We study the properties of the entanglement spectrum in gapped non-interacting non-Hermitian systems, and its relation to the topological properties of the system Hamiltonian. Two different families of entanglement Hamiltonians can be defined in non-Hermitian systems, depending on whether we consider only right (or equivalently only left) eigenstates or a combination of both left and right eigenstates. We show that their entanglement spectra can still be computed efficiently, as in the Hermitian limit. We discuss how symmetries of the Hamiltonian map into symmetries of the entanglement spectrum depending on the choice of the many-body state. Through several examples in one and two dimensions, we show that the biorthogonal entanglement Hamiltonian directly inherits the topological properties of the Hamiltonian for line gapped phases, with characteristic singular and energy zero modes. The right (left) density matrix carries distinct information on the topological properties of the many-body right (left) eigenstates themselves. In purely point gapped phases, when the energy bands are not separable, the relation between the entanglement Hamiltonian and the system Hamiltonian breaks down.
We study non-Hermitian spatial symmetries -- a class of symmetries that have no counterparts in Hermitian systems -- and study how normal and exceptional semimetals can be stabilized by these symmetries. Different from internal ones, spatial symmetries act nonlocally in momentum space and enforce global constraints on both band degeneracies and topological quantities at different locations. In deriving general constraints on band degeneracies and topological invariants, we demonstrate that non-Hermitian spatial symmetries are on an equal footing with, but are essentially different from Hermiti…
We study interaction-induced Mott insulators, and their topological properties in a 1D non-Hermitian strongly-correlated spinful fermionic superlattice system with either nonreciprocal hopping or complex-valued interaction. For the nonreciprocal hopping case, the low-energy neutral excitation spectrum is sensitive to boundary conditions, which is a manifestation of the non-Hermitian skin effect. However, unlike the single-particle case, particle density of strongly correlated system does not suffer from the non-Hermitian skin effect due to the Pauli exclusion principle and repulsive interactio…
We provide a systematic treatment of the tenfold way of classifying fermionic systems that naturally allows for the study of those with arbitrary $N$-body interactions. We identify four types of symmetries that such systems can possess, which consist of one ordinary type (usual unitary symmetries), and three non-ordinary symmetries (such as time reversal, charge conjugation and sublattice). Focusing on systems that possess no non-trivial ordinary symmetries, we demonstrate that the non-ordinary symmetries are strongly constrained. This approach not only leads very naturally to the tenfold clas…
In previous studies, the topological invariants of 1D non-Hermitian systems have been defined in open boundary condition (OBC) to satisfy the bulk-boundary correspondence. The extreme sensitivity of bulk energy spectra to boundary conditions has been attributed to the breakdown of the conventional bulk-boundary correspondence based on the topological invariants defined under periodic boundary condition (PBC). Here we propose non-Hermitian many-body polarization as a topological invariant for 1D non-Hermitian systems defined in PBC, which satisfies the bulk-boundary correspondence. Employing ma…
Non-Hermitian degeneracies are classified as defective and nondefective exceptional points~(EP). While in defective EPs, both eigenvalues and eigenenergies coalesce, nondefective EPs are characterized merely by the occurrence of degeneracies in their eigenvalues. It is also known that all degeneracies are either symmetry-protected or accidental. In this paper, I prove that anti-unitary symmetries protect all nondefective twofold EPs. By developing a 2D non-Hermitian tight-binding model, I have demonstrated that these symmetries are composite of various symmetry operations such as discrete or s…
We study a topological band degeneracy in non-Hermitian systems with parity-time ($PT$) and parity-particle-hole ($CP$) symmetries. In $d$-dimensional non-Hermitian systems, it is shown that $(d-1)$-dimensional exceptional surfaces can appear from band touching thanks to $PT$ or $CP$ symmetry. We investigate the topological stability and zero-gap quasiparticles for the exceptional surfaces due to the band degeneracy. We also demonstrate the band degeneracy by using lattice models of a topological semimetal and a superconductor.
