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:0803.4468 · Physical Review B

Breakdown of the Luttinger sum rule within the Mott-Hubbard insulator

J. Kokalj, Peter Prelovšek

Published Oct 9, 2008• 10 citations
arXivHTMLPDF

Abstract

The validity of the Luttinger sum rule is investigated within the prototype tight-binding model of interacting fermions in one dimension, i.e., the $t\text{\ensuremath{-}}V$ model including the next-nearest-neighbor hopping ${t}^{\ensuremath{'}}$, in order to break the particle-hole symmetry. Scaling analysis of finite-system results at half filling reveals evident breakdown of the sum rule in the large gap regime at $V⪢t$ while the sum rule appears to recover together with vanishing of the Mott-Hubbard gap.

DOI
10.1103/physrevb.78.153103

Related papers

Luttinger sum rule for finite systems of correlated electrons

J. Kokalj, Peter Prelovšek

The validity of the Luttinger sum rule is considered for finite systems of interacting electrons, where the Fermi volume is determined by location of zeros of Green's function. It is shown that the sum rule in the paramagnetic state is evidently violated within the planar $t\text{\ensuremath{-}}J$ model at low doping while for the related Hubbard model, even in the presence of next-nearest-neighbor hopping, no clear-cut exception is found.

arXiv:cond-mat/0609280•Physical Review B•Jan 10, 2007•20 citations

Comment on "Breakdown of the Luttinger sum rule within the Mott-Hubbard insulator", by J. Kokalj and P. Prelovsek [Phys. Rev. B 78, 153103 (2008), arXiv:arXiv:0803.4468]

Behnam Farid, Alexei M. Tsvelik

On the basis of an analysis of the numerical results corresponding to the half-filled 1D t-t'-V model on some finite lattices, Kokalj and Prelovsek (KP) have in a recent paper [Phys. Rev. B 78, 153103 (2008), arXiv:arXiv:0803.4468] concluded that the Luttinger theorem (LT) does not apply for the Mott-Hubbard (MH) insulating phase of this model (i.e. for V >> t) in the thermodynamic limit; KP even suggested, incorrectly, that failure of the LT were apparent for a half-filled finite system consisting of N=26 lattice sites. By employing a simple model for the self-energy Sigma of a MH state, we show that the finite-size-scaling approach of the type utilised by KP is not reliable for the system sizes considered by KP. On the basis of the equivalence of the model under consideration (at half-filling and for t'/t << 1) and the XXZ spin-chain Hamiltonian for SU(2) spins, we further show that for V > V_c(t,t') the system under consideration has a charge-density-wave (CDW) ground state (GS) in the thermodynamic limit, corresponding to a doubling of the unit cell in comparison with that specific to the underlying lattice. Although this GS is also insulating, its spectral gap is due to the broken translational symmetry of the GS; it is not a correlation-induced MH gap. The LT is therefore a priori valid for this GS. This fact establishes that the conclusion by KP is indeed erroneous. Finally, we present a heuristic argument due to Volovik that sheds light on the mechanism underlying the robustness of the LT. In an appendix, we present the details of the calculation of the single-particle Green function of the broken-symmetry GS of the model under consideration by means of bosonization and in terms of the form factors of a class of soliton-generating fields pertaining to the quantum sine-Gordon Hamiltonian. [Shortened abstract]

