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Physical review. B, Condensed matter

Weak-coupling approach to interaction-induced deformation of the Fermi surface in the two-dimensional Hubbard model

Shuntaro Yoda, Kōsaku Yamada

Published Sep 15, 1999• 7 citations
Publisher

Abstract

We investigate the interaction-induced deformation of the Fermi surface in the two-dimensional Hubbard model on the triangular lattice in the weak-coupling regime within the second-order perturbation theory. In systems without frustration, at the filling equal to or close to half-filling, the direction of the deformation of the Fermi surface is the direction toward the Fermi surface at half-filling with the smaller transfer amplitude set to zero, whose nesting vector is commensurate with the underlying lattice. On the other hand, in systems with frustration, the Fermi surface is not deformed by the interaction because of the frustration.

DOI
10.1103/physrevb.60.7886

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We investigate the interaction-induced deformation of the Fermi surface in the two-dimensional Hubbard model on the triangular lattice in the weak-coupling regime within the second-order perturbation theory. In systems without frustration, at the filling equal to or close to half-filling, the direction of the deformation of the Fermi surface is the direction toward the Fermi surface at half-filling with the smaller transfer amplitude set to zero, whose nesting vector is commensurate with the underlying lattice. On the other hand, in systems with frustration, the Fermi surface is not deformed by the interaction because of the frustration.
Divergencies appearing in perturbation expansions of interacting many-body systems can often be removed by expanding around a suitably chosen renormalized (instead of the noninteracting) Hamiltonian. We describe such a renormalized perturbation expansion for interacting Fermi systems, which treats Fermi surface shifts and superconductivity with an arbitrary gap function via additive counterterms. The expansion is formulated explicitly for the Hubbard model to second order in the interaction. Numerical solutions of the self-consistency condition determining the Fermi surface and the gap function are calculated for the two-dimensional case. For the repulsive Hubbard model close to half-filling we find a superconducting state with d-wave symmetry, as expected. For Fermi levels close to the van Hove singularity a Pomeranchuk instability leads to Fermi surfaces with a broken square lattice symmetry, whose topology can be closed or open. For the attractive Hubbard model the second-order calculation yields an s-wave superconductivity with a weakly momentum dependent gap, whose size is reduced compared to the mean-field result.
We consider serious conceptual problems with the application of standard perturbation theory, in its zero-temperature version, to the computation of the dressed Fermi surface for an interacting electronic system. In order to overcome these difficulties, we set up a variational approach which is shown to be equivalent to the renormalized perturbation theory where the dressed Fermi surface is fixed by recursively computed counterterms. The physical picture that emerges is that couplings that are irrelevant tend to deform the Fermi surface in order to become more relevant (irrelevant couplings being those that do not exist at vanishing excitation energy because of kinematical constraints attached to the Fermi surface). These insights are incorporated in a renormalization group approach, which allows for a simple approximate computation of Fermi surface deformation in quasi-one-dimensional electronic conductors. We also analyze flow equations for the effective couplings and quasiparticle weights. For systems away from half-filling, the flows show three regimes corresponding to a Luttinger liquid at high energies, a Fermi liquid, and a low-energy incommensurate spin-density wave. At half-filling Umklapp processes allow for a Mott insulator regime where the dressed Fermi surface is flat, implying a confined phase with vanishing effective transverse single-particle coherence. The boundary between the confined and Fermi liquid phases is found to occur for a bare transverse hopping amplitude of the order of the Mott charge gap of a single chain.
The deformation of the Fermi surface induced by Coulomb interactions is investigated in the $t\ensuremath{-}{t}^{\ensuremath{'}}$-Hubbard model. The interplay of the local U and extended V interactions is analyzed. It is found that the interactions V enhance small anisotropies producing deformations of the Fermi surface that break the point group symmetry of the square lattice at the Van Hove filling. This Pomeranchuk instability competes with magnetic instabilities and is suppressed at a critical value U(V).U(V).U(V). The interaction V effectively renormalizes the ${t}^{\ensuremath{'}}$ parameter to smaller values, which favors nesting. It also induces changes in the topology of the Fermi surface, which can go from holelike to electronlike. This may explain the results of recent angle-resolved photoemission spectroscopy experiments.
We investigate the nature of the interaction-driven Mott-Hubbard transition of the half-filled ${t}_{1}\ensuremath{-}{t}_{2}$ Hubbard model in one dimension, using a full-fledged variational Monte Carlo approach including a distance-dependent Jastrow factor and backflow correlations. We present data for the evolution of the magnetic properties across the Mott-Hubbard transition and on the commensurate to incommensurate transition in the insulating state. Analyzing renormalized excitation spectra, we find that the Fermi surface renormalizes to perfect nesting right at the Mott-Hubbard transition in the insulating state, with a first-order reorganization when crossing into the conducting state.
We apply the self-consistent renormalized perturbation theory to the Hubbard model on the square lattice at finite temperatures to study the evolution of the Fermi surface (FS) as a function of temperature and doping. Previously, a nematic phase for the same model has been reported to appear at weak coupling near a Lifshitz transition from closed to open FS at zero temperature where the self-consistent renormalized perturbation theory was shown to be sensitive to small deformations of the FS. We find that the competition with the superconducting order leads to a maximal nematic order appearing at nonzero temperature. We explicitly observe the two competing phases near the onset of nematic instability, and by comparing the grand canonical potentials, we find that the transitions are first order. We explain the origin of the interaction-driven spontaneous symmetry breaking to a nematic phase in a system with several symmetry-related Van Hove points and discuss the required conditions.
We investigate the metal-insulator transition in the half-filled Hubbard model on the two-dimensional triangular lattice using both the Kotliar-Ruckenstein slave-boson technique and an exact numerical diagonalization of finite clusters. Contrary to the case of a square lattice, where a perfect nesting of the Fermi surface leads to a metal-insulator transition at arbitrarily small values of U, always accompanied by antiferromagnetic ordering, on a triangular lattice, due to the lack of perfect nesting, the transition takes place at a finite value of U, and frustration induces a nontrivial competition among different magnetic phases. Indeed, within the mean-field approximation in the slave-boson approach, as the interaction grows the paramagnetic metal turns into a metallic phase with incommensurate spiral ordering. Increasing the interaction further, a linear spin-density wave is stabilized, and finally for strong coupling the latter phase undergoes a first-order transition toward an antiferromagnetic insulator. No trace of the intermediate phases is seen in the exact diagonalization results, indicating a transition between a paramagnetic metal and an antiferromagnetic insulator.