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arXiv:2307.14248 · Physical review. B./Physical review. B

Criteria and analytical results for the pseudogap at the Van Hove point in two dimensions

Yu. M. Vil’k

Published Dec 4, 2023• 6 citations
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Abstract

I established the criteria and obtained analytical results for the pseudogap at the Van Hove (antinodal) point on the Fermi surface in two dimensions. The original criterion $\ensuremath{\xi}\ensuremath{\gg}{\ensuremath{\xi}}_{th_db}={v}_{F}/\ensuremath{\pi}T$ is not applicable in this case since Fermi velocity ${v}_{F}=0$. It turns out that the characteristic length for the pseudogap crossover at the Van Hove point ${\ensuremath{\xi}}_{th_vh}\ensuremath{\propto}1/{T}^{1/2}$ is significantly shorter than the one at the regular Fermi surface points ${\ensuremath{\xi}}_{th_db}\ensuremath{\propto}1/T$. In particular, ${\ensuremath{\xi}}_{th_vh}$ is between one and two lattice spacings in the intermediate interaction regime of the Hubbard model. I have also identified the regime where there is still a single maximum in the spectral function, but single particle properties are abnormal. Specifically, the imaginary part of the self-energy has a minimum instead of maximum at the Fermi level and the slope of the real part of the self-energy is positive instead of negative. The important advantages of an analytical approach is that it provides results in both the Matsubara frequency representation and in the real frequencies representation. I compare Matsubara frequency results with the exact numerical results of the Monte Carlo methods in the Hubbard model and then show what they correspond to in the real frequency self-energy and spectral function.

