The bulk-edge correspondence in topological phases is extended to systems with the generalized chiral symmetry, where the conventional chiral symmetry is broken. In such systems, we find that the edge state exhibits an unconventional behavior in the presence of the symmetry breaking by the mass, which is explored explicitly in the case of a deformed Su-Schrieffer-Heeger model. The localization length of the edge states diverges at a certain critical mass, where the edge state touches to the bulk band. The edge state is specified by an imaginary wave vector that becomes real at the touching energy.
Surface physics dominated by bulk properties has been one of the central interests in modern condensed matter physics, from electric polarization to bulk-boundary correspondence of topological insulators and superconductors. Here, we extend theory of electric polarization to chirality polarizations, that is, surface charges corresponding to local antisymmetries characterized as a bulk property. Using the notion of chirality polarizations, we prove the recently proposed spectral bulk-boundary correspondence, a generalization of bulk-boundary correspondence in chiral symmetric systems into compl…
We consider a 3-dimensional (3D) non-Hermitian exceptional line semimetal model and take open boundary conditions in x, y, and z directions separately. In each case, we calculate the parameter regions where the bulk-boundary correspondence is broken. The breakdown of the bulk-boundary correspondence is manifested by the deviation from unit circles of generalized Brillouin zones (GBZ) and the discrepancy between spectra calculated with open boundary conditions (OBC) and periodic boundary conditions (PBC). The consistency between OBC and PBC spectra can be recovered if the PBC spectra are calcul…
Using the Laughlin's argument on a torus with two pin-holes, we numerically demonstrate that the discontinuities of the center-of-mass work well as an invariant of the pumping phenomena during the process of the flux-attachment, trading the magnetic flux for the statistical one. This is consistent with the bulk-edge correspondence of the fractional quantum Hall effect of anyons. We also confirm that the general feature of the edge states remains unchanged during the process while the topological degeneracy is discretely changed. This supports the stability of the quantum Hall edge states in th…
It was known that for non-Hermitian topological systems due to the non-Hermitian skin effect, the bulk-edge correspondence is broken down. In this paper, by using one-dimensional Su-SchriefferHeeger model and two-dimensional (deformed) Qi-Wu-Zhang model as examples, we focus on a special type of non-Hermitian topological system without non-Hermitian skin effect-topological systems under non-Hermitian similarity transformation. In these non-Hermitian systems, the defective edge states and the breakdown of bulk-edge correspondence are discovered. To characterize the topological properties, we in…
Non-Hermitian topological systems show quite different properties as their Hermitian counterparts. An important, puzzled issue on non-Hermitian topological systems is the existence of defective edge states beyond usual bulk-boundary correspondence (BBC) that localize either on the left edge or the right edge of the one-dimensional system. In this paper, to understand the existence of the defective edge states, the theory of anomalous bulk-boundary correspondence (A-BBC) is developed that distinguishes the non-Bloch bulk-boundary correspondence (NB-BBC) from non-Hermitian skin effect. By using …
We show that the flat bands in the chiral model of magic-angle twisted bilayer graphene remain exactly flat in the presence of a perpendicular magnetic field. This is shown by an exact mapping between the model and the lowest Landau level wavefunctions at an effective magnetic field, in which the external field is either augmented or reduced by one flux quantum per unit cell. When the external field reaches one flux quantum per unit cell, the model exhibits a topological phase transition. These findings allow us to analyze a Jain-series of Fractional Chern Insulators states in the exactly flat…
Majorana zero modes are well studied in the gapped phases of topological systems. We investigate Majorana zero modes at the topological quantum criticality in one dimensional topological superconducting model with longer range interaction. We identify stable localized Majorana zero modes appearing at criticality under certain conditions. Topological invariant number for these non-trivial criticalities is obtained from zeros of a complex function associated with the Hamiltonian. Behavior of parametric curve at criticalities validate the invariant obtained and account for the appearance of Major…
We derive a bulk-boundary correspondence for three-dimensional (3D) symmetry-protected topological (SPT) phases with unitary symmetries. The correspondence consists of three equations that relate bulk properties of these phases to properties of their gapped, symmetry-preserving surfaces. Both the bulk and surface data appearing in our correspondence are defined via a procedure in which we gauge the symmetries of the system of interest and then study the braiding statistics of excitations of the resulting gauge theory. The bulk data is defined in terms of the statistics of bulk excitations, whi…
The boundary of topological phases of matter can manifest its topology nature, which leads to the so-called boundary-bulk correspondence (BBC) of topological phases. In this Letter, we construct a one-to-one correspondence between the boundary theories of fermionic SPT (fSPT) phases protected by crystalline symmetry and on-site symmetry in 2D fermionic systems, which follow the so-called crystalline equivalence principle. We dub such correspondence crystalline equivalent BBC. We illustrate this correspondence by two simple examples and as an application, we discover the topological boundary th…
