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arXiv:cond-mat/0309120

Non-Abelian topological phases in an extended Hubbard model

Michael Hartley Freedman, Chetan Nayak, Kirill Shtengel

Published Sep 4, 2003• 6 citations
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Abstract

We describe four closely related Hubbard-like models (models A, B, C and D) of particles that can hop on a 2D Kagome lattice interacting via Coulomb repulsion. The particles can be either bosons (models A and B) or (spinless) fermions (models C and D). Models A and C also include a ring exchange term. In all four cases we solve equations in the model parameters to arrive at an exactly soluble point whose ground state manifold is the extensively degenerate ``d-isotopy space'' $\bar{V}_d$, 0

DOI
10.48550/arxiv.cond-mat/0309120

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We describe four closely related Hubbard-like models (models A, B, C and D) of particles that can hop on a 2D Kagome lattice interacting via Coulomb repulsion. The particles can be either bosons (models A and B) or (spinless) fermions (models C and D). Models A and C also include a ring exchange term. In all four cases we solve equations in the model parameters to arrive at an exactly soluble point whose ground state manifold is the extensively degenerate ``d-isotopy space'' Vˉd\bar{V}_dVˉd​, 0
We propose an extended Hubbard model on a 2D kagome lattice with an additional ring exchange term. The particles can be either bosons or spinless fermions. We analyze the model at the special filling fraction 1/6, where it is closely related to the quantum dimer model. We show how to arrive at an exactly soluble point whose ground state is the "d-isotopy" transition point into a stable phase with a certain type of non-Abelian topological order. Near the "special" values, d=2cos(pi/(k+2), this topological phase has anyonic excitations closely related to SU(2) Chern-Simons theory at level k.
We introduce a family of lattices for which the Hubbard model and its natural extensions can be quasiexactly solved, i.e., solved for the ground and low energy states. In particular, we show rigorously that the ground state of the Hubbard model with off-site Coulomb repulsions on a decorated Kagomè lattice is an ordered array of local currents. The low energy theory describing this chiral state is an S=1 / 2 XY model, where each spin degree of freedom represents the two possible chiralities of each local current.
We construct a supersymmetric model of interacting Majorana fermions on the kagome lattice. In the infinite-coupling limit, the model exhibits an extensively degenerate ground state manifold separated in two topological sectors, in addition to two parity supersectors. An exact solution for thin-torus geometries allows us to analytically construct the entire zero-energy ground state manifold. Upon inclusion of hopping terms with energy scale ggg, the supersymmetry is spontaneously broken with the ground state energy density scaling as g4{g}^{4}g4. We also briefly discuss the noninteracting limit of the model, which exhibits a zero-energy flat band and Majorana Chern bands.
We introduce a non-Abelian kagome lattice model that has both time-reversal and inversion symmetries and study the flat band physics and topological phases of this model. Due to the coexistence of both time-reversal and inversion symmetries, the energy bands consist of three doubly degenerate bands whose energy and conditions for the presence of flat bands could be obtained analytically, allowing us to tune the flat band with respect to the other two dispersive bands from the top to the middle and then to the bottom of the three bands. We further study the gapped phases of the model and show that they belong to the same phase as the band gaps only close at discrete points of the parameter space, making any two gapped phases adiabatically connected to each other without closing the band gap. Using the Pfaffian approach based on the time-reversal symmetry and parity characterization from the inversion symmetry, we calculate the bulk topological invariants and demonstrate that the unique gapped phases belong to the Z2{\mathbb{Z}}_{2}Z2​ quantum spin Hall phase, which is further confirmed by the edge state calculations.
Following an earlier construction of exactly soluble lattice models for Abelian fractional topological insulators in two and three dimensions, we construct here an exactly soluble lattice model for a non-Abelian $\ensuremath{\nu}=1$ quantum Hall state and a non-Abelian topological insulator in two dimensions. We show that both models are topologically ordered, exhibiting fractionalized charge, ground-state degeneracy on the torus, and protected edge modes. The models feature non-Abelian vortices which carry fractional electric charge in the quantum Hall case and spin in the topological insulator case. We analyze the statistical properties of the excitations in detail and discuss the possibility of extending this construction to three-dimensional non-Abelian topological insulators.
We investigate the problem of constructing solvable models of nonrelativistic spin-1/2 interacting fermions on an infinite lattice. The extended Hubbard model, which includes the direct and exchange intersite interaction, is considered. The hopping term in the Hamiltonian is not restricted to nearest-neighbor sites (the ``overjumping'' model), but the short-range character of the standard Hubbard model is retained. The solutions in one- and two-particle sectors are obtained in an explicit form by using the theory of Weierstrass elliptic functions. The possibilities of further development of models of that type are also considered.