**Ali Mostafazadeh**, (Koç University, Turkey), **07/05, 14:00 GMT**

Exactness of the Born approximation, broadband unidirectional invisibility, and implicit regularization of point scatterers in two dimensions

Abstract: Born approximation was proposed by Max Born in his monumental 1926 paper where he laid the foundations of quantum scattering theory. Because of its great theoretical and practical significance, this approximation scheme is covered in most standard textbooks on quantum mechanics. None of these, however, addresses the problem of finding potentials whose scattering problem is exactly solvable by the Born approximation. We give a simple condition under which a potential has this property in two dimensions. This in turn allows us to solve another important open problem of scattering theory, namely constructing potentials that enjoy perfect (non-approximate) unidirectional invisibility in a finite spectral band of arbitrary width. Our analysis provides the first examples of quasi-exactly solvable scattering potentials, i.e., potentials whose scattering problem is exactly solvable for energies not exceeding a given value. These results are particular outcomes of a recent dynamical formulation of time-independent scattering theory. We use this formulation to offer an explicit solution of the scattering problem for a delta-function potential in two dimensions. Our approach involves no divergent terms and reproduces the known expression for the scattering amplitude that is obtained by a regularization of the delta-function potential followed by a coupling constant renormalization. The implicit regularization property of our approach avoids such complications.

References

- F. Loran and A. Mostafazadeh, “
*Exact Solution of the Two-Dimensional Scattering Problem for a Class of δ-Function Potentials Supported on Subsets of a Line*,”J. Phys. A: Math. Theor.**51**, 335302 (2018); arXiv: 1708.06003. - F. Loran and A. Mostafazadeh, “
*Exactness of the Born Approximation and Broadband Unidirectional Invisibility in Two Dimensions*,”Phys. Rev. A**100**, 053846 (2019); arXiv: 1904.07737.

**Nick Mavromatos, (King’s College, Uni. of London, UK)**, **14/05, 14:00 GMT**

Non-Hermitian Yukawa Interactions: Consistency, Dynamical Mass Generation issues and Physics Motivation

Abstract: I discuss a fermion-(pseudo)scalar field theory model in (3+1)-dimensions, with anti hermitian Yukawa interactions, and demonstrate its consistency as a unitary quantum field theory. The model may be understood within a wider PT-symmetry framework, in the sense of identifying conditions for real energy eigenvalues of the corresponding Hamiltonian operators. Due to energetics, it is not possible to have dynamical mass generation for the fermions unless suitable attractive four fermion interactions are included, with sufficiently strong couplings. In the latter case, we estimate the dynamically generated masses using an appropriate Schwinger-Dyson treatment, for weak Yukawa couplings. Such models, with pseudoscalar-fermion Yukawa interactions, find partial motivation in neutrino physics, as providers of novel scenarios for Majorana neutrino mass generation, which could be of phenomenological interest.

**Phillip Mannheim, **(University of Connecticut, US), **21/05, 14:00 GMT**

Ghost Problems from Pauli-Villars to Fourth-Order Quantum Gravity and their Resolution

Abstract: We review the history of the ghost problem in quantum field theory from the Pauli-Villars regulator theory to currently popular fourth-order derivative quantum gravity theories. While these theories all appear to have unitarity-violating ghost states with negative norm, we show that in fact these ghost states only appear because the theories are being formulated in the wrong Hilbert space. In these theories the Hamiltonians are not Hermitian but instead possess an antilinear symmetry. Consequently, one cannot use inner products that are built out of states and their Hermitian conjugates. Rather, one must use inner products built out of states and their conjugates with respect to the antilinear symmetry, and these latter inner products are positive. In this way one can build quantum theories of gravity in four spacetime dimensions that are unitary.

**Fabio Bagarallo, **(University of Palermo, Italy), **04/06, 14:00 GMT**

Weak pseudo-bosons

Abstract: We show how the notion of pseudo-bosons, originally introduced as operators acting on some Hilbert space, can be extended to a distributional settings. In doing so, we are able to construct a rather general framework to deal with generalized eigenvectors of the multiplication and of the derivation operators. Connections with the quantum damped harmonic oscillator are also briefly discussed.

This talk is based on the following papers:

- F. Bagarello, Weak pseudo-bosons, J. Phys. A, 53, 135201 (2020)
- F. Bagarello, F. Gargano, F. Roccati, A no-go result for the quantum damped harmonic oscillator, Phys. Lett. A, 383, 2836-2838 (2019)
- F. Bagarello, F. Gargano, F. Roccati, Some remarks on few recent results on the damped quantum harmonic oscillator, Ann. of Phys., 414, 168091, (2020)

**Dorje Brody, **(University of Surrey, UK), **18/06, 14:00 GMT**

Evolution Speed of Open Quantum Dynamics

Abstract: Understanding the speed of the evolution of a quantum state is of interest for a variety of reasons in quantum information science. As well as being of interest in its own right, in implementing quantum algorithms for establishing communication and performing computation, the evolution speed determines how fast a given task can be processed. The speed also determines the sensitivity of quantum states against time evolution, and this information can be used to determine error bounds on quantum state estimation. In this talk I will explain how to derive a formula for the speed of evolution of a general open-system quantum dynamics. Of particular interest lies in PT-symmetric open quantum systems that exhibit PT phase transitions. It will be shown that the time dependence of the evolution speed (or the acceleration of the state) exhibits radically different behaviours in the two phases.

The talk is based on joint work with B. Longstaff.

**Savannah Garmon, **(Osaka Prefecture University, Japan), **02/07, 14:00 GMT**

Anomalous exceptional point and non-Markovian Purcell effect at threshold in 1-D continuum systems

Abstract: For a system consisting of a quantum emitter coupled near threshold to a 1-D continuum with a van Hove singularity in the density of states, we demonstrate general conditions such that a characteristic triple level convergence occurs directly on the threshold as the coupling g is shut off. For small g values the eigenvalue and norm of each of these states can be expanded in a Puiseux expansion in terms of powers of g^{2/3}, which suggests the influence of a third order exceptional point. However, in the actual g→0 limit, only two discrete states in fact coalesce as the system can be reduced to a 2 x 2 Jordan block; the third state instead merges with the continuum. The decay width of the resonance state involved in this convergence is proportional to g^{4/3}, significantly enhanced compared to the usual Fermi golden rule. However, non-Markovian dynamics due to the branch point effect are significantly enhanced near the threshold as well. Applying a perturbative analysis in terms of the Puiseux expansion that also takes into account the threshold influence, we show that the relaxation process of the quantum emitter actually results in an unusual 1 – C t^{3/2} decay on the key timescale during which most of the decay occurs.

