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arXiv:cond-mat/9604057

van Hove Singularity effects in Strongly Correlated Fermions

Pinaki Majumdar, H. R. Krishnamurthy

Published Apr 10, 1996• 0 citations
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Abstract

We examine the effects of a van Hove singularity (vHs) in the density of states (DOS) of a two dimensional, $t-t'-U$ Hubbard model within the local approximation. Within our approximation the non-interacting Fermi surface is always retained away from half-filling; and we find that the vHs in the DOS survives, and for hole doping is nearly "pinned" to the Fermi level due to the enhancement of the effective mass, $m^*$. We discuss the electron-hole asymmetry of this phenomena and its relevance to the electron and hole doped cuprates. At optimal (hole) doping, with the Fermi level at the vHs, we find that the single particle damping rate $Γ$, and the resistivity $ρ_{dc}$, scale as $T$ for moderately low temperatures ($T \gtrsim 100$K), while for the electron doped system $Γ$ and $ρ_{dc} \sim T^2$.

DOI
10.48550/arxiv.cond-mat/9604057

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Journal of the Physical Society of Japan•Aug 15, 1995•6 citations
We examine the effects of a van Hove singularity (vHs) in the density of states (DOS) of a two dimensional, t−t′−Ut-t'-Ut−t′−U Hubbard model within the local approximation. Within our approximation the non-interacting Fermi surface is always retained away from half-filling; and we find that the vHs in the DOS survives, and for hole doping is nearly "pinned" to the Fermi level due to the enhancement of the effective mass, m∗m^*m∗. We discuss the electron-hole asymmetry of this phenomena and its relevance to the electron and hole doped cuprates. At optimal (hole) doping, with the Fermi level at the vHs, we find that the single particle damping rate ΓΓΓ, and the resistivity ρdcρ_{dc}ρdc​, scale as TTT for moderately low temperatures (T≳100T \gtrsim 100T≳100K), while for the electron doped system ΓΓΓ and ρdc∼T2ρ_{dc} \sim T^2ρdc​∼T2.
Controlling the ordered phases of correlated electron systems remains a central challenge in quantum materials design. Divergences in the electronic density of states, known as Van Hove singularities (VHSs), are one obvious route to such control. It is clear from recent work that the exact functional form of these divergences can profoundly affect which phases are realized; a full picture, however, remains elusive. In this work, we use both the hot-spot parquet renormalization group and the truncated-unity functional renormalization group to theoretically study the emergent correlated states of a two-dimensional square-lattice Hubbard model with VHSs at or near the Fermi level. By varying a single hopping parameter, t3t_3t3​, we are able to change the strength of the VHS divergence in the density of states from logarithmic (for t3<t3ct_3 < t_{3c}t3​<t3c​) to power-law (for t3=t3ct_3 = t_{3c}t3​=t3c​). Further increase of t3t_3t3​ (t3>t3ct_3 > t_{3c}t3​>t3c​) causes each original Van Hove point to split into two, both of the conventional logarithmic type. We show that which of these regimes we are in strongly influences the predicted ordered states. We also study the dependence on doping, and find that the ferromagnetic state that occurs at Van Hove filling in these models is unstable to very small shifts in the Fermi level, often giving way to distinct ordered states depending on whether the model is electron- or hole-doped. These results highlight the importance of tuning VHS properties to control ordered states in correlated materials, and offer design rules to engineer these phases in novel systems.
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.
A quantum critical point is found in the phase diagram of the two-dimensional Hubbard model [Vidhyadhiraja et al., Phys. Rev. Lett. 102, 206407 (2009)]. It is due to the vanishing of the critical temperature associated with a phase-separation transition, and it separates the non-Fermi-liquid region from the Fermi liquid. Near the quantum critical point, the pairing is enhanced since the real part of the bare ddd-wave pairing susceptibility exhibits an algebraic divergence with decreasing temperature, replacing the logarithmic divergence found in a Fermi liquid [Yang et al., Phys. Rev. Lett. 106, 047004 (2011)]. In this paper, we explore the single-particle and transport properties near the quantum critical point using high-quality estimates of the self-energy obtained by direct analytic continuation of the self-energy from the continuous-time quantum Monte Carlo method. We focus mainly on a van Hove singularity coming from the relatively flat dispersion that crosses the Fermi level near the quantum critical filling. The flat part of the dispersion orthogonal to the antinodal direction remains pinned near the Fermi level