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arXiv:1510.06531 · Condensed Matter Physics

The β-expansion of the D=1 fermionic spinless Hubbard model off the half-filling regime

C.E.T. Gonçalves da Silva, M. T. Thomaz, Onofre Rojas

Published Sep 1, 2015• 0 citations
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Abstract

We found that when the spinless model is off the half-filling regime (µ V ), the Helmholtz free energy (HFE) can be written as two β-expansions: one expansion comes from the half-filling configuration and another one that depends on the parameter x = µ -V .We show numerically that the chemical potential as a function of temperature satisfies a relation similar to the one derived from the particle-hole symmetry of the fermionic spinless model.We extend the β-expansion of the HFE of the one-dimensional fermionic spinless Hubbard model up to order β 8 .

DOI
10.5488/cmp.18.33003

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We found that when the spinless model is off the half-filling regime (µ V ), the Helmholtz free energy (HFE) can be written as two β-expansions: one expansion comes from the half-filling configuration and another one that depends on the parameter x = µ -V .We show numerically that the chemical potential as a function of temperature satisfies a relation similar to the one derived from the particle-hole symmetry of the fermionic spinless model.We extend the β-expansion of the HFE of the one-dimensional fermionic spinless Hubbard model up to order β 8 .
We revisit the particle-hole symmetry of the one-dimensional (D = 1) fermionic spinless Hubbard model, associating that symmetry to the invariance of the Helmholtz free energy of the one-dimensional spin-1/2 X X Z Heisenberg model, under reversal of the longitudinal magnetic field and at any finite temperature.Upon comparing two regimes of that chain model so that the number of particles in one regime equals the number of holes in the other, one finds that, in general, their thermodynamics is similar, but not identical: both models share the specific heat and entropy functions, but not the internal energy per site, the first-neighbor correlation functions, and the number of particles per site.Due to that symmetry, the difference between the first-neighbor correlation functions is proportional to the z-component of magnetization of the X X Z Heisenberg model.The results presented in this paper are valid for any value of the interaction strength parameter V , which describes the attractive/null/repulsive interaction of neighboring fermions.
The phase diagram of the 2D t-J model has been investigated using high temperature expansions. Series for the Helmholtz free energy, the inverse compressibility, the chemical potential and the uniform spin susceptibility through tenth order on a square lattice have been calculated using the finite cluster method.1 The series are analytically continued beyond their radius of convergence by Padé and integral approximants. The most accurate extrapolations can be made for the Helmholtz free energy where for J/t≈0.3 and n≈0.9 we can reach T~t/5. We can test the accuracy of the continuation methods by comparing with the 1D results for the boundary of phase separation of Ogata et al.2 In 2D a region of phase separation was found at T=0 for J/t lying above a line extending from J/t=3.8 at zero filling to J/t=1.2 at half filling. No phase separation was found at very small J/t contrary to the earlier suggestion of Emery et al3 which was based on results from exact diagonalisation on 4×4 clusters but in agreement with Quantum Monte Carlo (QMC) calculations on the Hubbard model.4 For very small J/t near half filling where the Nagaoka effect is possible, we find a region of divergent uniform magnetic susceptibility at T=0. However the divergence is very weak when compared with the exponential behaviour expected from a 2D ferromagnet. This might imply a substantially reduced moment which is consistent with the recent QMC estimates of Zhang et al.5
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.
We calculate the thermodynamics of the one-dimensional spin-1/2 Ising model in the presence of a constant skew magnetic field. We obtain the high-temperature expansion of its Helmholtz free energy (HFE), for the ferromagnetic and antiferromagnetic cases, up to order beta7. This expansion permits us to obtain the behaviour of the model for $ |J| \beta \stackrel {<}{_\sim}} 1 $, when it cannot be described by its classical version. Among the calculated thermodynamical functions of the model, we have the diagonal elements of the magnetic susceptibility tensor for the transverse and logitudinal Ising models, obtained by taking the limits h z -> 0 and h y -> 0, respectively, of the beta-expansion of the HFE. The y-component of the magnetization and the chiyy component of the magnetic susceptibility tensor are almost the same for the antiferro- and ferromagnetic models, at least for $ |J| \beta \stackrel {<}{_\sim}} 1 $; and, c yy is practically independent of the direction of the external magnetic. We also show that, in this region of temperature, the thermodynamics of the Ising model with skew magnetic field and that of an XXZ model with longitudinal magnetic field are not similar.