arXiv:0909.2886•Sep 16, 2009•1 citation

Nonperturbative approach to full Mott behavior

Tudor D. Stanescu, Philip Phillips

Though most fermionic Mott insulators order at low temperatures, ordering is ancillary to their insulating behavior. Our emphasis here is on disentangling ordering from the intrinsic strongly correlated physics of a doped half-filled band. To this end, we focus on the two-dimensional Hubbard model. Because the charge gap arises from on-site correlations, we have been refining the nonperturbative approach of Matsumoto and Mancini [Phys. Rev. B 55, 2095 (1997)] which incorporates local physics. Crucial to this method is a self-consistent two-site dynamical cluster expansion which builds in the nearest-neighbor energy scale J. At half-filling, we find that the spectral function possesses a gap of order U and is devoid of any coherent quasiparticle peaks although ordering or charge fractionalization are absent. At low temperatures, local antiferromagnetic correlations emerge. In the doped case, we find that the Fermi surface exceeds the Luttinger volume. The breakdown of Luttinger's theorem in the underdoped regime is traced both to the dynamically generated Mott gap as well to a nonvanishing of the imaginary part of the self-energy at the Fermi level. Spectral weight transfer across the Mott gap also emerges as a ubiquitous feature of a doped Mott insulator and suggests that high- and low-energy scales are inseparable. Additionally in the underdoped regime, we find that a pseudogap exists in the single-particle density of states as well as in the heat capacity. The pseudogap (which is set by the energy scale ${t}^{2}/U)$ is argued to be a ubiquitous feature of a lightly doped Mott state and simply represents the fact that hole transport involves double occupancy. In analogy with the Mott gap and antiferromagnetism, we propose that ordering may also accompany the formation of a pseudogap. We suggest a current pattern within a one-band model that preserves translational but breaks time-reversal symmetry along the canonical x and y axes but not along $x=\ifmmode\pm\else\textpm\fi{}y$ that is consistent with the experimental observations. Finally, we show that the Hall coefficient in a doped Mott insulator must change sign at a doping level $x<1/3.$ The sign change is tied to a termination of strong correlation physics in the doped Mott state.

arXiv:cond-mat/0301254•Physical Review B•Jun 15, 2004•38 citations

Comment on "Violation of the Luttinger sum rule within the Hubbard model on a triangular lattice", by J. Kokalj and P. Prelovsek [Eur. Phys. J. B 63, 431 (2008), arXiv:arXiv:0709.0263]

Behnam Farid

Using the first-order series expansion of the function G(k;mu), in powers of mu' = mu - U/2, pertaining to the insulating ground state of a single-band Hubbard Hamiltonian at half-filling, Kokalj and Prelovsek have in a recent paper [Eur. Phys. J. B 63, 431 (2008), arXiv:arXiv:0709.0263] reported breakdown of the Luttinger theorem for the specific case where the lattice on which the Hubbard Hamiltonian is defined is a two-dimensional triangular lattice, for which the ground state is not invariant under particle-hole transformation. Here G(k;mu) is the single-particle Green function G(k;e) evaluated at e = mu, the zero-temperature limit of the chemical potential corresponding to half-filling, and U the on-site interaction energy. In this Comment we demonstrate that unless mu' = 0 (to be strictly distinguished from mu' small but non-vanishing), any finite-order series expansion for G(k;mu) in powers of mu' in general falsely signals breakdown of the Luttinger theorem. The violation of this theorem as asserted by Kokalj and Prelovsek is therefore an artifact of their first-order calculation.

arXiv:0909.2887•Sep 16, 2009•0 citations

Comment on `On the Luttinger theorem concerning the number of particles in the ground states of systems of interacting fermions', arXiv:0711.0952v1, by B. Farid

Achim Rosch

In his preprint, arXiv:0711.0952v1, Behnam Farid argues that the Luttinger theorem is valid not only for a metal but also for a Mott insulator if the chemical potential is calculated by taking the limit of vanishing temperature at fixed particle density. In contrast, we have found in our recent paper (Eur. Phys. J. B 59, 495 (2007) or arXiv:cond-mat/0602656) on the basis of a controlled strong coupling expansion that the Luttinger theorem is violated in this limit for a particle-hole asymmetric two-band Mott insulator. In an extensive discussion of our result Farid argues that an arbitrarily weak breaking of particle-hole symmetry leads to a destruction of the Mott insulating state at half filling. In this comment we point out that this is not correct.

