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Physical review. B./Physical review. B

Broken Luttinger theorem in the two-dimensional Fermi-Hubbard model

Ian S. Osborne, Thereza Paiva, Nandini Trivedi

Published Dec 10, 2021• 14 citations
Publisher

Abstract

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 $U$ 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 $\mathrm{Re}\phantom{\rule{0.16em}{0ex}}{G}_{\mathbf{k}}(E=0)$, 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.

DOI
10.1103/physrevb.104.235122

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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 \ensuremathμ\ensuremath{\mu}\ensuremathμ, 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 Ak(\ensuremathω=0){A}_{\mathbf{k}}(\ensuremath{\omega}=0)Ak​(\ensuremathω=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.
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.
Superconductivity results from an instability of the Fermi surface---contour of poles of the single-particle propagator---to an infinitesimally small attraction between electrons. Here, we instead discuss the analogous problem on a model Luttinger surface, or contour of zeros of the Green function. At zero temperature $(\ensuremath{\beta}\ensuremath{\rightarrow}\ensuremath{\infty})$ and a critical interaction strength $({u}_{c\ensuremath{\infty}})$ characterized by the residue of self-energy pole, we find that the pair susceptibility diverges leading to a superconducting instability. We evaluate the pair fluctuation partition function and find that the spectral density in the normal state has an interaction-driven, power-law $\frac{1}{\sqrt{\ensuremath{\omega}}}$ type, van-Hove singularity (vHS), indicating non-Fermi liquid (NFL) physics. Crucially, in the strong coupling limit $(\ensuremath{\beta}u\ensuremath{\gg}1)$, the leading-order fluctuation free-energy terms in the normal state of this NFL-SC transition resembles the equivalent [O(1)]\left[O(1)\right][O(1)] terms of the Sachdev-Ye-Kitaev (SYK) model. This free-energy contribution takes a simple form $\ensuremath{-}\ensuremath{\beta}F=\ensuremath{\beta}{u}_{c\ensuremath{\infty}}\ensuremath{-}\ensuremath{\gamma}\phantom{\rule{0.16em}{0ex}}\text{ln}\left(\ensuremath{\beta}{u}_{c\ensuremath{\infty}}\right),$ where $\ensuremath{\gamma}$ is a constant equal to 12\frac{1}{2}21​. Weak impurity scattering $(\ensuremath{\tau}\ensuremath{\gg}{\ensuremath{\beta}}^{\ensuremath{-}1})$ leaves the low-energy spectral density unaffected but leads to an interaction-driven enhancement of superconductivity. Our results shed light on the role played by order-parameter fluctuations in providing the key missing link between Mott physics and strongly coupled toy models exhibiting gravity duals.
Using the density matrix renormalization group method, we systematically investigate the evolution of the Luttinger integral in the one-dimensional generalized ttt-VVV model as a function of filling and interaction strength, and identify three representative phases. In the weak-coupling regime, the zero-frequency Green's function exhibits a branch-cut structure at the Fermi momentum, and the Luttinger integral accurately reflects the particle density, indicating that the Luttinger theorem holds. As the interaction increases, the spectral weight near the Fermi momentum is gradually suppressed. Interestingly, in the strong coupling regime near half-filling, this singularity is progressively destroyed, accompanied by the emergence of momentum-space zeros in the real part of the Green's function, leading to a novel non-Fermi liquid metallic phase beyond the classic Luttinger liquid paradigm, where the Luttinger surface is no longer defined by a single singularity. While finite spectral weight remains at the original Fermi momentum, the singularity gradually diminishes. Meanwhile, zeros with negligible spectral weight appear away from this momentum, significantly affecting the integral. At exact half-filling, a single-particle gap opens, and the Green's function becomes nearly vanishing across the entire momentum space, indicating the complete suppression of low-energy electronic states consistent with the nature of an insulating charge-density-wave phase. These results suggest that the breakdown of the Luttinger theorem is not triggered by a single mechanism, but rather results from the interplay between interaction-driven evolution of excitation modes and the breaking of particle-hole symmetry, ultimately leading to a continuous reconstruction of the generalized Fermi surface from topologically protected to correlation-driven.
