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Physical review. B./Physical review. B

Determination of the Fermi surface by charge density correlations

Zhipeng Sun

Published Apr 27, 2023• 0 citations
Publisher

Abstract

The Fermi surface topology plays a crucial role in the study of high-temperature superconductivity cuprates. The conventional method for determining the Fermi surface is the maximum spectral intensity method, which involves numerical analytical continuation of Matsubara data for the one-body Green's function. However, the numerical analytical continuation is sensitive to the noise or the precision of the Matsubara data, and hence is not always reliable. With this in mind, we propose a simple and specific notion as a reference for the Fermi surface. It is the derivative of the momentum distribution function ${n}_{\mathbit{k}}$ with respect to the chemical potential $\ensuremath{\mu}$. Our analysis of the noninteracting system shows that the momentum, at which $d{n}_{\mathbit{k}}/d\ensuremath{\mu}$ takes the maximum value, constitutes a surface, which coincides with the Fermi surface. The relationship between $d{n}_{\mathbit{k}}/d\ensuremath{\mu}$ and the Fermi surface in general cases is also analyzed. In order to numerically demonstrate this relationship, we calculate and compare on the two-dimensional Hubbard model within the HGW method. The results show that, at least in the weak and intermediate coupling regimes, using the notion of $d{n}_{\mathbit{k}}/d\ensuremath{\mu}$ to determine the Fermi surface is reliable. We further studied the cases with model parameters standing for realistic materials, and found the surface determined by $d{n}_{\mathbit{k}}/d\ensuremath{\mu}$ exhibits topological transition with the charge filling, similar to the Fermi surface in the cuprates. We believe that using the derivative of the momentum distribution function with respect to the chemical potential to determine the Fermi surface is a reasonable, efficient, and potentially valuable approach.

