Search

We introduce the concept of Almost-Companion Matrix (ACM) by relaxing the non-derogatory property of the standard Companion Matrix (CM). That is, we define an ACM as a matrix whose characteristic polynomial coincides with a given monic and generally complex polynomial. The greater flexibility inherent in the ACM concept, compared to CM, allows the construction of ACMs that have convenient matrix structures satisfying desired additional conditions, compatibly with specific properties of the polynomial coefficients. We demonstrate the construction of Hermitian and Unitary ACMs starting from appropriate third degree polynomials, with implications for their use in physical-mathematical problems such as the parameterization of the Hamiltonian, density, or evolution matrix of a qutrit. We show that the ACM provides a means of identifying properties of a given polynomial and finding its roots. For example, we describe the ACM-based solution of cubic complex algebraic equations without resorting to the use of the Cardano-Dal Ferro formulas. We also show the necessary and sufficient conditions on the coefficients of a polynomial for it to represent the characteristic polynomial of a unitary ACM. The presented approach can be generalized to complex polynomials of higher degree., Comment: 26 pages

Topological phases in non-Hermitian systems have become fascinating subjects recently. In this paper, we attempt to classify topological phases in 1D interacting non-Hermitian systems. We begin with the non-Hermitian generalization of the Su-Schrieffer-Heeger (SSH) model and discuss its many-body topological Berry phase, which is well defined for all interacting quasi-Hermitian systems (non-Hermitian systems that have real energy spectrum). We then demonstrate that the classification of topological phases for quasi-Hermitian systems is exactly the same as their Hermitian counterparts. Finally, we construct the fixed point partition function for generic 1D interacting non-Hermitian local systems and find that the fixed point partition function still has a one-to-one correspondence to their Hermitian counterparts. Thus, we conclude that the classification of topological phases for generic 1D interacting non-Hermitian systems is still exactly the same as Hermitian systems., 15 pages, 5 figures

We introduce the exceptional topological insulator (ETI), a non-Hermitian topological state of matter that features exotic non-Hermitian surface states which can only exist within the three-dimensional topological bulk embedding. We show how this phase can evolve from a Weyl semimetal or Hermitian three-dimensional topological insulator close to criticality when quasiparticles acquire a finite lifetime. The ETI does not require any symmetry to be stabilized. It is characterized by a bulk energy point gap, and exhibits robust surface states that cover the bulk gap as a single sheet of complex eigenvalues or with a single exceptional point. The ETI can be induced universally in gapless solid-state systems, thereby setting a paradigm for non-Hermitian topological matter., 6+ pages, 3 figures; 21 pages Supplemental Material

We associate the stationary harmonic oscillator with time-dependent systems exhibiting non-Hermiticity by means of point transformations. The new systems are exactly solvable, with all-real spectra, and transit to the Hermitian configuration for the appropriate values of the involved parameters. We provide a concrete generalization of the Swanson oscillator that includes the Caldirola–Kanai model as a particular case. Explicit solutions are given in both the classical and quantum pictures.

There is an ambiguity in choosing field-strength renormalization factors in the $ \overline{\text{MS}} $ scheme starting from the 3-loop order in perturbation theory. More concerning, trivially choosing Hermitian factors has been shown to produce divergent renormalization group functions, which are commonly understood to be finite quantities. We demonstrate that the divergences of the RG functions are such that they vanish in the RG equation due to the Ward identity associated with the flavor symmetry. It turns out that any such divergences can be removed using the renormalization ambiguity and that the use of the flavor-improved $ \beta $-function is preferred. We show how our observations resolve the issue of divergences appearing in previous calculations of the 3-loop SM Yukawa $ \beta $-functions and provide the first calculation of the flavor-improved 3-loop SM $ \beta $-functions in the gaugeless limit., Comment: 29 pages, not a single figure v2: Typos fixed, added paragraph regarding 1PR structures in N^I

Relation between the Virasoro constraints and KP integrability (determinant formulas) for matrix models is a lasting mystery. We elaborate on the claim that the situation is improved when integrability is enhanced to super-integrability, i.e. to explicit formulas for Gaussian averages of characters. In this case, the Virasoro constraints are equivalent to simple recursive formulas, which have appropriate combinations of characters as their solutions. Moreover, one can easily separate dependence on the size of matrix, and deduce superintegrability from the Virasoro constraints. We describe one of the ways to do so for the Gaussian Hermitian matrix model. The result is a spectacularly elegant reformulation of Virasoro constraints as identities for the Schur functions evaluated at appropriate loci in the space of time-variables., Comment: 6 pages

Discrete-time quantum walks are known to exhibit exotic topological states and phases. Physical realization of quantum walks in a noisy environment may destroy these phases. We investigate the behavior of topological states in quantum walks in the presence of a lossy environment. The environmental effects in the quantum walk dynamics are addressed using the non-Hermitian Hamiltonian approach. We show that the topological phases of the quantum walks are robust against moderate losses. The topological order in one-dimensional split-step quantum walk persists as long as the Hamiltonian is $\mathcal{PT}$-symmetric. Although the topological nature persists in two-dimensional quantum walks as well, the $\mathcal{PT}$-symmetry has no role to play there. Furthermore, we observe the noise-induced topological phase transition in two-dimensional quantum walks., Comment: 15 pages, 8 figures. Comments or suggestions are most welcome