The Fermi liquid theory is a central concept in modern condensed matter physics used to describe conventional metals. A state of this universality class consists of well-defined quasiparticles, which occupy a finite number of states within a Fermi volume defined in momentum space. According to the Luttinger’s theorem, such a state encloses a Fermi volume proportional to the electron density modulo the filled bands if no symmetry is broken. In the past decades strong deviations from Fermi liquid theory have been observed in the scaling behavior of thermodynamic observables in different materials, among which are for instance cuprates and iron pnictides. This thesis is about two prototypical systems that can not be described by Fermi liquid theory. The first part of this work investigates two-dimensional effective models, which are in a Mott insulating state at half-filling and have a finite conductivity when doped with holes. In such a Mott insulating state, the charge carriers are strongly localized due to the Coulomb repulsion. Hence, strong correlations are assumed to have a strong impact on the formation of the ground state, also in the regime of small hole concentration. In a first project we have used a so-called spinon dopon mean-field theory to represent the two-dimensional Fermi-Hubbard model with strong on-site repulsive interaction in effective degrees of freedom, in which the holes can be embedded into a quantum spin liquid. The corresponding SU(2) invariant ground state belongs to the class of fractionalized Fermi liquids. In a second project we investigate a quantum dimer model, an effective model based on a Hilbert space spanned by short range singlets and bound states of holes and spins. The focus here is on the calculation of the hole-part of the electron spectral function by using exact diagonalization and its comparison with two analytic methods, a diagrammatic computation based on the Bethe-Salpeter equation and a so-called two-mode approximation. The electron spectral function shows a similar analytic form in momentum space between nodal and antinodal point when compared to results from photoemission spectroscopy experiments on cuprates. Furthermore, in a subsequent work we calculate the exact ground state of the quantum dimer model along a certain parameter line. In order to analyze the behavior of the elctron spectral function when increasing the density of holes, we investigate the Fermi-Hubbard model in a current project using a dynamical mean-field approach. To solve the 4-site cluster impurity problem, we use a numerical renormalization group approach. However, numerical limitations force us to restrict the analysis to spin-polarized baths. According to the spectral data, the system is similar to the SU(2) invariant case at half-filling in a Mott insulating state and posseses a momentum-selective energy gap at finite doping. The topology of the Fermi surface shows a Lifshitz transition when increasing the hole concentration. Here, the curvature of the Fermi surface changes from electron- to hole-like. For comparison we apply the dynamical mean-field theory also to the quantum dimer model and observe that the electron spectral functions at finite doping are qualitatively similar to that of the two-dimensional Fermi-Hubbard model. The second half of the work is about Tomonaga-Luttinger liquid theory, which is used to describe the low-energy effective degrees of freedom of one-dimensional systems. Here, we first discuss a conceptional extension of the operator-based bosonization theory for one-dimensional systems. This extension is especially suited for inhomogeneous one-dimensional systems. First, we investigate a one-dimensional system with a local interation potential and compute an exact solution of the single particle propagator at T = 0. The critical exponent of the single particle propagator has an unconventional form as a function of the microscopic Tomonaga-Luttinger parameters, which is not covered by the original Luttinger paradigm postulated by F. Duncan M. Haldane. In a second project on one-dimensional systems, we study the impact of scattering processes among bosonic low-energy excitations on the thermalization process. Such scattering processes are irrelevant on large length scales, however strongly affect the dynamics. In our analytic analysis we focus on a experimental setup, where a one-dimensional Bose gas is instantaneously splitted in two identical, however strongly correlated, halves of one-dimensional electronic systems. In the following, the corresponding non-equilibrium state runs through multiple regimes in time, such as a metastable prethermalization regime. However, above a certain threshold in time such scattering processes cause an effective thermalization of the system. In order to demonstrate this, we compute the kinetic equation in the Keldysh field integral formalism from a diagrammatic expansion based on a self-consistent Born approximation.
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.