Relevant background:

1. S. Garmon, T. Petrosky, L. Simine, and D. Segal, Fortschr. Phys. 61, 261 (2013). https://doi.org/10.1002/prop.201200077

2. S. Garmon, M. Gianfreda and N. Hatano, Phys. Rev. A 92, 022125 (2015). http://dx.doi.org/10.1103/PhysRevA.92.022125

3. S. Garmon and G. Ordonez, J. Math. Phys. 58, 062101 (2017).

https://doi.org/10.1063/1.4983809

**Yogesh Joglekar, **(Indiana Uni. Purdue University, US), **16/07, 14:00 GMT**

Conserved quantities and their consequences in PT symmetric systems: theory and experiments [1]

Abstract: Conserved quantities such as energy or the electric charge of a closed system, or the Runge-Lenz vector in Kepler dynamics, are determined by its global, local, or accidental symmetries. They were instrumental in advances such as the prediction of neutrinos in the (inverse) beta decay process and the development of self-consistent approximate methods for isolated or thermal many-body systems. In contrast, little is known about conservation laws and their consequences in open systems in general, and parity-time (PT) systems in particular. Here, we present a complete set of conserved observables for a broad class of PT -symmetric Hamiltonians and experimentally demonstrate their properties and consequences by using a single-photon linear optical circuit. One surprising consequence is the generation of coherence between different eigenstates of a conserved observable — something that is forbidden in Hermitian quantum theories. Our results spell out nonlocal conservation laws in nonunitary dynamics and provide key elements that will underpin the self-consistent analyses of non-Hermitian quantum many-body systems that are forthcoming.

[1] Work done in collaboration with Xue Peng group, and Frantisek Ruzicka (Phys. Rev. Research 2, 022039(R) (2020)).

**Andrei Smilga, **(University of Nantes, France), **23/07, 14:00 GMT**

Classical and quantum dynamics of higher-derivative systems

Abstract: A brief review of the physics of systems including higher derivatives in the Lagrangian is given. We outline the Ostrogradsky formalism and prove simple theorems saying that all such systems involve ghosts. At the classical level, that means that the energy is not bounded neither from below, nor from above. At the quantum level, that means that the spectrum of the Hamiltonian is never bounded from below and the vacuum ground state is absent.

If the Hamiltonian includes nontrivial interactions, this usually

leads to collapse and loss of unitarity, but in certain special cases,

this does not happen, however: ghosts are benign.

We speculate that the Theory of Everything is a higher-derivative

field theory, characterized by the presence of such benign ghosts and

defined in a higher-dimensional bulk. Our Universe represents then a

classical solution in this theory, having the form of a 3-brane embedded

in the bulk.

**Eva-Maria Graefe, **(Imperial College London, UK), **30/07, 14:00 GMT**

A PT-symmetric kicked top

Abstract: A non-Hermitian PT-symmetric version of the kicked top is introduced to study the interplay of quantum chaos with balanced loss and gain. The classical dynamics arising from the quantum dynamics of the angular momentum expectation values are derived. It is demonstrated that the presence of PT-symmetry can lead to ”stable” mixed regular chaotic behaviour without sinks or sources for subcritical values of the gain-loss parameter. For large values of the kicking strength a strange attractor is observed that also persists if PT-symmetry is broken. Classical structures are also identified in the quantum dynamics. Finally, some of the statistics of the eigenvalues of the quantum system are analysed.

**Avadh Saxena, **(Los Alamos National Laboratory, US), **13/08, 14:00 GMT**

Anti-PT-Symmetric Qubit: Decoherence and Entanglement Entropy [1]

Abstract: We consider a two-level spin system based anti-parity-time (anti-PT)-symmetric qubit and study its decoherence as well as entanglement entropy properties. We compare our findings with that of the corresponding PT-symmetric and Hermitian qubits. First we consider the time-dependent Dyson map for the density matrix of a general non-Hermitian system and then specialize it to the case of the anti-PT-symmetric qubit. We find that the decoherence function for the anti-PT-symmetricqubit decays slower than the PT-symmetric and Hermitian qubits. For the von Neumann entropy we find that for the anti-PT-symmetric qubit it grows more slowly compared to the PT-symmetric and Hermitian qubits. Similarly, we find that the corresponding average Fisher information is much higher compared to the PT-symmetric and Hermitian qubits.

These results demonstrate that anti-PT-symmetric qubits may be better suited for quantum computing and quantum information processing applications than conventional Hermitian or even PT-symmetric qubits.

[1] Work done in collaboration with Julia Cen (LANL).

**Li Ge, **(City University of New York, US), **20/08, 14:00 GMT**

Pseudo-chirality and its conserved pseudo norm in non-Hermitian photonic systems

Abstract: Noether’s theorem relates constants of motion to the symmetries of the system. Here we investigate a manifestation of Noether’s theorem in non-Hermitian systems, where the inner product is defined differently from quantum mechanics. In this framework, a generalized symmetry which we term pseudo-chirality emerges naturally as the counterpart of symmetries defined by a commutation relation in quantum mechanics [1]. Using this observation, we reveal previously unidentified constants of motion in non-Hermitian systems with parity-time and chiral symmetries. We further elaborate the disparate implications of pseudo-chirality induced constant of motion: It signals the pair excitation of a generalized “particle” and the corresponding “hole” but vanishes universally when the pseudo-chiral operator is anti-symmetric. This disparity, when manifested in a non-Hermitian topological lattice with the Landau gauge, depends on whether the lattice size is even or odd. We further discuss previously unidentified symmetries of this non-Hermitian topological system, and we reveal how its constant of motion due to pseudo-chirality can be used as an indicator of whether a pure chiral edge state is excited.