Electromagnetic metasurfaces enable the advanced control of surface-wave propagation by spatially tailoring the local surface reactance. Interestingly, tailoring the surface resistance distribution in space provides new, largely unexplored degrees of freedom. Here, we show that suitable spatial modulations of the surface resistance between positive (i.e., loss) and negative (i.e., gain) values can induce peculiar dispersion effects, far beyond a mere compensation. Taking inspiration from the parity-time symmetry concept in quantum physics, we put forward and explore a class of non-Hermitian metasurfaces that may exhibit extreme anisotropy mainly induced by the gain-loss interplay. Via analytical modeling and full-wave numerical simulations, we illustrate the associated phenomenon of surface-wave canalization, explore nonlocal effects and possible departures from the ideal conditions, and address the feasibility of the required constitutive parameters. Our results suggest intriguing possibilities to dynamically reconfigure the surface-wave propagation, and are of potential interest for applications to imaging, sensing and communications., 11 pages; 10 figures

We propose an resistors, inductors and capacitors (RLC) electrical circuit to theoretically analyze and fully simulate a new type of non-Hermitian Su-Schrieffer-Heeger (SSH) model with complex hoppings. We formulate its construction and investigate its properties by taking advantage of the circuit's versatility. Rich physical properties can be identified in this system from the normal modes of oscillation of the RLC circuit, including the highly tunable bulk-edge correspondence between topological winding numbers and edge states and the non-Hermitian skin phenomenon originating from a novel complex energy plane topology. The present study is able to show the wide and appealing topological physics inherent to electric circuits, which is readily generalizable to a plenty of both Hermitian and non-Hermitian nontrivial systems.

If $A$ is an $n \times n$ Hermitian matrix with eigenvalues $\lambda_1(A),\dots,\lambda_n(A)$ and $i,j = 1,\dots,n$, then the $j^{\mathrm{th}}$ component $v_{i,j}$ of a unit eigenvector $v_i$ associated to the eigenvalue $\lambda_i(A)$ is related to the eigenvalues $\lambda_1(M_j),\dots,\lambda_{n-1}(M_j)$ of the minor $M_j$ of $A$ formed by removing the $j^{\mathrm{th}}$ row and column by the formula $$ |v_{i,j}|^2\prod_{k=1;k\neq i}^{n}\left(\lambda_i(A)-\lambda_k(A)\right)=\prod_{k=1}^{n-1}\left(\lambda_i(A)-\lambda_k(M_j)\right)\,.$$ We refer to this identity as the \emph{eigenvector-eigenvalue identity} and show how this identity can also be used to extract the relative phases between the components of any given eigenvector. Despite the simple nature of this identity and the extremely mature state of development of linear algebra, this identity was not widely known until very recently. In this survey we describe the many times that this identity, or variants thereof, have been discovered and rediscovered in the literature (with the earliest precursor we know of appearing in 1834). We also provide a number of proofs and generalizations of the identity., Comment: 28 pages, 1 figure, phase discussion expanded, matches published version

Basic matrices are defined which provide unique building blocks for the class of normal matrices which include the classes of unitary and Hermitian matrices. Unique builders for quantum logic gates are hence derived since a quantum logic gates is represented by, or is said to be, a unitary matrix. An efficient algorithm for expressing an idempotent as a unique sum of rank $1$ idempotents with increasing initial zeros is derived. This is used to derive a unique form for mixed matrices. A number of (further) applications are given: for example (i) $U$ is a symmetric unitary matrix if and only if it has the form $I-2E$ for a symmetric idempotent $E$, (ii) a formula for the pseudo inverse in terms of basic matrices is derived. Examples for various uses are readily available., Comment: Changes and reorganisation made due to suggestions from various people

This letter is a proof of concept for quantum power flow (QPF) algorithms which underpin various unprecedentedly efficient power system analytics exploiting quantum computing. Our contributions are three-fold: 1) Establish a quantum-state-based fast decoupled model empowered by Hermitian and constant Jacobian matrices; 2) Devise an enhanced Harrow-Hassidim-Lloyd (HHL) algorithm to solve the fast decoupled QPF; 3) Further improve the HHL efficiency by parameterizing quantum phase estimation and reciprocal rotation only at the beginning stage. Promising test results validate the accuracy and efficacy of QPF and demonstrate QPF's enormous potential in the era of quantum computing.

We consider the complex eigenvalues of a Wishart type random matrix model $X=X_1 X_2^*$, where two rectangular complex Ginibre matrices $X_{1,2}$ of size $N\times (N+\nu)$ are correlated through a non-Hermiticity parameter $\tau\in[0,1]$. For general $\nu=O(N)$ and $\tau$ we obtain the global limiting density and its support, given by a shifted ellipse. It provides a non-Hermitian generalisation of the Marchenko-Pastur distribution, which is recovered at maximal correlation $X_1=X_2$ when $\tau=1$. The square root of the complex Wishart eigenvalues, corresponding to the non-zero complex eigenvalues of the Dirac matrix $\mathcal{D}=\begin{pmatrix} 0 & X_1 \\ X_2^* & 0 \end{pmatrix},$ are supported in a domain parametrised by a quartic equation. It displays a lemniscate type transition at a critical value $\tau_c,$ where the interior of the spectrum splits into two connected components. At multi-criticality we obtain the limiting local kernel given by the edge kernel of the Ginibre ensemble in squared variables. For the global statistics, we apply Frostman's equilibrium problem to the 2D Coulomb gas, whereas the local statistics follows from a saddle point analysis of the kernel of orthogonal Laguerre polynomials in the complex plane., Comment: 30 pages, 4 figures, v2: references added, typos corrected

