The Seminars 2021: Term II

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: A minimum length scale of the order of Planck length is a feature of many models of quantum gravity that seek to unify quantum mechanics and gravitation. Recently, Perivolaropoulos in his seminal work [1] predicted the simultaneous existence of minimal and maximal length measurements. More recently, we have shown that both measurable lengths can be obtained from position-dependent noncommutativity [2]. In this talk, we present an alternative derivation of these lengths from non-Hermitian position-dependent noncommutativity [3]. We explore in one hand, the similarities between the maximal uncertainty measurement and the classical properties of gravity. Then, the connection between the minimal uncertainties and the non-Hermicity quantum mechanic scenarios. We will show that theexistence of minimal uncertainties are the consequences of non-Hermicities of some operators that are generators of the non-commutative algebra. With an appropriate Dyson map, we demonstrate by a similarity transformation that the physically meaningful of dynamical quantum systems is generated by a hidden Hermitian position-dependent non-commutative algebra. This transformation, neither changes the properties of gravity and nor removes the uncertainty measurements at this scale. It just gets rid of the fuzziness induced by minimal uncertainty measurements. Finally, we study eigenvalue problem of a free particle in a square-well potential in these new Hermitian variables.

[1] : L. Perivolaropoulos, Cosmological horizons, uncertainty principle, and maximum length quantum mechanics, Phys. Rev. D 95, 103523 (2017)

[2] : L. Lawson, Minimal and maximal lengths from position-dependent noncommutativity, J. Phys. A: Math. Theor. 53, 115303 (2020)

[3] L. Lawson, Minimal and maximal lengths from non-Hermitian position-dependent noncommutativity, arXiv: 2106.03586

Jacob Muldoon, (Indiana University Purdue University, US), 30/09, 14:00 London time (=14:00 GMT)

Floquet exceptional contours in Lindblad dynamics with time-periodic drive and dissipation

Abstract: The dynamics of an isolated quantum system is coherent and unitary. Weak coupling to the environment leads to decoherence, which is traditionally modeled with a Lindblad equation for the system’s density matrix. Starting from a pure state, such a system approaches a steady state (mixed or otherwise) in an underdamped or overdamped manner. This transition occurs at an eigenvalue degeneracy of a Lindblad superoperator, called an exceptional point (EP), where corresponding eigenvectors coalesce. Recent years have seen an explosion of interest in creating exceptional points in a truly
quantum domain, driven by the enhanced sensitivity and topological features EPs have shown in their classical realizations. Here, we present Floquet analysis of a prototypical qubit whose drive or dissipator strengths are varied periodically. We consider models with a single dissipator that generate global loss (phase damping) or mode-selective loss (spontaneous emission). In all cases, we find that periodic modulations lead to EP lines at small dissipator strengths and a rich EP structure in the parameter space. Our analytical and numerical results show that extending Lindblad Liouvillians to the Floquet domain is a potentially preferred route to accessing exceptional points in the transient dynamics towards the Lindblad steady state.

Ross Wakefield, (University of Bristol, UK), 30/09, 14:30 London time (=14:00 GMT)

Photonic quantum simulations of coupled PT-symmetric Hamiltonians

Abstract: Parity-time (PT) symmetric Hamiltonians are generally non-Hermitian and give rise to exotic behaviour in quantum systems at exceptional points, where eigenvectors coalesce. The recent realisation of PT-symmetric Hamiltonians in quantum systems has ignited efforts to simulate and investigate many-particle quantum systems across exceptional points. Here we use a programmable integrated photonic chip to simulate a model comprised of twin pairs of PT-symmetric Hamiltonians, with each the time reverse of its twin. We simulate quantum dynamics across exceptional points including two- and
three-particle interference, and a particle-trembling behaviour that arises due to interference between subsystems undergoing time-reversed evolutions. These results show how programmable quantum simulators can be used to investigate foundational questions in quantum mechanics.

