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]

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]

💡 Research Summary

The paper is a detailed comment on the claim made by Kokalj and Prelovsek (KP) that the Luttinger theorem (LT) fails in the Mott‑Hubbard insulating phase of the half‑filled one‑dimensional t‑t′‑V model when the interaction V is much larger than the hopping t. KP arrived at this conclusion by analysing numerical data obtained on finite lattices (N = 14, 18, 22, 26) and by performing a finite‑size scaling of the single‑particle Green function G(k, ω). They even suggested that the LT is already violated for a finite system of N = 26 sites.

The authors of the comment first point out that the finite‑size scaling employed by KP is unreliable for the modest system sizes considered. In small clusters the self‑energy Σ(k, ω) can display artificial singularities or non‑analytic features that do not survive in the thermodynamic limit. To demonstrate this, they construct a simple analytic model for the self‑energy of a Mott‑Hubbard insulator, Σ(ω) ≈ U²/(4 ω), and apply the same scaling procedure used by KP. The model reproduces an apparent “breakdown” of the LT even though the underlying physics is perfectly consistent with the theorem. This exercise shows that the observed violation is a finite‑size artefact rather than a genuine property of the infinite system.

Next, the authors exploit the exact mapping of the half‑filled t‑t′‑V model (with |t′| ≪ t) onto the anisotropic spin‑½ XXZ chain. Using the Bethe‑Ansatz solution of the XXZ model, they establish that for interaction strengths V exceeding a critical value V_c(t, t′) the ground state is not a correlation‑driven Mott insulator but a charge‑density‑wave (CDW) state. In this CDW phase the translational symmetry of the lattice is spontaneously broken, the unit cell doubles, and a gap opens in the single‑particle spectrum as a consequence of the enlarged Brillouin zone rather than of strong on‑site correlations. Consequently, the LT remains applicable: the Luttinger volume is now defined with respect to the reduced Brillouin zone, and the sum rule is satisfied exactly.

To substantiate this claim, the authors calculate the Green function of the broken‑symmetry ground state using bosonization and the quantum sine‑Gordon model. By expressing the electron operator in terms of soliton‑generating fields and employing known form‑factor results, they obtain an explicit expression for G(k, ω). The analysis shows that G(k, 0) possesses zeros precisely at the new zone boundaries, confirming that the Luttinger count is preserved when the proper Brillouin zone is used. The calculation also clarifies how the spectral gap originates from the CDW order parameter rather than from a Mott‑Hubbard mechanism.

Finally, the paper presents a heuristic argument due to Volovik, emphasizing that the LT is protected by topological invariants (the winding number of the Green function) which are insensitive to the detailed analytic structure of Σ(k, ω). Even if the self‑energy exhibits non‑analyticities or poles, as long as the overall topology of the Green function does not change, the Luttinger volume remains quantized. This perspective explains why the LT survives both strong correlation effects and symmetry‑breaking transitions.

In summary, the comment demonstrates that (i) the finite‑size scaling used by KP cannot reliably test the LT for the lattice sizes they considered, (ii) for V > V_c the model’s ground state is a CDW insulator with a doubled unit cell, and (iii) the LT is fully respected in this broken‑symmetry phase when the appropriate Brillouin zone is taken into account. Therefore, the conclusion drawn by Kokalj and Prelovsek—that the Luttinger theorem breaks down in the Mott‑Hubbard insulating regime of the t‑t′‑V model—is incorrect. The paper reaffirms the robustness of the Luttinger theorem across both Mott and CDW insulating phases.