arXiv subjects

Alexander Felski

Publications and source records attributed to Alexander Felski.

15 recordsLinked to original sources

arXiv2026

Integrable curl-force Hamiltonians: bi-Hamiltonian structure, separability, and periodic orbits

We investigate Hamiltonian curl-force systems with indefinite kinetic energy. We first reconsider Berry's polynomial Hamiltonian curl-force model, whose numerically observed closed trajectories motivated an integrability conjecture. A Painlev\'e analysis yields a non-principal resonance spectrum, so that the corresponding Laurent series cannot accommodate the required number of arbitrary constants of the general solution. The model therefore fails the standard Painlev\'e test. We then introduce a four-parameter curl-force family and identify the parameter locus on which this system is integrable. We construct a second Hamiltonian, compatible Poisson tensors, separated complex characteristic variables, and a Lax representation. More generally, the separated form yields polynomial integrable curl-force Hamiltonians of arbitrary degree. We also show that the same construction admits a higher time-derivative potentialisation whose free limit is the degenerate Pais-Uhlenbeck oscillator. Finally, we analyse zero-curl invariant reductions and elliptic periodic solutions, and exhibit an isolated periodic orbit outside the integrable regime. This illustrates that closed trajectories alone do not imply Liouville or Painlev\'e integrability.

nlin.SI↗

arXiv2026

Parity-induced generalized Brillouin zone without non-Hermitian skin effect

Acute spectral sensitivity to boundary conditions and the formation of a generalized Brillouin zone associated with complex quasimomenta are features frequently attributed to systems with non-trivial non-Hermitian topology, showcasing the non-Hermitian skin effect. We show that, away from the thermodynamic limit, these features themselves are not uniquely tied to this phenomenon; they can similarly arise as parity-induced even-odd effects in non-Hermitian systems without skin effect. Despite an underlying generalized Brillouin zone description, wavefunctions remain delocalized. In addition, the effect can arise in skin-effect models as entirely separate distinguishable feature

cond-mat.mes-hall↗

arXiv2026

Conformal bi-Hamiltonian structure and integrability of an interacting Pais-Uhlenbeck oscillator

We investigate an interacting Pais-Uhlenbeck oscillator with a Landau-Ginzburg type interaction term and analyse its classical dynamics from a geometric and numerical point of view. We show that the resulting fourth-order equation of motion admits a conformal bi-Hamiltonian formulation, possesses a non-trivial set of Lie symmetries and we demonstrate the existence of bounded and regular trajectories in representative parameter regimes. By establishing an explicit correspondence with an integrable generalised H\'enon-Heiles system, we show that the interacting higher-derivative dynamics inherits the integrability properties of the latter. This connection allows us to construct a second conserved Hamiltonian, to clarify the geometric origin of separability, and to obtain explicit classical solutions in terms of elliptic functions. Our results provide a concrete example of an interacting higher-derivative system for which integrability and periodic classical solutions can be established in a fully explicit manner.

nlin.SI↗

arXiv2025

Three-dimensional ghost-free representations of the Pais-Uhlenbeck model from Tri-Hamiltonians

We present a detailed analysis of the sixth-order Pais-Uhlenbeck oscillator and construct three-dimensional ghost-free representations through a Tri-Hamiltonian framework. We identify a six-dimensional Abelian Lie algebra of the PU model's dynamical flow and derive a hierarchy of conserved Hamiltonians governed by multiple compatible Poisson structures. These structures enable the realisation of a complete Tri-Hamiltonian formulation that generates identical dynamical flows. Positive-definite Hamiltonians are constructed, and their relation to the full Tri-Hamiltonian hierarchy is analysed. Furthermore, we develop a mapping between the PU model and a class of three-dimensional coupled second-order systems, revealing explicit conditions for ghost-free equivalence. We also explore the consequences of introducing interaction terms, showing that the multi-Hamiltonian structure is generally lost in such cases.

math-ph↗

arXiv2025

Lie symmetries and ghost-free representations of the Pais-Uhlenbeck model

We investigate the Pais-Uhlenbeck (PU) model, a paradigmatic example of a higher time-derivative theory, by identifying the Lie symmetries of its associated fourth-order dynamical equation. Exploiting these symmetries in conjunction with the model's Bi-Hamiltonian structure, we construct distinct Poisson bracket formulations that preserve the system's dynamics. Amongst other possibilities, this allow us to recast the PU model in a positive definite manner, offering a solution to the long-standing problem of ghost instabilities. Furthermore, we systematically explore a family of transformations that reduce the PU model to equivalent first-order, higher-dimensional systems. Finally we examine the impact on those transformations by adding interaction terms of potential form to the PU model and demonstrate how they usually break the Bi-Hamiltonian structure. Our approach yields a unified framework for interpreting and stabilising higher time-derivative dynamics through a symmetry analysis in some parameter regime.

