Spin-orbit coupling with bosons gives rise to novel properties that are absent in usual bosonic systems. Under very general conditions, the conventional ground state wavefunctions of bosons are constrained by the "no-node" theorem to be positive-definite. In contrast, the linear-dependence of spin-orbit coupling leads to complex-valued condensate wavefunctions beyond this theorem. In this article, we review the study of this class of unconventional Bose-Einstein condensations focusing on their topological properties. Both the 2D Rashba and 3D σ · p-type Weyl spin-orbit couplings give rise to Landau-level-like quantization of single-particle levels in the harmonic trap. The interacting condensates develop the half-quantum vortex structure spontaneously breaking time-reversal symmetry and exhibit topological spin textures of the skyrmion type. In particular, the 3D Weyl coupling generates topological defects in the quaternionic phase space as an SU(2) generalization of the usual U(1) vortices. Rotating spin-orbit coupled condensates exhibit rich vortex structures due to the interplay between vorticity and spin texture. In the Mott-insulating states in optical lattices, quantum magnetism is characterized by the Dzyaloshinskii-Moriya type exchange interactions.

The sign problem is a major obstacle in quantum Monte Carlo simulations for many-body fermion systems. We examine this problem with a new perspective based on the Majorana reflection positivity and Majorana Kramers positivity. Two sufficient conditions are proven for the absence of the fermion sign problem. Our proof provides a unified description for all the interacting lattice fermion models previously known to be free of the sign problem based on the auxiliary field quantum Monte Carlo method. It also allows us to identify a number of new sign-problem-free interacting fermion models including, but not limited to, lattice fermion models with repulsive interactions but without particle-hole symmetry and interacting topological insulators with spin-flip terms.

The theory of the focusing NLS equation under periodic boundary conditions, together with the Floquet spectral theory of its associated Zakharov-Shabat linear operator L, is developed in sufficient detail for later use in studies of perturbations of the NLS equation. "Counting lemmas" for the non-selfadjoint operator L, are established which control its spectrum and show that all of its eccentricities are finite in number and must reside within a finite disc D in the complex eigenvalue plane. The radius of the disc D is controlled by the H ι norm of the potential q. For this integrable NLS Hamiltonian system, unstable tori are identified, and Backlund transformations are then used to construct global representations of their stable and unstable manifolds -"whiskered tori" for the NLS pde.The Floquet discriminant Δ(λ; q) is used to introduce a natural sequence of NLS constants of motion, [¥j(q) = Δ(λ = λj(q);q), where λ^ denotes the j th critical point of the Floquet discriminant Δ(λ)]. A Taylor series expansion of the constants FjCg), with explicit representations of the first and second variations, is then used to study neighborhoods of the whiskered tori. In particular, critical tori with hyperbolic structure are identified through the first and second variations of F^Cg), which themselves are expressed in terms of quadratic products of eigenfunctions of L. The second variation permits identification, within the disc D, of important bifurcations in the spectral configurations of the operator L. The constant ¥j(q), as the height of the Floquet discriminant over the critical point λj, admits a natural interpretation as a Morse function for NLS isospectral level sets. This Morse interpretation is studied in some detail. It is valid globally for the infinite tail, {¥j(q)}^^> N , which is associated with critical points outside the disc D. Within this disc, the interpretation is only valid locally, with the same obstruction to its global validity as to a global ordering of the spectrum. Nevertheless, this local Morse theory, together with the Backlund representations of the whiskered tori, produces extremely clear pictures of the stratification of NLS invariant sets near these whiskered tori -pictures which are

We construct a minimal four-band model for the two-dimensional (2D) topological insulators and quantum anomalous Hall insulators based on the px-and py-orbital bands in the honeycomb lattice. The multiorbital structure allows the atomic spin-orbit coupling which lifts the degeneracy between two sets of on-site Kramers doublets jz = ± 3 2 and jz = ± 1 2 . Because of the orbital angular momentum structure of Bloch-wave states at Γ and K(K ) points, topological gaps are equal to the atomic spin-orbit coupling strengths, which are much larger than those based on the mechanism of the s-p band inversion. In the weak and intermediate regime of spin-orbit coupling strength, topological gaps are the global gap. The energy spectra and eigen wave functions are solved analytically based on Clifford algebra. The competition among spin-orbit coupling λ, sublattice asymmetry m and the Néel exchange field n results in band crossings at Γ and K(K ) points, which leads to various topological band structure transitions. The quantum anomalous Hall state is reached under the condition that three gap parameters λ, m, and n satisfy the triangle inequality. Flat bands also naturally arise which allow a local construction of eigenstates. The above mechanism is related to several classes of solid state semiconducting materials.

