[Paper Review] On the Luttinger theorem concerning number of particles in the ground states of systems of interacting fermions

Behnam Farid|arXiv (Cornell University)|Nov 6, 2007

Advanced Physical and Chemical Molecular Interactions10 references3 citations

TL;DR

This paper rigorously re-examines the Luttinger theorem for interacting fermions, proving its unconditional validity in systems with short-range interactions—such as lattice models—regardless of interaction strength. By demonstrating uniform convergence of the perturbative self-energy expansion in skeleton diagrams and analyzing zero-temperature limits, the author shows that apparent violations stem from flawed Green’s functions or misapplications, except in one case involving insulating states where a false zero-temperature limit may cause genuine failure.

ABSTRACT

We analyze the original proof by Luttinger and Ward of the Luttinger theorem, according to which for uniform ground states of systems of (interacting) fermions, which may be metallic or insulating, the number of k points corresponding to non-negative values of G_s(k;mu) is equal to the total number of particles with spin index s in these ground states. Here G_s(k;mu) is the single-particle Green function of particles with spin index s at the chemical potential mu. For the cases where the two-body interaction potential is short-range, and in particular for lattice models, we explicitly demonstrate that this theorem is unconditionally valid, irrespective of the strength of the bare interaction potential. We arrive at this conclusion by amongst other things demonstrating that the perturbation series expansion for self-energy in terms of skeleton diagrams, as encountered in the proof of the Luttinger-Ward identity, is uniformly convergent for almost all momenta and energies. We further investigate the mechanisms underlying some reported instances of failure of the Luttinger theorem. With one exception, for all the cases considered in this paper, we show that the apparent failures of the Luttinger theorem can be attributed either to shortcomings of the employed single-particle Green functions or to misapplication of this theorem. The one exceptional case brings to light the possibility of a genuine failure of the Luttinger theorem for insulating ground states, which we show to be brought about by a false limit that in principle can be reached on taking the zero-temperature limit without the value of mu coinciding with the zero-temperature limit of the chemical potential satisfying the equation of state at finite temperatures; no such ambiguity can arise for metallic states.

Motivation & Objective

• To re-express and rigorously validate the Luttinger theorem for interacting fermions in uniform ground states, including both metallic and insulating phases.
• To resolve reported instances of Luttinger theorem failure by identifying errors in Green’s function definitions or misapplications of the theorem.
• To investigate the exceptional case where a genuine failure may occur in insulating states due to a false zero-temperature limit.
• To establish the uniform convergence of the perturbative self-energy series in skeleton diagrams for almost all momenta and energies.
• To clarify the conditions under which the Luttinger number equals the sum of occupation numbers in the zero-temperature limit.

Proposed method

• Analyzes the original Luttinger-Ward proof using perturbation theory and skeleton diagram expansions for the self-energy.
• Demonstrates uniform convergence of the series $\sum_{\nu=1}^{\infty}\tilde{\Sigma}_{\sigma}^{(\nu)}(\mathbf{k};z)$ for almost all $\mathbf{k}$ and $z$, ensuring mathematical robustness.
• Applies contour integration techniques to evaluate the zero-temperature limit of the occupation number $\bar{\nu}_{\sigma}^{(1)}(\mathbf{k})$, focusing on analyticity and decay at infinity.
• Uses asymptotic expansions of the Green’s function $\tilde{G}_{\sigma}(\mathbf{k};z)$ and self-energy $\tilde{\Sigma}_{\sigma}(\mathbf{k};z)$ for large $|z|$ to assess integrand behavior.
• Investigates the behavior of $\tilde{G}_{\sigma}(\mathbf{k};z)\partial\tilde{\Sigma}_{\sigma}(\mathbf{k};z)/\partial z$ near $z = \mu$ to determine boundedness of the occupation number limit.
• Considers both symmetric and asymmetric cases in the local limit, particularly in the one-dimensional Luttinger model, to test the theorem’s robustness.

Experimental results

Research questions

• RQ1 Under what conditions is the Luttinger theorem unconditionally valid for interacting fermion systems with short-range interactions?
• RQ2 Why do some studies report apparent violations of the Luttinger theorem, and can these be traced to technical flaws in the Green’s function or self-energy definitions?
• RQ3 Can a genuine failure of the Luttinger theorem occur in insulating ground states, and if so, under what dynamical or thermodynamic conditions?
• RQ4 Is the perturbative expansion of the self-energy in skeleton diagrams uniformly convergent for all relevant momenta and energies in lattice models?
• RQ5 How does the zero-temperature limit of the occupation number $\bar{\nu}_{\sigma}^{(1)}(\mathbf{k})$ behave, and what ensures its boundedness despite singularities in the Green’s function?

Key findings

• The Luttinger theorem is unconditionally valid for systems with short-range interactions, including lattice models, regardless of the strength of the repulsive interaction.
• The perturbative series for the self-energy in skeleton diagrams converges uniformly for almost all $\mathbf{k}$ and $z$, ensuring the mathematical consistency of the Luttinger-Ward identity.
• Apparent violations of the Luttinger theorem in prior studies are traced to incorrect Green’s functions or misapplication of the theorem, not to a flaw in the theorem itself.
• In the one-dimensional Luttinger model, $\lim_{\beta\to\infty}\bar{\nu}_{\sigma}^{(2)}(\mathbf{k})$ is bounded for all $\mathbf{k}$, including at $k=0$ and $k\neq0$, due to integrand decay slower than $1/y$.
• For $k=0$, the leading-order asymptotic behavior of the integrand is $\sim - (1-2\gamma_0)/iy$, which is purely imaginary and integrable.
• The only case where a genuine failure may occur is in insulating states when the zero-temperature limit is taken incorrectly, where $\mu$ does not match the finite-temperature equation of state.