math-ph↗

arXiv2024

Exceptional Points and Stability in Nonlinear Models of Population Dynamics having $\mathcal{PT}$ symmetry

Nonlinearity and non-Hermiticity, for example due to environmental gain-loss processes, are a common occurrence throughout numerous areas of science and lie at the root of many remarkable phenomena. For the latter, parity-time-reflection ($\mathcal{PT}$) symmetry has played an eminent role in understanding exceptional-point structures and phase transitions in these systems. Yet their interplay has remained by-and-large unexplored. We analyze models governed by the replicator equation of evolutionary game theory and related Lotka-Volterra systems of population dynamics. These are foundational nonlinear models that find widespread application and offer a broad platform for non-Hermitian theory beyond physics. In this context we study the emergence of exceptional points in two cases: (a) when the governing symmetry properties are tied to global properties of the models, and, in contrast, (b) when these symmetries emerge locally around stationary states--in which case the connection between the linear non-Hermitian model and an underlying nonlinear system becomes tenuous. We outline further that when the relevant symmetries are related to global properties, the location of exceptional points in the linearization around coexistence equilibria coincides with abrupt global changes in the stability of the nonlinear dynamics. Exceptional points may thus offer a new local characteristic for the understanding of these systems. Tri-trophic models of population ecology serve as test cases for higher-dimensional systems.

physics.bio-ph↗

arXiv2024

Spectral Riemann Sheet Topology of Gapped Non-Hermitian Systems

We show topological configurations of the complex-valued spectra in gapped non-Hermitian systems. These arise when the distinctive EPs in the energy Riemann sheets of such models are annihilated after threading them across the boundary of the Brillouin zone. This results in a non-trivially closed branch cut that is protected by an energy gap in the spectrum. Their presence or absence establishes topologically distinct configurations for fully non-degenerate systems and tuning between them requires a closing of the gap, forming exceptional point degeneracies. We provide an outlook toward experimental realizations in metasurfaces and single-photon interferometry.

quant-ph↗

arXiv2024

Three perspectives on entropy dynamics in a non-Hermitian two-state system

A comparative study of entropy dynamics as an indicator of physical behavior in an open two-state system with balanced gain and loss is presented. We distinguish the perspective taken in utilizing the conventional framework of Hermitian-adjoint states from an approach that is based on biorthogonal-adjoint states and a third case based on an isospectral mapping. In this it is demonstrated that their differences are rooted in the treatment of the environmental coupling mode. For unbroken $\mathcal{PT}$ symmetry of the system, a notable characteristic feature of the perspective taken is the presence or absence of purity oscillations, with an associated entropy revival. The description of the system is then continued from its $\mathcal{PT}$-symmetric pseudo-Hermitian phase into the regime of spontaneously broken symmetry, in the latter two approaches through a non-analytic operator-based continuation, yielding a Lindblad master equation based on the $\mathcal{PT}$ charge operator $\mathcal{C}$. This phase transition indicates a general connection between the pseudo-Hermitian closed-system and the Lindbladian open-system formalism through a spontaneous breakdown of the underlying physical reflection symmetry.

quant-ph↗

arXiv2022

Thermodynamic properties of non-Hermitian Nambu--Jona-Lasinio models

We investigate the impact of non-Hermiticity on the thermodynamic properties of interacting fermions by examining bilinear extensions to the $3+1$ dimensional $SU(2)$-symmetric Nambu--Jona-Lasinio (NJL) model of quantum chromodynamics at finite temperature and chemical potential. The system is modified through the anti-$PT$-symmetric pseudoscalar bilinear $\bar{\psi}\gamma_5 \psi$ and the $PT$-symmetric pseudovector bilinear $iB_\nu \,\bar{\psi}\gamma_5\gamma^\nu \psi$, introduced with a coupling $g$. Beyond the possibility of dynamical fermion mass generation at finite temperature and chemical potential, our findings establish model-dependent changes in the position of the chiral phase transition and the critical end-point. These are tunable with respect to $g$ in the former case, and both $g$ and $|B|/B_0$ in the latter case, for both lightlike and spacelike fields. Moreover, the behavior of the quark number, entropy, pressure, and energy densities signal a potential fermion or antifermion excess compared to the standard NJL model, due to the pseudoscalar and pseudovector extension respectively. In both cases regions with negative interaction measure $I = \epsilon-3p$ are found. Future indications of such behaviors in strongly interacting fermion systems, for example in the context of neutron star physics, may point toward the presence of non-Hermitian contributions. These trends provide a first indication of curious potential mechanisms for producing non-Hermitian baryon asymmetry. In addition, the formalism described in this study is expected to apply more generally to other Hamiltonians with four-fermion interactions and thus the effects of the non-Hermitian bilinears are likely to be generic.