for a range of doping that increases when we include a negative next-near-neighbor hopping ${t}^{\ensuremath{'}}$ in the model. For comparison, we calculate the bare ddd-wave pairing susceptibility for noninteracting models with the usual two-dimensional tight-binding dispersion and a hypothetical quartic dispersion. We find that neither model yields a van Hove singularity that completely describes the critical algebraic behavior of the bare ddd-wave pairing susceptibility found in the numerical data. The resistivity, thermal conductivity, thermopower, and the Wiedemann-Franz law are examined in the Fermi liquid, marginal Fermi liquid, and pseudogap doping regions. A negative next-near-neighbor hopping ${t}^{\ensuremath{'}}$ increases the doping region with marginal Fermi liquid character. Both TTT and negative ${t}^{\ensuremath{'}}$ are relevant variables for the quantum critical point, and both the transport and the displacement of the van Hove singularity with filling suggest that they are qualitatively similar in their effect.
We study the weak-coupling limit of the $t\ensuremath{-}{t}^{\ensuremath{'}}\ensuremath{-}U$ Hubbard model on a two-dimensional square lattice using a direct perturbative approach. Aided by symbolic computational tools, we compute the longitudinal density-density correlation functions in the ${\ensuremath{\chi}}_{\ensuremath{\uparrow}\ensuremath{\uparrow}}$ and ${\ensuremath{\chi}}_{\ensuremath{\uparrow}\ensuremath{\downarrow}}$ basis from which we can obtain the dynamical spin and charge susceptibilities at arbitrary doping and temperature. We find that for nonzero ${t}^{\ensuremath{'}}$, the zero-frequency commensurate $\mathbf{q}=(\ensuremath{\pi},\ensuremath{\pi})$ spin and charge excitations are each strongest at different densities and we observe a clear behavioral change that appears tied to the van Hove singularity of the noninteracting dispersion upon which the perturbative expansion is built. We find a strongly reduced compressibility in the vicinity of the van Hove singularity as well as a behavioral change in the double occupancy. For finite ${t}^{\ensuremath{'}}$, the observed van Hove singularity occurs away from half filling, leading us to conclude that this reduction in compressibility is distinct from the Mott insulating physics that one expects in the strong-coupling regime. We compute the full dynamical spin and charge excitations and observe distinct structure for electron- and hole-doped scenarios, in agreement with experiments on cuprate materials. Finally, we observe a peculiar splitting in spin and charge excitations in the vicinity of the van Hove singularity, the origin of which is traced to a splitting near the bottom of the band.
Single-particle excitation spectra of the two-dimensional Hubbard model on the square lattice near half filling and at zero temperature are investigated on the basis of the self-consistent projection operator method. The method guarantees a high accuracy of the spectra with high energy and high momentum resolutions. It takes into account long-range intersite correlations as well as the strong on-site correlations. Effects of nonlocal excitations are clarified by comparing the results with those of the single-site approximation. The calculated spectra verify the quantum Monte-Carlo results for finite temperatures. The spectra at the Fermi level yield a hole-like Fermi surface in the underdoped region and an electron-like Fermi surface in the overdoped region. From a numerical analysis of the momentum dependent effective mass and self-energy, it is concluded that a marginal Fermi-liquid like state persists even at finite doping concentrations in the strongly correlated region because a van Hove singularity is pinned to the Fermi surface. It is also found that a kink structure appears in the quasiparticle energy band in the same region. The kink is shown to be caused by a mixing between the quasiparticle band and an excitation band with strong short-range antiferromagnetic correlations. These results suggest an explanation for some of the unusual properties of the normal state in high-TcT_{\rm c}Tc​ cuprates.
The density of states (DOS) of the two-dimensional Hubbard model is calculated in second-order perturbation with respect to the on-site Coulomb interaction U . The direct analytic continuation formula is presented to calculate the self-energy on the real-frequency axis. The momentum as well as the energy integral in the self-energy is seriously carried out with use of the Fast-Fourier-Transformation algorithm. Near the half-filling, the imaginary part of the self-energy becomes proportional to the energy in the vicinity of the Fermi level in our weak-coupling calculation. In the interacting case, the van Hove singularity (VHS) in DOS is smeared by the self-energy effect, but there still exists a sharp peak structure due to VHS.