Abdelhafid Bounames, (University of Jijel, Algeria), 30/09, 15:00 London time (=14:00 GMT)

A real expectation value of the time-dependent non-Hermitian Hamiltonians

Abstract: In order to solve the Schrödinger equation associated with an explicitly time-dependent (TD) non-Hermitian Hamiltonian, we introduce a unitary transformation that maps the initial Hamiltonian to a time-independent PT-symmetric one. Consequently, the solution of the time-dependent Schrodinger equation becomes easily deduced and the evolution preserves the C(t)PT-inner product, where C(t) is obtained from the usual charge conjugation operator C through a time-dependent unitary transformation. The expectation value of the TD non-Hermitian Hamiltonian in the C(t)PT normed states is guaranteed to be real. As an illustration, we consider a class of quantum time-dependent mass oscillators with a complex linear driving force. The expectation value of the Hamiltonian, the uncertainty relation and probability density have been calculated.

Himadri Barman, (Zhejiang University, Chinae), 30/09, 15:30 London time (=14:00 GMT)

A tale of two kinds of exceptional point in a hydrogen molecule

Abstract: We study the parity and time-reversal (PT) symmetric quantum physics in a non-Hermitian non-relativistic hydrogen molecule with local (Hubbard type) Coulomb interaction. We consider
non-Hermiticity generated from both kinetic and orbital energies of the atoms and encounter the existence of two different types of exceptional points (EPs) in pairs. These two kinds of EP are characteristically different and depend differently on the interaction strength. Our discovery may open the gates of a rich physics emerging out of a simple Hamiltonian resembling a two-site Hubbard model.

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

New classes of dynamical systems with benign ghosts

Abstract: A system with ghosts is a quantum system where the spectrum of the Hamiltonian has no bottom: there are states with arbitrary low and arbitrary high energies. In many such systems, the ghosts are “malignant”, bringing about the blow up of the classical
trajectories and quantum collapse associated with violation of unitarity. But there are also many systems where the ghosts are there, but they are benign: no blow up in the classical dynamics and no collapse.
We discuss three large classes of such benign systems:
1. The systems obtained by a variation of any ordinary Hamiltonian system with a double set of dynamic variables: (qi , pi , Qi = δqi , Pi = δpi).
2. The systems describing the motion over a Lorentzian manifold whose metric is not positive definite.
3. Exactly solvable systems. In particular, we discuss the 2-dimensional
KdV system with reversed time and spatial coordinates. The Lagrangian of such a system involves higher time derivatives.

Takanobu Taira, (City, University of London, UK), 14/10, 14:00 London time (=14:00 GMT)

Non-Hermitian gauge field theory and BPS solutions

Abstract: We present an overview of some key results in a recent series devoted to nonHermitian gauge field theories with SU(N) continuous symmetries and modified CPT symmetries. We demonstrate that Goldstones theorem and Higgs mechanism work conventionally in the CPT symmetric regime. However, it breaks down when the theory is in CPT broken regime. When the fields are in the adjoint representation of SU(N), we identify the t’Hooft-Polyakov monopoles using 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 and 3+1 dimensional Skyrme models for which we find new types of complex solutions. We will present the mechanism for which the energy of the complex soliton is real due to the CPT symmetry of the theory.

Rebecca Tenney, (City, University of London, UK), 14/10, 14:30 London time (=14:00 GMT)

New exact and approximation methods for time-dependent non-Hermitian quantum systems

Abstract: We present several new methods for determining the time-dependent metric for time-dependent non-Hermitian quantum systems. These methods include identifying a complex point transformation as a map from a solvable time-independent system to an explicitly time-dependent non-Hermitian system. This map can then be used to construct the time-dependent non-Hermitian invariant for the latter system, which in turn may be utilized in the construction of Dyson maps, hence metric operators, due to being pseudo Hermitian.

Akhil Kumar, (Indian Institute of Science Education and Research Kolkata, India), 14/10, 15:00 London time (=13:30 GMT)

Exceptional points and quantum entanglement via unitary and thermal dynamics

Abstract: We investigated the dynamics of a qubit undergoing periodic evolution on a two-dimensional temporal space, that is,
part thermal and part unitary. Properties of such evolution can be studied via Non-Hermitian Floquet Hamiltonians
that can be in the PT-symmetric or PT-broken phases. We map out the phase diagram of the system, along with
the exceptional point (EP) contours through analytical and numerical methods. We analyze the system analytically
for N-cycles to observe the stroboscopic behavior through the effective Floquet Hamiltonian. Our results suggest that
dynamicsin unitary and thermal systems are a new avenue to realize EP degeneracies. Moreover, we extend our single qubit
model to a two-qubit model, which is a Hermitian coupling of qubits in unitary and thermal dynamics, respectively.
We explore its dynamics with different initial configurations, andthe result shows a new possible way of creating
entanglement in a bipartite system.

