The selfconsistent high-density expansion for thermodynamic quantities of the quantum spin-polarized fermion lattice gas, derived in the preceding paper, is extended to higher orders. The (1/z)2 terms in the free energy are evaluated by summing two-loop diagrams and the corresponding equation for critical temperature in the classical limit is discussed. Ground state phase diagram of the system for arbitrary electron density is derived in the first and second order of the expansion. Properties of the half-filled, one-dimensional spinless fermion model at T = 0, obtained here in the first three orders in 1/z, are compared with the results of exact calculations and of other approximate methods. This provides a test for the unrenormalized high-density expansion method.
A diagrammatic technique for Hubbard's operators is employed to perform systematically the high-density expansion for the three-dimensional spinless fermion model. The molecular field theory is obtained by the zero-order renormalization of blocks. Summation of the first order diagrams is carried out in both selfconsistent and correctional way. It turns out that the charge ordering parameter, calculated self-consistently, has a jump for a certain medial temperature. We have also shown, that the Horwitz-Callen renormalization leads to the first or second order phase transition, depending on t/W and fails when this ratio is large enough. The phase diagrams of the system for the half-filled band case, derived in both unrenormalized and renormalized first order of high-density expansion are presented.
A first-order phase transition is a characteristic feature of the Gaussian approximation in spin-fluctuation theory. We describe a method for taking into account the fourth-order terms of the free energy expansion using partial averaging. In the example of the Ising model, we show that renormalization of the magnetic susceptibility leads to the second-order phase transition, which is experimentally observed in metals. Near the phase transition, we use the parameter substitution method to compute temperature dependencies. We perform a qualitative analysis and explain the mechanism of the renormalization.
To the lowest order approximation, the method of variational cumulant expansion has been successful in the determination of the critical point of a spin system on the Ising model. In the second order calculation, unphysical phase transition arises. In this letter, we first analyze the origin of unphysical phase transitions in high-order variational cumulant expansion calculations. We then explain the basis on which a conjecture is proposed that to any order m of the variational cumulant expansion, the critical point is given by the bifurcation point of the free energy. The theory is finally verified numerically for the two-dimensional case.
The partition function of the Ising model of ferromagnetism is examined in the limit of high density in the anticipation that in the limit of infinite density one recovers the Weiss molecular field. The formal parameter of expansion is
z1 where
z is the number of spins in the range of the exchange potential (not restricted to nearest neighbor interactions). In the absence of long-range order, only ring diagrams in the cluster expansion contribute. These give a divergence in the specific heat at $k{T}_{c}=\ensuremath{\Sigma}{j\ensuremath{\ne}i}^{}{v}_{\mathrm{ij}}$ where
vij is the exchange potential. This is the molecular field value for the Curie point
Tc. In the presence of a magnetic field the partition function is evaluated for fixed magnetic moment
M in the same approximation,
M being determined by minimization. This results in a susceptibility differing from the molecular field theory and hence an inconsistency in the theory.The inconsistency is traced back to the observation that the acceptance of ring diagrams is equivalent to the gaussian model of Kac and Berlin which violates the sum rule ${{\ensuremath{\Sigma}}_{i=1}}^{N}{{\ensuremath{\mu}}_{i}}^{2}=N$. Here ${\ensuremath{\mu}}_{i}$ is the "spin" per particle and
N is the total number of particles. This condition is reinstated by insuring the sum rule. The result leads to the spherical model. Thus, a consistent high density approximation to the Ising model is the spherical model. Below the Curie point or for fixed magnetic field,
M is again held fixed and only the Fourier components of the spin density with nonvanishing wave vector are "sphericalized." The result leads to a physically acceptable model which becomes the molecular field theory at low temperatures or high fields and deviates in
O(z1) in general. Formally, the results are simply expressed in terms of a temperature dependent Weiss field. These results differ from the ordinary spherical model which is physically unacceptable below the Curie point. However, a molecular field modification of the spherical model due to M. Lax yields the same result when properly interpreted.It is shown that the above results are also valid (to the same approximation) in the quantum mechanical Heisenberg model, for temperatures above the Curie point.
We compute the 2 n -point coupling constants in the high-temperature phase of the two-dimensional Ising model by using transfer-matrix techniques. This provides the first few terms of the expansion of the effective potential (Helmholtz free energy) and of the equation of state in terms of the renormalized magnetization. By means of a suitable parametric representation, we determine an analytic extension of these expansions, providing the equation of state in the whole critical region in the t , h plane.