Critical systems represent physical boundaries between different phases of matter and have been intensely studied for their universality and rich physics. Yet, with the rise of non-Hermitian studies, fundamental concepts underpinning critical systems - like band gaps and locality - are increasingly called into question. This work uncovers a new class of criticality where eigenenergies and eigenstates of non-Hermitian lattice systems jump discontinuously across a critical point in the thermodynamic limit, unlike established critical scenarios with spectrum remaining continuous across a transition. Such critical behavior, dubbed the “critical non-Hermitian skin effect”, arises whenever subsystems with dissimilar non-reciprocal accumulations are coupled, however weakly. This indicates, as elaborated with the generalized Brillouin zone approach, that the thermodynamic and zero-coupling limits are not exchangeable, and that even a large system can be qualitatively different from its thermodynamic limit. Examples with anomalous scaling behavior are presented as manifestations of the critical non-Hermitian skin effect in finite-size systems. More spectacularly, topological in-gap modes can even be induced by changing the system size. We provide an explicit proposal for detecting the critical non-Hermitian skin effect in an RLC circuit setup, which also directly carries over to established setups in non-Hermitian optics and mechanics., In non-Hermitian systems, fundamental concepts like bandgaps and locality cannot be applied as in Hermitian systems. Here, the authors introduce a class of non-Hermitian critical scenarios where the eigenstates and energies jump discontinuously across a critical point, with anomalous scaling properties

We study one-dimensional general non-Hermitian models with asymmetric long-range hopping and explore to analytically solve the systems under some specific boundary conditions. Although the introduction of long-range hopping terms prevents us from finding analytical solutions for arbitrary boundary parameters, we identify the existence of exact solutions when the boundary parameters fulfill some constraint relations, which give the specific boundary conditions. Our analytical results show that the wave functions take simple forms and are independent of hopping range, while the eigenvalue spectra display rich model-dependent structures. Particularly, we find the existence of a special point coined as pseudo-periodic boundary condition, for which the eigenvalues are the same as the periodical system when the hopping parameters fulfill certain conditions, whereas eigenstates display non-Hermitian skin effect., Comment: 7 pages, 4 figures

Non-Hermitian topological edge states have many intriguing properties, however, to date, they have mainly been discussed in terms of bulk–boundary correspondence. Here, we propose using a bulk property of diffusion coefficients for probing the topological states and exploring their dynamics. The diffusion coefficient was found to show unique features with the topological phase transitions driven by parity–time (PT)-symmetric non-Hermitian discrete-time quantum walks as well as by Hermitian ones, despite the fact that artificial boundaries are not constructed by an inhomogeneous quantum walk. For a Hermitian system, a turning point and abrupt change appears in the diffusion coefficient when the system is approaching the topological phase transition, while it remains stable in the trivial topological state. For a non-Hermitian system, except for the feature associated with the topological transition, the diffusion coefficient in the PT-symmetric-broken phase demonstrates an abrupt change with a peak structure. In addition, the Shannon entropy of the quantum walk is found to exhibit a direct correlation with the diffusion coefficient. The numerical results presented herein may open up a new avenue for studying the topological state in non-Hermitian quantum walk systems.

Synthetic crystal lattices provide ideal environments for simulating and exploring the band structure of solid-state materials in clean and controlled experimental settings. Physical realisations have, so far, dominantly focused on implementing irreversible patterning of the system, or interference techniques such as optical lattices of cold atoms. Here, we realise reprogrammable synthetic band-structure engineering in an all optical exciton-polariton lattice. We demonstrate polariton condensation into excited states of linear one-dimensional lattices, periodic rings, dimerised non-trivial topological phases, and defect modes utilising malleable optically imprinted non-Hermitian potential landscapes. The stable excited nature of the condensate lattice with strong interactions between sites results in an actively tuneable non-Hermitian analogue of the Su-Schrieffer-Heeger system., To simulate band structures of solid state materials synthetic lattices are usually generated by optical lattices or by irreversible patterning the system. Here, the authors present reconfigurable synthetic band-structures in optical exciton-polariton lattices and generate non-Hermitian topological phases.

Linear response theory lies at the heart of quantum many-body physics because it builds up connections between the dynamical response to an external probe and correlation functions at equilibrium. Here we consider the dynamical response of a Hermitian system to a non-Hermitian probe, and we develop a non-Hermitian linear response theory that can also relate this dynamical response to equilibrium properties. As an application of our theory, we consider the real-time dynamics of momentum distribution induced by one-body and two-body dissipations. We find that, for many cases, the dynamics of momentum occupation and the width of momentum distribution obey the same universal function, governed by the single-particle spectral function. We also find that, for critical state with no well-defined quasi-particles, the dynamics are slower than normal state and our theory provides a model independent way to extract the critical exponent. We apply our results to analyze recent experiment on the Bose-Hubbard model and find surprising good agreement between theory and experiment. We also propose to further verify our theory by carrying out a similar experiment on a one-dimensional Luttinger liquid., Comment: 6+3 pages, 3 figures; v2: supplementary material added