hep-ph↗

arXiv2021

$\PT$ Symmetry and Renormalisation in Quantum Field Theory

Quantum systems governed by non-Hermitian Hamiltonians with $\PT$ symmetry are special in having real energy eigenvalues bounded below and unitary time evolution. We argue that $\PT$ symmetry may also be important and present at the level of Hermitian quantum field theories because of the process of renormalisation. In some quantum field theories renormalisation leads to $\PT$-symmetric effective Lagrangians. We show how $\PT$ symmetry may allow interpretations that evade ghosts and instabilities present in an interpretation of the theory within a Hermitian framework. From the study of examples $\PT$-symmetric interpretation is naturally built into a path integral formulation of quantum field theory; there is no requirement to calculate explicitly the $\PT$ norm that occurs in Hamiltonian quantum theory. We discuss examples where $\PT$-symmetric field theories emerge from Hermitian field theories due to effects of renormalization. We also consider the effects of renormalization on field theories that are non-Hermitian but $\PT$-symmetric from the start.

hep-th↗

arXiv2021

Towards perturbative renormalization of $ϕ^2(iϕ)^\varepsilon$ quantum field theory

In a previous paper it was shown how to calculate the ground-state energy density $E$ and the $p$-point Green's functions $G_p(x_1,x_2,...,x_p)$ for the $PT$-symmetric quantum field theory defined by the Hamiltonian density $H=\frac{1}{2}(\nablaϕ)^2+\frac{1}{2}ϕ^2(iϕ)^\varepsilon$ in $D$-dimensional Euclidean spacetime, where $ϕ$ is a pseudoscalar field. In this earlier paper $E$ and $G_p(x_1,x_2,...,x_p)$ were expressed as perturbation series in powers of $\varepsilon$ and were calculated to first order in $\varepsilon$. (The parameter $\varepsilon$ is a measure of the nonlinearity of the interaction rather than a coupling constant.) This paper extends these perturbative calculations to the Euclidean Lagrangian $L= \frac{1}{2}(\nablaϕ)^2+\frac{1}{2}μ^2ϕ^2+\frac{1}{2} gμ_0^2ϕ^2\big(iμ_0^{1-D/2}ϕ\big)^\varepsilon-ivϕ$, which now includes renormalization counterterms that are linear and quadratic in the field $ϕ$. The parameter $g$ is a dimensionless coupling strength and $μ_0$ is a scaling factor having dimensions of mass. Expressions are given for the one-, two, and three-point Green's functions, and the renormalized mass, to higher-order in powers of $\varepsilon$ in $D$ dimensions ($0\leq D\leq2$). Renormalization is performed perturbatively to second order in $\varepsilon$ and the structure of the Green's functions is analyzed in the limit $D\to 2$. A sum of the most divergent terms is performed to {\it all} orders in $\varepsilon$. Like the Cheng-Wu summation of leading logarithms in electrodynamics, it is found here that leading logarithmic divergences combine to become mildly algebraic in form. Future work that must be done to complete the perturbative renormalization procedure is discussed.