We generalize the concept of Berry connection of the single-electron band structure to the twoparticle Cooper pair states between two Fermi surfaces with opposite Chern numbers. Because of underlying Fermi surface topology, the pairing Berry phase acquires non-trivial monopole structure. Consequently, pairing gap functions have the topologically-protected nodal structure as vortices in the momentum space with the total vorticity solely determined by the monopole charge qp. The pairing nodes behave as the Weyl-Majorana points of the Bogoliubov-de Gennes pairing Hamiltonian. Their relation with the connection patterns of the surface modes from the Weyl band structure and the Majorana surface modes inside the pairing gap is also discussed. Under the approximation of spherical Fermi surfaces, the pairing symmetry are represented by monopole harmonic functions. The lowest possible pairing channel carries angular momentum number j = |qp|, and the corresponding gap functions are holomorphic or anti-holomorphic functions on Fermi surfaces. The study of topological states has renewed our understanding of condensed matter physics. The discovery of two-dimensional integer quantum Hall states 1,2 initiated the exploration of novel states characterized by band topology rather than symmetry 3-8 , with magnetic band structures that possess non-trivial Chern numbers arising from broken time-reversal (TR) symmetry. The study of Berry curvature of Bloch bands in such lattice structures has led to to many results in anomalous Hall and quantum anomalous Hall physics 9-16 . The band structure topology has also been generalized to systems of topological insulators with TR symmetry [17][18][19][20][21][22][23][24][25][26][27] . The stable gapless surface modes which appear at the boundary of gapped topological systems have analogs in gapless semimetallic systems, which can also have non-trivial band topology. For example, topological Weyl semi-metals have been proposed and realized in three-dimensional (3D) systems in the absence of either TR or inversion symmetry . Their band structure is characterized by degenerate Weyl points in the Brillouin-zone (BZ), which can be understood as monopole sources and sinks of Berry-curvature flux in k-space.Topological phenomena are usually understood in terms of contributions from all the filled electronic states rather than the states in the vicinity of Fermi surfaces. The apparent disagreement with the central tenet of Fermi-liquid theory that all conduction processes can be understood at the Fermi level can be resolved by introducing the Berry phase of quasiparticles on the Fermi surface 13 . So far, the study of the Fermi surface topology and the associated Berry phase structure has mainly been discussed at the single-particle level 11-14 .Here we study a novel class of exotic superconductivity which can be realized in doped Weyl metals, and more generally in systems with topologically non-trivial Fermi surfaces. In superconductivity with pairing between states on two disjoint Fermi surface s...

We nonperturbatively investigate the ground state magnetic properties of the 2D half-filled SU(2N) Hubbard model in the square lattice by using the projector determinant quantum Monte Carlo simulations combined with the method of local pinning fields. Long-range Néel orders are found for both the SU(4) and SU(6) cases at small and intermediate values of U. In both cases, the long-range Néel moments exhibit nonmonotonic behavior with respect to U, which first grow and then drop as U increases. This result is fundamentally different from the SU(2) case in which the Néel moments increase monotonically and saturate. In the SU(6) case, a transition to the columnar dimer phase is found in the strong interaction regime.

We systematically generalize the exotic 3 He-B phase, which not only exhibits unconventional symmetry but is also isotropic and topologically non-trivial, to arbitrary partial-wave channels with multi-component fermions. The concrete example with four-component fermions is illustrated including the isotropic f , p and d-wave pairings in the spin septet, triplet, and quintet channels, respectively. The odd partial-wave channel pairings are topologically non-trivial, while pairings in even partial-wave channels are topologically trivial. The topological index reaches the largest value of N 2 in the p-wave channel (N is half of the fermion component number). The surface spectra exhibit multiple linear and even high order Dirac cones. Applications to multi-orbital condensed matter systems and multi-component ultra-cold large spin fermion systems are discussed. [5,6]. The p-wave superconductivity has also been extensively investigated in SrRu 2 O 4 [7-9], and heavy fermion systems including UGe 2 , URhGe, UCoGe [10]. The p-wave superfluid 3 He and superconductors exhibit rich topological structures of vortices and spin textures under rotations or in external magnetic fields, respectively [11,12]. In addition, experimental signatures of the possible nodal f -wave superconductivity have also been reported in UPt 3 [13,14].Among these unconventional pairing phases, the 3 He-B phase is distinct: in spite of its non-s-wave pairing symmetry and spin structure, the overall pairing structure remains isotropic and fully gapped. Its pairing exhibits the relative spin-orbit symmetry breaking fromwhere L, S, and J represent the orbital, spin, and total angular momentum, respectively. The relative spin-orbit symmetry-breaking has also been studied in the context of Pomeranchuk instability termed as unconventional magnetism leading to dynamic generation of spin-orbit coupling [15,16]. Furthermore, the 3 He-B phase possesses non-trivial topological properties [17][18][19]. Topological states of matter have become a major research focus since the discovery of the integer quantum Hall effect [20][21][22]. Recently, the study of topological band structures has extended from time-reversal (TR) breaking systems to TR invariant systems [23][24][25], from two to three dimensions [18,26,27], and from insulators to superconductors [17][18][19][28][29][30][31]. The 3 He-B phase is a 3D TR invariant topological Cooper pairing state. Its bulk Bogoliubov spectra are analogous to the 3D gapped Dirac fermions belonging to the DIII class characterized by an integer-valued index [18]. The non-trivial bulk topology gives rise to the gapless surface Dirac spectra of the mid-gap AndreevMajorana modes [32]. Evidence of these low energy states has been reported in recent experiments [33].Because the electron Cooper pair can only be either spin singlet or triplet, the p-wave 3 He-B phase looks the only choice of the unconventional 3D isotropic pairing state. In this article, we will show that actually there are much richer possibilities of this exotic c...

We investigate topological insulating states in both two and three dimensions with the harmonic potential and strong spin-orbit couplings breaking the inversion symmetry. Landau-level like quantization appear with the full 2D and 3D rotational symmetry and time-reversal symmetry. Inside each band, states are labeled by their angular momenta over which energy dispersions are strongly suppressed by spin-orbit coupling to nearly flat. The radial quantization generates energy gaps between neighboring bands at the order of the harmonic frequency. Helical edge or surface states appear on open boundaries characterized by the Z2 index. These Hamiltonians can be viewed from the dimensional reduction of the high dimensional quantum Hall states in 3D and 4D flat spaces. These states can be realized with ultra-cold fermions inside harmonic traps with the synthetic gauge fields.