We investigate the superfluid phases of a Rashba spin-orbit coupled Bose-Einstein condensate residing on a two dimensional square optical lattice in the presence of an effective Zeeman field Ω. At a critical value Ω = Ω c , the single-particle spectrum E k changes from having a set of four degenerate minima to a single minimum at k = 0, corresponding to condensation at finite or zero momentum, respectively. We describe this quantum phase transition and the symmetry breaking of the condensate phases. We use the Bogoliubov theory to treat the superfluid phases and determine the phase diagram, the excitation spectrum and the sound velocity of the phonon excitations. A novel dynamically unstable superfluid regime occurring when Ω is close to Ω c is analytically identified and the behavior of the condensate quantum depletion is discussed. Moreover, we show that there are two types of roton excitations occurring in the Ω < Ω c regime and obtain explicit values for the corresponding energy gaps. PACS numbers:Introduction. The recent realization of ultracold spin-orbit coupled (SOC) quantum gases [1] has attracted high interest and resulted in considerable research efforts both on the theoretical and experimental side [2][3][4][5][6], in part due to the possibility to tune the spin-orbit interactions [7] in contrast to solid state materials. Ultracold quantum gases with spin-orbit coupling manifest novel types of superfluid and magnetic groundstates and have also been predicted to host topological excitations like Majorana fermions [8].The SOC Bose-Einstein condensate (BEC) has intrinsic features that make it different from the standard BEC: the interaction among atoms make a SOC BEC stable since it cannot exist in the free regime [9], the SOC also breaks the Galileian invariance so that the superfluid properties change in different reference frames [10]; for a review see [11]. Several works have considered different types of SOC in the continuous limit: pure Rashba, mixed and symmetric Rashba-Dresselhaus, in two and three dimensions [12][13][14]. The exotic properties of the Mott insulating phase arising from the superfluid-Mott insulator (SF-MI) transition [15,16] were also considered in the case of an optically induced lattice. However, an analytical quantitative description of the SF phase for a SOC BEC in an optically induced lattice is still missing.In this work, we consider a Bose-Einstein condensate with Rashba SOC residing on a 2D square optical lattice and prove that the SOC qualitatively affects the features of the superfluid phase. The system's parameters are the Zeeman-coupling Ω, the strength of the spin-orbit coupling λ , the hopping t, and the intra-and interspecies interactions U,U . We discuss the origin and magnitude of these terms in more detail later on. We will in this paper show three main results: I) with λ t the existence of the SF is related to the ratio Ω/U and not to t/U like in the usual Bose-Hubbard models; II) Ω can trigger a breakdown of SF in a window near the critical value Ω c ≡ 2λ 2 /t...
The Bott index is an index that discerns among pairs of unitary matrices that can or cannot be approximated by a pair of commuting unitary matrices. It has been successfully employed to describe the approximate integer quantization of the transverse conductance of a system described by a short-range, bounded and spectrally gapped Hamiltonian on a finite two dimensional lattice on a torus and to describe the invariant of the Bernevig-Hughes-Zhang model even with disorder. This paper shows the constancy in time of the Bott index and the Chern number related to the time-evolved Fermi projection of a thermodynamically large system described by a short-range and time-dependent Hamiltonian that is initially gapped. The general situation of a ramp of a time-dependent perturbation is considered, a section is dedicated to time-periodic perturbations. arXiv:1708.05916v4 [cond-mat.mes-hall]
The Bott index of two unitary operators on an infinite dimensional Hilbert space is defined. Homotopic invariance with respect to multiplicative unitary perturbations of the type identity plus trace class and the "logarithmic" law for the index are proven. The index and its properties are then extended to the case of a pair of invertible operators. An application to the physics of two dimensional quantum systems proves that the index is equal to the Chern number therefore showing that the transverse Hall conductance is an integer.
This article reviews the foundations of the theory of the Bott index of a pair of unitary matrices in the context of condensed matter theory, as developed by Hastings and Loring (J. Math. Phys. 51, 015214 (2010), Ann. Phys. 326, 1699 (2011), providing a novel proof of the equality with the Chern number. The Bott index is defined for a pair of unitary matrices, then extended to a pair of invertible matrices and homotopic invariance of the index is proven. An insulator defined on a lattice on a two-torus, that is a rectangular lattice with periodic boundary conditions, is considered and a pair of quasi-unitary matrices associated to this physical system are introduced. It is shown that their Bott index is well defined and the connection with the transverse conductance, the Chern number, is established proving the equality of the two quantities, in certain units.
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