We propose a new theory to characterize equilibrium topological phase with non-equilibrium quantum dynamics by introducing the concept of high-order topological charges, with novel phenomena being predicted. Through a dimension reduction approach, we can characterize a $d$-dimensional ($d$D) integer-invariant topological phase with lower-dimensional topological number quantified by high-order topological charges, of which the $s$th-order topological charges denote the monopoles confined on the $(s-1)$th-order band inversion surfaces (BISs) that are $(d-s+1)$D momentum subspaces. The bulk topology is determined by the $s$th order topological charges enclosed by the $s$th-order BISs. By quenching the system from trivial phase to topological regime, we show that the bulk topology of post-quench Hamiltonian can be detected through a high-order dynamical bulk-surface correspondence, in which both the high-order topological charges and high-order BISs are identified from quench dynamics. This characterization theory has essential advantages in two aspects. First, the highest ($d$th) order topological charges are characterized by only discrete signs of spin-polarization in zero dimension (i.e. the $0$th Chern numbers), whose measurement is much easier than the $1$st-order topological charges that are characterized by the continuous charge-related spin texture in higher dimensional space. Secondly, a more striking result is that a first-order high integer-valued topological charge always reduces to multiple highest-order topological charges with unit charge value, and the latter can be readily detected in experiment. The two fundamental features greatly simplify the characterization and detection of the topological charges and also topological phases, which shall advance the experimental studies in the near future.
Motivated by previous efforts in detecting topological orders from the ground state(s) wave function, we introduce a new quantum information tool, coined the information convex, to capture the bulk and boundary topological excitations of a 2D topological order. Defined as a set of reduced density matrices that minimizes the energy in a subsystem, the information convex encodes not only the bulk anyons but also the gapped boundaries of 2D topological orders. Using untwisted gapped boundaries of non-Abelian quantum doubles as an example, we show how the information convex reveals and characteriz…
The Bose-Hubbard model subjected to an effective magnetic field hosts a plethora of phases with different topological orders when tuning the chemical potential. Using the density matrix renormalization group method, we identify several gapped phases near the first Mott lobe at strong interactions. They are connected by a particle-hole symmetry to a variety of quantum Hall states stabilized at low fillings. We characterize phases of both particle and hole type and identify signatures compatible with Laughlin, Moore-Read, and Bosonic Integer Quantum Hall states by calculating the quantized Hall …
One of the most striking features of quantum phases that exhibit topological order is the presence of long range entanglement that cannot be detected by any local order parameter. The formalism of projected entangled-pair states is a natural framework for the parameterization of the corresponding ground state wavefunctions, in which the full wavefunction is encoded in terms of local tensors. Topological order is reflected in the symmetries of these tensors, and we give a characterization of those symmetries in terms of matrix product operators acting on the virtual level. This leads to a set o…
Two established frameworks account for the onset of a gap in a superconducting system: one is based on spontaneous symmetry breaking via the Anderson-Higgs-Kibble mechanism, and the other is based on the recently developed paradigm of topological order. We show that, on manifolds with non trivial topology, both mechanisms yield a degeneracy of the ground state arising only from the {\it incompressibility} induced by the presence of a gap. We compute the topological entanglement entropy of a topological superconductor and argue that its measure allows to {\it distinguish} between the two mechan…
We study the quench dynamics of a topologically trivial one-dimensional gapless wire following its sudden coupling to topological bound states. We find that as the bound states leak into and propagate through the wire, signatures of their topological nature survive and remain measurable over a long lifetime. Thus, the quench dynamically induces topological properties in the gapless wire. Specifically, we study a gapless wire coupled to fractionally charged solitons or Majorana fermions and characterize the dynamically induced topology in the wire, in the presence of disorder and short-range in…
Kondo insulators combine strong electronic correlations with spin orbit coupling and thereby provide a potential realization of correlated topological insulators. We present model calculations which allow us to study the onset of bulk coherence and concomitant topological edge states from the mixed valence to local moment regimes. Our real-space dynamical mean-field results include the detailed temperature dependence of the single particle spectral function on slab geometries as well as the temperature dependence of the topological invariant. The relevance of our calculations for candidate mat…
Characterizing Long-Range Entanglement in a Mixed State Through an Emergent Order on the Entangling Surface
Perimeter Institute for Theoretical Physics（Perimeter理论物理研究所）；Department of Physics, University of California at San Diego（加州大学圣地亚哥分校物理系）；Department of Physics, University of California, Santa Barbara（加州大学圣塔芭芭拉分校物理系）
Given a microscopic lattice Hamiltonian for a topologically ordered phase, we describe a tensor network approach to characterize its emergent anyon model and, in a chiral phase, also its gapless edge theory. First, a tensor network representation of a complete, orthonormal set of ground states on a cylinder of infinite length and finite width is obtained through numerical optimization. Each of these ground states is argued to have a different anyonic flux threading through the cylinder. In a chiral phase, the entanglement spectrum of each ground state is seen to reveal a different sector of th…
Time-periodic driving of a quantum system can enable new dynamical topological phases of matter that could not exist in thermal equilibrium. We investigate two related classes of dynamical topological phenomena in 2D systems: Floquet symmetry protected topological phases (FSPTs), and Floquet enriched topological orders (FETs). By constructing solvable lattice models for a complete set of 2D bosonic FSPT phases, we show that bosonic FSPTs can be understood as topological pumps which deposit loops of 1D SPT chains onto the boundary during each driving cycle, which protects a non-trivial edge sta…
