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arXiv:1412.1578 · Physical Review B

Ground state degeneracy of interacting spinless fermions

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

Published Oct 9, 2015• 15 citations
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Abstract

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 $N$ is even, and is always doubly degenerate if $N$ is odd. This proof holds in all dimensions with arbitrary lattice structures.

DOI
10.1103/physrevb.92.161105

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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.
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 examine the spin-reflection positivity of the ground state of the Kondo lattice model at half-filling with the antiferromagnetic and ferromagnetic exchange couplings J≠0J\ne0J=0. For every positive U>0U>0U>0, where U is the Coulomb interaction between the conduction electrons, we can show that the ground state is unique.
We investigate the ground state of the Kondo-Hubbard model employing the recently introduced idea of spin-reflection positivity. It is shown exactly that the Kondo-Hubbard model for J<0J<0J<0 and U>∣J∣/4U>|J|/4U>∣J∣/4 at half filling has a unique ground state which can be ferromagnetic or singlet according to the lattice structure. In particular, for the Kondo lattice with U>∣J∣/4U>|J|/4U>∣J∣/4, the ground state is a singlet. We also show a theorem for the Kondo-Hubbard model with the anisotropic antiferromagnetic coupling ${J}_{\ensuremath{\perp}}>0$ and ${J}_{z}\ensuremath{\le}0$ for U>∣Jz∣/4U>|{J}_{z}|/4U>∣Jz​∣/4.
We introduce a local formalism to deal with the Hubbard model on an N × N square lattice (for even N ) in terms of eigenstates of number operators, having well defined point symmetry. For U →0, the low-lying shells of the kinetic energy are filled in the ground state. At half-filling, using the 2 N -2 one-body states of the partially occupied shell hf , we build a set of degenerate unperturbed ground states with S z = 0 which are then resolved by the Hubbard interaction Ŵ = U ∑ r r ↑ r ↓ . We study the many-body eigenstates in hf of the kinetic energy with vanishing eigenvalue of the Hubbard repulsion ( W = 0 states). In the S z = 0 sector, this is an N -times-degenerate multiplet. From the singlet component one obtains the ground state of the Hubbard model for U = 0 + , which is unique, in agreement with a theorem of Lieb. The wave function demonstrates an antiferromagnetic order, a lattice step translation being equivalent to a spin flip. We show that the total momentum vanishes, while the point symmetry is s or d for even or odd N /2, respectively.
In this work, we study interacting spinless fermions on a two-chain ladder with interchain pair tunneling while single-particle tunneling is suppressed at low energy. The model embodies a Z2{\mathbb{Z}}_{2}Z2​ symmetry associated with the fermion parity on each chain. We find that when the system is driven to the strong-coupling phase by the pair tunneling, Majorana excitations appear on the boundary. Such Majorana edge states correspond to two-fold degeneracy of ground states distinguished by different fermion parity on each chain, thus representing a generalization of one-dimensional topological superconductors. We also characterize the stability of the ground-state degeneracy against local perturbations. Lattice fermion models realizing such an effective field theory are discussed.
A simple one-dimensional model is proposed, in which N spinless repulsively interacting fermions occupy M > N degenerate states. It is argued that the energy spectrum and the wavefunctions of this system strongly resemble the spectrum and wavefunctions of 2D electrons in the lowest Landau level (the problem of the Fractional Quantum Hall Effect). In particular, Laughlin-type wavefunctions describe ground states at filling factors v = N/M = 1/ q , q odd.. Within this model the complimentary wavefunction for v =1−1/ q is found explicitly, and extremely simple ground state wavefunctions for arbitrary odd-denominator filling factors are proposed.