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Physical review. B, Condensed matter

Reply to ‘‘Comment on ‘Low-temperature properties of the Hubbard chain with an attractive interaction’ ’’

P. Schlottmann

Published May 1, 1991• 6 citations
Publisher

Abstract

In the preceding Comment, the energy of the system is expanded at a van Hove singularity of one band as a function of changes in the field, the magnetization, and the Fermi energy. Unfortunately, not all the derivatives are properly defined at a van Hove singularity. This leads to the erroneous conclusion that the susceptibility is finite at the critical point. The correct result is that M\ensuremath{\propto}(H-${\mathit{H}}_{\mathit{c}}$${)}^{1/2}$, in analogy to a Prokovsky-Talapov transition.

DOI
10.1103/physrevb.43.11451

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In the preceding Comment, the energy of the system is expanded at a van Hove singularity of one band as a function of changes in the field, the magnetization, and the Fermi energy. Unfortunately, not all the derivatives are properly defined at a van Hove singularity. This leads to the erroneous conclusion that the susceptibility is finite at the critical point. The correct result is that M\ensuremath{\propto}(H-{\mathit{H}}_{\mathit{c}}$${)}^{1/2}, in analogy to a Prokovsky-Talapov transition.
Using large-scale dynamical cluster quantum Monte Carlo simulations, we study the Lifshitz transition of the two-dimensional Hubbard model with next-nearest-neighbor hopping (${t}^{\ensuremath{'}}$), chemical potential, and temperature as control parameters. At ${t}^{\ensuremath{'}}\ensuremath{\le}0$, we identify a line of Lifshitz transition points associated with a change in the Fermi surface topology at zero temperature. In the overdoped region, the Fermi surface is complete and electron-like; across the Lifshitz transition, the Fermi surface becomes hole-like and develops a pseudogap. At (or very close to) the Lifshitz transition points, a van Hove singularity in the density of states crosses the Fermi level. The van Hove singularity occurs at finite doping due to correlation effects and becomes more singular when ${t}^{\ensuremath{'}}$ becomes more negative. The resulting temperature dependence on the bare ddd-wave pairing susceptibility close to the Lifshitz points is significantly different from that found in the traditional van Hove scenarios. Such unambiguous numerical observation of the Lifshitz transition at ${t}^{\ensuremath{'}}\ensuremath{\le}0$ extends our understanding of the quantum critical region in the phase diagram and shines lights on future investigations of the nature of the quantum critical point in the two-dimensional Hubbard model.
We consider the Heisenberg chain of spin 1/2 with a concentration x of impurity spins S. The thermodynamic Bethe ansatz equations are stated for this model. The ground-state equations are analyzed as a function of x and magnetic field. In zero field the ground state is a singlet for finite x, but non-Fermi-liquid-like for x\ensuremath{\rightarrow}0. Two energy bands play a role at T=0. The Van Hove singularities of the empty bands define two critical fields, Hc1{\mathit{H}}_{\mathit{c}1}Hc1​(x) and Hc2{\mathit{H}}_{\mathit{c}2}Hc2​(x), at which the susceptibility diverges. The non-Fermi-liquid behavior as x\ensuremath{\rightarrow}0 arises from nonanalyticities as Hc2{\mathit{H}}_{\mathit{c}2}Hc2​ vanishes.
When a van Hove singularity is located in the vicinity of the Fermi level, the electronic scattering rate acquires a nonanalytic contribution. This invalidates basic assumptions of Fermi liquid theory and within treatments based on perturbation theory leads to a non-Fermi liquid self-energy and transport properties. Such anomalies are shown to also occur in the strongly correlated metallic state within dynamical mean-field theory. We consider the Hubbard model on a two-dimensional square lattice with nearest- and next-nearest-neighbor hoppings within the single-site dynamical mean-field theory. At temperatures on the order of the low-energy scale T0{T}_{0}T0​ an unusual maximum emerges in the imaginary part of the self-energy which is renormalized toward the Fermi level for finite doping. At zero temperature this double-well structure is suppressed but an anomalous energy dependence of the self-energy remains. For the frustrated Hubbard model on the square lattice with next-nearest-neighbor hopping, the presence of the van Hove singularity changes the asymptotic low-temperature behavior of the resistivity from a Fermi liquid to non-Fermi liquid dependency as function of doping. The results of this work are discussed regarding their relevance for high-temperature cuprate superconductors.