The Seminars 2022: Term I

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

Non-Bloch band theory of non-Hermitian systems

Abstract: Non-Hermitian systems are nonequilibrium systems which can be described by a non-Hermitian Hamiltonian. Interestingly, non-Hermitian systems with periodic structure have the non-Hermitian skin effect which induces the localization of bulk eigenstates [1]. Then, the non-Hermitian skin effect exhibits the difference between eigenspectra under a periodic boundary condition and those under an open boundary condition. While the eigenspectra under a periodic boundary condition can be obtained from the conventional Bloch band theory, it is unclear how to calculate the eigenspectra under an open boundary condition. In this talk, we present the non-Bloch band theory which can produce the eigenvalues of non-Hermitian crystals with an open boundary condition [2]. We show that in the limit of a large system size, the eigenspectra can be calculated by the generalized Brillouin zone. Furthermore, we establish the bulk-edge correspondence between a topological invariant defined from the generalized Brillouin zone and existence of topological edge states. Finally, we apply the non-Bloch band theory to various physical systems [3,4].

[1] S. Yao and Z. Wang, Phys. Rev. Lett. 121, 086803 (2018).

[2] K. Yokomizo and S. Murakami, Phys. Rev. Lett. 123, 066404 (2019).

[3] K. Yokomizo and S. Murakami, Phys. Rev. B 103, 165123 (2021).

[4] K. Yokomizo, T. Yoda, and S. Murakami, arXiv:2112.02791

Hamed Ghaemidizicheh, (University of Texas Rio Grande Valley, US), 20/01/2022, 15:00 London time (=15:00 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 (=15:00 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 [VR~ -sech2(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

Aurelia Chenu, (University of Luxembourg, Luxembourg), 17/02/2022, 15:00 London time (=15:00 GMT)

From hybrid polariton to dipolariton using non-Hermitian Hamiltonians to handle particle lifetimes

Abstract: We consider photons strongly coupled to the excitonic excitations of a coupled quantum well, in the presence of an electric field.
We show how under a field increase, the hybrid polariton made of photon coupled to hybrid carriers lying in the two wells, transforms into a dipolariton made of photon coupled to direct and indirect excitons.
We also show how the cavity photon lifetime and the coherence time for carrier wave vectors, that we analytically handle through non-hermitian Hamiltonians, affect these polaritonic states.

Adolfo del Campo, (University of Luxembourg, Luxembourg), 03/03/2022, 15:00 London time (=15:00 GMT)

Spectral Filtering Induced by Non-Hermitian Evolution with Balanced Gain and Loss: Enhancing Quantum Chaos

Abstract: The dynamical signatures of an isolated quantum chaotic system are captured by the spectral form factor, which exhibits as a function of time a dip, a ramp, and a plateau, with the ramp being governed by the correlations in the level spacing distribution. These dynamical signatures are generally suppressed by decoherence. We consider the nonlinear non-Hermitian evolution associated with balanced gain and loss (BGL) in an energy-dephasing scenario and show that dissipation in this setting enhances manifestations of quantum chaos. Using the Sachdev-Ye-Kitaev model as a test-bed, BGL is shown to increase the span of the ramp, lowering the dip as well as the value of the plateau, providing an experimentally realizable physical mechanism for spectral filtering. The enhancement due to BGL is shown to be optimal with respect to the choice of the filter function.

Carl M. Bender, (Washington University in St. Louis, US), 17/03/2022, 15:00 London time (=15:00 GMT)

Dyson-Schwinger equations and PT symmetry

Abstract: The Dyson-Schwinger equations provide a nonperturbative approach to examining a Hermitian quantum field theory. Can we apply them to obtain accurate information about a non-Hermitian PT-symmetric quantum field theory? To answer this question we use them in very high order to study cubic and quartic PT-symmetric theories. The results are unexpected and surprising.

Micheline Soley, (Yale Quantum Institute, US), 31/03/2022, 15:00 London time (=14:00 GMT)

Experimentally-realizable PT phase transitions in reflectionless quantum scattering

Abstract: It has been shown previously that certain unbounded-below, PT-symmetric, purely real potentials (V(x)=-x^4, -x^6, and -x^8) have purely real Schrödinger spectra and support weakly “bound” states that die off as 1/x instead of exponentially asymptotically. Analysis of the analytic continuations of these potentials (with Hamiltonians H=p^2+x^2(ix)^epsilon and H=p^2+x^4(ix)^epsilon) show that for potentials V(x)=-x^p spectra are purely real for p=4,6,8 and feature no real eigenvalues for p=2. However, due to the infinite extent and unbounded energies of these potentials, these spectra are not experimentally measurable, and it is not clear how to adapt the potentials for implementation in experiment. Furthermore, the analytic continuations between these discrete integer p cases involve complex potentials that cannot be realized in closed quantum systems.

