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Journal of Physics Condensed Matter

Instability of the one-dimensional extended Hubbard model in a magnetic field

A. Kitazawa

Published Apr 22, 2003• 4 citations
Publisher

Abstract

We consider instabilities of two-component Tomonaga–Luttinger (TL) liquids for the one-dimensional extended Hubbard model in a magnetic field. When the density of electrons with spin σ is 1/2, electrons with spin σ tend to occupy every other site, and do not conduct. The instability stems from the umklapp scattering between same-spin electrons. But for the remaining electrons with spin −σ, an energy gap does not appear. We also consider the instability relating to the phase separation. This relates to the intrinsic stability condition of the TL liquid.

DOI
10.1088/0953-8984/15/17/313

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We consider instabilities of two-component Tomonaga–Luttinger (TL) liquids for the one-dimensional extended Hubbard model in a magnetic field. When the density of electrons with spin σ is 1/2, electrons with spin σ tend to occupy every other site, and do not conduct. The instability stems from the umklapp scattering between same-spin electrons. But for the remaining electrons with spin −σ, an energy gap does not appear. We also consider the instability relating to the phase separation. This relates to the intrinsic stability condition of the TL liquid.
We give details of the analysis of critical properties of spin-gap phases in one-dimensional lattice electron models. In Tomonaga-Luttinger (TL) liquid theory, the spin-gap instability occurs when the backward scattering changes from repulsive to attractive. This transition point is shown to be equivalent to that of the level-crossing of the singlet and triplet excitation spectra, using the c=1 conformal field theory and the renormalization group. Based on this notion, the transition point between the TL liquid and the spin-gap phases can be determined with high-accuracy from the numerical data of finite-size clusters. We also discuss the boundary conditions and discrete symmetries to extract these excitation spectra. This technique is applied to the extended Hubbard model, the $t\ensuremath{-}J$ model, and the $t\ensuremath{-}J\ensuremath{-}{J}^{\ensuremath{'}}$ model, and their phase diagrams are obtained. We also discuss the relation between our results and analytical solutions in weak-coupling and low-density limits.
We investigated the one-dimensional extended Hubbard model with the on-site interaction U and the nearest-neighbor interaction V by using the numerical diagonalization method. Assuming the Luttinger liquid relations, we obtained the phase diagram of the ground state on the U - V plane. In the weak coupling region, the phase diagram agrees with the result of g -ology. On the other hand, in the strong coupling region for V ≫∣ t ∣, a novel superconducting phase appears both for positive and negative U . We also investigated the charge gap and the spin gap. The results suggest that the superconducting phase extends over the Tomonaga-Luttinger region (without spin gap) and the Luther-Emery region (with spin gap).
We investigate numerically the ground state phase diagram of the one-dimensional extended Hubbard model, including an on-site interaction U and a nearest-neighbor interaction V. We focus on the ground state phases of the model in the $V\ensuremath{\gg}U$ region, where previous studies have suggested the possibility of dominant superconducting pairing fluctuations before the system phase separates at a critical value V=VPS.{V=V}_{\mathrm{PS}}.V=VPS​. Using quantum Monte Carlo methods on lattices much larger than in previous Lancz\"os diagonalization studies, we determine the boundary of phase separation, the Luttinger-liquid correlation exponent ${K}_{\ensuremath{\rho}},$ and other correlation functions in this region. We find that phase separation occurs for V significantly smaller than previously reported. In addition, for negative U, we find that a uniform state reenters from phase separation as the electron density is increased towards half filling. For V<VPS,V<{V}_{\mathrm{PS}},V<VPS​, our results show that superconducting fluctuations are not dominant. The system behaves asymptotically as a Luttinger liquid with ${K}_{\ensuremath{\rho}}<1,$ but we also find strong low-energy (but gapped) charge-density fluctuations at a momentum not expected for a standard Luttinger liquid.
The extended Hubbard model in one dimension is studied for half-filled bands using a continuum theory based on the Luttinger model. Interactions going beyond the Luttinger model are treated with a renormalization group. We compute correlation functions and use these to establish the phase diagram of the model. It compares favorably with earlier numerical work. Specifically, we identify additional interaction terms coupling charge and spin degrees of freedom; one of them is shown to mediate the crossover from a continuous to a first-order transition from spin- to charge-density waves at a finite value of U=2V. Our approach suggests viewing the model on this line as two weakly coupled interpenetrating systems of up-spin and down-spin electrons with strong backscattering interactions favoring charge-density waves separately in each of the subsystems. A first-order transition occurring for attractive interactions is interpreted as a characteristic instability of the Luttinger liquid towards phase separation. We finally discuss a possible equivalence to some recent studies of coupled two-dimensional XY models.
The spin-gap phase of the one-dimensional t-J-J' model is studied by the level-crossing of the singlet and the triplet excitation spectra. The phase boundary obtained between the Tomonaga-Luttinger and the spin-gap phases is remarkably consistent with the analytical results at the J,J′→0J,J'\to 0J,J′→0 and the low-density limits discussed by Ogata et al. The spin-gap phase has a single domain in the phase diagram even if the spin gap opens at half-filling. The phase boundary coincides with the Kρ=1K_ρ=1Kρ​=1 line where the Tomonaga-Luttinger liquid behaves as free electrons, in the low-density region. The relation between our method and the solution of the two-electron problem is also discussed.
We investigate the T=0T=0T=0 phase diagram of a variant of the one-dimensional extended Hubbard model where particles interact via a finite-range soft-shoulder potential. Using Density Matrix Renormalization Group (DMRG) simulations, we evidence the appearance of Cluster Luttinger Liquid (CLL) phases, similarly to what first predicted in a hard-core bosonic chain [M. Mattioli, M. Dalmonte, W. Lechner, and G. Pupillo, Phys. Rev. Lett. 111, 165302]. As the interaction strength parameters change, we find different types of clusters, that encode the order of the ground state in a semi-classical approximation and give rise to different types of CLLs. Interestingly, we find that the conventional Tomonaga Luttinger Liquid (TLL) is separated by a critical line with a central charge c=5/2c=5/2c=5/2, along which the two (spin and charge) bosonic degrees of freedom (corresponding to c=1c=1c=1 each) combine in a supersymmetric way with an emergent fermionic excitation (c=1/2c=1/2c=1/2). We also demonstrate that there are no significant spin correlations.