The Seminars 2021: Term I

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.   

The slides of the talk are here

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. 

The slides of the talk are here

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 < ψ1| ψ2 > → < ψ1|Θ| ψ2 > 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.

The slides of the talk are here

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.

The slides of the talk are here

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.

The slides of the talk are here

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


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.

Based on: A. Fring, Takanobu Taira arXiv:2103.13519

Andrew Harter, (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, (Czech Technical University Prague, Czech Republic), 03/06, 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, (Imperial College London,UK), 17/06, 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 xand -x4 and their related non-analytic segments.

Marta Reboiro, (University of La Plata, Argentina), 01/07, 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, (Tohoku University, Japan), 15/07, 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)