The paper continues previous works which study the behavior of second correlation function of characteristic polynomials of the special case of $n\times n$ one-dimensional Gaussian Hermitian random band matrices, when the covariance of the elements is determined by the matrix $J=(-W^2\triangle+1)^{-1}$. Applying the transfer matrix approach, we study the case when the bandwidth $W$ is proportional to the threshold $\sqrt{n}$, 21p

We compute the Hermitian sector of the relative BRST cohomology of the spacelike and timelike Liouville theories with generic real central charge $c_L$ in each case, coupled to a spacelike Coulomb gas and a generic transverse CFT. This paper is a companion of arXiv:1905.12689, and its main goal is to completely characterize the cohomology of the timelike theory with $c_L \le 1$ which was defined there. We also apply our formulas to revisit the BRST cohomology of the spacelike Liouville theory with $c_L > 1$, which includes generalized minimal gravity. We prove a no-ghost theorem for the Hermitian sector in the timelike theory and for some spacelike models., 25 pages; v2: 29 pages, minor improvements, match published version

For complex Wigner-type matrices, i.e. Hermitian random matrices with independent, not necessarily identically distributed entries above the diagonal, we show that at any cusp singularity of the limiting eigenvalue distribution the local eigenvalue statistics are universal and form a Pearcey process. Since the density of states typically exhibits only square root or cubic root cusp singularities, our work complements previous results on the bulk and edge universality and it thus completes the resolution of the Wigner-Dyson-Mehta universality conjecture for the last remaining universality type in the complex Hermitian class. Our analysis holds not only for exact cusps, but approximate cusps as well, where an extended Pearcey process emerges. As a main technical ingredient we prove an optimal local law at the cusp for both symmetry classes. This result is also used in the companion paper [arXiv:1811.04055] where the cusp universality for real symmetric Wigner-type matrices is proven., Comment: 58 pages, 2 figures. Updated introduction and references

Non-Hermitian systems characterized by suitable spatial distributions of gain and loss can exhibit "spectral singularities" in the form of zero-width resonances associated to real-frequency poles in the scattering operator. Here, we study this intriguing phenomenon in connection with cylindrical geometries, and explore possible applications to controlling and tailoring in unconventional ways the scattering response of sub-wavelength and wavelength-sized objects. Among the possible implications and applications, we illustrate the additional degrees of freedom available in the scattering-absorption-extinction tradeoff, and address the engineering of zero-forward-scattering, transverse scattering, and gain-controlled reconfigurability of the scattering pattern, also paying attention to stability issues. Our results may open up new vistas in active and reconfigurable nanophotonics platforms., 12 pages, 10 figures; some typos corrected

We study emergent behaviors of the Lohe hermitian sphere(LHS) model which is an aggregation model on ${\mathbb C}^d$. The LHS model is a complex analog of the Lohe sphere model on ${\mathbb R}^d$, and hermitian spheres are invariant sets for the LHS dynamics. For the derivation of the LHS model, we use a top-down approach, namely a reduction from a high-rank aggregation model, "$\textit{the Lohe tensor model}$." The Lohe tensor model is a first-order aggregation model on the space of tensors with the same rank and sizes, and it was first proposed by the authors in a recent work \cite{H-P}. In this work, we study how the LHS model appears as a special case of the Lohe tensor model and for the proposed model, we provide a cross-ratio like conserved quantity, a sufficient framework for the complete aggregation and a uniform $\ell^p$-stability estimate with respect to initial data.

In this manuscript, we present an explicit way to describe the violation of CP symmetry in the standard model (SM) of electroweak interactions. In such a way, complex Cabbibo-Kobayashi-Maskawa (CKM) matrices are achieved which stand for the violation of CP symmetry. At the beginning, two necessary but not sufficient conditions for yielding a complex CKM matrix are stated as criteria. Then we found an interesting condition between the real and imaginary parts of a Hermitian $3\times 3$ matrix may provide extra relations among its parameters and reduce the number of them from eighteen down to five. In previous investigations, this can be done only down to nine. With another assumption among some of those parameters, the mass-matrix pattern is further simplified so as to be diagonalized analytically and in consequence four matrices which reveal $S_N$ symmetries among or between quark generations are obtained. In some of such $S_N$-symmetric cases, the derived CKM matrices are complex which indicate that CP symmetry is violated accordingly. Taking the Jarlskog invariant as an estimate of the CPV strength, the value predicted by this model is orders stronger than the empirical value detected experimentally. However, that happens to fill partly the gap between the cosmologically observed amount of Baryon Asymmetry of the Universe (BAU) and that current Standard Model of particle physics predicts. It also proves the long suspected existence of BAU-productive eras in early universe if some fermions were indistinguishable, i.e., $S_N$-symmetric, under circumstances of extremely high temperatures., Comment: 7 pages, 0 figures