Bulk-boundary correspondence, a central principle in topological matter relating bulk topological invariants to edge states, breaks down in a generic class of non-Hermitian systems that have so far eluded experimental effort. Here we theoretically predict and experimentally observe non-Hermitian bulk-boundary correspondence, a fundamental generalization of the conventional bulk-boundary correspondence, in discrete-time non-unitary quantum-walk dynamics of single photons. We experimentally demonstrate photon localizations near boundaries even in the absence of topological edge states, thus conf…
Topological edge modes are excitations that are localized at the materials' edges and yet are characterized by a topological invariant defined in the bulk. Such bulk-edge correspondence has enabled the creation of robust electronic, electromagnetic and mechanical transport properties across a wide range of systems, from cold atoms to metamaterials, active matter and geophysical flows. Recently, the advent of non-Hermitian topological systems---wherein energy is not conserved---has sparked considerable theoretical advances. In particular, novel topological phases that can only exist in non-Herm…
Vortices in chiral topological superconductors are known for trapping Majorana zero modes as a signature response to their bulk-state topologies. In this work, we establish a similar bulk-vortex correspondence for two-dimensional $C_n$-protected class D higher-order topological superconductors, in which a quantum vortex will universally trap a pair of $C_n$-protected Majorana bound states with $\hat{z}$-directional angular momenta $J_z=0$ and $J_z=n/2$, respectively. This intriguing one-to-one mapping between anomalous vortex modes and higher-order topology can be systematically derived throug…
We present a detailed microscopic study of edge excitations for n filled Landau levels. We show that the higher-level wavefunctions possess a non-trivial radial dependence that should be integrated over for properly defining the edge conformal field theory. This analysis let us clarify the role of the electron orbital spin s in the edge theory and to discuss its universality, thus providing a further instance of the bulk-boundary correspondence. We find that the values s_i for each level, i=1,...,n, parameterize a Casimir effect or chemical potential shift that could be experimentally observed…
We present a Green's function formalism to investigate the topological properties of weakly interacting one-dimensional topological insulators, including the bulk-edge correspondence and the quantum criticality near topological phase transitions, and using interacting Su-Schrieffer-Heeger model as an example. From the many-body spectral function, we find that closing of the bulk gap remains a defining feature even if the topological phase transition is driven by interactions. The existence of edge state in the presence of interactions can be captured by means of a T-matrix formalism combined w…
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…
We consider the existence of bulk chiral fermions around points of symmetry in the Brillouin zone of nonmagnetic 3D crystals with negligible spin-orbit interactions. We use group theory to show that this is possible, but only for a reduced number of space groups and points of symmetry that we tabulate. Moreover, we show that for a handful of space groups the existence of bulk chiral fermions is not only possible but unavoidable, irrespective of the concrete crystal structure. Thus our tables can be used to look for bulk chiral fermions in a specific class of systems, namely that of nonmagnetic…
We discuss the dispersionless spectrum with zero energy in the linear topological defects - vortices. The flat band emerges inside the vortex living in the bulk medium containing topologically stable Fermi points in momentum space. The boundaries of the flat band in the vortex are determined by projections of the Fermi points in bulk to the vortex axis. This bulk-vortex correspondence for flat band is similar to the bulk-surface correspondence discussed earlier in the media with topologically protected lines of zeroes. In the latter case the flat band emerges on the surface of the system, and …
We discuss (2+1)-dimensional gapless surface theories of bulk (3+1)-dimensional topological phases, such as the BF theory at level $\mathrm{K}$, and its generalization. In particular, we put these theories on a flat (2+1) dimensional torus $T^3$ parameterized by its modular parameters, and compute the partition functions obeying various twisted boundary conditions. We show the partition functions are transformed into each other under $SL(3,\mathbb{Z})$ modular transformations, and furthermore establish the bulk-boundary correspondence in (3+1) dimensions by matching the modular $\mathcal{S}$ a…
The criteria for strong correlations on surfaces of three-dimensional topological insulators are discussed. Usually, the Coulomb repulsion at such surfaces is too weak for driving a phase transition to a strongly correlated regime. I discuss a mechanism and possibilities of its experimental implementation by which the strength of the Coulomb interaction can be tuned over a wide range. In the strongly interacting regime, the surface states are gapped, even though the topological classification of the bulk band structure predicts gapless surface states.