Classifications of symmetry-protected topological (SPT) phases provide a framework to systematically understand the physical properties and potential applications of topological systems. While such classifications have been widely explored in the context of Hermitian systems, a complete understanding of the roles of more general non-Hermitian symmetries and their associated classification is still lacking. Here, we derive a periodic table for non-interacting SPTs with general non-Hermitian symmetries. Our analysis reveals novel non-Hermitian topological classes, while also naturally incorporat…
Topological phases are enriched in non-equilibrium open systems effectively described by non-Hermitian Hamiltonians. While several properties unique to non-Hermitian topological systems were uncovered, the fundamental role of symmetry in non-Hermitian physics has yet to be fully understood, and it has remained unclear how symmetry protects non-Hermitian topological phases. Here we show that two fundamental anti-unitary symmetries, time-reversal and particle-hole symmetries, are topologically equivalent in the complex energy plane and hence unified in non-Hermitian physics. A striking consequen…
Hawking Radiation in non-Hermitian Microscopic Analogues
Institute for Theoretical Physics, Utrecht University（乌得勒支大学理论物理研究所）；Utrecht University（乌得勒支大学）；Department of Physics and Astronomy, Uppsala University（乌普萨拉大学物理与天文系）；Uppsala University（乌普萨拉大学）
Symmetry plays fundamental role in physics and the nature of symmetry changes in non-Hermitian physics. Here the symmetry-protected scattering in non-Hermitian linear systems is investigated by employing the discrete symmetries that classify the random matrices. The even-parity symmetries impose strict constraints on the scattering coefficients: the time-reversal (C and K) symmetries protect the symmetric transmission or reflection; the pseudo-Hermiticity (Q symmetry) or the inversion (P) symmetry protects the symmetric transmission and reflection. For the inversion-combined time-reversal symm…
Recently, topological phases in non-Hermitian systems have attracted much attention because non-Hermiticity sometimes gives rise to unique phases with no Hermitian counterparts. Non-Hermitian Bloch Hamiltonians can always be mapped to doubled Hermitianized Hamiltonians with chiral symmetry, which enables us to utilize the existing framework for Hermitian systems into the classification of non-Hermitian topological phases. While this strategy succeeded in the topological classification of non-Hermitian Bloch Hamiltonians in the presence of internal symmetries, the generalization of symmetry ind…
In free fermion systems with given symmetry and dimension, the possible topological phases are labeled by elements of only three types of Abelian groups, Z_1, Z_2, or Z. For example non-interacting 1D fermionic superconducting phases with S_z spin rotation and time-reversal symmetries are classified by Z. We show that with weak interactions, this classification reduces to Z_4. Using group cohomology, one can additionally show that there are only four distinct phases for such 1D superconductors even with strong interactions. Comparing their projective representations, we find all these four sym…
We propose an ultracold-atom setting where a fermionic superfluidity with attractive s-wave interaction is uploaded in a non-Hermitian Lieb optical lattice. The existence of a real-energy flat band solution is revealed. We show that the interplay between the skin effect and flat-band localization leads to exotic localization properties. We develop a multiband mean-field description of this system and use both order parameters and superfluid weight to describe the phase transition. A relation between the superfluid weight and non-Hermitian quantum metric of the quantum states manifold is built.…
Constraints of internal symmetry on the non-Hermitian skin effect and bidirectional skin effect under the action of the Hermitian conjugate of time-reversal symmetry
We study the characterization and realization of higher-order topological Anderson insulator (HOTAI) in non-Hermitian systems, where the non-Hermitian mechanism ensures extra symmetries as well as gain and loss disorder.We illuminate that the quadrupole moment $Q_{xy}$ can be used as the real space topological invariant of non-Hermitian higher-order topological insulator (HOTI). Based on the biorthogonal bases and non-Hermitian symmetries, we prove that $Q_{xy}$ can be quantized to $0$ or $0.5$. Considering the disorder effect, we find the disorder-induced phase transition from normal insulato…
Symmetry Protected Topological (SPT) phases are a minimal generalization of the concept of topological insulators to interacting systems. In this paper we describe the classification and properties of such phases for three dimensional(3D) electronic systems with a number of different symmetries. For symmetries representative of all classes in the famous 10-fold way of free fermion topological insulators/superconductors, we determine the stability to interactions. By combining with results on bosonic SPT phases we obtain a classification of electronic 3D SPT phases for these symmetries. In case…
We analyze the Chiral Magnetic Effect for non-Hermitian fermionic systems using the biorthogonal formulation of quantum mechanics. In contrast to the Hermitian chiral counterparts, we show that the Chiral Magnetic Effect may take place in thermal equilibrium of an open non-Hermitian system with, generally, massive fermions. The key observation is that for non-Hermitian charged systems, there is no strict charge conservation as understood in the Hermitian case, so the Bloch theorem preventing currents in the thermodynamic limit in equilibrium does not apply.