DOI
10.1103/physrevb.107.165154

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•Apr 16, 2020•1 citation
The Fermi surface topology plays a crucial role in the study of high-temperature superconductivity cuprates. The conventional method for determining the Fermi surface is the maximum spectral intensity method, which involves numerical analytical continuation of Matsubara data for the one-body Green's function. However, the numerical analytical continuation is sensitive to the noise or the precision of the Matsubara data, and hence is not always reliable. With this in mind, we propose a simple and specific notion as a reference for the Fermi surface. It is the derivative of the momentum distribution function n\mathbitk{n}_{\mathbit{k}}n\mathbitk​ with respect to the chemical potential \ensuremathμ\ensuremath{\mu}\ensuremathμ. Our analysis of the noninteracting system shows that the momentum, at which dn\mathbitk/d\ensuremathμd{n}_{\mathbit{k}}/d\ensuremath{\mu}dn\mathbitk​/d\ensuremathμ takes the maximum value, constitutes a surface, which coincides with the Fermi surface. The relationship between dn\mathbitk/d\ensuremathμd{n}_{\mathbit{k}}/d\ensuremath{\mu}dn\mathbitk​/d\ensuremathμ and the Fermi surface in general cases is also analyzed. In order to numerically demonstrate this relationship, we calculate and compare on the two-dimensional Hubbard model within the HGW method. The results show that, at least in the weak and intermediate coupling regimes, using the notion of dn\mathbitk/d\ensuremathμd{n}_{\mathbit{k}}/d\ensuremath{\mu}dn\mathbitk​/d\ensuremathμ to determine the Fermi surface is reliable. We further studied the cases with model parameters standing for realistic materials, and found the surface determined by dn\mathbitk/d\ensuremathμd{n}_{\mathbit{k}}/d\ensuremath{\mu}dn\mathbitk​/d\ensuremathμ exhibits topological transition with the charge filling, similar to the Fermi surface in the cuprates. We believe that using the derivative of the momentum distribution function with respect to the chemical potential to determine the Fermi surface is a reasonable, efficient, and potentially valuable approach.
The Fermi surface topology in the two-dimensional Hubbard model is particularly relevant for the high-temperature superconductors, whereas its theoretical research encounters with the difficulty of the analytical continuation problem. To this end, we proposed the concept of the momentum-dependent compressibility, defined as the variation of the momentum distribution function with respect to the chemical potential. The surface determined by the maximum of the momentum-dependent compressibility is nearly identical to the Fermi surface in the weakly and intermediate coupling regions according to our numerical results. In the correlated region, this surface also exhibits pocket and arc features, just like the Fermi surface in high-temperature superconductors. Therefore, for theoretical studies, this surface can be used as an alternative to determine the underlying Fermi surface. Considering that the momentum-dependent compressibility is closely related to the charge density correlations, our work also shows a connection between the Fermi surface topology and the charge density fluctuations.
One of the fundamental questions about high-temperature cuprate superconductors is the size of the Fermi surface underlying the superconducting state. By analyzing the single-particle spectral function for the Fermi-Hubbard model as a function of repulsion UUU and chemical potential $\ensuremath{\mu}$, we find that the Fermi surface in the normal state undergoes a transition from a large Fermi surface matching the Luttinger volume as expected in a Fermi liquid, to a Fermi surface that encloses fewer electrons that we dub the ``Luttinger breaking'' phase, as the Mott insulator is approached. This transition into a non-Fermi-liquid phase that violates the Luttinger count occurs at a critical density in the absence of any other broken symmetry. We obtain the Fermi-surface contour from the spectral weight ${A}_{\mathbf{k}}(\ensuremath{\omega}=0)$ and from an analysis of the singularities of the Green's function ReGk(E=0)\mathrm{Re}\phantom{\rule{0.16em}{0ex}}{G}_{\mathbf{k}}(E=0)ReGk​(E=0), calculated using determinantal quantum Monte Carlo and analytic continuation methods. We discuss our numerical results in connection with experiments on Hall measurements, scanning tunneling spectroscopy, and angle-resolved photoemission spectroscopy.
One of the fundamental questions about the high temperature cuprate superconductors is the size of the Fermi surface (FS) underlying the superconducting state. By analyzing the single particle spectral function for the Fermi Hubbard model as a function of repulsion UUU and chemical potential μμμ, we find that the Fermi surface in the normal state reconstructs from a large Fermi surface matching the Luttinger volume as expected in a Fermi liquid, to a Fermi surface that encloses fewer electrons that we dub the "Luttinger Breaking" (LB) phase, as the Mott insulator is approached. This transition into a non-Fermi liquid phase that violates the Luttinger count, is a continuous phase transition at a critical density in the absence of any other broken symmetry. We obtain the Fermi surface contour from the spectral weight Ak⃗(ω=0)A_{\vec{k}}(ω=0)Ak​(ω=0) and from an analysis of the poles and zeros of the retarded Green's function Gk⃗ret(E=0)G_{\vec{k}}^{ret}(E=0)Gkret​(E=0), calculated using determinantal quantum Monte Carlo and analytic continuation methods.We discuss our numerical results in connection with experiments on Hall measurements, scanning tunneling spectroscopy and angle resolved photoemission spectroscopy.
The question of determining the underlying Fermi surface (FS) that is gapped by superconductivity (SC) is of central importance in strongly correlated systems, particularly in view of angle-resolved photoemission experiments. Here we explore various definitions of the FS in the superconducting state using the zero-energy Green's function, the excitation spectrum, and the momentum distribution. We examine (a) d-wave SC in high-Tc cuprates, and (b) the s-wave superfluid in the BCS-Bose-Einstein condensation (BEC) crossover. In each case we show that the various definitions agree, to a large extent, but all of them violate the Luttinger count and do not enclose the total electron density. We discuss the important role of chemical potential renormalization and incoherent spectral weight in this violation.
We calculate the single-particle Green's function for a contact interaction with nearest-neighbor hopping on a square lattice as a function of chemical potential $\ensuremath{\mu}.$ This allows us to investigate the dependence of the leading Fermi-liquid dependencies on the band structure as the Fermi surface evolves from a circle at $\ensuremath{\mu}\ensuremath{\sim}\ensuremath{-}4t$ to a square at $\ensuremath{\mu}=0.$ The form of the single-particle self-energy $\ensuremath{\Sigma}(\stackrel{\ensuremath{\rightarrow}}{p},E)$ is determined by the density-density correlation function $\ensuremath{\chi}(\stackrel{\ensuremath{\rightarrow}}{q},\ensuremath{\omega})$ which develops two peaks for $\ensuremath{\mu}\ensuremath{\gtrsim}\ensuremath{-}2.5t$ unlike the parabolic band case. Near half-filling, $\ensuremath{\chi}(\stackrel{\ensuremath{\rightarrow}}{q},\ensuremath{\omega})$ becomes independent of $\ensuremath{\omega},$ one-dimensional behavior, at intermediate values of $\ensuremath{\omega},$ which leads to the one-dimensional behavior in $\ensuremath{\Sigma}(\stackrel{\ensuremath{\rightarrow}}{p},E).$ However, with $\ensuremath{\mu}<~\ensuremath{-}0.1t$ there is no influence on the Fermi-liquid dependencies from the spin-density wave instability. We find that throughout the doping region $\ensuremath{\Sigma}(\stackrel{\ensuremath{\rightarrow}}{p},E)$ remains qualitatively the same as for the isotropic Fermi surface with quantitative differences. The strong $\stackrel{\ensuremath{\rightarrow}}{p}$ and E dependence of the off-shell self-energy $\ensuremath{\Sigma}(p,E)$ found earlier for the parabolic band is recovered for $\ensuremath{\mu}\ensuremath{\lesssim}\ensuremath{-}t$ but deviates from this develop for $\ensuremath{\mu}\ensuremath{\gtrsim}\ensuremath{-}0.1t.$ The resonance peak width of the spectral function $A(\stackrel{\ensuremath{\rightarrow}}{p}E),$ has a linear dependence in ${\ensuremath{\xi}}_{\stackrel{\ensuremath{\rightarrow}}{p}}$ due to the E dependence of the imaginary part of $\ensuremath{\Sigma}(\stackrel{\ensuremath{\rightarrow}}{p},E).$ We point out that an accurate detailed form for $\ensuremath{\Sigma}(\stackrel{\ensuremath{\rightarrow}}{p},E)$ would be very difficult to recover from angle-resolved photoemission spectroscopy data for the spectral density. Since the leading corrections are determined by the long wavelength particle-hole excitations, the results found here for the Hubbard model carry over to Hamiltonians with finite range interactions.
We take an in depth look into the celebrated Fermi-Hubbard model for strongly interacting fermions, beginning with a two-site exact diagonalization demonstration of the Mott ground state. We continue with measurements of compressibility and susceptibility using Determinantal Quantum Monte Carlo (DQMC) methods on a larger square lattice 6x6 over the parameter space of interaction strength U/t and doping, controlled by the chemical potential, mu/t. We construct the Fermi surface for a 16x16 cluster size using the retarded Green's function and the single particle spectral function (derived from the Green's function using an analytic continuation procedure) and find conclusive evidence of Fermi surface restructuring (FSR) as a result of strong interactions freezing out states from contributing to the Luttinger count, n, at intermediate temperatures, T = t/2. FSR is found for strongly interacting, doped systems for dopings p < 0.2. This phase is dubbed the Luttinger's rule breaking (LB) phase, and we provide a discussion of the consequences of the deviation from Luttinger's theorem. We strongly suggest a topological nature underlying the restructuring of the Fermi surface by eliminating competing ordering in spin or charge channels.