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Geometric Phase–Driven Transport in Non-Hermitian Floquet Lattices Near Exceptional Points

Francis Procaccia

Published May 9, 2026• 0 citations
Publisher

Abstract

This work presents a comprehensive study of geometric transport in non-Hermitian Floquet lattices. We derive the non-Hermitian Berry curvature using the biorthogonal formalism and prove an exact algebraic no-go theorem: perfectly anti-correlated driving (φ=0) yields zero net displacement for any wavepacket. When driving symmetry is broken (φ=π/2), a Gaussian wavepacket acquires finite directional displacement per cycle. Using adaptive peak-centered quadrature with analytic derivatives and direct time-dependent wavepacket simulations on finite chains, we demonstrate transport enhancement near the exceptional-point manifold, accompanied by non-monotonic behavior due to moving curvature hotspots. The paper includes full analytical characterization of the EP manifold topology, explicit convergence tests, finite-size scaling, and clear separation between geometric pumping and non-Hermitian skin effect. All Python scripts, raw outputs, and high-resolution figures are provided for full reproducibility. Keywords: non-Hermitian Floquet systems, exceptional points, geometric transport, Berry curvature, no-go theorem, adiabaticity, wavepacket dynamics

DOI
10.5281/zenodo.20103204

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This work presents a comprehensive study of geometric transport in non-Hermitian Floquet lattices. We derive the non-Hermitian Berry curvature using the biorthogonal formalism and prove an exact algebraic no-go theorem: perfectly anti-correlated driving (φ=0) yields zero net displacement for any wavepacket. When driving symmetry is broken (φ=π/2), a Gaussian wavepacket acquires finite directional displacement per cycle. Using adaptive peak-centered quadrature with analytic derivatives and direct time-dependent wavepacket simulations on finite chains, we demonstrate transport enhancement near the exceptional-point manifold, accompanied by non-monotonic behavior due to moving curvature hotspots. The paper includes full analytical characterization of the EP manifold topology, explicit convergence tests, finite-size scaling, and clear separation between geometric pumping and non-Hermitian skin effect. All Python scripts, raw outputs, and high-resolution figures are provided for full reproducibility. Keywords: non-Hermitian Floquet systems, exceptional points, geometric transport, Berry curvature, no-go theorem, adiabaticity, wavepacket dynamics
We show that the analog of the geometric phase for non-Hermitian coupled waveguides with PT\mathcal{PT}PT symmetry and at least one periodically varying parameter can be purely imaginary, is hence no longer a true phase and instead a real multiplier, and will consequently result in the amplification of Floquet sidebands in the system. The sideband peaks seen in the spectrum of the system's eigenstates after evolution along the waveguides can be directly mapped to the spectrum of the derivative of the geometric function. The sidebands are magnified (becoming virtually unstable) as the exceptional point of the system is approached, and nonadiabatic effects begin to appear. Because the system cannot evolve adiabatically in the vicinity of the exceptional point, PT\mathcal{PT}PT symmetry will be observed breaking earlier than theoretically predicted.
We develop a comprehensive, gauge-covariant geometric framework for non-Hermitian quantum systems in the quasi-Hermitian regime, that is, the region of parameter space where the non-Hermitian Hamiltonian admits a real spectrum and a positive-definite metric operator. We build this framework by elevating the Dyson map to a central geometric object. This map is the transformation that converts a non-Hermitian Hamiltonian into an equivalent Hermitian one. From it we construct the Dyson connection and decompose it into Hermitian and anti-Hermitian parts, identified respectively as {\it stretching } and {\it rotation } components. This decomposition cleanly separates the genuine physical metric deformations from the unitary gauge redundancies. Working with manifestly gauge-covariant states, we then derive the complex non-Hermitian Berry phase and the quantum geometric tensor (QGT), and show that the non-Hermitian geometric curvature originates from the non-commutativity of the stretching components at the operator level. We further analyse the geometric singularities near an exceptional point (EP) and uncover a distinct hierarchy of divergences. For a general two-level non-Hermitian model, the quantum metric tensor (QMT) exhibits a leading-order divergence ∼∣εμ∣−2\sim |ε_μ|^{-2}∼∣εμ​∣−2, while the Berry curvature shows a weaker, subleading divergence ∼∣εμ∣−3/2\sim |ε_μ|^{-3/2}∼∣εμ​∣−3/2, with εμε_μεμ​ denoting the parameter displacement from the EP along an individual parameter axis μμμ. Finally, we examine physical realizations of this model, including the non-Hermitian Su--Schrieffer--Heeger (SSH) and Hatano--Nelson (HN) models, where exact analytical results confirm the predicted critical scaling laws and illustrate the metric-deformation-driven non-Hermitian geometries.
