Detection of topological symmetry in the one-dimensional Afeck-Kennedy-Lieb-Tasaki model of integer spin with ultracold spinor atoms in optical lattices

A scheme is proposed for detection of the topology in the one-dimensional Afeck-Kennedy-Lieb-Tasaki model,based on ultracold spinor atomic gas in an optical lattice.For this purpose,a global operation O(θ)is introduced with respect to the breaking of spinrotational symmetry.Consequently,the topology can be manifested unambiguously by identifying the special values ofθwhere the expectation value of the global operator with degenerate ground states is vanishing.Furthermore,experimentallyθcan be detected readily by the interference of ultracold atomic gases.This scheme can be implemented readily in experiment since it does not need the addressing of individual atoms or the probing of a boundary.

Two-dimensional topological insulators with large bulk energy gap

Two-dimensional（2D） topological insulators（TTs,or quantum spin Hall insulators） are special insulators that possess bulk 2D electronic energy gap and time-reversal symmetry protected one-dimensional（1D） edge state.Carriers in the edge state have the property of spin-momentum locking,enabling dissipation-free conduction along the 1D edge.The existence of 2D TIs was confirmed by experiments in semiconductor quantum wells.However,the 2D bulk gaps in those quantum wells are extremely small,greatly limiting potential application in future electronics and spintronics.Despite this limitation,2D TIs with a large bulk gap attracted plenty of interest.In this paper,recent progress in searching for TIs with a large bulk gap is reviewed briefly.We start by introducing some theoretical predictions of these new materials and then discuss some recent important achievements in crystal growth and characterization.

Topological nodal line semimetals

We review the recent,mainly theoretical,progress in the study of topological nodal line semimetals in three dimensions.In these semimetals,the conduction and the valence bands cross each other along a one-dimensional curve in the three-dimensional Brillouin zone,and any perturbation that preserves a certain symmetry group（generated by either spatial symmetries or time-reversal symmetry） cannot remove this crossing line and open a full direct gap between the two bands.The nodal line（s） is hence topologically protected by the symmetry group,and can be associated with a topological invariant.In this review,（ⅰ） we enumerate the symmetry groups that may protect a topological nodal line;（ⅱ） we write down the explicit form of the topological invariant for each of these symmetry groups in terms of the wave functions on the Fermi surface,establishing a topological classification;（ⅲ） for certain classes,we review the proposals for the realization of these semimetals in real materials;（ⅳ） we discuss different scenarios that when the protecting symmetry is broken,how a topological nodal line semimetal becomes Weyl semimetals,Dirac semimetals,and other topological phases;and（ⅴ） we discuss the possible physical effects accessible to experimental probes in these materials.

Electronic properties of SnTe-class topological crystalline insulator materials

The rise of topological insulators in recent years has broken new ground both in the conceptual cognition of condensed matter physics and the promising revolution of the electronic devices.It also stimulates the explorations of more topological states of matter.Topological crystalline insulator is a new topological phase,which combines the electronic topology and crystal symmetry together.In this article,we review the recent progress in the studies of SnTe-class topological crystalline insulator materials.Starting from the topological identifications in the aspects of the bulk topology,surface states calculations,and experimental observations,we present the electronic properties of topological crystalline insulators under various perturbations,including native defect,chemical doping,strain,and thickness-dependent confinement effects,and then discuss their unique quantum transport properties,such as valley-selective filtering and helicity-resolved functionalities for Dirac fermions.The rich properties and high tunability make SnTe-class materials promising candidates for novel quantum devices.

Weak antilocalization and interaction-induced localization of Dirac and Weyl Fermions in topological insulators and semimetals

Weak localization and antilocalization are quantum transport phenomena that arise from the quantum interference in disordered metals.At low temperatures,they can give distinct temperature and magnetic field dependences in conductivity,allowing the symmetry of the system to be explored.In the past few years,they have also been observed in newly emergent topological materials,including topological insulators and topological semimetals.In contrast from the conventional electrons,in these new materials the quasiparticles are described as Dirac or Weyl fermions.In this article,we review our recent efforts on the theories of weak antilocalization and interaction-induced localization for Dirac and Weyl fermions in topological insulators and topological semimetals.

Topological hierarchy matters — topological matters with superlattices of defects

Topological insulators/superconductors are new states of quantum matter with metallic edge/surface states.In this paper,we review the defects effect in these topological states and study new types of topological matters — topological hierarchy matters.We find that both topological defects（quantized vortices） and non topological defects（vacancies） can induce topological mid-gap states in the topological hierarchy matters after considering the superlattice of defects.These topological mid-gap states have nontrivial topological properties,including the nonzero Chern number and the gapless edge states.Effective tight-binding models are obtained to describe the topological mid-gap states in the topological hierarchy matters.

Symmetry protected topological Luttinger liquids and the phase transition between them

We show that a doped spin-1/2 ladder with antiferromagnetic intra-chain and ferromagnetic inter-chain coupling is a symmetry protected topologically non-trivial Luttinger liquid.Turning on a large easy-plane spin anisotropy drives the system to a topologically-trivial Luttinger liquid.Both phases have full spin gaps and exhibit power-law superconducting pair correlation.The Cooper pair symmetry is singlet dxy in the non-trivial phase and triplet Sz? 0 in the trivial phase.The topologically non-trivial Luttinger liquid exhibits gapless spin excitations in the presence of a boundary,and it has no non-interacting or mean-field theory analog even when the fluctuating phase in the charge sector is pinned.As a function of the strength of spin anisotropy there is a topological phase transition upon which the spin gap closes.We speculate these Luttinger liquids are relevant to the superconductivity in metalized integer spin ladders or chains.

Classification of topological phases in one dimensional interacting non-Hermitian systems and emergent unitarity

Topological phases in non-Hermitian systems have become fascinating subjects recently.In this paper,we attempt to classify topological phases in 1D interacting non-Hermitian systems.We begin with the non-Hermitian generalization of the Su-Schrieffer-Heeger(SSH)model and discuss its many-body topological Berry phase,which is well defined for all interacting quasi-Hermitian systems(non-Hermitian systems that have real energy spectrum).We then demonstrate that the classification of topological phases for quasi-Hermitian systems is exactly the same as their Hermitian counterparts.Finally,we construct the fixed point partition function for generic 1D interacting non-Hermitian local systems and find that the fixed point partition function still has a one-to-one correspondence to their Hermitian counterparts.Thus,we conclude that the classification of topological phases for generic 1D interacting non-Hermitian systems is still exactly the same as Hermitian systems.

Node line semimetal states in tungsten-based materials

Node line semimetals(NLSMs) were characterized by one-dimensional band crossings in their bulk electronic structures.The nontrivial band topology of NLSM gives rise to "drumhead" surface electronic excitations that exhibits exotic physical properties.The symmetries of crystalline provide the needed protection of node line from being gapped out by the perturbations that preserve the symmetry.The progress of NLSM in tungsten-based materials is reviewed with an emphasis on their symmetry-based protection,characteristic electronic band structures and their response to the spin-orbit coupling(SOC)and breaking of time-reversal symmetry.The potential exploration directions of tungsten-based NLSM in the future are also discussed.