In Hermitian systems, according to the bulk-edge correspondence interfacing two topological optical media with different bulk topological numbers implies the existence of edge states, which can trap light at the interface. However, such a general scenario can be violated when dealing with non-Hermitian systems. Here we show that interfacing two semi-infinite Hatano-Nelson chains with different bulk topological numbers can result in the existence of infinitely many edge (interface) states, however light waves cannot be rather generally trapped at the interface.
The bulk-boundary correspondence, a topic of intensive research interest over the past decades, is one of the quintessential ideas in the physics of topological quantum matter. Nevertheless, it has not been proven in all generality and has in certain scenarios even been shown to fail, depending on the boundary profiles of the terminated system. Here, we introduce bulk numbers that capture the exact number of in-gap modes, without any such subtleties in one spatial dimension. Similarly, based on these 1D bulk numbers, we define a new 2D winding number, which we call the pole winding number, tha…
The fermionic Shastry-Sutherland model has a rich phase diagram, including phases with massless Dirac fermions, a quadratic band crossing point, and a pseudospin-1 Weyl fermion. Berry phases defined by the one-dimensional momentum as a parameter are quantized into 0 or pi due to the inversion symmetry combined with the time reversal, or existence of the glide plane, which also protects the massless Dirac cones with continuous parameters. This is the symmetry protected Z2 quantization. We have further demonstrated the Z2 Berry phases generically determine the existence of edge states in various…
The bulk-boundary correspondence is an integral feature of topological analysis and the existence of boundary or interface modes offers direct insight into the topological structure of the Bloch wave function. While only the topology of the wave function has been considered relevant to boundary modes, we demonstrate that another geometric quantity, the so-called quantum distance, can also host a bulk-interface correspondence. We consider a generic class of two-dimensional flat band systems, where the flat band has a parabolic band-crossing with another dispersive band. While such flat bands ar…
We substantiate a complete picture of the "bulk-edge correspondence" conjecture for topological phases. By studying the eigenstates in the entanglement spectrum for both the ideal and realistic Coulomb ground state of the fractional quantum Hall system, it is verified that the eigenstates in the universal part of the entanglement spectrum purely lie in the Hilbert space of the edge excitations projected onto the physical Hilbert space of the subsystem itself. Hence, not only are the eigenlevels in the entanglement spectrum in one-to-one correspondence with the eigenenergies of an effective dyn…
We study bulk-boundary correspondences and related surface phenomena stabilized by the second Chern number in three-dimensional insulators driven in adiabatic cycles. Magnetic fields and disorder effects are incorporated in our analysis using operator algebraic methods. We use the connecting maps between the $K$-theories of bulk and boundary algebras as engines for the bulk-boundary correspondences. We discovered that both the exponential and the index connecting maps are relevant for the context considered here as they lead to distinct experimentally observable surface phenomena, such as pump…
We study topological phases in the hyperbolic plane using noncommutative geometry and T-duality, and show that fractional versions of the quantised indices for integer, spin and anomalous quantum Hall effects can result. Generalising models used in the Euclidean setting, a model for the bulk-boundary correspondence of fractional indices is proposed, guided by the geometry of hyperbolic boundaries.
