Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:cond-mat/9906349 · International Journal of Modern Physics B

VIOLATION OF LUTTINGER'S THEOREM IN STRONGLY CORRELATED ELECTRONIC SYSTEMS WITHIN A 1/N EXPANSION

E. Cappelluti, R. Zeyher

Published Aug 10, 1999• 6 citations
arXivHTMLPDF

Abstract

We study the 1/N expansion of a generic, strongly correlated electron model ( SU (N) symmetric Hubbard model with U=∞ and N degrees of freedom per lattice site) in terms of X operators. The leading order of the expansion describes a usual Fermi liquid with renormalized, stable particles. The next-to-leading order violates Luttinger's theorem if a finite convergence radius for the 1/N expansion for a fixed and non-vanishing doping away from half-filling is assumed. We find that the volume enclosed by the Fermi surface, is at large, but finite N's and small dopings larger than at N=∞. As a by-product an explicit expression for the electronic self-energy in O(1/N) is given which cannot be obtained by factorization or mode-coupling assumptions but contains rather sophisticated vertex corrections.

DOI
10.1142/s021797929900254x

Related papers

Luttinger's theorem in the presence of Luttinger surfaces

Jan Skolimowski, Michele Fabrizio

Breakdown of Landau's hypothesis of adiabatic continuation from noninteracting to fully interacting electrons is commonly believed to bring about a violation of Luttinger's theorem. Here, we elucidate what may go wrong in the proof of Luttinger's theorem. The analysis provides a simple way to correct Luttinger's expression of the electron number in single-band models where perturbation theory breaks down through the birth of a Luttinger surface without symmetry breaking. In those cases, we find that the Fermi volume only accounts for the doping away from half-filling. In the hypothetical circumstance of a non-symmetry-breaking Mott insulator with a Luttinger surface, our analysis predicts the noteworthy existence of quasiparticles whose Fermi surface is just the Luttinger one. Therefore, those quasiparticles can be legitimately regarded as spinons, and the Mott insulator with a Luttinger surface as realization of a spin-liquid insulator.

arXiv:2202.00426•Physical review. B./Physical review. B•Jul 8, 2022•32 citations

Strong renormalization of the Fermi-surface topology close to the Mott transition

Luca F. Tocchio, Federico Becca, Claudius Gros

The underlying Fermi surface is a key concept for strongly interacting electron models and has been introduced to generalize the usual notion of the Fermi surface to generic (superconducting or insulating) systems. By using improved correlated wave functions that contain backflow and Jastrow terms, we examine the two-dimensional $t$-${t}^{\ensuremath{'}}$ Hubbard model and find a nontrivial renormalization of the topology of the underlying Fermi surface close to the Mott insulator. Moreover, we observe a sharp crossover region, which arises from the metal-insulator transition, from a weakly interacting metal at small coupling to a resonating valence-bond superconductor at intermediate coupling. A violation of the Luttinger theorem is detected at low hole dopings.

arXiv:1201.2298•Physical Review B•Jul 5, 2012•25 citations

Screened-interaction expansion for the Hubbard model and determination of the quantum Monte Carlo Fermi surface

Gökhan Esirgen, H.-B. Schüttler, C. Gröber +1 more

We develop a systematic self-consistent perturbative expansion for the self-energy of Hubbard-like models. The interaction lines in the Feynman diagrams are dynamically screened by the charge fluctuations in the system. Although the formal expansion is exact---assuming that the model under the study is perturbative---only if diagrams to all orders are included, it is shown that for large-on-site-Coulomb-repulsion-$U$ systems weak-coupling expansions to a few orders may already converge. In order to test the approximation at intermediate-to-high temperatures, we use the exact charge-fluctuation susceptibility from quantum Monte Carlo (QMC) simulation studies as input, which determines the exact screened interaction, and compare our results for the self-energy to the QMC results. We also make comparisons with fluctuation-exchange approximation. We show that the screened interaction for the large-$U$ system can be vanishingly small at a certain intermediate electron filling, and it is found that our approximation for the imaginary part of the one-particle self-energy agrees well with the QMC results in the low-energy scales at this particular filling. But the usefulness of the approximation is hindered by the fact that it has the incorrect filling dependence when the filling deviates from this value. We also calculate the exact QMC Fermi surfaces for the two-dimensional (2D) Hubbard model for several fillings. Our results near half filling show extreme violation of the concepts of the band theory; in fact, instead of growing, the Fermi surface vanishes when doped toward the half-filled Mott-Hubbard insulator. Sufficiently away from half filling, noninteractinglike Fermi surfaces are recovered. These results combined with the Luttinger theorem might show that diagrammatic expansions for the nearly-half-filled Hubbard model are unlikely to be possible; however, the nonperturbative part of the solution seems to be less important as the filling gradually moves away from one half. Results for the 2D one-band Hubbard model for several hole dopings are presented. Implications of this study for the high-temperature superconductors are also discussed.

arXiv:cond-mat/0105514•Physical review. B, Condensed matter•Oct 15, 2001•6 citations

