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International Journal of Modern Physics B

HIGH-DENSITY EXPANSION FOR THE SPINLESS FERMION MODEL I: FIRST ORDER IN 1/z AND HORWITZ-CALLEN RENORMALIZATION

M. Bartkowiak, E. Peter Münger, R. Micnas

Published Aug 1, 1988• 4 citations
Publisher

Abstract

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.

DOI
10.1142/s0217979288000342

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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.
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.
The Green’s-function diagrammatic high-density expansion technique for Hubbard’s operators is constructed and applied to the extended Hubbard model. The electron Green’s functions and the band structure of the ground state for arbitrary electron density are calculated in the zeroth-order in 1/z. The phase diagram, staggered magnetic susceptibility and the Néel temperature for the half-filled band case are derived in the first order of the high-density expansion. Connections of the high-density and linked cluster expansions are also discussed.
We study the nature of the ground state of a one-dimensional electron-phonon model for molecular crystals: The phonons are assumed dispersionless and couple to the local electronic density. We consider the half-filled-band sector and discuss the stability of the Peierls-dimerized ground state as a function of the phonon frequency ($\ensuremath{\omega}$), electron-phonon coupling constant ($\ensuremath{\lambda}$), and number of components of the electron spin (nnn). First, we discuss the properties of the model in the limiting cases of zero frequency and infinite frequency. We then perform a strong coupling expansion which maps the system onto a spinless fermion model with nearest-neighbor repulsion for both spinless and spin-\textonehalf{} electrons, but with different parameters in both cases. Finally, we perform a numerical study of the system using a Monte Carlo technique. We study the behavior of the order parameter and of correlation functions for various points in parameter space. We also perform a finite-size---scaling analysis of the numerical data. The conclusions of our study are the following: For the case of spinless electrons, there exists always a disordered phase for small coupling constant, and the system undergoes an infinite-order transition to a Peierls-dimerized state as the coupling constant increases beyond a critical value. The phase diagram is divided between a disordered and an ordered region by a line that connects the points ($\ensuremath{\omega}=0$, $\ensuremath{\lambda}=0$) and ($\ensuremath{\omega}=\ensuremath{\infty}$, $\ensuremath{\lambda}=\ensuremath{\infty}$). For the case of spin-\textonehalf{} electrons, the system is dimerized for arbitrary $\ensuremath{\lambda}$ and $\ensuremath{\omega}$ except in the limit $\ensuremath{\omega}=\ensuremath{\infty}$.
We determine the ground-state phase diagram of the one-dimensional half-filled Hubbard model with on-site (nearest-neighbor) repulsive interaction U (V) and nearest-neighbor hopping t using the density-matrix renormalization group technique. Based on the results of the excitation gaps, Luttinger-liquid exponents, and bond-order-wave (BOW) order parameter, we confirm that the BOW phase appears in a substantial region between the charge-density-wave (CDW) and spin-density-wave phases. Each phase boundary is determined by multiple means and it allows us to make a cross-check on the validity of our estimations. We also find that the BOW-CDW transition changes from continuous to first order at the tricritical point (U(t),V(t)) approximately (5.89 t,3.10 t) and the BOW phase shrinks to zero at the critical end point (U(c),V(c)) approximately (9.25 t,4.76 t).
We examine the phase diagram of the half-filled one-dimensional extended Hubbard model using quantum information entropies within the density-matrix renormalization group. It is well known that there is a charge-density-wave phase at large nearest-neighbor and a small on-site Coulomb repulsion and a spin-density wave at small nearest-neighbor and large on-site Coulomb repulsion. At intermediate Coulomb interaction strength, we find an additional narrow region of a bond-order phase between these two phases. The phase transition line for the transition out of the charge-density-wave phase changes from first order at strong coupling to second order in a parameter regime where all three phases are present. We present evidence that the additional phase-transition line between the spin-density-wave and bond-order phases is infinite order. While these results are in agreement with recent numerical work, our study provides an independent unbiased means of determining the phase boundaries by using quantum information analysis, yields values for the location of some of the phase boundaries that differ from those previously found, and provides insight into the limitations of numerical methods in determining phase boundaries, especially those of infinite-order transitions.
We establish the ground-state phase diagram of a spinless fermion model with the nearest and next-nearest-neighbor repulsions via the density-matrix-renormalization-group method. The bond-order (BO) correlation function and its static structure factor can be employed to accurately determine the BO phase boundaries. The first derivative of the charge-density wave (CDW) structure factor and the second derivative of the energy are expect to diverge at the 2k(F)-CDW-BO transition point, which indicates that a second-order transition occurs. We determine the metallic-4k(F)-CDW boundary by analyzing the decay behavior of the charge correlation function near the transition point.