Kaustubh Argarwal, (Indiana University Purdue University, US), 14/10, 15:30 London time (=14:00 GMT)

Exploring the PT-symmetry breaking threshold in a Kitaev chain with one pair of gain-loss potentials

Abstract: Over the past decade, it has become clear that Parity-Time reversal (PT) symmetric systems represent open, classical and quantum systems with balanced, spatially separated gain and loss that are represented by complex real-space potentials. The dynamics of these gain-loss systems aregoverned by non-Hermitian Hamiltonians with exceptional-point (EP) degeneracies. The eigenvalues change from real to complex conjugates at a critical value of gain-loss strength that is called the PTbreaking threshold. We explore the PT-threshold for a one-dimensional, finite Kitaev chain—a prototype for a p-wave superconductor— in the presence of a single pair of gain and loss potentials. We investigate the fate of this PT-threshold as a function of the superconducting order parameter, on-site potential, and the distance between the gain and loss sites. Along with a robust threshold, we find a rich phase diagram that can be qualitatively understood in terms of the band-structure of the Hermitian Kitaev model. Along with a rich and robust PT-threshold, we point out persistent differences between even and odd parity lattices. The Kitaev chain we have considered, for an even chain with edge gain-loss potentials and superconducting couplingδ&1, we discover re-entrant PT-symmetric phase, and PT-phase boundaries that contain both second and third order EPs [1].

[1] K. S. Agarwal and Y. N. Joglekar, Physical Review A 022218 (2021)

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)

Low-energy scattering and non-Hermitian Hamiltonians

Abstract: Scattering of low-energy/frequency waves has numerous applications in different areas of physics and engineering. This has motivated the development of a rigorous mathematical theory of low-energy quantum scattering in one dimension which provides a highly elaborate iterative construction of the low-energy series expansions for reflection and transmission amplitudes of exponentially decaying potentials. We offer a much simpler method of constructing these series by expressing the transfer matrix of the potential in terms of the evolution operator for a non-stationary non-Hermitian Hamiltonian and employing its Dyson series expansion. Our results find an interesting application in the study of the transmission of scalar waves through a large class of wormholes. We use a similar approach to study the low-frequency scattering defined by the Helmholtz equation where the standard results on low-energy quantum scattering fail to apply. For effectively one-dimensional non-homogeneous optical slabs we obtain explicit formulas for the coefficients of the low-frequency series expansion of the transfer matrix. This in turn allow for the determination of the low-frequency expansions of the reflection, transmission, and absorption coefficients. Our results extend to the scattering problems defined in the half-line and reveal a number of interesting aspects of low-frequency scattering particularly in relation to permittivity profiles having balanced gain and loss. 


B. Azad, F. Loran, and A. Mostafazadeh, “Transmission of low-energy scalar waves through a traversable wormhole,” Eur. Phys. J. C 80, 197 (2020); arXiv:2010.15023. 

F. Loran, and A. Mostafazadeh, “Dynamical formulation of low-energy scattering in one dimension,” J. Math. Phys. 62, 042103 (2021); arXiv:2102.06084

F. Loran and A. Mostafazadeh, Low-frequency scattering defined by the Helmholtz equation in one dimension, J. Phys. A: Math. Theor. 54, 315204 (2021); arXiv: 2105.07895

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

On-demand Parity-Time symmetry in a lone oscillator through complex, synthetic gauge fields [1]

Abstract: What is the fate of an oscillator when its inductance and capacitance are varied while its frequency is kept constant?
Inspired by this question, we propose a protocol to implement parity-time (PT) symmetry in a lone oscillator. Different forms
of constrained variations lead to static, periodic, or arbitrary balanced gain and loss profiles, that can be interpreted as purely imaginary gauge fields. With a state-of-the-art, dynamically tunable LC oscillator comprising synthetic circuit elements, we demonstrate static and Floquet PT breaking transitions. Concurrently, we derive and observe conserved quantities in this open, balanced gain-loss system, both in the static and Floquet cases. Distinct from material or parametric gain and loss mechanisms, our protocol enables on-demand parity-time symmetry in a minimal classical system — a single oscillator — and may be ported to other realizations including metamaterials and optomechanical systems.