Violation of Luttinger's theorem in one-dimensional interacting fermions

Meng Gao, Yin Zhong

Using the density matrix renormalization group method, we systematically investigate the evolution of the Luttinger integral in the one-dimensional generalized $t$-$V$ 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.

arXiv:2506.04064•Jun 4, 2025•0 citations

Violation of Luttinger's theorem in the simplest doped Mott insulator: Falicov-Kimball model in the strong-correlation limit

Wei-Wei Yang, Qiaoni Chen, Hong‐Gang Luo +1 more

Luttinger's theorem has long been taken as the key feature of Landau's Fermi liquid, which signals the presence of quasiparticles. Here, by the unbiased Monte Carlo method, violation of Luttinger's theorem is clearly revealed in the Falicov-Kimball (FK) model, indicating a robust correlation-driven non-Fermi-liquid characteristic under any electron density. Introducing hole carriers to the half-filled FK model leads to Mott insulator-metal transition, where the Mott quantum criticality manifests unconventional scaling behavior in transport properties. Further insight on the violation of Luttinger's theorem is examined by combining Hubbard-I approximation with a composite fermion picture, which emphasizes the importance of a mixed excitation of the itinerant electron and the composite fermion. Interestingly, comparing the FK model with a binary disorder system suggests that the two-peak band structure discovered by Monte Carlo and Hubbard-I approaches is underlying the violation of Luttinger's theorem.

arXiv:2204.01203•Physical review. B./Physical review. B•Nov 9, 2022•6 citations

Successive opening of the Fermi surface in dopedN-leg Hubbard ladders

Urs Ledermann, Karyn Le Hur, Thomas Maurice Rice

We study the effect of doping away from half-filling in weakly (but finitely) interacting N-leg Hubbard ladders using renormalization-group and bosonization techniques. For a small on-site repulsion U, the N-leg Hubbard ladders are equivalent to an N-band model, where at half-filling the Fermi velocities are ${v}_{1}{=v}_{N}<{v}_{2}{=v}_{N\ensuremath{-}1}<\ensuremath{\cdot}\ensuremath{\cdot}\ensuremath{\cdot}.$ We then obtain a hierarchy of energy scales, where the band pairs $(j,N+1\ensuremath{-}j)$ are successively frozen out. The low-energy Hamiltonian is then the sum of $N/2$ [or $(N\ensuremath{-}1)/2$ for N odd] two-leg ladder Hamiltonians without gapless excitations (plus a single chain for N odd with one gapless spin mode)---similar to the N-leg Heisenberg spin ladders. The energy scales lead to a hierarchy of gaps. Upon doping away from half-filling, the holes first enter the band(s) with the smallest gap: For odd N, the holes enter first the nonbonding band $(N+1)/2$ and the phase is a Luttinger liquid, while for even N, the holes enter first the band pair $(N/2,N/2+1)$ and the phase is a Luther-Emery liquid, similar to numerical treatments of the $t\ensuremath{-}J$ model, i.e., at and close to half-filling, the phases of the Hubbard ladders for small and large U are the same. For increasing doping, hole pairs subsequently enter at critical dopings the other band pairs $(j,N+1\ensuremath{-}j)$ (accompanied by a diverging compressibility): The Fermi surface is successively opened by doping, starting near the wave vector $(\ensuremath{\pi}/2,\ensuremath{\pi}/2).$ Explicit calculations are given for the cases $N=3$ and 4.