The bulk-boundary correspondence relates quantized edge states to bulk topological invariants in topological phases of matter. In one-dimensional symmetry-protected topological systems (SPTs), quantized topological Thouless pumps directly reveal this principle and provide a sound mathematical foundation. Symmetry-protected higher-order topological phases of matter (HOSPTs) also feature a bulk-boundary correspondence, but its connection to quantized charge transport remains elusive. Here we show that quantized Thouless pumps connecting $C_4$-symmetric HOSPTs can be described by a tuple of four …
We derive a series of quantitative bulk-boundary correspondences for 3D bosonic and fermionic symmetry-protected topological (SPT) phases under the assumption that the surface is gapped, symmetric and topologically ordered, i.e., a symmetry-enriched topological (SET) state. We consider those SPT phases that are protected by the mirror symmetry and continuous symmetries that form a group of $U(1)$, $SU(2)$ or $SO(3)$. In particular, the fermionic cases correspond to a crystalline version of 3D topological insulators and topological superconductors in the famous ten-fold-way classification, with…
The Haldane phase is the prototype of symmetry protected topological (SPT) phases of spin chain systems. It can be protected by several symmetries having in common the degeneracy of the entanglement spectrum. Here we explore in depth this degeneracy for the spin-1 AKLT and bilinear-biquadratic Hamiltonians and show the emergence of a bulk-edge correspondence that relates the low energy properties of the entanglement Hamiltonian of a periodic chain and that of the physical Hamiltonian of an open chain. We find that the entanglement spectrum can be described in terms of two spins-1/2 behaving as…
An extended Bose-Hubbard model (EBHM) with three- and four-body constraints can be feasible in cold atoms in an optical lattice. A rich phase structure including various symmetry-protected topological (SPT) phases is obtained numerically with suitable parameter settings and particle filling. The SPT phase is characterized by the Berry phase as a local topological order parameter and the structure of the entanglement spectrum (ES). Based on the presence of various topological phases, separated by gapless phase boundaries, the EBHM exhibits various bosonic topological pumps, which are constructe…
We present a method for deriving bulk and edge invariants for interacting, many-body localized Floquet systems in two spatial dimensions. This method is based on a general mathematical object which we call a flow. As an application of our method, we derive bulk invariants for Floquet systems without symmetry, as well as for systems with $U(1)$ symmetry. We also derive new formulations of previously known single-particle and many-body invariants. For bosonic systems without symmetry, our invariant gives a bulk counterpart of the rational-valued GNVW index $\frac{p}{q}$ quantifying transport of …
We present the Fermi-surface map of the spin-chiral bulk states for the non-centrosymmetric semiconductor BiTeI using de Haas-van Alphen and Shubnikov-de Haas oscillations. We identify two distinct Fermi surfaces with a unique spindle-torus-type topology and the non-trivial Berry phases, confirming the spin chirality with oppositely circulating spin-texture. Near the quantum limit at high magnetic fields, we find a substantial Zeeman effect with an effective g-factor of ~ 60 for the Rashba-split Fermi surfaces. These findings provide clear evidence of strong Rashba and Zeeman coupling in the b…
Simulating Bulk Gap in Chiral Projected Entangled-Pair States
Center for Neutron Science and Technology, Guangdong Provincial Key Laboratory of Magnetoelectric Physics and Devices, School of Physics, Sun Yat-sen University（中山大学物理学院）；Department of Physics, Southern University of Science and Technology（南方科技大学物理系）；Laboratoire de Physique Théorique, Université de Toulouse, CNRS, UPS（图卢兹大学理论物理实验室）
We determine the effective field theory in (2+1)-dimensional space and time that captures the long wave length and low-energy limit of fermions hopping on a honeycomb lattice at half-filling when both a dominant intrinsic and subdominant Rashba spin-orbit couplings are present. This effective field theory for a Z_2 topological insulator (the Kane-Mele model at vanishing uniform and staggered chemical potentials) is a perturbation around a double Chern-Simons theory, with the U(1) gauge invariance associated to spin conservation explicitly broken due to the Rashba spin orbit coupling. Nonethele…
A $(2+1)$-dimensional gapped quantum many-body system can have a topologically protected energy current at its edge. The magnitude of this current is determined entirely by the temperature and the chiral central charge, a quantity associated with the effective field theory of the edge. We derive a formula for the chiral central charge that, akin to the topological entanglement entropy, is completely determined by the many-body ground state wave function in the bulk. According to our formula, nonzero chiral central charge gives rise to a topological obstruction that prevents the ground state wa…
Using holographic renormalization coupled with the Caffarelli/Silvestre\cite{caffarelli} extension theorem, we calculate the precise form of the boundary operator dual to a bulk scalar field rather than just its average value. We show that even in the presence of interactions in the bulk, the boundary operator dual to a bulk scalar field is an anti-local operator, namely the fractional Laplacian. The propagator associated with such operators is of the general power-law (fixed by the dimension of the scalar field) type indicative of the absence of particle-like excitations at the Wilson-Fisher …
We study the Loschmidt echo for quenches in open one-dimensional lattice models with symmetry protected topological phases. For quenches where dynamical quantum phase transitions do occur we find that cusps in the bulk return rate at critical times tc are associated with sudden changes in the boundary contribution. For our main example, the Su-Schrieffer-Heeger model, we show that these sudden changes are related to the periodical appearance of two eigenvalues close to zero in the dynamical Loschmidt matrix. We demonstrate, furthermore, that the structure of the Loschmidt spectrum is linked to…