[1] J. Rivero and L. Ge, Phys. Rev. Letts., in press (2020).

**Carl Bender, **(Washington University in St. Louis, US), **27/08, 14:00 GMT**

Unmasking classical PT symmetry

Abstract: This talk reports the latest results of an ongoing research program to study the classical trajectories of the family of PT-symmetric Hamiltonians H = p² +x² (ix)^ε (ε≥0). A rich variety of phenomena, heretofore overlooked, have been discovered, such as the existence of infinitely many separatrix trajectories, infinite sequences of critical initial values associated with limiting classical orbits, regions of broken PT-symmetric classical trajectories, and a

topological transition at ε = 2.

**Vladimir Konotop, **(Universidade de Lisboa, Portugal), **10/09, 14:00 GMT**

The universal form of one-dimensional potentials with spectral singularities

Abstract: We establish necessary and sufficient conditions for localized complex potentials in the Schrödinger equation to enable spectral singularities. All such potentials have the universal form. We describe an algorithm of constructing potentials featuring two or three coexisting spectral singularities, as well as potentials with a second order spectral singularity. The analysis is generalized to the discrete Schrödinger equation: all lattice potentials with spectral singularities have the universal form of the gain-and-loss distribution. Discrete potentials characterized by several spectral singularities at prescribed wavelengths, as well as potentials with second-order spectral singularities in their spectra are constructed.

[*] Work done in collaboration with Dmitry A. Zezyulin

[1] D A Zezyulin and V. V. Konotop New J. Phys. 22, 013057 (2020)

[2] V V Konotop and D A Zezyulin J. Phys. A 53 305202 (2020)

[3] D A Zezyulin and V. V. Konotop, Opt. Lett. 45 3447 (2020)

**Joshua Feinberg, **(University of Haifa and Technion, Israel), **24/09, 14:00 GMT**

Dynamics of Disordered Mechanical Systems with Large Connectivity, Free

Probability Theory, and Quasi-Hermitian Random Matrices

Abstract: Disordered mechanical systems with high connectivity represent a limit opposite to the more familiar case of disordered crystals. Individual

ions in a crystal are subjected to nearest-neighbor interactions. In

contrast, the systems discussed in this talk have all their degrees of

freedom coupled to each other: Each momentum is typically coupled to all

other momenta, and similarly for the coordinates. Thus, the problem of

linearized small oscillations of such systems involves two full

positive-definite and non-commuting matrices, as opposed to the sparse

matrices associated with disordered crystals. Consequently, the familiar

methods for determining the averaged vibrational spectra of disordered

crystals, introduced many years ago by Dyson and Schmidt, are

inapplicable for highly connected disordered systems. In this talk we

apply random matrix theory (RMT) to calculate the averaged vibrational

spectra of such systems, in the limit of infinitely large system size.

At the heart of our analysis lies a calculation of the average spectrum

of the product of two positive definite random matrices by means of free

probability theory techniques. We also show that this problem is

intimately related with quasi-hermitian random matrix theory (QHRMT),

which means that the `Hamiltonian’ matrix is hermitian with respect to a

non-trivial metric. This extends ordinary hermitian matrices, for which

the metric is simply the unit matrix. The analytical results we obtain

for the spectrum agree well with our numerical results. The latter also

exhibit oscillations at the high-frequency band edge, which fit well the

Airy kernel pattern. We also compute inverse participation ratios of the

corresponding amplitude eigenvectors and demonstrate that they are all

extended, in contrast with conventional disordered crystals. Finally, we

compute the thermodynamic properties of the system from its spectrum of

vibrations.

This talk is based on work done jointly with Roman Riser at the

University of Haifa, which will be published in a couple of upcoming

papers.

**Bhabani Prasad Mandal, **(BHU, Varanasi, India), **08/10, 14:00 GMT**

PT phase transitions: QM to QFT

Abstract: In this talk I will summarize some of our previous works on PT phase transition in quantum mechanical (QM) systems and will talk about PT phase transition in gauge field theoretic models. I will start with a non-Hermitian QES model in 1-dimensional QM systems to demonstrate PT phase transition which we realized way back in the year 2000. Next I will discuss a couple of exactly solvable non-Hermitian models in higher dimensional QM systems in this context. Dirac oscillator in the background of an external magnetic field with imaginary spin orbit coupling will be considered in (2+1) dimension next to talk about PT phase transition in the relativistic system. Lastly by considering natural Hermiticity properties of the ghost fields we cast SU(N) QCD in a newly introduced quadratic gauge as non-Hermitian but symmetric under combined Parity and Time Reversal transformation. As the ghost fields condensate PT symmetry is broken spontaneously . This leads to realization of the transition from deconfined phase to confined phase as PT phase transitions in QCD.

**Stéphane Boris Tabeu, **(University of Yaounde, Cameroon), **15/10, 14:00 GMT**

Non-Hermitian two-level systems with imaginary components in electronics

Abstract: The systems having real spectra without being Hermitian are in expansion today in many fields of physics. The most investigated are Parity-Time Symmetric systems. Their Hamiltonians commute with the joint parity operator P and the time reversal T operator such that [PT, H]=0. Anti-Parity-Time symmetry is the counterpart of Parity-Time symmetry and is very emerging. Its Hamiltonians anti-commute with the PT operator. Generally, they are obtained in the form of matrices after a coupled mode theory, adiabatic elimination or rotating wave approximations. Electronics is presented nowadays as a test bed of non-Hermitian systems due the easy manipulation of components such resistors, inductors and capacitors. The imaginary components bring new configurations in the design of two-level systems without any approximation. The most powerful applications are listed as telemetry and sensing, wireless power transfer, topological circuits, new waveguides for asymmetric and non-reciprocal transport of information.

Our results open the road for the design of new systems for possible used in the generation and manipulation of quantum bits at room temperature, and for the introduction of electronics in the complex quaternionic space.

**Naomichi Hatano, **(University of Tokyo, Japan), **22/10, 14:00 GMT**

What is the “environment” in open quantum systems?