arXiv:cond-mat/0002445•Physical review. B, Condensed matter•Dec 15, 2000•36 citations
We study the 1/N expansion of a generic, strongly correlated electron model ( SU (N) symmetric Hubbard model with U=∞ and N degrees of freedom per lattice site) in terms of X operators. The leading order of the expansion describes a usual Fermi liquid with renormalized, stable particles. The next-to-leading order violates Luttinger's theorem if a finite convergence radius for the 1/N expansion for a fixed and non-vanishing doping away from half-filling is assumed. We find that the volume enclosed by the Fermi surface, is at large, but finite N's and small dopings larger than at N=∞. As a by-product an explicit expression for the electronic self-energy in O(1/N) is given which cannot be obtained by factorization or mode-coupling assumptions but contains rather sophisticated vertex corrections.
Breakdown of Landau's hypothesis of adiabatic continuation from noninteracting to fully interacting electrons is commonly believed to bring about a violation of Luttinger's theorem. Here, we elucidate what may go wrong in the proof of Luttinger's theorem. The analysis provides a simple way to correct Luttinger's expression of the electron number in single-band models where perturbation theory breaks down through the birth of a Luttinger surface without symmetry breaking. In those cases, we find that the Fermi volume only accounts for the doping away from half-filling. In the hypothetical circumstance of a non-symmetry-breaking Mott insulator with a Luttinger surface, our analysis predicts the noteworthy existence of quasiparticles whose Fermi surface is just the Luttinger one. Therefore, those quasiparticles can be legitimately regarded as spinons, and the Mott insulator with a Luttinger surface as realization of a spin-liquid insulator.
The underlying Fermi surface is a key concept for strongly interacting electron models and has been introduced to generalize the usual notion of the Fermi surface to generic (superconducting or insulating) systems. By using improved correlated wave functions that contain backflow and Jastrow terms, we examine the two-dimensional ttt-${t}^{\ensuremath{'}}$ Hubbard model and find a nontrivial renormalization of the topology of the underlying Fermi surface close to the Mott insulator. Moreover, we observe a sharp crossover region, which arises from the metal-insulator transition, from a weakly interacting metal at small coupling to a resonating valence-bond superconductor at intermediate coupling. A violation of the Luttinger theorem is detected at low hole dopings.
We develop a systematic self-consistent perturbative expansion for the self-energy of Hubbard-like models. The interaction lines in the Feynman diagrams are dynamically screened by the charge fluctuations in the system. Although the formal expansion is exact---assuming that the model under the study is perturbative---only if diagrams to all orders are included, it is shown that for large-on-site-Coulomb-repulsion-UUU systems weak-coupling expansions to a few orders may already converge. In order to test the approximation at intermediate-to-high temperatures, we use the exact charge-fluctuation susceptibility from quantum Monte Carlo (QMC) simulation studies as input, which determines the exact screened interaction, and compare our results for the self-energy to the QMC results. We also make comparisons with fluctuation-exchange approximation. We show that the screened interaction for the large-UUU system can be vanishingly small at a certain intermediate electron filling, and it is found that our approximation for the imaginary part of the one-particle self-energy agrees well with the QMC results in the low-energy scales at this particular filling. But the usefulness of the approximation is hindered by the fact that it has the incorrect filling dependence when the filling deviates from this value. We also calculate the exact QMC Fermi surfaces for the two-dimensional (2D) Hubbard model for several fillings. Our results near half filling show extreme violation of the concepts of the band theory; in fact, instead of growing, the Fermi surface vanishes when doped toward the half-filled Mott-Hubbard insulator. Sufficiently away from half filling, noninteractinglike Fermi surfaces are recovered. These results combined with the Luttinger theorem might show that diagrammatic expansions for the nearly-half-filled Hubbard model are unlikely to be possible; however, the nonperturbative part of the solution seems to be less important as the filling gradually moves away from one half. Results for the 2D one-band Hubbard model for several hole dopings are presented. Implications of this study for the high-temperature superconductors are also discussed.
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.
Luttinger's theorem has long been taken as the key feature of Landau's Fermi liquid, which signals the presence of quasiparticles. Here, by the unbiased Monte Carlo method, violation of Luttinger's theorem is clearly revealed in the Falicov-Kimball (FK) model, indicating a robust correlation-driven non-Fermi-liquid characteristic under any electron density. Introducing hole carriers to the half-filled FK model leads to Mott insulator-metal transition, where the Mott quantum criticality manifests unconventional scaling behavior in transport properties. Further insight on the violation of Luttinger's theorem is examined by combining Hubbard-I approximation with a composite fermion picture, which emphasizes the importance of a mixed excitation of the itinerant electron and the composite fermion. Interestingly, comparing the FK model with a binary disorder system suggests that the two-peak band structure discovered by Monte Carlo and Hubbard-I approaches is underlying the violation of Luttinger's theorem.
We study the effect of doping away from half-filling in weakly (but finitely) interacting N-leg Hubbard ladders using renormalization-group and bosonization techniques. For a small on-site repulsion U, the N-leg Hubbard ladders are equivalent to an N-band model, where at half-filling the Fermi velocities are ${v}_{1}{=v}_{N}<{v}_{2}{=v}_{N\ensuremath{-}1}<\ensuremath{\cdot}\ensuremath{\cdot}\ensuremath{\cdot}.$ We then obtain a hierarchy of energy scales, where the band pairs $(j,N+1\ensuremath{-}j)$ are successively frozen out. The low-energy Hamiltonian is then the sum of N/2N/2N/2 [or $(N\ensuremath{-}1)/2$ for N odd] two-leg ladder Hamiltonians without gapless excitations (plus a single chain for N odd with one gapless spin mode)---similar to the N-leg Heisenberg spin ladders. The energy scales lead to a hierarchy of gaps. Upon doping away from half-filling, the holes first enter the band(s) with the smallest gap: For odd N, the holes enter first the nonbonding band (N+1)/2(N+1)/2(N+1)/2 and the phase is a Luttinger liquid, while for even N, the holes enter first the band pair (N/2,N/2+1)(N/2,N/2+1)(N/2,N/2+1) and the phase is a Luther-Emery liquid, similar to numerical treatments of the $t\ensuremath{-}J$ model, i.e., at and close to half-filling, the phases of the Hubbard ladders for small and large U are the same. For increasing doping, hole pairs subsequently enter at critical dopings the other band pairs $(j,N+1\ensuremath{-}j)$ (accompanied by a diverging compressibility): The Fermi surface is successively opened by doping, starting near the wave vector $(\ensuremath{\pi}/2,\ensuremath{\pi}/2).$ Explicit calculations are given for the cases N=3N=3N=3 and 4.