We examine the Schrödinger equation for a class of completely real potentials, V=-|x|^p for all real p, truncated in length -L/2<x<L/2 and constant beyond that length; thus corresponding to a scattering geometry. Such potentials have no bound states and a continuous spectrum, but they can support discrete, above-barrier reflectionless states, which are expected to exhibit properties related to the unbounded cases, which also map to a certain kind of reflectionless behavior. Although the corresponding Hamiltonian has both P and T symmetry, once the reflectionless boundary condition is imposed the operator only has PT symmetry. Moreover, these potentials are non-analytic at the origin (except at the even integers) and cannot be treated by analytic continuation in general. Nonetheless, with minimal smoothing they are realizable in experiments, e.g., in cold-atom systems. We use the theory of reflectionless scattering modes and Wentzel-Kramers-Brillouin (WKB) forces to study the reflectionless states of this class of Schrödinger Hamiltonian.

We find that for these experimentally realizable potentials it is possible to identify low-energy reflectionless eigenvalues that agree with analytic results with 7-8 digits of accuracy for the even integers and exhibit conventional PT-symmetry breaking as p is varied, even though the potential is purely real. In addition, we identify signatures of exceptional points, including the quartic energy variation of the reflection coefficient. These scattering systems provide the first example of a fundamental quantum mechanical Hamiltonian where PT transitions are measurable experimentally.

Michael Lubasch, (Quantinuum Ltd, UK), 14/04/2022, 15:00 London time (=14:00 GMT)

Diagonalization algorithms for complex and symmetric matrices with applications to PT-symmetric Hamiltonians

Abstract: Efficient and accurate algorithms for the diagonalization of complex and symmetric matrices are presented and applied to PT-symmetric Hamiltonians [1, 2].

[1] J. H. Noble, M. Lubasch, and U. D. Jentschura, “Generalized Householder transformations for the complex symmetric eigenvalue problem”, Eur. Phys. J. Plus 128, 93 (2013)

[2] J. H. Noble, M. Lubasch, J. Stevens, and U. D. Jentschura, “Diagonalization of complex symmetric matrices: Generalized Householder reflections, iterative deflation and implicit shifts”, Comp. Phys. Comm. 221, 304 (2017)

Karl Landsteiner, (Inst. de Fisica Teorica UAM/CSIC, Spain), 28/04/2022, 15:00 London time (=14:00 GMT)


Abstract: Holography (also called the Gauge/Gravity) duality allows to described strongly correlated quantum field theories as a gravitational theory in higher dimensions. This idea orginated in string theory but has developed its own life with applications to physical systems such as the quark gluon plasma and certain condensed matter systems. After a brief review and introduction I will show how the concept of non-Hermitian PT-symmetric Hamiltonians can be incorporated into Holography. I will develop a simple model, show that it has a PT-symmetric and a PT-broken regime separated by an exceptional point. Then will go on a study what happens if one makes the couplings time dependent (quantum quenches). A rich “phenomenology” arises in such holographic PT-quenches. In particular one can quite explicitly contrast non-unitary against unitary time evolution. I’ll also try to draw some lessons for conventional, weakly coupled PT-symmetric quantum field theories.

Ion Cosma Fulga, (Leibniz Inst. Dresden, Germany), 12/05/2022, 15:00 London time (=14:00 GMT)

Non-Hermitian physics without gain or loss: the skin effect of reflected waves

Abstract: Physically, one tends to think of non-Hermitian systems in terms of gain and loss: the decay or amplification of a mode is given by the imaginary part of its energy. Here, we introduce an alternative way of realizing non-Hermitian physics, which involves neither gain nor loss. Instead, complex eigenvalues emerge from the amplitudes and phase-differences of waves backscattered from the boundary of insulators. We show that for any strong topological insulator in a Wigner-Dyson class, the reflected waves are characterized by a reflection matrix exhibiting the non-Hermitian skin effect. This leads to an unconventional Goos-Hänchen effect: due to non-Hermitian topology, waves undergo a lateral shift upon reflection, even at normal incidence. Going beyond systems with gain and loss expands the set of experimental platforms that can access non-Hermitian physics and show signatures associated to non-Hermitian topology.

Barry Sanders + (Abhijeet Alase, Salini Karuvade) (University of Calgary, Canada), 26/05/2022, 15:00 London time (=14:00 GMT)

Observing a changing Hilbert-space inner product

Abstract: In quantum mechanics, physical states are represented by rays in Hilbert space H, which is a vector space imbued by an inner product ⟨|⟩, whose physical meaning arises as the overlap ⟨ϕ|ψ⟩ for |ψ⟩ a pure state (description of preparation) and ⟨ϕ| a projective measurement. However, current quantum theory does not formally address the consequences of a changing inner product during the interval between preparation and measurement. We establish a theoretical framework for such a changing inner product, which we show is consistent with standard quantum mechanics. Furthermore, we show that this change is described by a quantum operation, which is tomographically observable, and we elucidate how our result is strongly related to the exploding topic of PT-symmetric quantum mechanics. We explain how to realize experimentally a changing inner product for a qubit in terms of a qutrit protocol with a unitary channel.