We describe a diagrammatic technique for non-Hermitian fermionic systems that is applicable in the steady state, and which allows addressing correlations effects by systematic expansion. Applying this method to exceptional points or rings, we find that nodal objects in non-Hermitian systems are generically displaced in momentum-space due to interactions. This in turn can be connected to the fact that exceptional points invariably break a class of orthonormal symmetries that are generally present for nodal points in Hermitian systems, and which protect the integrity of the node at low energy sc…
We employ the fermionic and bosonic replicated nonlinear sigma models to treat Ginibre unitary, symplectic, and orthogonal ensembles of non-Hermitian random matrix Hamiltonians. Using saddle point approach combined with Borel resummation procedure we derive the exact large-N results for microscopic density of states in all three ensembles. We also obtain tails of the density of states as well the two-point function for the unitary ensemble.
We classify topological defects in non-Hermitian systems with point gap, real gap and imaginary gap for all the Bernard-LeClair classes or generalized Bernard-LeClair classes in all dimensions. The defect Hamiltonian $H(\bf{k}, {\bf r})$ is described by a non-Hermitian Hamiltonian with spatially modulated adiabatical parameter ${\bf r}$ surrounding the defect. While the non-Hermitian system with point gap belongs to any of 38 symmetry classes (Bernard-LeClair classes), for non-Hermitian systems with line-like gap we get 54 non-equivalent generalized Bernard-LeClair classes as a natural general…
In non-Hermitian systems, the defective band degeneracies, so-called exceptional points (EPs), can form robust exceptional lines (ELs) in 3D momentum space in the absence of any symmetries. Here, we show that a natural orientation can be assigned to every EL according to the eigenenergy braiding around it, and prove the source-free principle of ELs as a corollary of the generalized Fermion doubling theorem for EPs on an arbitrary closed oriented surface, which indicates that if several ELs flow into a junction, the same number of outflow ELs from the junction must exist. Based on this principl…
We introduce a one dimensional non-Hermitian four band tight binding lattice system. We find stable topological edge states protected by particle-hole and parity-time symmetries. We show that topological phase appears in the system. We discuss that the system can also be used as a topological laser if the gain and loss positions in a unit cell are judiciously arranged.
We relate topological properties of non-Hermitian systems and observables of quantum open systems by using the Keldysh path-integral method. We express Keldysh Green's functions in terms of effective non-Hermitian Hamiltonians that contain all the relevant topological information. We arrive at a frequency dependent topological index that is linked to the response of the system to perturbations at a given frequency. We show how to detect a transition between different topological phases by measuring the response to local perturbations. Our formalism is exemplified in a 1D Hatano-Nelson model, h…
Symmetries associated with complex conjugation and Hermitian conjugation, such as time-reversal symmetry and pseudo-Hermiticity, have great impact on eigenvalue spectra of non-Hermitian random matrices. Here, we show that time-reversal symmetry and pseudo-Hermiticity lead to universal level statistics of non-Hermitian random matrices on and around the real axis. From the extensive numerical calculations of large random matrices, we obtain the five universal level-spacing and level-spacing-ratio distributions of real eigenvalues, each of which is unique to the symmetry class. Furthermore, we an…
In this paper, with the help of Hermitian/non-Hermitian dynamic strings, the theory of non-Hermitian topological order is developed based on a non-Hermitian Wen-plaquette model. The effective models for bosonic topological excitations (e-particle and m-particle) are Hermitian tight-binding lattice model; the effective model for fermionic topological excitation (f-particle) becomes a non-Hermitian tight-binding lattice model. In addition, the effective pseudo-spin model for topologically degenerate ground states is derived by calculating the expectation values of Hermitian/non-Hermitian topolog…
The topological classification of hermitian operators is solely determined by the presence or absence of certain discrete symmetries. For non-hermitian operators we in addition need to specify the type of spectral gap. They come in the flavor of a point gap or a line gap. Since the presence of a line gap implies the existence of a point gap, there is usually more than one mathematical classification applicable to a physical system. That raises the question: which of these gap-type classifications is physically meaningful? To decide this question, I propose a simple criterion, namely the choice…
We present a new class of hermitian one-matrix models originated in the W-infinity algebra: more precisely, the polynomials defining the W-infinity generators in their fermionic bilinear form are shown to expand the orthogonal basis of a class of random hermitian matrix models. The corresponding potentials are given, and the thermodynamic limit interpreted in terms of a simple plasma picture. The new matrix models can be successfully applied to the full bosonization of interesting one-dimensional systems, including all the perturbative orders in the inverse size of the system. As a simple appl…