We present a method of extracting information about topological order from the ground state of a strongly correlated two-dimensional system represented by an infinite projected entangled pair state (iPEPS). As in Phys. Rev. B 101, 041108 (2020) and 102, 235112 (2020) we begin by determining symmetries of the iPEPS represented by infinite matrix product operators (iMPO) that map between the different iPEPS transfer matrix fixed points, to which we apply the fundamental theorem of MPS to find zipper tensors between products of iMPO's that encode fusion properties of the anyons. The zippers can b…
We review the basic properties of the Haldane phase in spin-1 Heisenberg antiferromagnetic chains, including its persistence in quasi-one-dimensional geometries. Using large-scale numerical simulations, we map out the phase diagram for a realistic model applicable to experimental Haldane compounds. We also investigate the effect of different chain coupling geometries and confirm a general mean field universality of the critical coupling times the coordination number of the lattice. Inspired by the recent development of characterization of symmetry protected topological states, of which the Hal…
We show that certain singularities of the Hamiltonian in the complex wave vector space can be used to identify topological quantum phase transitions for $1D$ chiral topological superconductors/superfluids in the BDI class. These singularities fall into the category of the so-called exceptional points ($EP$'s) studied in the context of non-Hermitian Hamiltonians describing open quantum systems. We also propose a generic formula in terms of the properties of the $EP$'s to quantify the exact number of Majorana zero modes in a particular chiral topological superconducting phase, given the values o…
Dramatically different from the Hermitian systems, the conventional Bulk-Boundary Correspondence (BBC) is broken in the non-Hermitian systems. In this article, we use edge entanglement entropy to characterize the topological properties of non-Hermitian Su-Schrieffer-Heeger Hubbard model. For free Fermions, we study the scaling behavior of entanglement entropy and demonstrate that the edge entanglement entropy is a good indicator to delimit different phases of non-Hermitian systems. We further generalize the edge entanglement entropy to the non-Hermitian interacting Hubbard chain, and obtain th…
Hamiltonian engineering is an important approach for quantum information processing, when appropriate materials do not exist in nature or are unstable. So far there is no stable material for the Kitaev spin Hamiltonian with anisotropic interactions on a honeycomb lattice (A. Kitaev, Annals of Physics vol 321, 2 (2006)), which plays a crucial role in the realization of both Abelian and non-Abelian anyons. Here, we show how to dynamically realize the Kitaev spin Hamiltonian from the conventional Heisenberg spin Hamiltonian using a pulse-control technique. By repeating the same pulse sequence, th…
The discovery by Kane and Mele of a model of spinful electrons characterized by a $\mathcal{Z}_2$ topological invariant had a lasting effect on the study of electronic band structures. Given this, it is natural to ask whether similar topology can be found in the band-like excitations of magnetic insulators, and recently models supporting $\mathcal{Z}_2$ topological invariants have been proposed for both magnon [Kondo et al. Phys. Rev. B 99, 041110(R) (2019)] and triplet [D. G. Joshi and A. P. Schnyder, Phys. Rev. B 100, 020407 (2019)] excitations. In both cases, magnetic excitations form time-…
We discuss detection strategies for fractional topological insulators (FTIs) realizing time-reversal invariant analogues of fractional quantum Hall systems in the Laughlin universality class. Focusing on transport measurements, we study the effect of magnetic perturbations on the edge modes. We find that the modes show unexpected robustness against magnetic backscattering for moderate couplings and edge interactions, allowing for various phase transitions signaling the FTI phase. We also describe protocols for extracting the universal integer m characterizing the phase and the edge interaction…
We report on the dynamical formation of exceptional degeneracies in basic correlation functions of non-integrable one- and two-dimensional systems quenched to the vicinity of a critical point. Remarkably, fine-tuned semi-metallic points in the phase diagram of the considered systems are thereby promoted to topologically robust non-Hermitian (NH) nodal phases emerging in the coherent long-time evolution of a dynamically equilibrating system. In the framework of non-equilibrium Green's function methods within the conserving second Born approximation, we predict observable signatures of these nov…
In topologically ordered quantum states of matter in 2+1D (space-time dimensions), the braiding statistics of anyonic quasiparticle excitations is a fundamental characterizing property which is directly related to global transformations of the ground-state wavefunctions on a torus (the modular transformations). On the other hand, there are theoretical descriptions of various topologically ordered states in 3+1D, which exhibit both point-like and loop-like excitations, but systematic understanding of the fundamental physical distinctions between phases, and how these distinctions are connected …
We investigate the interplay of the topological and Kondo effects in a one-dimensional Kondo-Heisenberg model with nontrivial conduction band using the density matrix renormalization group method. By analyzing the density profile, the local hybridization, and the spin/charge gap, we find that the Kondo effect can be destructed at the edges of the chain by the topological end state below a finite critical Kondo coupling $J_{K}^{c}$. We construct a phase diagram characterizing the transition of the end states.
Hall conductance of noninteracting fermions filling a certain number of Landau levels can be written as a topological invariant. A particular version of this invariant when expressed in terms of the single particle Green's functions directly generalizes to cases when interactions are present including those of fractional Hall states, although in those cases this invariant no longer corresponds to Hall conductance. We argue that when evaluated in fractional Hall states this invariant gives twice the total orbital spin of fermions which in turn is closely related to the Hall viscosity, a quantit…
We consider the coexistence of superconductivity(SC) and spin-density wave(SDW). The SC is presumed to be of $d_{x^2-y^2}+id_{xy} (d_1 + i d_2) $ type whereas the SDW order parameter is of $BCS$/$d_{x y} $ symmetry. The Hamiltonian having such a structure is shown to have topological coexistence phases in addition to the conventional one. It is shown that the amplitudes of both the order parameters determine the nature of topological phases. A phase diagram characterizing different topological phases with their Chern numbers are obtained. The experimental realization of such topological phases…