Sujit Sarkar (Poornaprajna Institute of Scientific Research, India), 9/06/2022, 15:00 London time (=14:00 GMT)

Topological quantum criticality in non-Hermitian extended Kitaev chain

Abstract: An attempt is made to study the quantum criticality in non-Hermitian system with topological characterization. We use the zero mode solutions to characterize the topological phases and, criticality and also to construct the phase diagram. The Hermitian counterpart of the model Hamiltonian possess quite a few interesting features such as Majorana zero modes (MZMs) at criticality, unique topological phase transition on the critical line and hence these unique features are of an interest to study in the non-Hermitian case also. We observe a unique behavior of critical lines in presence of non-Hermiticity. We study the topological phase transitions in the non-Hermitian case using parametric curves which also reveal the gap closing point through exceptional points. We study bulk and edge properties of the system where at the edge, the stability dependence behavior of MZMs at criticality is studied and at the bulk we study the effect of non-Hermiticity on the topological phases by investigating the behavior of the critical lines. The study of non-Hermiticity on the critical lines revels the rate of receding of the topological phases with respect to the increase in the value of non-Hermiticity. This work gives a new perspective on topological quantum criticality in non-Hermitian quantum system.

Gernot Akemann (Universität Bielefeld, Germany), 23/06/2022, 15:00 London time (=14:00 GMT)

Non-Hermitian random matrices with complex eigenvalues – how to characterise their local statistics

Abstract: Hermitian random matrices have been classified and their spectral statistics is very well understood. They fall into different universality classes regarding the local statistics of their real eigenvalues, in the limit of large matrix size. Here, universality means independence from the distribution of matrix elements. For their non-Hermitian counterparts having complex eigenvalues in general, much less is known to date. While a classification exists, only in few examples the spectral statistics is understood: the three Ginibre ensembles and their chiral counterparts. In this talk I will review some of these recent results and indicate, how a unifying picture of different spectral statistics in the complex plane starts to emerge, based on the spacing distribution in the complex plane.

Chia-Yi Ju (National Sun Yat-sen University, Taiwan), 07/07/2022, 15:00 London time (=14:00 GMT)

Einstein’s quantum elevator: Hermitization of non-Hermitian Hamiltonians via a generalized vielbein formalism

Abstract: Studies have shown that non-Hermitian quantum mechanics can be made self-consistent by introducing a metric for the Hilbert space. However, not only can the metric be daunting, but it can also obscure the underlying physics. Generalizing the vielbein formalism in (pseudo-) Riemannian geometry, we can transform the Hilbert space of a non-Hermitian system into another Hilbert space through generalized vielbein operators. It can be shown that the induced Hamiltonians for the new spaces are always Hermitian. Hence, the generalized vielbein operators can be considered a bridge between non-Hermitian quantum mechanics and its Hermitian counterpart.

based on: C.Y. Ju, A. Miranowicz, F. Minganti, C.T. Chan, G.Y. Chen, F. Nori
Einstein’s quantum elevator: Hermitization of non-Hermitian Hamiltonians via a generalized vielbein formalism Phys. Rev. Research 4, 023070 (2022)

Eric Pap (University of Groningen, The Netherlands), 21/07/2022, 15:00 London time (=14:00 GMT)

A unified parallel transport description of non-cyclic states, exceptional points and complex geometric phases

Abstract: Right after Berry discovered geometric phases in Hermitian adiabatic quantum mechanics, a parallel transport interpretation of adiabatic state evolution was reported by Simon. Soon after, also adiabatic state evolution for non-Hermitian Hamiltonians was studied, e.g. by Garrison and Wright. However, a parallel transport interpretation for this had not been found. The main challenge is to include non-cyclic states, which can appear around exceptional points in the parameter space. We show that such a parallel transport description does exist by using a bundle with non-standard geometry. Using this rigorous framework, we have a more general perspective on e.g. the definition of exceptional points, the quantum geometric tensor, and the case of topological geometric phases.

Jan Wiersig (Otto-von-Guericke University Magdeburg, Germany), 04/08/2022, 15:00 London time (=14:00 GMT)

Quantifying the response of open systems at exceptional points

Abstract: One reason for the considerable attention of exceptional points in photonics, plasmonics, and acoustics is the strong response of open systems to external perturbations and excitations at such degeneracies. In the first part of the talk I introduce the spectral response strength and the eigenstate response strength to quantify the response to generic perturbations in terms of energy splittings and eigenstate changes. The second part of this talk deals with the concept of the distance of a given exceptional point in matrix space to the set of diabolic points. This distance determines an upper bound for the spectral response strength. A small distance therefore implies a weak spectral response to perturbations. This finding has profound consequences for physical realizations of exceptional points that rely on perturbing a diabolic point.