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Ground State Degeneracy of a Class of Interacting Fermion Models

Zhongchao Wei, Xing-Jie Han, Z. Y. Xie, Tao Xiang

Published Dec 4, 2014• 0 citations
OpenAlex

Abstract

We prove rigourously that the ground state of a class of interacting Majorana fermion models and its equivalent spinless fermion models is either unique or doubly degenerate if the lattice size N is even, and is always doubly degenerate if N is odd. Our proof is based on the reflection positivity of the system in the eigen-operator representation and holds in all dimensions with arbitrary lattice structures. Understanding of physical properties of strongly correlated quantum lattice models has long been a challenging problem. Because of the subtlety of interactions, it is generally difficult to solve these models. Rigorous results and exact solutions are clearly useful bench marks, but these are rare. In 1989, Lieb proved that the ground state of the Hubbard model is unique by utilizing a Perron-Frobenius-type argument and the spinreflection positivity of the system[1]. This extended the proof for the uniqueness of the ground state from an antiferromagnetic Heisenberg model[2] to an interacting fermion system and led to a more general application of reflection positivity in quantum flux phases[3], frustrated Heisenberg antiferromagnets[4], and other quantum lattice models[5]. The spin-reflection positivity is intimately connected with the sign rule of the groundstate wave function, which can be used to understand the minus sign problem caused by the fermion characteristics of itinerant electrons in quantum Monte Carlo simulations[5]. In this letter, we present a rigorous proof for two theorems on the degeneracy of ground states for a class of interacting quantum lattice models of two-species Majorana fermions. We will show that the ground state is doubly degenerate when the lattice size N is odd and at most doubly degenerate when N is even. Our proof utilizes a new kind of reflection positivity between the two-species Majorana fermions. These theorems can be applied to a number of interacting spinless fermion models which are equivalent to the Majorana fermion model. It sheds light on the understanding of topological degeneracy and symmetry-protected topological phases of Majorana fermions[6, 7], and the minus sign problem in an interacting spinless fermion model that was recently studied by a number of groups[8–10]. The model studied here is defined on a lattice with N sites,