[1] In collaboration with Robert Leon Montiel’s group at UNAM, Mexico. arXiv:2109.03846

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

Ladder operators in compatible spaces and the BCH formula

Abstract: My talk is divided in two different parts: in the first part, I consider a large class of ladder operators of pseudo-bosonic type and I construct the eigenstates of their connected number operators. These functions are not necessarily square-integrable but still their product is integrable. This suggests the introduction of a “compatibility form” which extends the scalar product in L2(R). I will show some examples and construct bi-coherent states as weak eigenvectors of the pseudo-bosonic annihilation operators.

In the second part I will comment on the role of unbounded operators in the analysis of the displacement operator. In particular, I will briefly describe some subtle aspects and some results on the Baker-Campbell-Hausdorff formula, with an eye on applications to coherent and bi-coherent states.

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

Symmetry and Topology in Non-Hermitian Physics

Abstract: Non-Hermiticity enriches topological phases beyond the existing framework for Hermitian topological phases. Here, we develop a general theory of symmetry and topology in non-Hermitian physics. We demonstrate that non-Hermiticity ramifies and unifies the 10-fold internal symmetry for insulators and superconductors, leading to 38-fold symmetry [1]. Moreover, we find two types of complex-energy gaps, both of which constitute non-Hermitian topology. On the basis of these fundamental insights in non-Hermitian physics, we classify topological phases of non-Hermitian systems. Our theoretical framework also enables classification of exceptional points and non-Hermitian topological semimetals [2]. Furthermore, we show that intrinsic non-Hermitian topology leads to the non-Hermitian skin effect (i.e., extreme sensitivity to boundary conditions due to non-Hermiticity) [3]. We also develop a topological field theory of non-Hermitian systems [4]. Because of the dissipative and nonequilibrium nature of non-Hermiticity, our theory is formulated solely in terms of spatial degrees of freedom, which contrasts with the conventional theory defined in spacetime.

[1] K. Kawabata, K. Shiozaki, M. Ueda, and M. Sato, Phys. Rev. X 9, 041015 (2019).

[2] K. Kawabata, T. Bessho, and M. Sato, Phys. Rev. Lett. 123, 066405 (2019).

[3] N. Okuma, K. Kawabata, K. Shiozaki, and M. Sato, Phys. Rev. Lett. 124, 086801 (2020).

[4] K. Kawabata, K. Shiozaki, and S. Ryu, Phys. Rev. Lett. 126, 216405 (2021).

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

Exceptional points under symmetry and correlations

Abstract: Recent extensive studies elucidated that non-Hermiticity induces novel topological phenomena which do not have Hermitian counterparts[1]. An representative example is the emergence of exceptional points on which the non-Hermitian band touching occurs due to the violation of diagonalizability.In this talk, we discuss effects of symmetry and correlations on exceptional points. In the first part, we point out that local symmetry may enrich the topological structure of exceptional points by demonstrating the emergence of symmetry-protected exceptional rings with chiral symmetry[2]. In the second part, we show that spatial symmetry may provide efficient tools for searching exceptional points. Specifically, we introduce discriminant indicators for generalized inversion symmetry[3]. In the last part, we discuss correlation effects on the point-gap topology in zero dimension. Our analysis clarifies that correlationsdestroy an exceptional point separating two distinct topological phases whose topological invariant differs by two(N=0 and N=2). This result indicates that correlations changethe topological classification fromZ to Z2[4].


[1] E. J. Bergholtz, J. C. Budich, F. K. Kunst, Rev. Mod. Phys. 93, 015005 (2021)

[2] T. Yoshida, R. Peters, N. Kawakami, and Y. Hatsugai, Phys. Rev. B 99,121101(R) (2019).

[3] T. Yoshida, R. Okugawa, and Y. Hatsugai, arXiv:2111.07077

[4] T. Yoshida,and Y. Hatsugai, Phys. Rev. B 104, 075106 (2021)