DOI
10.1103/physrevb.108.235110

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I established the criteria and obtained analytical results for the pseudogap at the Van Hove (antinodal) point on the Fermi surface in two dimensions. The original criterion \ensuremathξ\ensuremath≫\ensuremathξthdb=vF/\ensuremathπT\ensuremath{\xi}\ensuremath{\gg}{\ensuremath{\xi}}_{th_db}={v}_{F}/\ensuremath{\pi}T\ensuremathξ\ensuremath≫\ensuremathξthd​b​=vF​/\ensuremathπT is not applicable in this case since Fermi velocity vF=0{v}_{F}=0vF​=0. It turns out that the characteristic length for the pseudogap crossover at the Van Hove point \ensuremathξthvh\ensuremath∝1/T1/2{\ensuremath{\xi}}_{th_vh}\ensuremath{\propto}1/{T}^{1/2}\ensuremathξthv​h​\ensuremath∝1/T1/2 is significantly shorter than the one at the regular Fermi surface points \ensuremathξthdb\ensuremath∝1/T{\ensuremath{\xi}}_{th_db}\ensuremath{\propto}1/T\ensuremathξthd​b​\ensuremath∝1/T. In particular, \ensuremathξthvh{\ensuremath{\xi}}_{th_vh}\ensuremathξthv​h​ is between one and two lattice spacings in the intermediate interaction regime of the Hubbard model. I have also identified the regime where there is still a single maximum in the spectral function, but single particle properties are abnormal. Specifically, the imaginary part of the self-energy has a minimum instead of maximum at the Fermi level and the slope of the real part of the self-energy is positive instead of negative. The important advantages of an analytical approach is that it provides results in both the Matsubara frequency representation and in the real frequencies representation. I compare Matsubara frequency results with the exact numerical results of the Monte Carlo methods in the Hubbard model and then show what they correspond to in the real frequency self-energy and spectral function.
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 consider many-fermion systems with singular Fermi surfaces, which contain Van Hove points where the gradient of the band function k ↦ e(k) vanishes. In a previous paper, we have treated the case of spatial dimension d ≥ 3. In this paper, we focus on the more singular case d = 2 and establish properties of the fermionic self-energy to all orders in perturbation theory. We show that there is an asymmetry between the spatial and frequency derivatives of the self-energy. The derivative with respect to the Matsubara frequency diverges at the Van Hove points, but, surprisingly, the self-energy is C1 in the spatial momentum to all orders in perturbation theory, provided the Fermi surface is curved away from the Van Hove points. In a prototypical example, the second spatial derivative behaves similarly to the first frequency derivative. We discuss the physical significance of these findings.
The two-dimensional attractive Hubbard model is studied in the weak-to-intermediate-coupling regime by employing a nonperturbative approach. It is shown that this approach is in quantitative agreement with Monte Carlo calculations for both single-particle and two-particle quantities. Both the density of states and the single-particle spectral weight show a pseudogap at the Fermi energy below some characteristic temperature T∗,{T}^{*},T∗, also in good agreement with quantum Monte Carlo calculations. The pseudogap is caused by critical pairing fluctuations in the low-temperature renormalized classical regime $(\ensuremath{\Elzxh}\ensuremath{\omega}<{k}_{B}T)$ of the two-dimensional system. With increasing temperature the spectral weight fills in the pseudogap instead of closing it, and the pseudogap appears earlier in the density of states than in the spectral function. Small temperature changes around T∗{T}^{*}T∗ can modify the spectral weight over frequency scales much larger than temperature. Several qualitative results for the s-wave case should remain true for d-wave superconductors.
To clarify the origin of the pseudogap and strange metal states as well as their mutual relationship in cuprate superconductors, a comprehensive study on the spectral function, Fermi surface, resistivity and dynamical spin susceptivity of the Hubbard model on the square lattice has been conducted by means of the ladder dual-fermion approximation with an electron self-energy correction. It is found that the appearance of these two states requires that the characteristic hole concentration below which the Mott-Heisenberg and Slater mechanisms of electron localization occurs pMSp_{\rm MS}pMS​ nearly coincides with the hole concentration where the Van Hove singularity (VHS) point is placed at the vicinity of the Fermi level. When this condition is met the VHS point is pinned at which the nesting condition of the antiferromagnetic (AFM) fluctuation is fulfilled almost everywhere on the Fermi surface in wide range of the hole concentration in a metallic state, i.e., the strange metal state. The spin fluctuation of the strange metal state is nearly quantum critical and the dynamical spin susceptivity is well described by overdamped spin wave having the ω/Tω/Tω/T scaling with the relaxation rate at the Planckian limit. Because of these distinctive features of the strange metal state, the kkk dependence of scattering rate of electrons is small and electrons behave as the marginal Fermi liquid, resulting in TTT-linear resistivity. In contrast, the pseudogap state is magnetically in the renormalized classical regime and the pseudogap is formed near the X point where the nesting condition of the short-range AFM order is fulfilled.
Exact diagrammatic quantum Monte Carlo (DiagMC) results for the nearest-neighbor Hubbard model motivate a closer study of the pseudogap. Using the improved two-particle self-consistent approach (TPSC+), we analyze Energy (EDC) and Momentum Distribution Curves (MDC) simultaneously. We show that Fermi-liquid terminology breaks down in the pseudogap regime, requiring a distinction between true and false Fermi surfaces and zero-energy quasiparticle (ZEQ) lines. A false Fermi surface has a momentum-space spectral maximum but a frequency-space depression at zero energy, while a false ZEQ line violates the standard quasiparticle condition ∂Σ′(k,ω)/∂ω∣ω=0<0\partial Σ'(\mathbf{k},ω)/\partial ω|_{ω=0}<0∂Σ′(k,ω)/∂ω∣ω=0​<0.The pseudogap is driven by critical thermal spin fluctuations, which occur in two dimensions because of the Mermin-Wagner theorem. Commensurate fluctuations first open an antinodal pseudogap, leaving true Fermi arcs. With decreasing temperature, the Fermi surface evolves into hole- and electron-like false Fermi surfaces. Incommensurate fluctuations generate hot spots near the diagonal, where the pseudogap persists to the quantum critical point (QCP), whereas at kAN\mathbf{k}_{AN}kAN​ it disappears before the QCP doping. There, the spectrum has two precursor antiferromagnetic (AFM) bands, both in the unoccupied (ω>0ω>0ω>0) region. We benchmark TPSC+ against DiagMC results for the Matsubara spectral proxy −Im[G(k,iπT)]/π-\mathrm{Im}[\mathcal{G}(\mathbf{k},iπT)]/π−Im[G(k,iπT)]/π. TPSC+ underestimates pseudogap suppression in the strong-interaction regime, reproducing DiagMC behavior at lower temperatures or doping. This proxy is equivalent to the spectral function with thermal broadening η=πTη=πTη=πT, which obscures features when πTπTπT is not the smallest energy scale. Using A(k,0)A(\mathbf{k},0)A(k,0), we find that hole-like Fermi surfaces emerge at any interaction strength at low temperature, even at weak coupling.
We have studied the pseudogap formation in the single-particle spectra of the half-filling two-dimensional Hubbard model. Using a Green's function with the one-loop self-energy correction of the spin and charge fluctuations, we have numerically calculated the self-energy, the spectral function, and the density of states in the weak-coupling regime at finite temperature. Pseudogap formations have been observed in both the density of states and the spectral function at the Fermi level. The pseudogap in the spectral function is explained by the non-Fermi-liquid-like nature of the self-energy. The anomalous behavior in the self-energy is caused by both the strong antiferromagnetic spin fluctuation and the nesting condition on the non-interacting Fermi surface. In the present approximation, we find a logarithmic singularity in the integrand of the imaginary part of the self-energy. Concerning the energy dependence of the spectral function and the self-energy, two theorems are proved. They give a necessary condition in the self-energy to produce the pseudogap at the Fermi level. The pseudogap in the spectral function is highly momentum dependent on the Fermi surface. It opens initially in the (±π,0)(\pm π,0)(±π,0), (0,±π)(0,\pm π)(0,±π) regions as the normal state pseudogap observed in the high-TcT_cTc​ superconductors and if the interaction is increased, it spreads to other Fermi surface sectors. The anisotropy of the pseudogap is produced by the low-energy enhancement of the spin excitation around Q=(π,π){\bf Q}=(π,π)Q=(π,π) and the flatness of the band dispersion around the saddle point.