We investigate both analytically and numerically the wavepacket's dynamics in momentum space for a Floquet non-Hermitian system with a periodically kicked driven potential. We have deduced the exact expression of a time-evolving wavepacket under the condition of quantum resonance. With this analytical expression, we can investigate thoroughly the temporal behaviors of the directed transport, mean energy, and quantum scrambling. We find interestingly that, by tuning the relative phase between the real part and imaginary part of the kicking potential, one can manipulate the directed transport, mean energy, and quantum scrambling efficiently: When the phase equals to π / 2 , we observe a maximum directed transport and mean energy, while a minimum scrambling phenomenon protected by the PT symmetry; when the phase is π , both the directed transport and the time dependence of the energy are suppressed; in contrast, the quantum scrambling is enhanced by the non-Hermiticity. For the quantum nonresonance case, we numerically find that the quantum interference effects lead to dynamical localization, characterized by the suppression of the directed transport, the time dependence of the energy, and quantum scrambling. Interestingly, these suppression effects can be adjusted by the phase of the non-Hermitian kicking potential. Possible applications of our findings are discussed. Published by the American Physical Society 2024
We consider the space of $n\ifmmode\times\else\texttimes\fi{}n$ non-Hermitian Hamiltonians (n=2n=2n=2, 3, ...) that are equivalent to a single $n\ifmmode\times\else\texttimes\fi{}n$ Jordan block. We focus on adiabatic transport around a closed path (i.e., a loop) within this space, in the limit as the time scale $T=1/\ensuremath{\varepsilon}$ taken to traverse the loop tends to infinity. We show that, for a certain class of loops and a choice of initial state, the state returns to itself and acquires a complex phase that is ${\ensuremath{\varepsilon}}^{\ensuremath{-}1}$ times an expansion in powers of ${\ensuremath{\varepsilon}}^{1/n}$. The exponential of the term of nthn\mathrm{th}nth order (which is equivalent to the ``geometric'' or Berry phase modulo $2\ensuremath{\pi}$) is thus independent of $\ensuremath{\varepsilon}$ as $\ensuremath{\varepsilon}\ensuremath{\rightarrow}0$; it depends only on the homotopy class of the loop and is an integer power of ${e}^{2\ensuremath{\pi}i/n}$. One of the conditions under which these results hold is that the state being transported is, for all points on the loop, that of slowest decay.
A wave function picks up, in addition to the dynamic phase, the geometric (Berry) phase when traversing adiabatically a closed cycle in parameter space. We develop a general multidimensional theory of the geometric phase for (double) cycles around exceptional degeneracies in non-Hermitian Hamiltonians. We show that the geometric phase is exactly $\ensuremath{\pi}$ for symmetric complex Hamiltonians of arbitrary dimension and for nonsymmetric non-Hermitian Hamiltonians of dimension 2. For nonsymmetric non-Hermitian Hamiltonians of higher dimension, the geometric phase tends to $\ensuremath{\pi}$ for small cycles and changes as the cycle size and shape are varied. We find explicitly the leading asymptotic term of this dependence, and describe it in terms of interaction of different energy levels.
Exceptional degeneracies and generically complex spectra of non-Hermitian systems are at the heart of numerous phenomena absent in the Hermitian realm. Recently, it was suggested that Floquet dissipative coupling in the space-time domain may provide a novel mechanism to drive intriguing spectral topology with no static analogs, though its experimental investigation in quantum systems remains elusive. We demonstrate such Floquet dissipative coupling in an ensemble of thermal atoms interacting with two spatially separated optical beams, and observe an anomalous anti-parity-time symmetry phase transition at an exception point far from the phase-transition threshold of the static counterpart. Our protocol sets the stage for Floquet engineering of non-Hermitian topological spectra, and for engineering new quantum phases that cannot exist in static systems.