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...
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.