arXiv:0711.3093•Nov 20, 2007•0 citations

Phase diagram of the one-dimensional Hubbard model with next-nearest-neighbor hopping

Satoshi Nishimoto, Kazuhiro Sano, Y. Ohta

We study the one-dimensional Hubbard model with nearest-neighbor and next-nearest-neighbor hopping integrals by using the density-matrix renormalization group method and Hartree-Fock approximation. Based on the calculated results for the spin gap, total-spin quantum number, and Tomonaga-Luttinger-liquid parameter, we determine the ground-state phase diagram of the model in the entire filling and wide parameter region. We show that, in contrast to the weak-coupling regime where a spin-gapped liquid phase is predicted in the region with four Fermi points, the spin gap vanishes in a substantial region in the strong-coupling regime. It is remarkable that a large variety of phases, such as the paramagnetic metallic phase, spin-gapped liquid phase, singlet and triplet superconducting phases, and fully polarized ferromagnetic phase, appear in such a simple model in the strong-coupling regime.

arXiv:0710.2274•Physical Review B•Feb 26, 2008•21 citations
The validity of the Luttinger sum rule is investigated within the prototype tight-binding model of interacting fermions in one dimension, i.e., the t\ensuremath-Vt\text{\ensuremath{-}}Vt\ensuremath-V model including the next-nearest-neighbor hopping t\ensuremath′{t}^{\ensuremath{'}}t\ensuremath′, in order to break the particle-hole symmetry. Scaling analysis of finite-system results at half filling reveals evident breakdown of the sum rule in the large gap regime at V⪢tV⪢tV⪢t while the sum rule appears to recover together with vanishing of the Mott-Hubbard gap.
On the basis of an analysis of the numerical results corresponding to the half-filled 1D t-t'-V model on some finite lattices, Kokalj and Prelovsek (KP) have in a recent paper [Phys. Rev. B 78, 153103 (2008), arXiv:arXiv:0803.4468] concluded that the Luttinger theorem (LT) does not apply for the Mott-Hubbard (MH) insulating phase of this model (i.e. for V >> t) in the thermodynamic limit; KP even suggested, incorrectly, that failure of the LT were apparent for a half-filled finite system consisting of N=26 lattice sites. By employing a simple model for the self-energy Sigma of a MH state, we show that the finite-size-scaling approach of the type utilised by KP is not reliable for the system sizes considered by KP. On the basis of the equivalence of the model under consideration (at half-filling and for t'/t << 1) and the XXZ spin-chain Hamiltonian for SU(2) spins, we further show that for V > V_c(t,t') the system under consideration has a charge-density-wave (CDW) ground state (GS) in the thermodynamic limit, corresponding to a doubling of the unit cell in comparison with that specific to the underlying lattice. Although this GS is also insulating, its spectral gap is due to the broken translational symmetry of the GS; it is not a correlation-induced MH gap. The LT is therefore a priori valid for this GS. This fact establishes that the conclusion by KP is indeed erroneous. Finally, we present a heuristic argument due to Volovik that sheds light on the mechanism underlying the robustness of the LT. In an appendix, we present the details of the calculation of the single-particle Green function of the broken-symmetry GS of the model under consideration by means of bosonization and in terms of the form factors of a class of soliton-generating fields pertaining to the quantum sine-Gordon Hamiltonian. [Shortened abstract]
Though most fermionic Mott insulators order at low temperatures, ordering is ancillary to their insulating behavior. Our emphasis here is on disentangling ordering from the intrinsic strongly correlated physics of a doped half-filled band. To this end, we focus on the two-dimensional Hubbard model. Because the charge gap arises from on-site correlations, we have been refining the nonperturbative approach of Matsumoto and Mancini [Phys. Rev. B 55, 2095 (1997)] which incorporates local physics. Crucial to this method is a self-consistent two-site dynamical cluster expansion which builds in the nearest-neighbor energy scale J. At half-filling, we find that the spectral function possesses a gap of order U and is devoid of any coherent quasiparticle