Gapless Luttinger liquid is conventionally viewed as topologically trivial, unless it hosts degenerate ground states and or entanglement spectrum, which necessitates partial bulk degree of freedom to be gapped out. Here we predict an emergent gapless
topological Luttinger liquid which is beyond the conventional scenarios and is characterized by the nontrivial many-body bulk spin texture, and propose feasible scheme for experimental observation. We consider a one-dimensional spin-orbit coupled Fermi-Hubbard model with fractional filling, whose low-energy physics is effectively described by a spinless Luttinger liquid and is trivial in the conventional characterization. We show that, as being tuned by the filling factor and interaction strength, the many-body ground state may exhibit nontrivial winding in its bulk spin texture in the projected momentum space, manifesting an emergent topological phase. A topological transition occurs when the projected spin-state at a high symmetry momentum becomes fully mixed one, resulting from competing processes of particle scattering to the lower and higher subbands, for which the spin texture at such momentum point is ill-defined, but the Luttinger liquid keeps gapless through the transition. Surprisingly, at relatively small filling the Luttinger liquid remains topologically nontrivial even at infinitely strong interaction. The results can be generalized to finite temperature which facilitates the real experimental detection. This work shows a novel gapless topological Luttinger liquid whose characterization is beyond the low-energy effective theory, and can be verified based on current experiments.
We uncover an edge geometric phase mechanism to realize the second-order topological insulators and topological superconductors (SCs), and predict realistic materials for the realization. The theory is built on a novel result shown here that the nont
rivial pseudospin textures of edge states in a class of two-dimensional (2D) topological insulators give rise to the geometric phases defined on the edge, for which the effective edge mass domain walls are obtained across corners when external magnetic field or superconductivity is considered, and the Dirac or Majorana Kramers corner modes are resulted. Remarkably, with this mechanism we predict the Majorana Kramers corner modes by fabricating 2D topological insulator on only a uniform and conventional $s$-wave SC, in sharp contrast to the previous proposals which applies unconventional SC pairing or SC $pi$-junction. We find that Au/GaAs(111) can be a realistic material candidate for realizing such Majorana Kramers corner modes.
Topological phases of the famous Altland-Zirnbauer (AZ) tenfold classes are defined on the equilibrium ground states. Whether such equilibrium topological phases have universal correspondence to far-from-equilibrium quantum dynamics is a fundamental
issue of both theoretical and experimental importance. Here we uncover the universal topological quench dynamics linking to the equilibrium topological phases for the complete AZ tenfold classes, with a general framework being established. We show a fundamental result that a $d$-dimensional topological phase of the tenfold class, with an integer invariant or $mathbb{Z}_{2}$ index defined on high symmetry momenta, is generically characterized by topology reduced to the highest-order band-inversion surfaces located at arbitrary discrete momenta of Brillouin zone. Such dimension-reduced topology is further captured by universal topological patterns emerging in far-from-equilibrium quantum dynamics by quenching the system from trivial phase to the topological regime, rendering the dynamical hallmark of the equilibrium topological phase. This work establishes a universal dynamical characterization for the complete AZ symmetry classes of topological phases, which has broad applications in theory and experiment.
We propose a general analytic method to study the localization transition in one-dimensional quasicrystals with parity-time ($mathcal{PT}$) symmetry, described by complex quasiperiodic mosaic lattice models. By applying Avilas global theory of quasip
eriodic Schrodinger operators, we obtain exact mobility edges and prove that the mobility edge is identical to the boundary of $mathcal{PT}$-symmetry breaking, which also proves the existence of correspondence between extended (localized) states and $mathcal{PT}$-symmetry ($mathcal{PT}$-symmetry-broken) states. Furthermore, we generalize the models to more general cases with non-reciprocal hopping, which breaks $mathcal{PT}$ symmetry and generally induces skin effect, and obtain a general and analytical expression of mobility edges. While the localized states are not sensitive to the boundary conditions, the extended states become skin states when the periodic boundary condition is changed to open boundary condition. This indicates that the skin states and localized states can coexist with their boundary determined by the mobility edges.
Equilibrium topological phases are robust against weak static disorder but may break down in the strong disorder regime. Here we explore the stability of the quench-induced emergent dynamical topology in the presence of dynamical noise. We develop an
analytic theory and show that for weak noise, the quantum dynamics induced by quenching an initial trivial phase to Chern insulating regime exhibits robust emergent topology on certain momentum subspaces called band inversion surfaces (BISs). The dynamical topology is protected by the minimal oscillation frequency over the BISs, mimicking a bulk gap of the dynamical phase. Singularities emerge in the quench dynamics, with the minimal oscillation frequency vanishing on the BISs if increasing noise to critical strength, manifesting a dynamical topological transition, beyond which the emergent topology breaks down. Two types of dynamical transitions are predicted. Interestingly, we predict a sweet spot in the critical transition when noise couples to all three spin components in the same strength, in which case the dynamical topology survives at arbitrarily strong noise regime. This work unveils novel features of the dynamical topology under dynamical noise, which can be probed with control in experiment.