Abstract: I will tutorially review what I had talked about resonant states with complex eigenvalues in the previous meetings of the PHHQP series, and particularly try to reveal what the “environment” of open quantum systems does mean in the actual physical setups. There are two ways of computing resonant states, namely the Siegert (out-going wave) boundary condition and the Feshbach formalism. The former is a direct consequence of finding zeros of the S matrix. Here the environment is in actual physical setups detectors and other experimental apparatus, which dictates the absorbing boundary condition.The spatial divergence of the resonant wave function means that macroscopic part of the wave amplitudes is absorbed in the detectors. This makes the system non-Hermitian. In the Feshbach formalism, on the other hand, the environmental part of the effective Hamiltonian produces the non-Hermiticity when we define the retarded and advanced Green’s functions of the environment. The choice of the retarded Green’s function corresponds to the out-going wave boundary condition, whereas the choice of the advanced one to the in-coming wave boundary condition. The introduction of ±iε in the denominator of the Green’s function produces the resonant and anti-resonant states. I will finally mention the proper choice of the right-eigenvector in analyzing non-Hermitian systems.

**Stefan Rotter, **(Vienna University of Technology, Austria), **05/11, 15:00 GMT**

Non-Hermitian channels of invisibility across complex media

Abstract: Waves typically propagate very differently through a homogeneous medium like free space than through an inhomogeneous medium like a complex dielectric structure. It has thus been quite surprising to find that one-dimensional scattering systems can be engineered in such a way that by way of the gain and loss added to them, they become not only reflection-less, but entirely scattering-free [1,2], even for pulses propagating through them [3]. Our most recent insight is that a straightforward way to extend these non-Hermitian features to two-dimensional space is to map the wave solutions in free space to those inside a suitably designed non-Hermitian potential landscape such that both solutions share the same spatial distribution of their wave intensity [4]. This mapping turns out to be broadly applicable as a design protocol for a special class of non-Hermitian media across which specific incoming waves form scattering-free propagation channels. This protocol naturally enables the design of structures with a broadband unidirectional invisibility for which outgoing waves are indistinguishable from those of free space. We illustrate this concept through the example of a beam that maintains its Gaussian shape while passing through a randomly assembled distribution of scatterers with gain and loss.

[1] K. G. Makris, A. Brandstötter, P. Ambichl, Z. H. Musslimani, and S. Rotter, Light Sci. Appl. 6, e17035 (2017)

[2] E. Rivet, A. Brandstötter, K. G. Makris, H. Lissek, S. Rotter, and R. Fleury, Nature Physics 14, 942 (2018)

[3] A. Brandstötter, K. G. Makris, and S. Rotter, Phys. Rev. B 99, 115402 (2019)

[4] K. G. Makris, I. Krešić, A. Brandstötter, and S. Rotter, Optica 7, 619 (2020)

**Maxim Chernodub, **(Institut Denis Poisson, CNRS, Tours, France), **12/11, 15:00 GMT**

Spontaneous Non-Hermiticity in Nambu-Jona-Lasinio model and Non-Hermitian Chiral Magnetic Effect

Abstract: This talk discusses two topics. First, 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. The NJL model is used to model the chiral symmetry breaking in the model of strong interactions, and we argue that its non-Hermitian gound state may potentially be realized in an off-equilibrium environment of quark-gluon plasma created in the heavy-ion collisions. Second, we discuss the emergence of the chiral magnetic effect — anomalous generation of electric current along the axis of the background magnetic field — in the context of a non-Hermitian fermion system.

This talk is based on the following papers:

1. Alberto Cortijo and M.C., Non-Hermitian Chiral Magnetic Effect in Equilibrium, Symmetry 12, 761 (2020).

2. Alberto Cortijo, Marco Ruggieri, and M.C., Spontaneous Non-Hermiticity in Nambu-Jona-Lasinio model, ArXiv:2008.11629.

**Pijush Ghosh, **(Siksha-Bhavana, Visva-Bharati, India), **19/11, 15:00 GMT**

Taming Hamiltonian systems with balanced loss and gain via Lorentz interaction

Abstract: We present a Hamiltonian formulation of generic many-particle systems with space-dependent balanced loss and gain. The instability of such system, both at the classical as well as quantum level, can be tamed by the introduction of additional Lorentz interaction. The Landau problem with balanced loss and gain is analysed to exemplify the idea. Further, we present non-PT-symmetric Hamiltonian systems with balanced loss and gain which admits periodic solutions. The examples include a mechanical model of

coupled Duffing oscillators with balanced loss and gain exhibiting Hamiltonian chaos, an exactly solvable dimer model with balanced loss and gain that describes the time evolution of the amplitudes of the former model and a vector nonlinear Schrodinger equation with balanced loss and gain. It appears that the criteria for the existence of periodic solutions in classical systems with balanced loss and gain can not be specified only in terms of PT-symmetry. Finally, a method is given to construct exact solutions of a class of cubic vector nonlinear Schrodinger equations with balanced loss and gain by the use of pseudo-unitary transformations.

References:

(1) Pijush K. Ghosh, Taming Hamiltonian systems with balanced loss and gain via Lorentz interaction : General results and a case study with Landau Hamiltonian, J. Phys. A: Math. Theor. 52 (2019) 415202 (24pp) arXiv:1810.04137

(2) Pijush K. Ghosh and Puspendu Roy, On regular and chaotic dynamics of a non-PT-symmetric Hamiltonian system of a coupled Duffing oscillator with balanced loss and gain, J. Phys. A: Math. Theor. 53 (2020) 475202 (27pp) arXiv:2007.07286

(3) Pijush K. Ghosh, Constructing Solvable Models of Vector Non-linear Schrodinger Equation with Balanced Loss and Gain via Non-unitary transformation, arXiv:2008.12252

**Mikhail Plyushchay, **(Universidad de Santiago, Chile), **03/12, 15:00 GMT**

Conformal symmetry plus PT-symmetry for perfect invisibility and related exotics

Abstract: We show how perfectly invisible quantum mechanical zero-gap systems can be obtained by exploiting conformal and PT symmetry. Such systems give rise naturally to the exotic supersymmetric structure and allow ones to generate extreme waves via their intimate relation to the classical KdV equation. We also briefly discuss another application of conformal symmetry, the so-called conformal bridge transformation, by which asymptotically free and harmonically trapped conformal quantum systems can be related, and speculate on a possible relevance of the construction to PT-symmetry.