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We prove rigourously that the ground state of a class of interacting Majorana fermion models and its equivalent spinless fermion models is either unique or doubly degenerate if the lattice size N is even, and is always doubly degenerate if N is odd. Our proof is based on the reflection positivity of the system in the eigen-operator representation and holds in all dimensions with arbitrary lattice structures. Understanding of physical properties of strongly correlated quantum lattice models has long been a challenging problem. Because of the subtlety of interactions, it is generally difficult to solve these models. Rigorous results and exact solutions are clearly useful bench marks, but these are rare. In 1989, Lieb proved that the ground state of the Hubbard model is unique by utilizing a Perron-Frobenius-type argument and the spinreflection positivity of the system[1]. This extended the proof for the uniqueness of the ground state from an antiferromagnetic Heisenberg model[2] to an interacting fermion system and led to a more general application of reflection positivity in quantum flux phases[3], frustrated Heisenberg antiferromagnets[4], and other quantum lattice models[5]. The spin-reflection positivity is intimately connected with the sign rule of the groundstate wave function, which can be used to understand the minus sign problem caused by the fermion characteristics of itinerant electrons in quantum Monte Carlo simulations[5]. In this letter, we present a rigorous proof for two theorems on the degeneracy of ground states for a class of interacting quantum lattice models of two-species Majorana fermions. We will show that the ground state is doubly degenerate when the lattice size N is odd and at most doubly degenerate when N is even. Our proof utilizes a new kind of reflection positivity between the two-species Majorana fermions. These theorems can be applied to a number of interacting spinless fermion models which are equivalent to the Majorana fermion model. It sheds light on the understanding of topological degeneracy and symmetry-protected topological phases of Majorana fermions[6, 7], and the minus sign problem in an interacting spinless fermion model that was recently studied by a number of groups[8–10]. The model studied here is defined on a lattice with N sites,
We propose an eigenoperator scheme to study the lattice model of interacting spinless fermions at half filling and show that this model possesses a hidden form of reflection positivity in its Majorana fermion representation. Based on this observation, we prove rigourously that the ground state of this model is either unique or doubly degenerate if the lattice size NNN is even, and is always doubly degenerate if NNN is odd. This proof holds in all dimensions with arbitrary lattice structures.
In the present chapter we review some rigorous results derived with Lieb’s spin-reflectionpositivity method in a pedagogical manner. To begin with, we emphasize the important role played by the sign rule satisfied by the ground-state wave function of a quantum many-body system. For example, we study some localized spin systems, such as the antiferromagnetic Heisenberg model, in detail. Then we argue why such a rule is difficult to establish for the ground-state wave functions of itinerant electron models. By using the spin-reflection-positivity method, we show that this problem can be partially remedied. A weaker form of the sign rule is uncovered. It enables us to prove several inequalities satisfied by either an electron pairing correlation function or a magnetic correlation function of the Hubbard model on a bipartite lattice. These inequalities imply that, if a bipartite lattice has macroscopically unequal numbers of sublattice sites, then the Hubbard model at half-filling is either a supersolid or a ferrimagnet. Similar inequalities can also be proven for the periodic Anderson model and the Kondo lattice model. Finally, by applying a generalized version of the spin-reflection-positivity method, we further establish an inequality satisfied by the charged and spin gaps of these models. Based on them, we make some audacious conjectures, and then we summarize this chapter.
The sign problem is a major obstacle in quantum Monte Carlo simulations for many-body fermion systems. We examine this problem with a new perspective based on the Majorana reflection positivity and Majorana Kramers positivity. Two sufficient conditions are proven for the absence of the fermion sign problem. Our proof provides a unified description for all the interacting lattice fermion models previously known to be free of the sign problem based on the auxiliary field quantum Monte Carlo method. It also allows us to identify a number of new sign-problem-free interacting fermion models including, but not limited to, lattice fermion models with repulsive interactions but without particle-hole symmetry, and interacting topological insulators with spin-flip terms.
Theory of spin-reflection positivity developed in recent years is reviewed. This theory makes use of symmetries in an electron system and theory of matrix to investigate the ground state properties. Existences of anti- and ferromagnetic long-range orders in itinerant electron systems, and of off-diagonal long-range order are two successful applications of the theory. In this article, the author attempt to summarize exact results proved by utilizing this theory and related topics. First a general theory and basic theorems are introduced. Second, based on the band structures of conduction electrons, existences of a singlet state with strongly antiferromagnetic correlation, a state with both anti- and ferromagnetic long-range orders, and a fully saturated ferromagnetic state are proved. The theory is applied to several of the main theoretical models for strongly correlated electron systems, such as the Heisenberg model, the Hubbard model, the Anderson model, the single- and multichannel Kondo model, and the generalized Hubbard model, and a series of rigorous results are found in these models. Third, it is proved that off-diagonal long-range order and charge-density wave exist in the ground states of the attractive Hubbard model and the generalized Hubbard model. A relation between pseudospin symmetry and the uniform density theorem is introduced. Fourth, the theory is applied successfully to explain experimental observations of oscillatory interlayer magnetic coupling in ultrathin magnetic films. Finally several unsolved problems are discussed. All results introduced in this article are mathematically exact.
We introduce a frustration-free, one-dimensional model of spinless fermions with hopping, ppp-wave superconducting pairing and alternating chemical potentials. The model possesses two exactly degenerate ground states even for finite system sizes. We present analytical results for the strong Majorana zero modes, the phase diagram, and the topological order. Furthermore, we generalize our results to include interactions.
We present rigorous and universal results for the ground states of the f=2 spinor Bose-Hubbard model. The model includes three two-body on-site interaction terms, two of which are spin dependent while the other one is spin independent. We prove that, depending only on the coefficients of the two spin-dependent terms, the ground state exhibits maximum or minimum total spin or SU(5) ferromagnetism. Exact ground-state degeneracies and the forms of ground-state wave function are also determined in each case. All these results are valid regardless of dimension, lattice structure, or particle number. Our approach takes advantage of the symmetry of the Hamiltonian and employs mathematical tools including the Perron-Frobenius theorem and the Lie algebra so(5).