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arXiv:1108.4996 · Physical Review B

Topological phase in a one-dimensional interacting fermion system

Huaiming Guo, Shun-Qing Shen

Published Nov 4, 2011• 67 citations
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Abstract

We study a one-dimensional (1D) interacting topological model by means of the exact diagonalization method. The topological properties are first examined with the existence of the edge states at half-filling. We find that the topological phases are not only robust to small repulsive interactions, but also are stabilized by small attractive interactions, and also finite repulsive interaction can drive a topological nontrivial phase into a trivial one while the attractive interaction can drive a trivial phase into a nontrivial one. Next we calculate the Berry phase and parity of the bulk system and find that they are equivalent in characterizing the topological phases. With them we obtain the critical interaction strengths and construct part of the phase diagram in the parameters' space. Finally we discuss the effective Hamiltonian at the large-$U$ limit and provide an additional understanding of the numerical results. These results could be realized experimentally using cold atoms trapped in the 1D optical lattice.

DOI
10.1103/physrevb.84.195107

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We study a one-dimensional (1D) interacting topological model by means of the exact diagonalization method. The topological properties are first examined with the existence of the edge states at half-filling. We find that the topological phases are not only robust to small repulsive interactions, but also are stabilized by small attractive interactions, and also finite repulsive interaction can drive a topological nontrivial phase into a trivial one while the attractive interaction can drive a trivial phase into a nontrivial one. Next we calculate the Berry phase and parity of the bulk system and find that they are equivalent in characterizing the topological phases. With them we obtain the critical interaction strengths and construct part of the phase diagram in the parameters' space. Finally we discuss the effective Hamiltonian at the large-UUU limit and provide an additional understanding of the numerical results. These results could be realized experimentally using cold atoms trapped in the 1D optical lattice.
The one-dimensional interacting topological insulator is studied by means of exact diagonalization method. The topological properties are examined with the existence of the edge states and the quantized Berry phase at half-filling. It is found that the topological phases are not only robust to small repulsive interactions but also are stabilized by small attractive interactions, and also finite repulsive interaction can drive a topological non-trivial phase into a trivial one while the attractive interaction can drive a trivial phase into a non-trivial one. These results could be realized experimentally using cold atoms trapped in the 1D optical lattice.
The one-dimensional interacting topological insulator is studied by means of exact diagonalization method. The topological properties are examined with the existence of the edge states and the quantized berry phase at half-filling. It is found that the topological phases are not only robust to small repulsive interactions but also are stabilized by small attractive interactions and also finite repulsive interaction can drive a topological nontrivial phase into a trivial one while the attractive interaction can drive a trivial phase into a nontrivial one. These results could be realized experimentally using cold atoms trapped in the 1D optical lattice.
We show the existence of the fractional topological phase (FTP) in a one-dimensional interacting fermion model using exact diagonalization, in which the noninteracting part has flat bands with nontrivial topology. In the presence of the nearest-neighboring interaction V1{V}_{1}V1​, the FTP at filling factor $\ensuremath{\nu}=1/3$ appears. It is characterized by the threefold degeneracy and the quantized total Berry phase of the ground states. The FTP is destroyed by a next-nearest-neighboring interaction V2{V}_{2}V2​, and the phase diagrams in the (V1,V2)({V}_{1},{V}_{2})(V1​,V2​) plane are determined. We also present a physical picture of the phase and discuss its existence in the nearly flat band. Within the picture, we argue that the FTP at other filling factors can be generated by introducing proper interactions. The present study contributes to a systematic understanding of the FTPs and can be realized in cold-atom experiments.
We study strong correlation effects in topological insulators via the Lanczos algorithm, which we utilize to calculate the exact many-particle ground-state wave function and its topological properties. We analyze the simple, noninteracting Haldane model on a honeycomb lattice with known topological properties and demonstrate that these properties are already evident in small clusters. Next, we consider interacting fermions by introducing repulsive nearest-neighbor interactions. A first-order quantum phase transition was discovered at finite interaction strength between the topological band insulator and a topologically trivial Mott insulating phase by use of the fidelity metric and the charge-density-wave structure factor. We construct the phase diagram at T=0T=0T=0 as a function of the interaction strength and the complex phase for the next-nearest-neighbor hoppings. Finally, we consider the Haldane model with interacting hard-core bosons, where no evidence for a topological phase is observed. An important general conclusion of our work is that despite the intrinsic nonlocality of topological phases their key topological properties manifest themselves already in small systems and therefore can be studied numerically via exact diagonalization and observed experimentally, e.g., with trapped ions and cold atoms in optical lattices.
We study the topological properties of a one-dimensional (1D) hardcore Bose—Fermi mixture using the exact diagonalization method. We firstly add a hardcore boson to a fermionic system and by examining the edge states we find that the quasi-particle manifests the topological properties of the system. Then we study a mixture with 7 fermions and 1 boson. We find that the mixture also exhibits topological properties and its behaviors are similar to that of the corresponding fermionic system. We present a qualitative explanation to understand such behaviors using the mapping between a hardcore boson and a spinless fermion. These results show the existence of topological properties in a 1D hardcore Bose—Fermi mixture and may be realized using cold atoms trapped in optical lattices experimentally.
We investigate the stability of helical superfluid phase in a spin-orbit coupled Fermi gas loaded in a bilayer optical lattice. The phase diagram of the system is constructed in the mean field framework. We investigate the topological properties of the superfluid phases by a nontrivial application of the Fermi surface topological invariant to our time-reversal invariant system with degeneracies on the Fermi surface. We find that there is a first-order phase boundary in the phase diagram of half filling case and the superfluid phases are all topological trivial. The superfluid phase is topological nontrivial when the filling fraction deviates from the half filling. In the topological nontrivial superfluid phase, a full pairing gap exists in the bulk and gapless helical Majorana edge states exist at the boundary.