We investigate the low-energy physics of non-Hermitian quantum spin models with $PT$-symmetry. To this end we consider the one-dimensional Ising chain and the two-dimensional toric code in a non-Hermitian staggered field. For both systems dual descriptions in terms of non-Hermitian staggered Ising interactions in a conventional transverse field exist. We perform high-order series expansions about the high- and low-field limit for both systems to determine the ground-state energy per site and the one-particle gap. The one-dimensional non-Hermitian Ising chain is known to be exactly solvable. Its ground-state phase diagram consists of second-order quantum phase transitions, which can be characterized by logarithmic singularities of the second derivative of the ground-state energy and, in the symmetry-broken phase, the gap closing of the low-field gap. In contrast, the gap closing from the high-field phase is not accessible perturbatively due to the complex energy and the occurrence of exceptional lines in the high-field gap expression. For the two-dimensional toric code in a non-Hermitian staggered field we study the quantum robustness of the topologically ordered phase by the gap closing of the low-field gap. We find that the well-known second-order quantum phase transition of the toric code in a uniform field extends into a large portion of the non-Hermitian parameter space. However, the series expansions become unreliable for a dominant anti-Hermitian field. Interestingly, the analysis of the high-field gap reveals the potential presence of an intermediate region., Comment: 15 pages, 9 figures

We develop the perturbation theory of the fidelity susceptibility in biorthogonal bases for arbitrary interacting non-Hermitian many-body systems with real eigenvalues. The quantum criticality in the non-Hermitian transverse field Ising chain is investigated by the second derivative of ground-state energy and the ground-state fidelity susceptibility. We show that the system undergoes a second-order phase transition with the Ising universal class by numerically computing the critical points and the critical exponents from the finite-size scaling theory. Interestingly, our results indicate that the biorthogonal quantum phase transitions are described by the biorthogonal fidelity susceptibility instead of the conventional fidelity susceptibility., 7 pages, 4 figures

We introduce a continuous one-dimensional non-Hermitian matrix gauge potential and study its effect on dynamics of a two-component field. The model is emulated by a system of evanescently coupled nonlinear waveguides with distributed gain and losses. The considered gauge fields lead to a variety of unusual physical phenomena in both linear and nonlinear regimes. In the linear regime, the field may undergo superexponential convective amplification. A total power of an input Gaussian beam may exhibit a finite-distance blowup, which manifests itself in absolute delocalization of the beam at a finite propagation distance, where the amplitude of the field remains finite. The defocusing Kerr nonlinearity initially enhances superexponential amplification, while at larger distances it suppresses the growth of the total power. The focusing nonlinearity at small distances slows down the power growth and eventually leads to the development of the modulational instability. Complex periodic gauge fields lead to the formation of families of stable fundamental and dipole solitons., Comment: accepted for Phys. Rev. A (Letters)

We propose a notion of functor of points for noncommutative spaces, valued in categories of bimodules, and endowed with an action functional determined by a notion of hermitian structures and height functions, modeled on an interpretation of the classical functor of points as a physical sigma model. We discuss different choices of such height functions, based on different notions of volumes and traces, including one based on the Hattori-Stallings rank. We show that the height function determines a dynamical time evolution on an algebra of observables associated to our functor of points. We focus in particular the case of noncommutative arithmetic curves, where the relevant algebras are sums of matrix algebras over division algebras over number fields, and we discuss a more general notion of noncommutative arithmetic spaces in higher dimensions, where our approach suggests an interpretation of the Jones index as a height function., Comment: 43 pages, LaTeX

Eigenmode coalescence imparts remarkable properties to non-Hermitian time evolution, culminating in a purely non-Hermitian spectral degeneracy known as an exceptional point (EP). Here, we revisit time evolution around EPs, looking at both static and periodically modulated non-Hermitian Hamiltonians. We connect a Möbius group classification of two-level non-Hermitian Hamiltonians with the theory of Hill's equation, which unlocks a large class of analytical solutions. Together with the classification, this allows us to investigate the impact of the shape of the temporal modulation on the long-term dynamics of the system. In particular, we find that EP encircling does not predict the temporal class, and that the elaborate interplay between non-Hermitian and modulation instabilities is better understood through the lens of parametric resonance. Finally, we identify specific signatures of complex parametric resonance by exhibiting stability diagrams with features that cannot occur in traditional parametric resonance.

Exceptional points (EPs), at which both eigenvalues and eigenvectors coalesce, are ubiquitous and unique features of non-Hermitian systems. Second-order EPs are by far the most studied due to their abundance, requiring only the tuning of two real parameters, which is less than the three parameters needed to generically find ordinary Hermitian eigenvalue degeneracies. Higher-order EPs generically require more fine-tuning, and are thus assumed to play a much less prominent role. Here, however, we illuminate how physically relevant symmetries make higher-order EPs dramatically more abundant and conceptually richer. More saliently, third-order EPs generically require only two real tuning parameters in the presence of either a parity-time (PT) symmetry or a generalized chiral symmetry. Remarkably, we find that these different symmetries yield topologically distinct types of EPs. We illustrate our findings in simple models, and show how third-order EPs with a generic $\sim k^{1/3}$ dispersion are protected by PT symmetry, while third-order EPs with a $\sim k^{1/2}$ dispersion are protected by the chiral symmetry emerging in non-Hermitian Lieb lattice models. More generally, we identify stable, weak, and fragile aspects of symmetry-protected higher-order EPs, and tease out their concomitant phenomenology., minor typos fixed

Finding four six-dimensional mutually unbiased bases (MUBs) containing the identity matrix is a long-standing open problem in quantum information. We show that if they exist, then the $H_2$-reducible matrix in the four MUBs has exactly nine $2\times2$ Hadamard submatrices. We apply our result to exclude from the four MUBs some known CHMs, such as symmetric $H_2$-reducible matrix, the Hermitian matrix, Dita family, Bjorck's circulant matrix, and Szollosi family. Our results represent the latest progress on the existence of six-dimensional MUBs., 15 pages, be published on quantum information processing. arXiv admin note: text overlap with arXiv:1904.10181