[1] J. M. Guilarte, M. S. Plyushchay, JHEP 12 (2017) 061.

[2] J. M. Guilarte, M. S. Plyushchay, JHEP 01 (2019) 194.

[3] L. Inzunza, M. S. Plyushchay, A. Wipf, Phys. Rev. D 101 (2020) 10, 105019.

[4] L. Inzunza, M. S. Plyushchay, A. Wipf, JHEP 04 (2020) 028.

**Qing-hai Wang, (National University of Singapore, Singapore**), **17/12, 15:00 GMT**

Reinterpret Non-Hermitian Hamiltonians using time-dependent PT-symetric quantum mechanics

Abstract: In this talk, I will reinterpret an arbitrary finite-dimensional non-Hermitian Hamiltonian using time-dependent PT-symetric quantum mechanics. The conceptual framework is built upon a nontrivial time-dependent metric operator. All the ingredients of our framework, such as

the time-dependent Hilbert space, the observable, and the measurement postulate, can be “realized” by means of dilating and reinterpreting the non-Hermitian system in question as a part of a larger Hermitian system. Aided by our metric operator, we formulate the concepts of stable and unstable phases for generic non-Hermitian systems and argue that they, respectively, generalize the notions of unbroken and broken phases in time-independent

PT -symmetric systems.

**Gonzalo Muga, (University of the Basque Country, Spain**) , **14/01, 15:00 GMT**

Symmetry and asymmetry in nonhermitian Physics

Abstract: I shall first review the importance of devices with asymmetrical responses in many areas of biology, or technology. In the microscopic world and in the context of the second quantum revolution we expect such response asymmetries to be needed to build different devices such as diodes or rectifiers. Hermitian Hamiltonians do not allow for them at least at a basic level of description. Different Non-Hermitian (NH) Hamiltonians allow for different asymmetric responses (eg in reflection or transmission) that depend on the symmetry properties of the Hamiltonian through selection rules. Scattering Hamiltonians are classified and studied according to symmetry operations that form a group (E8). PT symmetry is just one of the possible symmetries in this group. The analysis is finally generalized to discrete finite Hamiltonian matrices, whose symmetry operations form larger groups.

**Stella Schindler, (Massachusetts Institute of Technology, US)** **21/01, 15:00 GMT**

PT and patterns in finite-density QCD

Abstract: A PT-symmetric extension of the Φ⁴- model exhibits patterned field configurations near its critical point. Analytic results and lattice simulations associate this patterning with tachyonic instabilities. Patterning is not confined, however, to just this two-component PT-symmetric scalar model with a Z(2) symmetry. Rather, patterned critical behavior occurs in broad universality classes of PT-symmetric theories with two or more components and any symmetry group. Finite-density QCD is a member of the PT-symmetric Z(2) universality class. This raises the possibility that stable patterns of confined and deconfined phase occur near the critical endpoint in the μ-T plane.

**Miloslav Znojil, (Nucl. Phys. Inst. of the AS, Czech Republic**), **28/01, 15:00 GMT**

Domains of unitarity close to exceptional points

Abstract: The description of unitary evolution using non-Hermitian but “hermitizable” Hamiltonians H is feasible via an ad hoc metric Θ = Θ(H) and a (non-unique) amendment < ψ_{1}| ψ_{2} > → < ψ_{1}|Θ| ψ_{2} > of the inner product in Hilbert space. Via a proper fine-tuning of Θ(H) this opens the possibility of

reaching the boundaries of stability (i.e., exceptional points, EPs) in many quantum systems. Several aspects of the problem will be discussed.

**Roman Riser, (University of Haifa, Israel**), **11/02, 15:00 GMT**

A quasi-hermitian random matrix model

Abstract: We present a quasi-Hermitian random matrix model which shows an eigenvalue distribution in the complex plane. We provide an analytical solution of its density by using a diagrammatic approach in the planar limit. This model shows various phase transitions, in the complex plane when complex eigenvalues flow to the real axis and also on the real axis itself when intervals of disjoint eigenvalues merge.

**Duncan O’Dell, (McMaster University, Canada)**, **25/02, 15:00 GMT**

Schwinger pair production as a non-Hermitian problem

Abstract: The creation and annihilation of particles is a fundamental feature of relativistic quantum fields. A famous example of this is provided by Schwinger’s 1951 prediction that the vacuum is unstable to particle-antiparticle production if a static electric field is applied to it. In this talk we examine the classical field limit of Schwinger pair production by mapping the Klein-Gordon equation with an electric field onto the non-relativistic Schrödinger equation with a 1/r^2 potential. This latter problem has two regimes depending on the depth of the potential: a sub-critical regime where PT symmetry is preserved and a supercritical regime where PT symmetry is broken and one finds “fall-to-the-centre’’ where probability is absorbed at the origin. Schwinger pair production occurs in the latter case and from this point can be described in terms of non-Hermitian quantum mechanics.

**Mustapha Maamache, (Ferhat Abbas University, Algeria), 11/03, 15:00 GMT**

Time dependent pseudo-squeezed coherent states

Abstract: We discuss the extension of non-Hermitian integrals of motion method to cases where the quantum systems are desribed by time-dependent non-Hermitian Hamiltonians. We introduce a time dependent metric in order to construct time-dependent non-Hermitian pseudo-bosonic coherent states as eigenstates of pseudo-Hermitian boson annihilation operators that are time-dependent non-Hermitian linear invariants. As illustration, we study the time-dependent non-Hermitian Swanson Hamiltonian.

**Andreas Ruschhaupt, (University College Cork, Ireland), 01/04, 15:00 London time (=14:00 GMT) **

Physical implementation of asymmetric scattering devices

Abstract: Non-local, non-Hermitian, one-dimensional potentials allow, compared to local ones, for richer asymmetric transmission and reflection responses to the incidence of a particle from left or right. We will first identify six basic device types, based on non-local, non-Hermitian potentials, for which the squared moduli of scattering amplitudes adopt zero/one values, and transmission and/or reflection are asymmetric. Then, we will put forward a possible, simple, quantum-optical realization of non-local potentials to implement some of these devices.

