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arXiv:cond-mat/9701141 · Physical Review Letters

Nonperturbative Approach to Luttinger's Theorem in One Dimension

Masanori Yamanaka, Masaki Oshikawa, Ian K. Affleck

Published Aug 11, 1997• 176 citations
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Abstract

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 ${2k}_{F}$, where ${k}_{F}$ is the Fermi momentum of the noninteracting model, corresponding to Luttinger's theorem. For the Kondo lattice model, our result implies that ${k}_{F}$ must be calculated by regarding the localized spins as additional electrons.

DOI
10.1103/physrevlett.79.1110

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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.
The nature of the insulating phases of the SU(N\mathrm{SU}(NSU(N)-generalization of the one-dimensional Kondo lattice model is investigated by means of nonperturbative approaches. By extending the Lieb-Schultz-Mattis (LSM) argument to multicomponent fermion systems with translation and global SU(N\mathrm{SU}(NSU(N) symmetries, we derive two indices which depend on the filling and the ``SU(N\mathrm{SU}(NSU(N)-spin'' (representation) of the local moments. These indices strongly constrain possible insulating phases; for instance, when the local moments transform in the NNN-dimensional (defining) representation of SU(N\mathrm{SU}(NSU(N), a featureless Kondo insulator is possible only at filling $f=1\ensuremath{-}1/N$. To obtain further insight into the insulating phases suggested by the LSM argument, we derive low-energy effective theories by adding an antiferromagnetic Heisenberg exchange interaction among the local moments [the SU(N\mathrm{SU}(NSU(N) Kondo-Heisenberg model]. A conjectured global phase diagram of the SU(N\mathrm{SU}(NSU(N) Kondo lattice model as a function of the filling and the Kondo coupling is then obtained by a combination of different analytical approaches.
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.
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A theorem about the spin properties of the ground state and the ordering of the energy levels for different spins of an interacting-electron model including an arbitrary diagonal interacting potential, an antiferromagnetic exchange, and an electron pair-hopping term for particles on a one-dimensional lattice is stated and proved. We show that when the number of electrons N=4n+2 (n an integer) with periodic boundary conditions or N=4n with antiperiodic boundary conditions, the ground state is a nondegenerate singlet. This theorem generalizes a similar theorem Lieb and Mattis proved for the one-dimensional interacting electron system.
A theorem about the spin properties of the ground state and the ordering of the energy levels for different spins of an interacting-electron model including an arbitrary diagonal interacting potential, an antiferromagnetic exchange, and an electron pair-hopping term for particles on a one-dimensional lattice is stated and proved. We show that when the number of electrons N = 4n + 2 (n an integer) with periodic boundary conditions or N = 4n with antiperiodic boundary conditions, the ground state is a nondegenerate singlet. This theorem generalizes a similar theorem Lieb and Mattis proved for the one-dimensional interacting electron system.
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.