We demonstrate that crystal defects can act as a probe of intrinsic non-Hermitian topology. In particular, in point-gapped systems with periodic boundary conditions, a pair of dislocations may induce a non-Hermitian skin effect, where an extensive number of Hamiltonian eigenstates localize at only one of the two dislocations. An example of such a phase are two-dimensional systems exhibiting weak non-Hermitian topology, which are adiabatically related to a decoupled stack of Hatano-Nelson chains. Moreover, we show that strong two-dimensional point-gap topology may also result in a dislocation response, even when there is no skin effect present with open boundary conditions. For both cases, we directly relate their bulk topology to a stable dislocation non-Hermitian skin effect. Finally, and in stark contrast to the Hermitian case, we find that gapless non-Hermitian systems hosting bulk exceptional points also give rise to a well-localized dislocation response., 6 pages, 4 figures, supplement included, accepted manuscript

In this paper, we accomplish the complete one-loop matching of the type-I seesaw model onto the Standard Model Effective Field Theory (SMEFT), by integrating out three heavy Majorana neutrinos with the functional approach. It turns out that only 31 dimension-six operators (barring flavor structures and Hermitian conjugates) in the Warsaw basis of the SMEFT can be obtained, and most of them appear at the one-loop level. The Wilson coefficients of these 31 dimension-six operators are computed up to $\mathcal{O}\left( M^{-2}\right)$ with $M$ being the mass scale of heavy Majorana neutrinos. As the effects of heavy Majorana neutrinos are encoded in the Wilson coefficients of these higher-dimensional operators, a complete one-loop matching is useful to explore the low-energy phenomenological consequences of the type-I seesaw model. In addition, the threshold corrections to the couplings in the Standard Model and to the coefficient of the dimension-five operator are also discussed., Comment: 36 pages, 2 tables, a typo in Eq. (123) corrected

We explore the physical consequences of a scenario when the standard Hermitian Nambu--Jona-Lasinio (NJL) model spontaneously develops a non-Hermitian PT-symmetric ground state via dynamical generation of an anti-Hermitian Yukawa coupling. We demonstrate the emergence of a noncompact non-Hermitian (NH) symmetry group which characterizes the NH ground state. We show that the NH group is spontaneously broken both in weak- and strong-coupling regimes. In the chiral limit at strong coupling, the NH ground state develops inhomogeneity, which breaks the translational symmetry. At weak coupling, the NH ground state is a spatially uniform state, which lies at the boundary between the PT-symmetric and PT-broken phases. Outside the chiral limit, the minimal NJL model does not possess a stable non-Hermitian ground state., 15 pages, 1 figure; v2: minor corrections, clarifications and references added; published version

The energy bands of non-Hermitian systems exhibit nontrivial topological features that arise from the complex nature of the energy spectrum. Under periodic boundary conditions (PBC), the energy spectrum describes rather generally closed loops in complex plane, characterized by integer nonzero winding numbers. Such nontrivial winding provides the topological signature of the non-Hermitian skin effect (NHSE), i.e. the macroscopic condensation of bulk states at the lattice edges under open boundary conditions (OBC). In spite of the great relevance of band winding in the non-Hermitian topological band theory and the related NHSE, most of current results rely on tight-binding models of non-Hermitian systems, while exact Bloch wave function analysis of the NHSE and related topological band theory is still lacking. While tight-binding models can correctly describe narrow-band electronic states with a relatively weak degree non-Hermiticity, they are not suited to describe high-energy wide-band electronic states and/or regimes corresponding to strong non-Hermiticity. Here we consider the single-particle continuous Schr\"odinger equation in a periodic potential, in which non-Hermiticity is introduced by an imaginary vector potential in the equation, and show that the NHSE is ubiquitous under OBC and characterized by a non-vanishing integer winding number, even thought the energy spectrum under PBC always comprises an open curve, corresponding to high-energy electronic states. We also show that the interior of the PBC energy spectrum corresponds to the complex eigenenergies sustaining localized (edge) states under semi-infinite boundary conditions., Comment: 12 pages, 8 figures, to appear in Phys. Rev. B

The simple update (SU) and full update (FU) are the two paradigmatic time evolution algorithms for a tensor network known as the infinite projected entangled pair state (iPEPS). They differ by an error measure that is either, respectively, local or takes into account full infinite tensor environment. In this paper we test an intermediate neighborhood tensor update (NTU) accounting for the nearest neighbor environment. This small environment can be contracted exactly in a parallelizable way. It provides an error measure that is Hermitian and non-negative down to machine precision. In the 2D quantum Ising model NTU is shown to yield stable unitary time evolution following a sudden quench. It also yields accurate thermal states despite correlation lengths that reach up to 20 lattice sites. The latter simulations were performed with a manifestly Hermitian purification of a thermal state. Both were performed with reduced tensors that do not include physical (and ancilla) indices. This modification naturally leads to two other schemes: a local SVD update (SVDU) and a full tensor update (FTU) being a variant of FU., Comment: 11 pages, 12 figures