peaks although ordering or charge fractionalization are absent. At low temperatures, local antiferromagnetic correlations emerge. In the doped case, we find that the Fermi surface exceeds the Luttinger volume. The breakdown of Luttinger's theorem in the underdoped regime is traced both to the dynamically generated Mott gap as well to a nonvanishing of the imaginary part of the self-energy at the Fermi level. Spectral weight transfer across the Mott gap also emerges as a ubiquitous feature of a doped Mott insulator and suggests that high- and low-energy scales are inseparable. Additionally in the underdoped regime, we find that a pseudogap exists in the single-particle density of states as well as in the heat capacity. The pseudogap (which is set by the energy scale t2/U){t}^{2}/U)t2/U) is argued to be a ubiquitous feature of a lightly doped Mott state and simply represents the fact that hole transport involves double occupancy. In analogy with the Mott gap and antiferromagnetism, we propose that ordering may also accompany the formation of a pseudogap. We suggest a current pattern within a one-band model that preserves translational but breaks time-reversal symmetry along the canonical x and y axes but not along $x=\ifmmode\pm\else\textpm\fi{}y$ that is consistent with the experimental observations. Finally, we show that the Hall coefficient in a doped Mott insulator must change sign at a doping level x<1/3.x<1/3.x<1/3. The sign change is tied to a termination of strong correlation physics in the doped Mott state.
Using the first-order series expansion of the function G(k;mu), in powers of mu' = mu - U/2, pertaining to the insulating ground state of a single-band Hubbard Hamiltonian at half-filling, Kokalj and Prelovsek have in a recent paper [Eur. Phys. J. B 63, 431 (2008), arXiv:arXiv:0709.0263] reported breakdown of the Luttinger theorem for the specific case where the lattice on which the Hubbard Hamiltonian is defined is a two-dimensional triangular lattice, for which the ground state is not invariant under particle-hole transformation. Here G(k;mu) is the single-particle Green function G(k;e) evaluated at e = mu, the zero-temperature limit of the chemical potential corresponding to half-filling, and U the on-site interaction energy. In this Comment we demonstrate that unless mu' = 0 (to be strictly distinguished from mu' small but non-vanishing), any finite-order series expansion for G(k;mu) in powers of mu' in general falsely signals breakdown of the Luttinger theorem. The violation of this theorem as asserted by Kokalj and Prelovsek is therefore an artifact of their first-order calculation.
In his preprint, arXiv:0711.0952v1, Behnam Farid argues that the Luttinger theorem is valid not only for a metal but also for a Mott insulator if the chemical potential is calculated by taking the limit of vanishing temperature at fixed particle density. In contrast, we have found in our recent paper (Eur. Phys. J. B 59, 495 (2007) or arXiv:cond-mat/0602656) on the basis of a controlled strong coupling expansion that the Luttinger theorem is violated in this limit for a particle-hole asymmetric two-band Mott insulator. In an extensive discussion of our result Farid argues that an arbitrarily weak breaking of particle-hole symmetry leads to a destruction of the Mott insulating state at half filling. In this comment we point out that this is not correct.
We study the one-dimensional Hubbard model with nearest-neighbor and next-nearest-neighbor hopping integrals by using the density-matrix renormalization group method and Hartree-Fock approximation. Based on the calculated results for the spin gap, total-spin quantum number, and Tomonaga-Luttinger-liquid parameter, we determine the ground-state phase diagram of the model in the entire filling and wide parameter region. We show that, in contrast to the weak-coupling regime where a spin-gapped liquid phase is predicted in the region with four Fermi points, the spin gap vanishes in a substantial region in the strong-coupling regime. It is remarkable that a large variety of phases, such as the paramagnetic metallic phase, spin-gapped liquid phase, singlet and triplet superconducting phases, and fully polarized ferromagnetic phase, appear in such a simple model in the strong-coupling regime.