Non-Hermitian Hamiltonians are generally sensitive to boundary conditions, and their spectra and wave functions under open boundary conditions are not necessarily predicted by the Bloch band theory for periodic boundary conditions. To elucidate such a non-Bloch feature, recent works have developed a non-Bloch band theory that works even under arbitrary boundary conditions. Here, it is demonstrated that the standard non-Bloch band theory breaks down in the symplectic class, in which non-Hermitian Hamiltonians exhibit Kramers degeneracy because of reciprocity. Instead, a modified non-Bloch band theory for the symplectic class is developed in a general manner, as well as illustrative examples. This nonstandard non-Bloch band theory underlies the $\mathbb{Z}_{2}$ non-Hermitian skin effect protected by reciprocity.
We investigate novel features of three dimensional non-Hermitian Weyl semimetals, paying special attention to its unconventional bulk-boundary correspondence. We use the non-Bloch Chern numbers as the tool to obtain the topological phase diagram, which is also confirmed by the energy spectra from our numerical results. It is shown that, in sharp contrast to Hermitian systems, the conventional (Bloch) bulk-boundary correspondence breaks down in non-Hermitian topological semimetals, which is caused by the non-Hermitian skin effect. We establish the non-Bloch bulk-boundary correspondence for non-…
In conventional Hermitian systems with the open boundary condition, Bloch's theorem is perturbatively broken down, which means although the crystal momentum is not a good quantum number, the eigenstates are the superposition of several extended Bloch waves. In this paper, we show that Bloch's theorem can be non-perturbatively broken down in some Hermitian Bosonic systems. The quasiparticles of the system are the superposition of localized non-Bloch waves, which are characterized by the complex momentum whose imaginary part determines the localization properties. Our work is a Hermitian general…
Bulk-boundary correspondence is the cornerstone of topological physics. In some non-Hermitian topological system this fundamental relation is broken in the sense that the topological number calculated for the Bloch energy band under the periodic boundary condition fails to reproduce the boundary properties under the open boundary. To restore the bulk-boundary correspondence in such non-Hermitian systems a framework beyond the Bloch band theory is needed. We develop a non-Hermitian Bloch band theory based on a modified periodic boundary condition that allows a proper description of the bulk of …
A hallmark of topological band theory in periodic media is that bulk properties are not affected by boundary conditions. Remarkably, in certain non-Hermitian lattices the bulk properties are largely affected by boundaries, leading to such major effects as the non-Hermitian skin effect and violation of the bulk-boundary correspondence. Here we unveil that non-unitary discrete-time quantum walks of photons in systems involving gain and loss show rather generally non-Bloch parity-time ($\mathcal{PT}$) symmetry breaking phase transitions, and suggest a bulk probing method to detect such boundary-d…
In this paper, we review our non-Bloch band theory in one-dimensional non-Hermitian tight-binding systems. In our theory, it is shown that in non-Hermitian systems, the Brillouin zone is determined so as to reproduce continuum energy bands in a large open chain. By using simple models, we explain the concept of the non-Bloch band theory and the method to calculate the Brillouin zone. In particular, for the non-Hermitian Su-Schrieffer-Heeger model, the bulk-edge correspondence can be established between the topological invariant defined from our theory and existence of the topological edge stat…
In spatially periodic Hermitian systems, such as electronic systems in crystals, the band structure is described by the band theory in terms of the Bloch wave functions, which reproduce energy levels for large systems with open boundaries. In this paper, we establish a generalized Bloch band theory in one-dimensional spatially periodic tight-binding models. We show how to define the Brillouin zone in non-Hermitian systems. From this Brillouin zone, one can calculate continuum bands, which reproduce the band structure in an open chain. As an example, we apply our theory to the non-Hermitian Su-…
Knots have a twisted history in quantum physics. They were abandoned as failed models of atoms. Only much later was the connection between knot invariants and Wilson loops in topological quantum field theory discovered. Here we show that knots tied by the eigenenergy strings provide a complete topological classification of one-dimensional non-Hermitian (NH) Hamiltonians with separable bands. A $\mathbb{Z}_2$ knot invariant, the global biorthogonal Berry phase $Q$ as the sum of the Wilson loop eigenphases, is proved to be equal to the permutation parity of the NH bands. We show the transition b…