Non-Hermitian skin effect exhibits the collapse of the extended bulk modes into the extensive number of localized boundary states in open boundary conditions. Here we demonstrate the disorder-driven phase transition of the trivial non-Hermitian system to the higher-order non-Hermitian skin effect phase. In contrast to the clean systems, the disorder-induced boundary modes form an arc in the complex energy plane, which is the manifestation of the disorder-driven dynamical phase transition. At the phase transition, the localized corner modes and bulk modes characterized by trivial Hamiltonian coexist within the single-band but are separated in the complex energy plane. This behavior is analogous to the mobility edge phenomena in the disordered Hermitian systems. Using effective medium theory and numerical diagonalizations, we provide a systematic characterization of the disorder-driven phase transitions., 4+7 pages

The geometry of Hamiltonian's eigenstates is encoded in the quantum geometric tensor (QGT). It contains both the Berry curvature, central to the description of topological matter and the quantum metric. So far the full QGT has been measured only in Hermitian systems, where the role of the quantum metric is mostly shown to determine corrections to physical effects. On the contrary, in non-Hermitian systems, and in particular near exceptional points, the quantum metric is expected to diverge and to often play a dominant role, for example on the enhanced sensing and on wave packet dynamics. In this work, we report the first experimental measurement of the quantum metric in a non-Hermitian system. The specific platform under study is an organic microcavity with exciton-polariton eigenstates, which demonstrate exceptional points. We measure the quantum metric's divergence and we determine the scaling exponent $n=-1.01\pm0.08$, which is in agreement with theoretical predictions for the second-order exceptional points.

The effects of gain and loss on the band structures of a bulk topological dielectric photonic crystal (PC) with $C_{6v}$ symmetry and the PC-air-PC interface are studied based on first-principle calculation. To illustrate the importance of parity-time (PT) symmetry, three systems are considered, namely the PT-symmetric, PT-asymmetric, and lossy systems. We find that the system with gain and loss distributed in a PT symmetric manner exhibits a phase transition from a PT exact phase to a PT broken phase as the strength of the gain and loss increases, while for the PT-asymmetric and lossy systems, no such phase transition occurs. Furthermore, based on the Wilson loop calculation, the topology of the PT-symmetric system in the PT exact phase is demonstrated to keep unchanged as the Hermitian system. At last, different kinds of edge states in Hermitian systems under the influences of gain and loss are studied and we find that while the eigenfrequencies of nontrivial edge states become complex conjugate pairs, they keep real for the trivial defect states., 6 pages, 6 figures

In this work we study topological properties of the one-dimensional Creutz ladder model with different non-Hermitian asymmetric hoppings and on-site imaginary potentials, and obtain phase diagrams regarding the presence and absence of an energy gap and in-gap edge modes. The non-Hermitian skin effect (NHSE), which is known to break the bulk-boundary correspondence (BBC), emerges in the system only when the non-Hermiticity induces certain unbalanced non-reciprocity along the ladder. The topological properties of the model are found to be more sophisticated than that of its Hermitian counterpart, whether with or without the NHSE. In one scenario without the NHSE, the topological winding is found to exist in a two-dimensional plane embedded in a four-dimensional space of the complex Hamiltonian vector. The NHSE itself also possesses some unusual behaviors in this system, including a high spectral winding without the presence of long-range hoppings, and a competition between two types of the NHSE, with the same and opposite inverse localization lengths for the two bands, respectively. Furthermore, it is found that the NHSE in this model does not always break the conventional BBC, which is also associated with whether the band gap closes at exceptional points under the periodic boundary condition., 15 pages, 15 figures, 1 table

In this paper, we study connections on hermitian modules, and show that metric connections exist on regular hermitian modules; i.e. finitely generated projective modules together with a non-singular hermitian form. In addition, we develop an index calculus for such modules, and provide a characterization in terms of the existence of a pseudo-inverse of the matrix representing the hermitian form with respect to a set of generators. As a first illustration of the above concepts, we find metric connections on the fuzzy sphere. Finally, the framework is applied to a class of noncommutative minimal surfaces, for which there is a natural concept of torsion, and we prove that there exist metric and torsion free connections for every minimal surface in this class.

We analyze the well-known equivalence between the quadratic Kontsevich-Penner and Hermitian matrix models from the point of view of superintegrability relations, i.e. explicit formulas for character averages. This is not that trivial on the Kontsevich side and seems important for further studies of various deformations of Kontsevich models. In particular, the Brezin-Hikami extension of the above equivalence becomes straightforward., 4 pages

Knot matrix models are defined so that the averages of characters are equal to knot polynomials. From this definition one can extract single trace averages and generation functions for them in the group rank – which generalize the celebrated Harer-Zagier formulas for Hermitian matrix model. We describe the outcome of this program for HOMFLY-PT polynomials of various knots. In particular, we claim that the Harer-Zagier formulas for torus knots factorize nicely, but this does not happen for other knots. This fact is mysteriously parallel to existence of explicit β=1 eigenvalue model construction for torus knots only, and can be responsible for problems with construction of a similar model for other knots.

