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arXiv:0909.2886

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

Published Sep 16, 2009• 1 citation
arXivHTMLPDF

Abstract

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]

DOI
10.48550/arxiv.0909.2886

Related papers

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 "Absence of Luttinger's Theorem", by Kiaran B. Dave, Philip W. Phillips and Charles L. Kane, arXiv:1207.4201

Behnam Farid

In this Comment, we first present general arguments showing that the absence of the Luttinger theorem (LT) for the SU(N) model of Dave, Phillips and Kane (DPK) is rooted in the non-uniqueness of the ground state (GS) of this model for 0 < n < N, where n denotes the number of particles in the GS; the validity of the Luttinger theorem for n = N/2, when N even, is accidental, a consequence of particle-hole symmetry. Consequently, by supplementing the Hamiltonian of the SU(N) model with a perturbation Hamiltonian that removes the GS degeneracy, the LT is to apply also for the SU(N) model in the limit of the coupling constant, λ, of this perturbation approaching zero, where the limit λ--> 0 is clearly to be taken subsequent to taking the zero-temperature limit of the thermal single-particle Green function in the expression for the Luttinger number N_L. We explicitly establish the validity of this statement for the case of N=4. The details of the relevant calculations being distinctly transparent, one can readily convince oneself that our observation is valid for arbitrary N. It follows that the issues raised by DPK, such as non-existence of the Luttinger-Ward functional and "breakdown of the elemental particle picture in strongly correlated electron matter", are all inessential to the observed failure of the LT. As regards the singularity of the self-energy Σ(ω) on the real ω-axis, observed by DPK, we demonstrate that this also is a direct consequence of the non-uniqueness of the GS of the SU(N) model for 0 < n < N. In the light of the above observations, we are in a position to state that to this date no case has come to light indicative of the failure of the LT under the conditions for which it has been deduced. [Abridged Abstract]

arXiv:1211.5612•Nov 23, 2012•0 citations

Reply to ``Comment on `On the Luttinger theorem concerning number of particles in the ground states of systems of interacting fermions','' arXiv:0711.3093v1, by A. Rosch

Behnam Farid

We reply to the Comment by Achim Rosch [1] (arXiv:0711.3093v1) who challenges our finding in Ref. [2] (arXiv:0711.0952v1) with regard to the validity of the Luttinger theorem in the cases of Mott insulating N-particle ground states (even) when the chemical potential used in applying this theorem coincides with the zero-temperature limit of the chemical potential satisfying the equation of state corresponding to N particles. Rosch further argues that the strong-coupling expression for the single-particle Green function presented in Ref. [3] (arXiv:cond-mat/0602656v2) and analyzed in Ref. [2] does not imply destruction of the Mott insulating state at half-filling as a result of an arbitrary weak hopping contribution that breaks particle-hole symmetry and therefore suggests that our conclusion in Ref. [2] to the contrary were incorrect. Here we show the shortcomings of Rosch's arguments.

arXiv:0711.3195•Nov 20, 2007•0 citations

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 `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

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

J. Kokalj, Peter Prelovšek

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.

arXiv:0803.4468•Physical Review B•Oct 9, 2008•10 citations
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]
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 this Comment, we first present general arguments showing that the absence of the Luttinger theorem (LT) for the SU(N) model of Dave, Phillips and Kane (DPK) is rooted in the non-uniqueness of the ground state (GS) of this model for 0 < n < N, where n denotes the number of particles in the GS; the validity of the Luttinger theorem for n = N/2, when N even, is accidental, a consequence of particle-hole symmetry. Consequently, by supplementing the Hamiltonian of the SU(N) model with a perturbation Hamiltonian that removes the GS degeneracy, the LT is to apply also for the SU(N) model in the limit of the coupling constant, λ, of this perturbation approaching zero, where the limit λ--> 0 is clearly to be taken subsequent to taking the zero-temperature limit of the thermal single-particle Green function in the expression for the Luttinger number N_L. We explicitly establish the validity of this statement for the case of N=4. The details of the relevant calculations being distinctly transparent, one can readily convince oneself that our observation is valid for arbitrary N. It follows that the issues raised by DPK, such as non-existence of the Luttinger-Ward functional and "breakdown of the elemental particle picture in strongly correlated electron matter", are all inessential to the observed failure of the LT. As regards the singularity of the self-energy Σ(ω) on the real ω-axis, observed by DPK, we demonstrate that this also is a direct consequence of the non-uniqueness of the GS of the SU(N) model for 0 < n < N. In the light of the above observations, we are in a position to state that to this date no case has come to light indicative of the failure of the LT under the conditions for which it has been deduced. [Abridged Abstract]
We reply to the Comment by Achim Rosch [1] (arXiv:0711.3093v1) who challenges our finding in Ref. [2] (arXiv:0711.0952v1) with regard to the validity of the Luttinger theorem in the cases of Mott insulating N-particle ground states (even) when the chemical potential used in applying this theorem coincides with the zero-temperature limit of the chemical potential satisfying the equation of state corresponding to N particles. Rosch further argues that the strong-coupling expression for the single-particle Green function presented in Ref. [3] (arXiv:cond-mat/0602656v2) and analyzed in Ref. [2] does not imply destruction of the Mott insulating state at half-filling as a result of an arbitrary weak hopping contribution that breaks particle-hole symmetry and therefore suggests that our conclusion in Ref. [2] to the contrary were incorrect. Here we show the shortcomings of Rosch's arguments.
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.
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.
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⪢tV⪢tV⪢t while the sum rule appears to recover together with vanishing of the Mott-Hubbard gap.