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Physical review. B./Physical review. B

Probing a modified Luttinger sum rule in the strongly interacting one-dimensional Fermi-Hubbard model

Annika Böhler, Henning Schlömer, Ulrich Schollwöck, Annabelle Bohrdt, Fabian Grusdt

Published Jan 3, 2025• 2 citations
Publisher

Abstract

Fermi surface reconstruction in cuprates can lead to an abrupt change in the Fermi momentum ${k}_{F}$ between different phases. This phenomenon remains a subject of debate and is at the heart of an ongoing discussion about the nature of the metallic state in the pseudogap regime. Here we study a minimal model of a ${k}_{F}$ changing crossover in the one-dimensional Fermi-Hubbard model, where a tuning of the on-site interaction leads to a crossover between a spin-$1/2$ Luttinger liquid with small Fermi momentum and a spinless chargon liquid with large Fermi momentum. We attribute this to an emergent $U(1)$ symmetry in the strongly correlated limit, which can be used to derive a modified Luttinger sum rule recovering the large Fermi momentum. We analyze Friedel oscillations at the edge of a system to directly probe the change of Fermi momentum at zero and nonzero temperature. This paves the way for a direct experimental observation of changes of the Fermi momentum using ultracold fermions in a quantum gas microscope, with possible extensions to higher dimensional systems.

DOI
10.1103/physrevb.111.045107

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Physical review. B, Condensed matter•Aug 1, 1993•10 citations
Fermi surface reconstruction in cuprates can lead to an abrupt change in the Fermi momentum kF{k}_{F}kF​ between different phases. This phenomenon remains a subject of debate and is at the heart of an ongoing discussion about the nature of the metallic state in the pseudogap regime. Here we study a minimal model of a kF{k}_{F}kF​ changing crossover in the one-dimensional Fermi-Hubbard model, where a tuning of the on-site interaction leads to a crossover between a spin-1/21/21/2 Luttinger liquid with small Fermi momentum and a spinless chargon liquid with large Fermi momentum. We attribute this to an emergent U(1)U(1)U(1) symmetry in the strongly correlated limit, which can be used to derive a modified Luttinger sum rule recovering the large Fermi momentum. We analyze Friedel oscillations at the edge of a system to directly probe the change of Fermi momentum at zero and nonzero temperature. This paves the way for a direct experimental observation of changes of the Fermi momentum using ultracold fermions in a quantum gas microscope, with possible extensions to higher dimensional systems.
Fermi surface reconstruction in cuprates can lead to an abrupt change in the Fermi momentum kFk_FkF​ between different phases. This phenomenon remains subject of debate and is at the heart of an ongoing discussion about the nature of the metallic state in the pseudogap regime. Here we study a minimal model of a kFk_FkF​ changing crossover in the one-dimensional Fermi-Hubbard model, where a tuning of the onsite interaction leads to a crossover between a spin-1/21/21/2 Luttinger liquid with small Fermi momentum and a spinless chargon liquid with large Fermi momentum. We attribute this to an emergent U(1)U(1)U(1) symmetry in the strongly correlated limit, which can be used to derive a modified Luttinger sum rule recovering the large Fermi momentum. We analyse Friedel oscillations at the edge of a system to directly probe the change of Fermi momentum at zero and non-zero temperature. This paves the way for a direct experimental observation of changes of the Fermi momentum using ultracold fermions in a quantum gas microscope, with possible extensions to higher dimensional systems.
The Luttinger sum rule is usually considered for Landau Fermi liquids in which the single particle Green’s function G(k, 0) changes sign at the the Fermi surface by passing through infinity. However the general proof allows also for a sign change at which G has a zero. A recent analysis by Konik and coworkers considers a model of 2‐leg Hubbard ladders weakly coupled by a small long range interladder tunneling. At half‐filling a semimetallic state with small Fermi pockets is induced beyond a threshold tunneling strength. The sign changes in G(k, 0) relevent for the Luttinger sum rule now take place at surfaces with both zeros and infinities. The zero surfaces differ from the minimum gap surfaces. The latter are often used in ARPES experiments on underdoped cuprates to obtain an underlying Fermi surface but this procedure leads to problems with the Luttinger sum rule. Some speculations on how the Luttinger sum rule should be applied to the pseudogap phase of the underdoped cuprates are included.
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 quantum phase transition from a spin-Peierls phase with a small Fermi surface to a paramagnetic Luttinger-liquid phase with a large Fermi surface is studied in the framework of a one-dimensional Kondo-Heisenberg model that consists of an electron gas away from half filling, coupled to a spin-1/2 chain by Kondo interactions. The Kondo spins are further coupled to each other with isotropic nearest-neighbor and next-nearest-neighbor antiferromagnetic Heisenberg interactions which are tuned to the Majumdar-Ghosh point. Focusing on three-eighths filling and using the density-matrix renormalization-group (DMRG) method, we show that the zero-temperature transition between the phases with small and large Fermi momenta appears continuous, and involves a new intermediate phase where the Fermi surface is not well defined. The intermediate phase is spin gapped and has Kondo-spin correlations that show incommensurate modulations. Our results appear incompatible with the local picture for the quantum phase transition in heavy fermion compounds, which predicts an abrupt change in the size of the Fermi momentum.
We study a one-dimensional system that consists of an electron gas coupled to a spin-1/2 chain by Kondo interaction away from half filling. We show that zero-temperature transitions between phases with ``small'' and ``large'' Fermi momenta can be continuous. Such a continuous but Fermi-momentum-changing transition arises in the presence of spin anisotropy, from a Luttinger liquid with a small Fermi momentum to a Kondo-dimer phase with a large Fermi momentum. We have also added a frustrating next-nearest-neighbor interaction in the spin chain to show the possibility of a similar Fermi-momentum-changing transition, between the Kondo phase and a spin-Peierls phase, in the spin isotropic case. This transition, however, appears to involve a region in which the two phases coexist.
The crossover from conventional Fermi-liquid (FL) behavior to the Luttinger-liquid-like behavior in the two-dimensional single-band small-U Hubbard model near half filling is demonstrated. The applicability of the FL picture is discussed. This crossover is reproduced in the fermionic channel of the t-J model, if the fermion-fermion interaction is taken into account. The results are in qualitative agreement with the normal-state resistivity and photoemission data on the high-Tc{\mathit{T}}_{\mathit{c}}Tc​ cuprates.