**Hamidreza Ramezani, (Univ. of Texas Rio Grande Valley, US), 08/04, 15:00 London time (=14:00 GMT) **

Non-Hermiticity and topology in synthetic non-Hermitian lattices

Abstract:

In this talk, I review briefly some of my recent contributions in the field of synthetic non-Hermitian lattices. More specifically, I will discuss obtaining constant intensity waves [1], flat bands [2], and localization from exceptional points [3]. If time permits, I will talk about the dynamical approach to shortcut to adiabaticity [4], unidirectional lasing [5], tunable filters [6], non-reciprocal localization [7], robust exceptional points [8], and topology at will [9]. I will show how non-Hermiticity can help us to achieve all these interesting effects in synthetic lattices.

[1] M. Dehghani, et al., Optics Letters 45 (1) (2020)

[2] H. Ramezani, Phys. Rev. A 96, 011802 (2017)

[3] C. Yuce, H. Ramezani, Optics Letters 46 (4), (2021)

[4] F. Mostafavi, L. Yuan, and H. Ramezani, PRL, 122, 050404 (2019)

[5] H. Ramezani, et al., PRL 113 (26), 263905 (2014), H.Ramezani, et al., PRL 112 (4), 043904 (2014)

[6] S. Puri, et al. (submitted, 2021)

[7] H. Ramezani, et al PRL 120 (4), 043901 (2018)

[8] C. Yuce, H Ramezani, EPL 126 (1), 17002 (2019)

[9] F. Mostafavi, et al., Phys. Rev. Research 2, 032057(R) (2020)

**Andreas Fring, (City, University of London, UK), 22/04, 15:00 London time (=14:00 GMT) **

Non-Hermitian gauge field theories and BPS limits

Abstract: We present an overview of some key results obtained in a recent series devoted to non-Hermitian quantum field theories for which we systematically modify the underlying symmetries. Particular attention is placed on the interplay between the continuous symmetry group that we alter from global to local, from Abelian to non-Abelian, from rank one to generic rank N, and the discrete anti-linear modified CPT-symmetries. The presence of the latter guarantees the reality of the mass spectrum in a certain parameter regime. We investigate the extension of Goldstone’s theorem and the Higgs mechanism, which we demonstrate to work in the conventional fashion in the CPT-symmetric regime, but which needs to be modified technically at the standard exceptional points of the mass spectrum and entirely fails at what we refer to as zero exceptional points as well as in the broken CPT-symmetric regime. In the full non-Hermitian non-Abelian gauge theory we identify the t’Hooft-Polyakov monopoles by means of a fourfold Bogomol’nyi-Prasad-Sommerfield (BPS) limit. We investigate this limit further for other types of non-Hermitian field theories in 1+1 dimensions that possess complex super-exponential and inverse hyperbolic kink/anti-kink solutions and for 3+1 dimensional Skyrme models for which we find new types of complex solutions, that all have real energies due to the presence of different types of CPT-symmetries.

**Andrew Harte r, **(University of Tokyo, Japan), 06/05,

**15:00 London time (=14:00 GMT**)

Induced Symmetries and Topological States of a Floquet PT-symmetric lattice

Abstract: Previously, it has been shown [1] that certain PT-symmetric [2] lattices can exhibit a topologically non-trivial phase. However, the PT-symmetric phase never coincides with the topoligcal phase, and the topological states are associated to the imaginary eigenvalues of the Hamiltonian. By introducing periodic time-dependent (Floquet) driving to the system, we can externally move the system between configurations which correspond to different static phases. By analyzing the Floquet effective Hamiltonian, we can gain insight into the long-term dynamics of the system and its effective, time-independent properties. Such driving has previously been shown [3] to induce PT-symmetric topological states when the driving frequency is above a certain high-frequency threshold. In our study, by using a simple driving model with discrete steps, we explore the full range of driving frequencies to highlight similar robust phases which are explicitly below the high-frequency regime. We also analyze the symmetries of the Floquet effective Hamiltonian to better understand these phenomena.

[1] Rudner, M. S. and Levitov, L. S. Phys. Rev. Lett. 102, 065703 (2009)

[2] Bender, C. and Boettcher, S. Phys. Rev. Lett. 80, 5243 (1998)

[3] C. Yuce, Eur. Phys. J. D 69, 184 (2015)

**Özlem Yeşiltaş, (Gazi University, Turkey), 20/05, 15:00 London time (=14:00 GMT) **

Pseudo-Hermitian Dirac Operator On the Torus For The Massless Fermions

Abstract: The Dirac equation in (2+1)-dimensions on the toroidal surface is studied for a massless fermion particle. Using the covariant approach based in general relativity, the Dirac operator stemming from a metric related to the strain tensor

is discussed within the Pseudo-Hermitian operator theory.

** Iveta Semorádová**, (

**), 03/06,**

**Czech Technical University Prague, Czech Republic****15:00 London time (=14:00 GMT**)

Diverging eigenvalues in domain truncations of Schrödinger operators with complex potentials

Abstract: Domain truncations of Schrödinger operators with complex potentials are known to be spectrally exact. However, several examples suggest that additional eigenvalues escaping to infinity seem to be a generic feature. We find conditions on the presence of such eigenvalues and obtain their asymptotic expansions. Our approach also yields asymptotic formulas for diverging eigenvalues in strong coupling regime for the imaginary part of the potential.

**Hugh Jones**, (

**), 17/06,**

**Imperial College London,UK****15:00 London time (=14:00 GMT**)

Scattering and bound states of bottomless potentials

Abstract: We explore the interconnections between the scattering states and bound states of different non-analytic segments (depending on |x|) of the exponential potential, and elucidate the status of the special scattering states found in an earlier publication by Ahmed et al. We then go on to consider the nature of the scattering and bound states of bottomless power potentials such as x^{3 }and -x^{4} and their related non-analytic segments.