Fracton topological order (FTO) is a new classification of correlated phases in three spatial dimensions with topological ground state degeneracy (GSD) scaling up with system size, and fractional excitations which are immobile or have restricted mobi
lity. With the topological origin of GSD, FTO is immune to local perturbations, whereas a strong enough global external perturbation is expected to break the order. The critical point of the topological transition is however very challenging to identify. In this work, we propose to characterize quantum phase transition of the type-I FTOs induced by external terms and develop a theory to study analytically the critical point of the transition. In particular, for the external perturbation term creating lineon-type excitations, we predict a generic formula for the critical point of the quantum phase transition, characterized by the breaking-down of GSD. This theory applies to a board class of FTOs, including X-cube model, and for more generic FTO models under perturbations creating two-dimensional (2D) or 3D excitations, we predict the upper and lower limits of the critical point. Our work makes a step in characterizing analytically the quantum phase transition of generic fracton orders.
There is an immense effort in search for various types of Weyl semimetals, of which the most fundamental phase consists of the minimal number of i.e. two Weyl points, but is hard to engineer in solids. Here we demonstrate how such fundamental Weyl se
mimetal can be realized in a maneuverable optical Raman lattice, with which the three-dimensional (3D) spin-orbit (SO) coupling is synthesised for ultracold atoms. In addition, a new novel Weyl phase with coexisting Weyl nodal points and nodal ring is also predicted here, and is shown to be protected by nontrivial linking numbers. We further propose feasible techniques to precisely resolve 3D Weyl band topology through 2D equilibrium and dynamical measurements. This work leads to the first realization of the most fundamental Weyl semimetal band and the 3D SO coupling for ultracold quantum gases, which are respectively the significant issues in the condensed matter and ultracold atom physics.
Cold atoms with laser-induced spin-orbit (SO) interactions provide promising platforms to explore novel quantum physics, in particular the exotic topological phases, beyond natural conditions of solids. The past several years have witnessed important
progresses in both theory and experiment in the study of SO coupling and novel quantum states for ultracold atoms. Here we review the physics of the SO coupled quantum gases, focusing on the latest theoretical and experimental progresses of realizing SO couplings beyond one-dimension (1D), and the further investigation of novel topological quantum phases in such systems, including the topological insulating phases and topological superfluids. A pedagogical introduction to the SO coupling for ultracold atoms and topological quantum phases is presented. We show that the so-called optical Raman lattice schemes, which combine the creation of the conventional optical lattice and Raman lattice with topological stability, can provide minimal methods with high experimental feasibility to realize 1D to 3D SO couplings. The optical Raman lattices exhibit novel intrinsic symmetries, which enable the natural realization of topological phases belonging to different symmetry classes, with the topology being detectable through minimal measurement strategies. We introduce how the non-Abelian Majorana modes emerge in the SO coupled superfluid phases which can be topologically nontrivial or trivial, for which a few fundamental theorems are presented and discussed. The experimental schemes for achieving non-Abelian superfluid phases are given. Finally, we point out the future important issues in this rapidly growing research field.
We predict a linear logarithmical scaling law of Bloch oscillation dynamics in Weyl semimetals (WSMs), which can be applied to detect Weyl nodal points. Applying the semiclassical dynamics for quasiparticles which are accelerated bypassing a Weyl poi
nt, we show that transverse drift exhibits asymptotically a linear log-log relation with respect to the minimal momentum measured from the Weyl point. This linear scaling behavior is a consequence of the monopole structure nearby the Weyl points, thus providing a direct measurement of the topological nodal points, with the chirality and anisotropy being precisely determined. We apply the present results to two lattice models for WSMs which can be realized with cold atoms in experiment, and propose realistic schemes for the experimental detection. With the analytic and numerical results we show the feasibility of identifying topological Weyl nodal points based on the present prediction.
We find an optical Raman lattice without spin-orbit coupling showing chiral topological orders for cold atoms. Two incident plane-wave lasers are applied to generate simultaneously a double-well square lattice and periodic Raman couplings, the latter
of which drive the nearest-neighbor hopping and create a staggered flux pattern across the lattice. Such a minimal setup is can yield the quantum anomalous Hall effect in the single particle regime, while in the interacting regime it achieves the $J_1$-$J_2$-$K$ model with all parameters controllable, which supports a chiral spin liquid phase. We further show that heating in the present optical Raman lattice is reduced by more than one order of magnitude compared with the conventional laser-assisted tunneling schemes. This suggests that the predicted topological states be well reachable with the current experimental capability.
