dimensional Ising system with binary bond mixture is given. This method is simpler and more systematic compared with that used in a previous paper by the author and Takano. By using this new method, the expansion is carried out up to the order of c3, where c is the concentration of impurity bonds. The result is shown to be consistent with the assumption on functional form of the internal energy proposed in the previous paper. This assumption leads us to a conclusion that the specific heat anomaly in the quenched system is a cusp as in the annealed system.
Two-dimensional Ising systems with bond-impurities are treated exactly by using the method of Gibberd and Hurst. First we derive recurrence equations by which a Green's function with n scattering centers can be obtained from that with n-1 scattering centers. Using these recurrence equations, we are able to expand the free energy systematically in power series of concentration c of impure bonds. This concentration expansion is carried out up to the order of c2 for quenched and annealed systems, and we discuss critical properties of the quenched system comparing with those of the annealed system. The difference between the two systems is seen to arise first in the term of the order c2, but the singularities in these terms are almost the same for both systems. This result leads to an conjecture that critical properties of a quenched system are the same as those of an annealed system in the critical region. The shift of the critical temperature in the quenched system is also calculated up to the order of c2 in the bond-dilution case (the strength of the impurity bonds is zero).
The two-dimensional Ising model with two kinds of bonds –ferro and antiferromagnetic– is investigated in both quenched and annealed approximations. High temperature series expansion technique is used to get the respective free energies and susceptibilities. Form the analysis of the ratio method results, the phase diagrams are drawn and critical concentrations are estimated. It is shown that in both approximations the system behaves similarly, although in the annealed case the magnetically ordered phases are more stable against increasing impurity concentration. It is also proved that for Bethe lattices initial susceptibilities for quenced and annealed systems coincide.
The techniques developed by Edwards and Anderson (1975) for the spin-glass problem are here applied to the bond-diluted quenched Ising model (S= 1 / 2 ). The n to 0 trick is used to express exactly the free energy in terms of a generalized Hamiltonian representing n different but coupled replicas. The only approximation of the theory consists of decoupling these replicas in a way that reduces the free energy of the diluted system to that of a pure system with renormalized, concentration-dependent parameters. To first order in the concentration of missing bonds the author obtains all the results previously derived by Harris (1974) using a cumulant expansion and which are exact in this limit. More generally, the theory allows a discussion of the concentration dependence of the critical temperature; in particular, the values deduced for the critical concentration agree rather well with those obtained exactly or through series expansions. The behaviour of the system near the transition is also studied and the critical exponents are found to be Fisher-renormalized.
Journal of the Physical Society of Japan 64 (1995) pp. 3699-3706 Specific Heat of the Two-Dimensional ±J Ising Model Makoto Inoue Progress of Theoretical Physics Vol. 47 No. 4 (1972) pp. 1438-1440 Impurity Problem in the Ising Model Katuro Sawada and Takeo Osawa Progress of Theoretical Physics Vol. 49 No. 1 (1973) pp. 83-88 Critical Properties of Ising Models Containing Dilute Impurities Takeo Osawa and Katuro Sawada Progress of Theoretical Physics Vol. 64 No. 4 (1980) pp. 1212-1232 Concentration Expansion for Random Ising Systems Tsuguhiro Tamaribuchi and Fumihiko Takano
The problem of the effect of inhomogeneous impurities in a lattice structure on the critical behaviour of systems which undergo a second-order phase transition is studied on a model system, the 2D Ising Model with impurity bonds randomly distributed over the lattice. It is known that in the critical region the 2D Ising Model is equivalent to the model of free fermions. We show that the effect of impurities is to add a four-fermion interaction with the corresponding charge proportional to the concentration of impurities. The resulting fermion model is simple enough and can be studied exactly by renormalization group methods. We show that for any small concentration of impurity bonds a new critical regime is established if we go sufficiently close to the phase-transition point. In particular we find that the specific heat singularity changes from C ˜ ln 1/|τ|(τ=(Τ − Τc/Τc to C ˜ ln ln 1/|τ| for τ≪τi ˜ exp(−const/c i), c i being the concentration of impurities, while the spin-spin correlation function in the critical point changes more seriously: from <σ R σΟ> ˜ 1/R1/4 to <σ R σΟ> ˜ exp { − (const/c i)(ln ln R2 )} for R≫R i˜ exp (const/c i), which corresponds to a change of the critical exponent η from 1/4 to 0.