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arXiv:cond-mat/9803035 · Physical review. B, Condensed matter

Generalization of the Luttinger theorem for fermionic ladder systems

Patrick Gagliardini, Stephan Haas, Thomas Maurice Rice

Published Oct 15, 1998• 18 citations
arXivHTMLPDF

Abstract

We apply a generalized version of the Lieb-Schultz-Mattis theorem to fermionic ladder systems to show the existence of a low-lying excited state (except for some special fillings). This can be regarded as a nonperturbative proof for the conservation under interaction of the sum of the Fermi wave vectors of the individual channels, corresponding to a generalized version of the Luttinger theorem to fermionic ladder systems. We conclude by noticing that the Lieb-Schultz-Mattis theorem is not applicable in this form to show the existence of low-lying excitations in the limit that the number of legs goes to infinity, e.g., in the limit of a 2D plane.

DOI
10.1103/physrevb.58.9603

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We apply a generalized version of the Lieb-Schultz-Mattis theorem to fermionic ladder systems to show the existence of a low-lying excited state (except for some special fillings). This can be regarded as a nonperturbative proof for the conservation under interaction of the sum of the Fermi wave vectors of the individual channels, corresponding to a generalized version of the Luttinger theorem to fermionic ladder systems. We conclude by noticing that the Lieb-Schultz-Mattis theorem is not applicable in this form to show the existence of low-lying excitations in the limit that the number of legs goes to infinity, e.g., in the limit of a 2D plane.
The Lieb-Schultz-Mattis theorem for spin chains is generalized to a wide range of models of interacting electrons and localized spins on a one dimensional lattice. The existence of a low-energy state is generally proved except for special commensurate fillings where a gap may occur. Moreover, the crystal momentum of the constructed low-energy state is 2kF{2k}_{F}2kF​, where kF{k}_{F}kF​ is the Fermi momentum of the noninteracting model, corresponding to Luttinger's theorem. For the Kondo lattice model, our result implies that kF{k}_{F}kF​ must be calculated by regarding the localized spins as additional electrons.
Abstract. For a large class of finite-range quantum spin models with half-integer spins, we prove that uniqueness of the ground state implies the existence of a low-lying excited state. For systems of linear size L, of arbitrary finite dimension, we obtain an upper bound on the excitation energy (i.e., the gap above the ground state) of the form (C log L)/L. This result can be regarded as a multi-dimensional Lieb-Schultz-Mattis theorem [7] and provides a rigorous proof of the main result in [4]. 1. Introduction and
Nonperturbative constraints on many-body physics, such as the famous Lieb-Schultz-Mattis theorem, are valuable tools for studying strongly correlated systems. To this end, we present a number of nonperturbative results that constrain the low-energy physics of systems having conserved dipole moments. We find that for these systems, a unique translationally invariant gapped ground state is only possible if the polarization of the system is integer. Furthermore, if a lattice system also has U(1)\text{U}(1)U(1) subsystem charge conservation symmetry, a unique gapped ground state is only possible if the particle filling along these subsystems is integer. We also apply these methods to spin systems, and determine criteria for the existence of a different type of magnetic response plateau in the presence of a nonuniform magnetic field. Finally, we formulate a version of Luttinger's theorem for one-dimensional systems consisting of dipoles.
A generalization of the Lieb-Schultz-Mattis theorem to higher-dimensional spin systems is shown. The physical motivation for the result is that such spin systems typically either have long-range order, in which case there are gapless modes, or have only short-range correlations, in which case there are topological excitations. The result uses a set of loop operators, analogous to those used in gauge theories, defined in terms of the spin operators of the theory. We also obtain various cluster bounds on expectation values for gapped systems. These bounds are used, under the assumption of a gap, to rule out the first case of long-range order, after which we show the existence of a topological excitation. Compared to the ground state, the topologically excited state has, up to a small error, the same expectation values for all operators acting within any local region, but it has a different momentum.
We consider a general one-dimensional tight-binding electron model which has a period PPP. For any filling factor ννν such that PνPνPν is non-integral, we prove that the model in the infinite volume limit has either a symmetry breaking or a unique ground state with gapless excitations. The proof is based on the idea of Yamanaka, Oshikawa and Affleck (cond-mat/9701141), who extended the Lieb-Schultz-Mattis argument to electron systems.
Lieb, Schultz and Mattis (LSM) (1961 Ann. Phys., NY 16 407 ) studied the S = 1/2 XXZ spin chain. The theorems of LSM's paper can be applied to broader models. In the original LSM theorem the nonfrustrating system was assumed. However, reconsidering the LSM theorem, we can extend the LSM theorem for frustrating systems. Next, several researchers tried to extend the LSM theorem for excited states. In the cases , the lowest energy eigenvalues are continuous for wave number q . But we found that their proofs were insufficient, and improve upon them. In addition, we can prove the LSM theory without the assumption of the discrete symmetry, which means that LSM-type theorems are applicable for Dzyaloshinskii–Moriya type interactions or other nonsymmetric models.