Bulk-boundary correspondence, connecting the bulk topology and the edge states, is an essential principle of the topological phases. However, the bulk-boundary correspondence is broken down in general non-Hermitian systems. In this paper, we construct one-dimensional non-Hermitian Su-Schrieffer-Heeger model with periodic driving that exhibits non-Hermitian skin effect: all the eigenstates are localized at the boundary of the systems, whether the bulk states or the zero and the $π$ modes. To capture the topological properties, the non-Bloch winding numbers are defined by the non-Bloch periodize…
Topological features embedded in ancient braiding and knotting arts endow significant impacts on our daily life and even cutting-edge science. Recently, fast growing efforts are invested to the braiding topology of complex Bloch bands in non-Hermitian systems. This new classification of band topology goes far beyond those established in Hermitian counterparts. Here, we present the first acoustic realization of the topological non-Hermitian Bloch braids, based on a two-band model easily accessible for realizing any desired knot structure. The non-Hermitian bands are synthesized by a simple bina…
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…
Nonsmooth Bloch Oscillations in Non-Hermitian Systems
Advanced Institute for Materials Research (WPI-AIMR), Tohoku University（东北大学先进材料研究所）；RIKEN Center for Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS), RIKEN（理化学中心交叉理论与数学科学中心）
In recent research, it has been shown that non-Hermitian systems exhibit sensitivity to boundaries, and it is caused by the non-Hermitian skin effect. In this work, we construct the non-Bloch band theory in bosonic Bogoliubov--de Gennes (BdG) systems. From our theory, we can calculate the generalized Brillouin zone and the energy spectrum in such systems with open boundary conditions in the thermodynamic limit, and we can thus discuss its non-Hermitian nature, despite Hermiticity of an original Hamiltonian. In fact, we find that the bosonic Kitaev-Majorana chain exhibits rich aspects of the no…
The relation between chiral edge modes and bulk Chern numbers of quantum Hall insulators is a paradigmatic example of bulk-boundary correspondence. We show that the chiral edge modes are not strictly tied to the Chern numbers defined by a non-Hermitian Bloch Hamiltonian. This breakdown of conventional bulk-boundary correspondence stems from the non-Bloch-wave behavior of eigenstates (non-Hermitian skin effect), which generates pronounced deviations of phase diagrams from the Bloch theory. We introduce non-Bloch Chern numbers that faithfully predict the numbers of chiral edge modes. The theory …
The bulk-boundary correspondence is among the central issues of non-Hermitian topological states. We show that a previously overlooked `non-Hermitian skin effect' necessitates redefinition of topological invariants in a generalized Brillouin zone. The resultant phase diagrams dramatically differ from the usual Bloch theory. Specifically, we obtain the phase diagram of non-Hermitian Su-Schrieffer-Heeger model, whose topological zero modes are determined by the non-Bloch winding number instead of the Bloch-Hamiltonian-based topological number. Our work settles the issue of the breakdown of conve…
We study the quench dynamics of non-Hermitian topological models with non-Hermitian skin effects. Adopting the non-Bloch band theory and projecting quench dynamics onto the generalized Brillouin zone, we find that emergent topological structures, in the form of dynamic skyrmions, exist in the generalized momentum-time domain, and are correlated with the non-Bloch topological invariants of the static Hamiltonians. The skyrmion structures anchor on the fixed points of dynamics whose existence are conditional on the coincidence of generalized Brillouin zones of the pre- and post-quench Hamiltonia…
Non-Bloch topological invariants preserve the bulk-boundary correspondence in non-Hermitian topological systems, and are a key concept in the contemporary study of non-Hermitian topology. Here we report the dynamic detection of non-Bloch topological invariants in single-photon quantum walks, revealed through the biorthogonal chiral displacement, and crosschecked with the dynamic spin textures in the generalized quasimomentum-time domain following a quantum quench. Both detection schemes are robust against symmetry-preserving disorders, and yield consistent results with theoretical predictions.…