**Marta Reboiro**, (

**), 01/07,**

**University of La Plata, Argentina****15:00 London time (=14:00 GMT**)

Swanson Model beyond the PT symmetry phase

Abstract: The Swanson Model has been introduced some time ago as an example of a non-Hermitian Hamiltonian that obey Parity-Time Reversal (PT) symmetry. It is well known that it admits real eigenvalues for a well defined region of the parameter space of the model. The similarity between the Swanson Hamiltonian and the Harmonic Oscillator in the PT-symmetry regions well as the dynamic of physical observables have been extensively analyzed. However, to our knowledge, much less has been investigated in region of PT-broken symmetry. We study the spectrum and the generalized eigenfunctions in the non-PT symmetry phase. Our interest is both from the physical and from the mathematical point of view. We show that, depending on the region on the parameter model-space, the Swanson model is similar to different physical systems, ie parabolic barrier, or harmonic oscillator with negative mass. From the mathematical point of view, we deal with a system of infinite dimension whose eigenfunctions, in the non-PT symmetry phase, does not belong to the Hilbert space. We shall use the formalism of rigged Hilbert spaces to construct the spectrum an the generalized eigenfunctions of the problem. We compute mean values of different observables. Also, we discuss the time evolution of a given initial states under the action of the Swanson Hamiltonian.

**Ken Mochizuki**, (

**), 15/07,**

**Tohoku University, Japan****15:00 London time (=14:00 GMT**)

Statistical Properties of the Non-Hermitian SSH Model and Symmetry Inheritance owing to Real Spectra

Abstract: Non-Hermitian Hamiltonians have been studied extensively during the last couple of decades theoretically and experimentally. This renewed interest stems from two exceptional results; localization-delocalization transitions in one-dimensional non-Hermitian disordered systems [1] and entirely real spectra in non-Hermitian Hamiltonians which satisfy parity and time-reversal (PT) symmetry [2]. While disorder and PT symmetry would conflict since spatial disorder does not maintain the parity operation in position space, some non-Hermitian Hamiltonians with disorder can possess entirely real spectra. This is because Hamiltonians without PT symmetry may have pseudo-Hermiticity ensuring entirely real spectra [3], which is irrelevant to the parity operation in general and can be robust against disorder. The spectral statistics of random Hermitian Hamiltonians usually exhibits universal behavior, depending only on symmetries of the system. An interesting question then naturally arises whether the spectral statistics of disordered non-Hermitian Hamiltonians also exhibits universal behavior.

In the present work [4], we explore the spectral statistics of the non-Hermitian disordered Su-Schrieffer-Heeger (SSH) model. This model without disorder is one of the most vigorously studied non-Hermitian models, in the research field of topological phases. In contrast, its spectral statistics has not received attention thus far. We find that owing to the structure of the Hamiltonian, eigenvalues can be purely real in a certain range of parameters, even in the absence of PT symmetry. We reveal that, in this case of purely real spectrum, the level statistics is that of the Gaussian orthogonal ensemble (GOE). We clarify the origin of GOE, by proving a general feature that a non-Hermitian Hamiltonian whose eigenvalues are purely real can be mapped to a Hermitian Hamiltonian which inherits the symmetries of the original Hamiltonian. When the spectrum contains imaginary eigenvalues, we find that the density of states (DOS) vanishes at the origin and diverges at the spectral edges on the imaginary axis. We show that the divergence of the DOS originates from the Dyson singularity in chiral-symmetric one-dimensional Hermitian systems.

[1] N. Hatano and D. R. Nelson, Physical Review Letters 77, 570 (1996).

[2] C. M. Bender and S. Boettcher, Physical Review Letters 80, 5243 (1998).

[3] A. Mostafazadeh, Journal of Mathematical Physics 43, 205 (2002).

[4] K. Mochizuki, N. Hatano, J. Feinberg, and H. Obuse,

Physical Review E 102, 012101 (2020).

Daniel Dizdarevic**, (Universität Stuttgart, Germany), 29/07, 15:00 London time (=14:00 GMT) **

Symmetries and symmetrisation in quantum and electromagnetic multi-mode systems for balancing gain and loss

Abstract: Losses usually are an undesirable effect in physics. However, in combination with gain, novel and unexpected features occur. By balancing gain and loss, stable stationary states with intriguing properties can be realised. Balanced gain and loss occurs in combination with anti-unitary symmetries, which are related to time reversal. The simplest and most powerful symmetry in this regard is PT symmetry. Due to the generality of the PT operator, PT symmetry is applicable to almost any physical system, though, it is broken even for small perturbations.

In the absence of symmetries, balanced gain and loss can still be achieved by means of symmetrisation or semi-symmetrisation. This allows for the description of physical multi-well potentials with gain and loss. The relations between symmetries and symmetrisation are discussed and both concepts are applied to one-dimensional multi-mode quantum systems and spatially extended Gaussian multi-well potentials, which can be used in experimental realisations with Bose-Einstein condensates involving non-linear contact interactions. Such non-linear systems can be used to realise a self-stabilising mechanism of stationary states, thus making them robust with respect to small perturbations.

By deriving a mathematically equivalent model for inductively coupled electric resonant circuits, the concepts of symmetries and symmetrisation can be transferred from the quantum realm to the classical field of electrodynamics. While this provides a simple and, in particular, accessible platform for experiments, this also allows for applications in wireless power transfer.