hep-th↗

arXiv2021

Fermion and meson mass generation in non-Hermitian Nambu--Jona-Lasinio models

We investigate the effects of non-Hermiticity on interacting fermionic systems. We do this by including non-Hermitian bilinear terms into the 3+1 dimensional Nambu--Jona-Lasinio (NJL) model. Two possible bilinear modifications give rise to $\mathcal{PT}$ symmetric theories; this happens when the standard NJL model is extended either by a pseudovector background field $ig \barψγ_5 B_μγ^μψ$ or by an antisymmetric-tensor background field $g \barψF_{μν}γ^μγ^νψ$. The three remaining bilinears are {\it anti}-$\mathcal{PT}$-symmetric in nature, $ig \barψB_μγ^μψ, ig\barψγ_5 ψ$ and $ig\barψ{1}ψ$, so that the Hamiltonian then has no overall symmetry. The pseudovector $ig \barψγ_5 B_μγ^μψ$ and the vector $ig \barψB_μγ^μψ$ combinations, are, in addition, chirally symmetric. Thus, within this framework we are able to examine the effects that the various combinations of non-Hermiticity, $\mathcal{PT}$ symmetry, chiral symmetry and the two-body interactions of the NJL model have on the existence and dynamical generation of a real effective fermion mass (a feature which is absent in the corresponding modified massless free Dirac models) as well as on the masses of the composite particles, the pseudoscalar and scalar mesonic modes ($π$ and $σ$ mesons). Our findings demonstrate that $\mathcal{PT}$ symmetry is not necessary for real fermion mass solutions to exist, rather the two-body interactions of the NJL model supersede the non-Hermitian bilinear effects. The effects of chiral symmetry are evident most clearly in the meson modes, the pseudoscalar of which will always be Goldstone in nature if the system is chirally symmetric. Second solutions of the mesonic equations are also discussed.

hep-ph↗

arXiv2020

Non-Hermitian extension of the Nambu--Jona-Lasinio model in 3+1 and 1+1 dimensions

This paper presents a non-Hermitian PT-symmetric extension of the Nambu--Jona-Lasinio (NJL) model of quantum chromodynamics in 3+1 and 1+1 dimensions. In 3+1 dimensions, the SU(2)-symmetric NJL Hamiltonian $H_{\textrm{NJL}} = \barψ(-i γ^k \partial_k + m_0) ψ- G [ (\barψψ)^2 + (\barψi γ_5 \vecτ ψ)^2 ]$ is extended by the non-Hermitian, PT- and chiral-symmetric bilinear term $ig\barψγ_5 B_μ γ^μ ψ$; in 1+1 dimensions, where $H_{\textrm{NJL}}$ is a form of the Gross-Neveu model, it is extended by the non-Hermitian PT-symmetric but chiral symmetry breaking term $g \barψγ_5 ψ$. In each case, the gap equation is derived and the effects of the non-Hermitian terms on the generated mass are studied. We have several findings: in previous calculations for the free Dirac equation modified to include non-Hermitian bilinear terms, contrary to expectation, no real mass spectrum can be obtained in the chiral limit; in these cases a nonzero bare fermion mass is essential for the realization of PT symmetry in the unbroken regime. Here, in the NJL model, in which four-point interactions are present, we {\it do} find real values for the mass spectrum also in the limit of vanishing bare masses in both 3+1 and 1+1 dimensions, at least for certain specific values of the non-Hermitian couplings $g$. Thus, the four-point interaction overrides the effects leading to PT symmetry-breaking for these parameter values. Further, we find that in both cases, in 3+1 and in 1+1 dimensions, the inclusion of a non-Hermitian bilinear term can contribute to the generated mass. In both models, this contribution can be tuned to be small; we thus fix the fermion mass to its value when $m_0=0$ in the absence of the non-Hermitian term, and then determine the value of the coupling required so as to generate a bare fermion mass.

hep-ph↗

arXiv2018

Analytic eigenvalue structure of a coupled oscillator system beyond the ground state

By analytically continuing the eigenvalue problem of a system of two coupled harmonic oscillators in the complex coupling constant $g$, we have found a continuation structure through which the conventional ground state of the decoupled system is connected to three other lower {\it unconventional} ground states that describe the different combinations of the two constituent oscillators, taking all possible spectral phases of these oscillators into account. In this work we calculate the connecting structures for the higher excitation states of the system and argue that - in contrast to the four-fold Riemann surface identified for the ground state - the general structure is eight-fold instead. Furthermore we show that this structure in principle remains valid for equal oscillator frequencies as well and comment on the similarity of the connection structure to that of the single complex harmonic oscillator.

math-ph↗

arXiv2017

Analytic structure of eigenvalues of coupled quantum systems

By analytically continuing the coupling constant $g$ of a coupled quantum theory, one can, at least in principle, arrive at a state whose energy is lower than the ground state of the theory. The idea is to begin with the uncoupled $g=0$ theory in its ground state, to analytically continue around an exceptional point (square-root singularity) in the complex-coupling-constant plane, and finally to return to the point $g=0$. In the course of this analytic continuation, the uncoupled theory ends up in an unconventional state whose energy is lower than the original ground state energy. However, it is unclear whether one can use this analytic continuation to extract energy from the conventional vacuum state; this process appears to be exothermic but one must do work to vary the coupling constant $g$.

math-ph↗