A striking feature of non-Hermitian tight-binding Hamiltonians is the high sensitivity of both spectrum and eigenstates to boundary conditions. Indeed, if the spectrum under periodic boundary conditions is point gapped, by opening the lattice the non-Hermitian skin effect will necessarily occur. Finding the exact skin eigenstates may be demanding in general, and many methods in the literature are based on ansatzes and on recurrence equations for the eigenstates' components. Here we devise a general procedure based on the Green's function method to calculate the eigenstates of non-Hermitian tight-binding Hamiltonians under open boundary conditions. We apply it to the Hatano-Nelson and non-Hermitian SSH models and finally we contrast the edge states localization with that of bulk states., Comment: 8 pages, 4 figures

Time-periodic driving fields could endow a system with peculiar topological and transport features. In this work, we find dynamically controlled localization transitions and mobility edges in non-Hermitian quasicrystals via shaking the lattice periodically. The driving force dresses the hopping amplitudes between lattice sites, yielding alternate transitions between localized, mobility edge and extended non-Hermitian quasicrystalline phases. We apply our Floquet engineering approach to five representative models of non-Hermitian quasicrystals, obtain the conditions of photon-assisted localization transitions and mobility edges, and find the expressions of Lyapunov exponents for some models. We further introduce topological winding numbers of Floquet quasienergies to distinguish non-Hermitian quasicrystalline phases with different localization nature. Our discovery thus extend the study of quasicrystals to non-Hermitian Floquet systems, and provide an efficient way of modulating the topological and transport properties of these unique phases., 17 pages, 7 figures, 6 tables, version accepted by Phys. Rev. Res

Non-Hermitian topological phases exhibit a number of exotic features that have no Hermitian counterparts, including the skin effect and breakdown of the conventional bulk-boundary correspondence. Here, we implement the non-Hermitian Su-Schrieffer-Heeger (SSH) Hamiltonian, which is a prototypical model for studying non-Hermitian topological phases, with a solid-state quantum simulator consisting of an electron spin and a $^{13}$C nuclear spin in a nitrogen-vacancy (NV) center in a diamond. By employing a dilation method, we realize the desired non-unitary dynamics for the electron spin and map out its spin texture in the momentum space, from which the corresponding topological invariant can be obtained directly. Our result paves the way for further exploiting and understanding the intriguing properties of non-Hermitian topological phases with solid-state spins or other quantum simulation platforms., 10 pages, 6 figures

We propose and explore a scheme that leads to an infinite series of time- dependent Dyson maps which associate different Hermitian Hamiltonians to a uniquely specified time-dependent non-Hermitian Hamiltonian. We identify the underlying sym- metries responsible for this feature respected by various Lewis-Riesenfeld invariants. The latter are used to facilitate the explicit construction of the Dyson maps and metric oper- ators. As a concrete example for which the scheme is worked out in detail we present a two-dimensional system of oscillators that are coupled to each other in a non-Hermitian PT -symmetrical fashion, 16 pages

We derive the planar limit of 2- and 3-point functions of single-trace chiral primary operators of ${\cal N}=2$ SQCD on $S^4$, to all orders in the 't Hooft coupling. In order to do so, we first obtain a combinatorial expression for the planar free energy of a hermitian matrix model with an infinite number of arbitrary single and double trace terms in the potential; this solution might have applications in many other contexts. We then use these results to evaluate the analogous planar correlation functions on ${\mathbb R}^4$. Specifically, we compute all the terms with a single value of the $\zeta$ function for a few planar 2- and 3-point functions, and conjecture general formulas for these terms for all 2- and 3-point functions on ${\mathbb R}^4$., Comment: 25 pages. v2 : added reference

Unconventional Weyl points with topological charges higher than 1 can transform into various complex unconventional Weyl exceptional contours under non-Hermitian perturbations. However, theoretical studies of these exceptional contours have been limited to tight-binding models. Here, we propose to realize unconventional Weyl exceptional contours in photonic continua -- non-Hermitian anisotropic chiral plasma, based on ab initio calculation by Maxwell's equations. By perturbing in-plane permittivity, an unconventional Weyl point can transform into a quadratic Weyl exceptional circle, a Type-I Weyl exceptional chain with one chain point, a Type-II Weyl exceptional chain with two chain points, or other forms. Realistic metamaterials with effective constitutive parameters are proposed to implement these unconventional Weyl exceptional contours. Our work paves a way toward exploration of exotic physics of unconventional Weyl exceptional contours in non-Hermitian topological photonic continua., 12 pages, 4 figures, and 1 table

The random variable $1+z_1+z_1z_2+\dots$ appears in many contexts and was shown by Kesten to exhibit a heavy tail distribution. We consider natural extensions of this variable and its associated recursion to $N \times N$ matrices either real symmetric $\beta=1$ or complex Hermitian $\beta=2$. In the continuum limit of this recursion, we show that the matrix distribution converges to the inverse-Wishart ensemble of random matrices. The full dynamics is solved using a mapping to $N$ fermions in a Morse potential, which are non-interacting for $\beta=2$. At finite $N$ the distribution of eigenvalues exhibits heavy tails, generalizing Kesten's results in the scalar case. The density of fermions in this potential is studied for large $N$, and the power-law tail of the eigenvalue distribution is related to the properties of the so-called determinantal Bessel process which describes the hard edge universality of random matrices. For the discrete matrix recursion, using free probability in the large $N$ limit, we obtain a self-consistent equation for the stationary distribution. The relation of our results to recent works of Rider and Valk\'o, Grabsch and Texier, as well as Ossipov, is discussed., Comment: 44 pages, 5 figures, 6 appendices