Abstract
Topological physics in optical lattices have attracted much attention in recent years. The nonlinear effects on such optical systems remain well-explored and a large amount of progress has been achieved. In this paper, under the mean-field approximation for a nonlinearly optical coupled boson–hexagonal lattice system, we calculate the nonlinear Dirac cone and discuss its dependence on the parameters of the system. Due to the special structure of the cone, the Berry phase (two-dimensional Zak phase) acquired around these Dirac cones is quantized, and the critical value can be modulated by interactions between different lattices sites. We numerically calculate the overall Aharonov-Bohm (AB) phase and find that it is also quantized, which provides a possible topological number by which we can characterize the quantum phases. Furthermore, we find that topological phase transition occurs when the band gap closes at the nonlinear Dirac points. This is different from linear systems, in which the transition happens when the band gap closes and reopens at the Dirac points.
Highlights
Entropy 2021, 23, 1404. https://The topological concepts developed in condensed matter theory have a profound impact on physics [1,2,3,4,5,6,7]
Topological invariants play an important role; for example, quantized Hall conductances [18] are represented in terms of Chern numbers associated with the Berry phase [19,20,21,22], while its extension to the case of four-dimensional quantum Hall conductances is described by second Chern numbers [23]
We find that there are two nonlinear Dirac cone (NDC) in the Brillouin Zone (BZ), and the phase acquired by adiabatically transporting the system on the lowest band around the two NDCs in the BZ is zero, while the phase acquired in the same processing is quantized around only one of the NDCs
Summary
The topological concepts developed in condensed matter theory have a profound impact on physics [1,2,3,4,5,6,7]. We aim to study a bosonic version of hexagonal lattice model [46] with nonlinear couplings (see Figure 1) by NSE. This interesting, two-dimensional hexagonal lattice was first purposed in condensed matter physics [47,48] and is attractive for both experimental and theoretical study due to the novel features of the system. The NDCs and teh effective Hamiltonian around the NDCs are given and discussed The realization of this model in an experiment, for example, in a two-dimensional hexagonal array of waveguides with refractive indices, nearest neighbor couplings and on-site Kerr nonlinearity [53], is suggested.
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