More information:

– D. Dizdarevic, “Symmetries and symmetrisation in quantum and electromagnetic multi-mode systems for balancing gain and loss”, Ph.D. thesis, University of Stuttgart (2021)

– D. Dizdarevic, H. Cartarius, J. Main and G. Wunner, “Balancing gain and loss in symmetrised multi-well potentials”, J. Phys. A: Math. Theor. 53, 405304 (2020)

– S. Altinisik, D. Dizdarevic and J. Main, “Balanced gain and loss in spatially extended non-PT-symmetric multiwell potentials”, Phys. Rev. A 100, 063639 (2019)

Philip Mannheim**, (University of Connecticut, US), 09/09, 15:00 London time (=14:00 GMT) **

Goldstone bosons and the Englert-Brout-Higgs mechanism in non-Hermitian theories

Abstract: In recent work, Alexandre, Ellis, Millington and Seynaeve have extended the Goldstone theorem to non- Hermitian Hamiltonians that possess a discrete antilinear symmetry such as PT and possess a continuous global symmetry. They restricted their discussion to those realizations of antilinear symmetry in which all the energy eigenvalues of the Hamiltonian are real. Here, we extend the discussion to the two other realizations possible with antilinear symmetry, namely energies in complex conjugate pairs or Jordan-block Hamiltonians that are not diagonalizable at all. In particular, we show that under certain circumstances it is possible for the Goldstone boson mode itself to be one of the zero-norm states that are characteristic of Jordan-block Hamiltonians. While we discuss the same model as Alexandre, Ellis, Millington and Seynaeve, our treatment is quite different, though their main conclusion that one can have Goldstone bosons in the non-Hermitian case remains intact. We extend our analysis to a continuous local symmetry and find that the gauge boson acquires a nonzero mass by the Englert-Brout-Higgs mechanism in all realizations of the antilinear symmetry, except the one where the Goldstone boson itself has zero norm, in which case, and despite the fact that the continuous local symmetry has been spontaneously broken, the gauge boson remains massless.

Latevi Lawson**, (African Inst. for Math. Sciences, Ghana), 23/09, 15:00 London time (=14:00 GMT) **

Minimal and maximal lengths from non-Hermitian position-dependent noncommutativity

Abstract: TBA

Andrei Smilga**, (University of Nantes, France), 07/10, 15:00 London time (=14:00 GMT) **

New classes of dynamical systems with benign

ghosts

Abstract: TBA

Federico Roccati**, (Università degli Studi di Palermo, Italy), 21/10, 15:00 London time (=14:00 GMT) **

Non-Hermitian skin effect as an impurity problem

Abstract: 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.

Ali Mostafazadeh**, (Koç University, Turkey), 04/11, 15:00 London time (=14:00 GMT) **

TBA

Abstract: TBA

Yogesh Joglekar**, (Indiana Uni. Purdue University, US), 18/11, 15:00 London time (=14:00 GMT) **

Non-Hermiticity and PT symmetry via complex, synthetic gauge fields*

Abstract: TBA

Fabio Bagarello**, (Università degli Studi di Palermo, Italy), 02/12, 15:00 London time (=14:00 GMT) **

TBA

Abstract: TBA

Kohei Kawabata**, (University of Tokyo, Japan), 09/12, 15:00 London time (=14:00 GMT) **

TBA

Abstract: TBA

Tsuneya Yoshida**, (University of Tsukuba, Japan), 16/12, 15:00 London time (=14:00 GMT) **

TBA

Abstract: TBA

Kazuki Yokomizo**, (RIKEN National Science Institute, Japan), 06/01/2022, 15:00 London time (=14:00 GMT) **

TBA

Abstract: TBA

Hamed Ghaemidizicheh**, (University of Texas Rio Grande Valley, US), 20/01/2022, 15:00 London time (=14:00 GMT) **

Transport Effects in Nonreciprocal Tight Binding Models with Gain/Loss

Abstract: Based on a general transport theory for non-reciprocal non-Hermitian systems and a topological model that encompasses a wide range of previously studied models, we (i) provide conditions for effects such as reflectionless and transparent transport, lasing, and coherent perfect absorption, (ii) identify which effects are compatible and linked with each other, and (iii) determine by which levers they can be tuned independently. For instance, the directed amplification inherent in the non-Hermitian skin effect does not enter the spectral conditions for reflectionless transport, lasing, or coherent perfect absorption, but allows to adjust the transparency of the system. In addition, in the topological model the conditions for reflectionless transport depend on the topological phase, but those for coherent perfect absorption do not. This then allows us to establish a number of distinct transport signatures of non-Hermitian, nonreciprocal, and topological behaviour, in particular (I) reflectionless transport in a direction that depends on the topological phase, (II) invisibility coinciding with the skin-effect phase transition of topological edge states, and (III) coherent perfect absorption in a system that is transparent when probed from one side.

Géza Lévai**, (Institute for Nuclear Research, Hungary), 03/02/2022, 15:00 London time (=14:00 GMT) **

Do similar potential shapes lead to similar physical results?

A case study with two PT-symmetric potentials

Abstract: It is often possible to guess the main characteristics of quantum mechanical potentials without solving them exactly or numerically. For example, the shape of the potential gives a hint of where the maximum of the wave functions can be expected; its asymptotic behavior is a telling sign on whether the number of bound states is finite or infinite, etc. This intuitive approach works for most real potentials, however, the situation changes for complex potentials.

Here we discuss two such potentials that have similar shape: the PT-symmetric Rosen-Morse II and the finite PT-symmetric square well potentials. Their real compoment is the Pöschl-Teller hole [V_{R}~ -sech^{2}(x)] and the finite real square well, while their imaginary component is the tanh(x) function and the constant function outside the boundaries of the real square well. The question we address is whether the physical characteristics of the two potential show any similarity. The PT-symmetric Rosen-Morse II potential has the special feature that it supports exclusively real energy eigenvalues [2], no matter how large the coupling coefficient of its imaginary component is. Increasing it leds to rapidly increasing real eigenvalues, rather than to their complexification. This finding has been attributed to the asymptotically non-vanishing imaginary potential component. The finite PT-symmetric square well was discussed only recently [2], inspired by this unusual finding.

We present an analytical proof that the energy eigenvalues

of the PT-symmetric square well potential are also real. We

also derive the transmission and reflection coefficients,

demonstrate that they exhibit handedness and link them with

the bound states. Some important differences are also pointed

out. As a consistency check, we also analyse the common limit

of the two potentials: the potential with a Dirac-delta and

the step function as its real, and imaginary component,

respectively [3].

[1] G. Lévai and E. Magyari,

J. Phys. A:Math. Theor. 42 (2009) 195302

[2] G. Lévai and J. Kovács,

J. Phys. A:Math. Theor. 52 (2019) 025302

[3] J. Kovács and G. Lévai,

Acta Polytechnica 57 (2017) 412