Bosonization study of a generalized statistics model with four Fermi points

Abstract

We study a one-dimensional lattice model of fractional statistics in which particles have next-nearest-neighbor hopping between sites which depends on the occupation number at the intermediate site and a statistical parameter φ. The model breaks parity and time-reversal symmetries and has four-fermion interactions if φ = 0. We first analyze the model using mean field theory and find that there are four Fermi points whose locations depend on φ and the filling η. We then study the modes near the Fermi points using the technique of bosonization. Based on the quadratic terms in the bosonized Hamiltonian, we find that the low-energy modes form two decoupled Tomonaga-Luttinger liquids with different values of the Luttinger parameters which depend on φ and η; further, the right and left moving modes of each system have different velocities. A study of the scaling dimensions of the cosine terms in the Hamiltonian indicates that the terms appearing in one of the Tomonaga-Luttinger liquids will flow under the renormalization group and the system may reach a non-trivial fixed point in the long distance limit. We examine the scaling dimensions of various charge density and superconducting order parameters to find which of them is the most relevant for different values of φ and η. Finally we look at two-particle bound states that appear in this system and discuss their possible relevance to the properties of the system in the thermodynamic limit. Our work shows that the low-energy properties of this model of fractional statistics have a rich structure as a function of φ and η.

Chiral Spin-Chain Interfaces Exhibiting Event-Horizon Physics

“…The MF faithfully reproduces many features of the full model, especially for jvj < juj suggesting the interactions are not significant here. We now investigate the validity of the MF analytically by bosonizing the spin Hamiltonian for the jvj < juj phase, and employing Luttinger liquid theory as an alternative derivation of the Fermi velocities of the model [38][39][40]. After a Jordan-Wigner transformation, the spin Hamiltonian takes the form H ¼ H MF þ H int , where H MF is the quadratic MF Hamiltonian of Eq.…”

“…( 3) suggests the spin model has two Fermi points located at p R;L ¼ AEπ=2, with Fermi velocities v R;L ¼ 2ðAEu − vÞ. By expanding around these Fermi points and bosonizing the interaction terms using the methods of [38][39][40], the fully interacting Hamiltonian is mapped to the free boson Hamiltonian…”

Chiral spin chain interfaces as event horizons

“…As we have seen numerically, the mean field approximation can faithfully reproduce many of the features of the fully interacting model, especially for |v| < |u|, suggesting that the interactions are not significant. We investigate this further by bosonising the spin Hamiltonian for |v| < |u|, mapping it to a Luttinger liquid [18][19][20]. After a Jordan-Wigner transformation, the spin Hamiltonian takes the form H = H MF + H int , where H MF is the quadratic mean-field Hamiltonian of Eq.…”

“…( 4) suggests that the spin model has two Fermi points located at p R,L = ±π/2, with the Fermi velocities v R,L = 2(±u − v). By expanding around these Fermi points and bosonising the interaction terms using the methods of [18][19][20], we find the fully interacting Hamiltonian can be mapped to the free boson Hamiltonian…”