Green's functions are fundamental quantities that determine the linear responses of physical systems. The recent developments of non-Hermitian systems, therefore, call for Green's function formulas of non-Hermitian bands. This task is complicated by the high sensitivity of energy spectrums to boundary conditions, which invalidates the straightforward generalization of Hermitian formulas. Here, based on the non-Bloch band theory, we obtain simple Green's function formulas of general one-dimensional non-Hermitian bands. Furthermore, in the large-size limit, these formulas dramatically reduce to …
We construct a two-dimensional, discrete-time quantum walk exhibiting non-Hermitian skin effects under open-boundary conditions. As a confirmation of the non-Hermitian bulk-boundary correspondence, we show that the emergence of topological edge states are consistent with Floquet winding numbers calculated using a non-Bloch band theory invoking time-dependent generalized Billouin zones. Further, the non-Bloch topological invariants associated with quasienergy bands are captured by a non-Hermitian local Chern marker in real space, defined through local biorthogonal eigen wave functions of the no…
We introduce a non-Hermitian approximation of Bloch optical equations. This approximation provides a complete description of the excitation, relaxation and decoherence dynamics of ensembles of coupled quantum systems in weak laser fields, taking into account collective effects and dephasing. In the proposed method one propagates the wave function of the system instead of a complete density matrix. Relaxation and dephasing are taken into account via automatically-adjusted time-dependent gain and decay rates. As an application, we compute the numerical wave packet solution of a time-dependent no…
The breakdown of the conventional bulk-boundary correspondence due to non-Hermitian skin effect leads to the non-Bloch bulk-boundary correspondence in the generalized Brillouin zone. Inspired by the case of the equivalence between the non-reciprocal hopping and imaginary gauge field, we propose a method to construct the topological equivalent models of the non-Hermitian dimerized lattices with the similarity transformations. The idea of the constructions is from that the imaginary magnetic flux vanishes under the open boundary condition and the period boundary spectra can be well approximated …
We solve the complex extension of the chiral Gaussian Symplectic Ensemble, defined as a Gaussian two-matrix model of chiral non-Hermitian quaternion real matrices. This leads to the appearance of Laguerre polynomials in the complex plane and we prove their orthogonality. Alternatively, a complex eigenvalue representation of this ensemble is given for general weight functions. All k-point correlation functions of complex eigenvalues are given in terms of the corresponding skew orthogonal polynomials in the complex plane for finite-N, where N is the matrix size or number of eigenvalues, respecti…
In the presence of non-Hermitian skin effect, non-Hermitian lattices generally have complex-valued eigenenergies under periodic boundary condition, but they can have non-Bloch PT symmetry and therefore completely real eigenenergies under open boundary condition. This novel PT symmetry and its breaking have been experimentally observed in one dimension. Here, we find that non-Bloch PT symmetry in two and higher dimensions exhibits drastically different behaviors compared to its one-dimensional counterpart. Whereas Bloch PT breaking and one-dimensional non-Bloch PT breaking generally have nonzer…
Rabi flopping between Bloch bands induced by a weak ac resonant field is a coherent effect involving interband transitions. Here we consider the fundamental processes of emission/absorption of quanta and Rabi oscillations in non-Hermitian two-band lattices exhibiting unbalanced non-Hermitian skin effect, and unveil an unprecedented scenario of Rabi flopping. The effective dipole moment of the transition - usually considered a bulk property - is however strongly dependent on boundary conditions, being greatly enhanced with increased Rabi frequency only when open boundaries are present. As the f…
Machine learning topological energy braiding of non-Bloch bands
School of Physics and Electronic Engineering, Jiangsu University（江苏大学物理与电子工程学院）；Jiangsu Engineering Research Center on Quantum Perception and Intelligent Detection of Agricultural Information（江苏省农业信息量子感知与智能检测工程研究中心）
This work unveils how geometric features of two-band non-Hermitian Hamiltonians can completely classify the topology of their eigenstates and energy manifolds. Our approach generalizes the Bloch sphere visualization of Hermitian systems to a ``Bloch torus'' picture for non-Hermitian systems, where a singularity ring (SR) captures the degeneracy structure of generic exceptional points. The SR picture affords convenient visualization of various symmetry constraints and reduces their topological characterization to the classification of simple intersection or winding behavior, as detailed by our …
Skin effect, where macroscopically many bulk states are aggregated towards the system boundary, is one of the most important and distinguishing phenomena in non-Hermitian quantum systems. We discuss a new aspect of this effect whereby, despite its topological origin, applying magnetic field can largely suppress it. Skin states are pushed back into the bulk and the skin topological area, which we define, is sharply reduced. As seen from exact solutions of representative models this is fundamentally rooted in the fact that the applied magnetic field restores the validity of the low-energy descri…
Josephson-Phase Reversal of Non-Bloch Andreev Propagation
College of Physics and Materials Science, Tianjin Normal University（天津师范大学物理与材料科学学院）；Interdisciplinary Center, Tianjin Normal University（天津师范大学交叉研究中心）；Center for Joint Quantum Studies & Tianjin Key Laboratory of Low Dimensional Materials Physics and Preparing Technology, Department of Physics, School of Science, Tianjin University（天津大学理学院物理系量子联合研究中心及天津市低维材料物理与制备技术重点实验室）
The topology of non-Hermitian systems is drastically shaped by the non-Hermitian skin effect, which leads to the generalized bulk-boundary correspondence and non-Bloch band theory. The essential part in formulations of bulk-boundary correspondence is a general and computable definition of topological invariants. In this paper, we introduce a construction of non-Hermitian topological invariants based directly on real-space wavefunctions, which provides a general and straightforward approach for determining non-Hermitian topology. As an illustration, we apply this formulation to several represen…
We use the idea of a Wigner surmise to compute approximate distributions of the first eigenvalue in chiral Random Matrix Theory, for both real and complex eigenvalues. Testing against known results for zero and maximal non-Hermiticity in the microscopic large-N limit we find an excellent agreement, valid for a small number of exact zero-eigenvalues. New compact expressions are derived for real eigenvalues in the orthogonal and symplectic classes, and at intermediate non-Hermiticity for the unitary and symplectic classes. Such individual Dirac eigenvalue distributions are a useful tool in Latti…
Although the non-Bloch band theory is a milestone in elaborating bulk energy bands of non-Hermitian systems under the open-boundary condition (OBC), vital issues related to multivalued functions of non-Hermitian energy bands remain unsolved. In this paper, we anatomize the bulk properties of one-dimensional multiband non-Hermitian systems under OBC. We put forward the energy-band branches (EBBs) to settle the multivalued functions of non-Hermitian energy bands, which become gapped or gapless corresponding to disconnected or connected EBBs in the complex energy plane, where the branch points an…
Asymmetric coupling amplitudes effectively create an imaginary gauge field, which induces a non-Hermitian Aharonov-Bohm (AB) effect. Nonzero imaginary magnetic flux invalidates the bulk-boundary correspondence and leads to a topological phase transition. However, the way of non-Hermiticity appearance may alter the system topology. By alternatively introducing the non-Hermiticity under symmetry to prevent nonzero imaginary magnetic flux, the bulk-boundary correspondence recovers and every bulk state becomes extended; the bulk topology of Bloch Hamiltonian predicts the (non)existence of edge sta…
The generalized Brillouin zone (GBZ), which is the core concept of the non-Bloch band theory to rebuild the bulk boundary correspondence in the non-Hermitian topology, appears as a closed loop generally. In this work, we find that even if the GBZ itself collapses into a point, the recovery of the open boundary energy spectrum by the continuum bands remains unchanged. Contrastively, if the bizarreness of the GBZ occurs, the winding number will become illness. Namely, we find that the bulk boundary correspondence can still be established whereas the GBZ has singularities from the perspective of …
In some non-Hermitian systems, the eigenstates in the bulk are localized at the boundaries of the systems. This is called the non-Hermitian skin effect, and it has been studied mostly in discrete systems. In the present work, we study the non-Hermitian skin effect in a continuous periodic model. In a one-dimensional system, we show that the localization lengths are equal for all the eigenstates. Moreover, the localization length and the eigenspectra in a large system are independent of the types of open boundary conditions. These properties are also found in a non-Hermitian photonic crystal. S…
