non-Hermitian Theory Research Articles

Non-hermitian Dirac theory from Lindbladian dynamics

This study investigates the intricate relationship between dissipative processes of open quantum systems and the non-Hermitian quantum field theory of relativistic fermionic systems. By examining the influence of dissipative effects on Dirac fermions via Lindblad formalism, we elucidate the effects of coupling relativistic Dirac particles with the environment and show the lack of manifest Lorentz invariance. Employing rigorous theoretical analysis, we generalize the collisionless Boltzmann equations for the relativistic dissipation-driven fermionic system and find the Lyapunov equation, which governs the stationary solutions. Using our formalism, one presents a simple non-Hermitian model that the relativistic fermionic particles and anti-particles are stable. Going further, using the solution to the Lyapunov equations, one analyses the effect of dissipation on the stationary charge imbalance of this non-Hermitian model and finds that the dissipation can induce a new kind of charge imbalance compared with the collisionless equilibrium case.

Temporary Anions of Benzoxazole in Charge-Transfer Cluster Photodetachment.

The photoelectron spectra of cluster anions of superoxide (O2-) solvated by one molecule of benzoxazole (BzOx) reveal two competing photodetachment mechanisms: a direct photoemission from the solvated cluster core and an indirect pathway involving temporary anion states of benzoxazole accessed via the O2-·BzOx → O2·BzOx- charge-transfer transitions. Benzoxazole is a bicyclic unsaturated organic molecule that does not form permanent anions. However, its low-lying vacant π* orbitals permit a resonant capture of the electron emitted from the O2- cluster core. The non-Hermitian theory using a complex absorbing potential predicts the existence of two BzOx- π* resonances within the experimental energy range: resonance A (π1*) at 0.891 eV and resonance B (π2*) at 1.76 eV, relative to the onset of the BzOx + e- continuum at the ground-state geometry of neutral BzOx. Within the clusters, the O2·BzOx- charge-transfer states are partially stabilized relative to the free-electron limit by interactions with the O2 molecule. These interactions depend on the electronic states of both species. The theory predicts that at the O2-·BzOx cluster geometry, the O2(X3Σg-)·BzOx-(A) and O2(a1Δg)·BzOx-(A) states lie at 0.56 and 0.47 eV (vertically) above the respective neutral states. The O2(3Σg-)·BzOx-(B) resonance is found 1.43 eV (vertically) above O2(X3Σg-)·BzOx. Intense signatures of both BzOx- resonances and the three above-mentioned charge-transfer cluster states, O2(X3Σg-)·BzOx-(A), O2(a1Δg)·BzOx-(A), and O2(3Σg-)·BzOx-(B) are observed in the 355 nm (3.495 eV) and 532 nm (2.330 eV) photoelectron spectra of the O2-·BzOx cluster anion.

Power-law correlation in the homogeneous disordered state of anisotropically self-propelled systems

Self-propelled particles display unique collective phenomena, due to the intrinsic coupling of density and polarity. For instance, the giant number fluctuation appears in the orientationally ordered state, and the motility-induced phase separation appears in systems with repulsion. Effects of strong noise typically lead to a homogeneous disordered state, in which the coupling of density and polarity can still play a significant role. Here we study universal properties of the homogeneous disordered state in two-dimensional systems with uniaxially anisotropic self-propulsion. Using hydrodynamic arguments, we propose that the density correlation and polarity correlation generically exhibit power-law decay with distinct exponents (−2 and −4, respectively) through the coupling of density and polarity. Simulations of self-propelled lattice gas models indeed show the predicted power-law correlations, regardless of whether the interaction type is repulsion or alignment. Further, by mapping the model to a two-component boson system and employing non-Hermitian perturbation theory, we obtain the analytical expression for the structure factors, the Fourier transform of the correlation functions. This reveals that even the first order of the interaction strength induces the power-law correlations. Published by the American Physical Society 2024

Phase transitions and thermodynamic cycles in the broken PT-regime

We propose a new type of quantum thermodynamic cycle whose efficiency is greater than the one of the classical Carnot cycle for the same conditions for a system when viewed as homogeneous. In our model, this type of cycle only exists in the low-temperature regime in the spontaneously broken parity-time-reversal (PT) symmetry regime of a non-Hermitian quantum theory and does not manifest in the PT-symmetric regime. We discuss this effect for an ensemble based on a model of a single boson coupled in a non-Hermitian way to a bath of different types of bosons with and without a time-dependent boundary. The cycle cannot be set up when considering our system as heterogeneous, i.e. undergoing a first-order phase transition. Within that interpretation, we find that the entropy is vanishing throughout the spontaneously broken PT-regime.

Non-Hermitian linear-response theory for spin diffusion in quantum systems

Spin transport theory of non-Hermitian quantum systems, where the non-Hermiticity led to a series of exotic phenomena when the system experiences dissipation to an environment is proposed. The goal is a better understanding of the spin diffusion in the one-dimensional non-Hermitian XXZ model which allows the transport coefficients be computed analytically. Moreover, we analyzed the electric transport in the one-dimensional non-Hermitian Hubbard model which is a very important model of electrons strongly correlated, where we have investigated the effect of non-Hermitian parameters like the imaginary hopping on AC and DC conductivities of the system. We analyzed the large U limit of this model, where such non-Hermiticity contributes with a minus sign in the virtual exchange of quasiparticles and the behavior of the ground state energy and low-lying excitations is reversed.

Non-Hermitian topological theory of finite-lifetime quasiparticles: Prediction of bulk Fermi arc due to exceptional point

Non-Hermitian topological theory of finite-lifetime quasiparticles: Prediction of bulk Fermi arc due to exceptional point

Anomalous dispersion, superluminality, and instabilities in two-flavor theories with local non-Hermitian mass mixing

Pseudo-Hermitian field theories possess a global continuous “similarity” symmetry, interconnecting the theories with the same physical particle content and an identical mass spectrum. In their regimes with real spectra, within this family of similarity transformations, there is a map from the non-Hermitian theory to its Hermitian similarity partner. We promote the similarity transformation to a local symmetry, which requires the introduction of a new vector similarity field as a connection in the similarity space of non-Hermitian theories. In the case of non-Hermitian two-flavor scalar or fermion mixing and by virtue of a novel IR/UV mixing effect, the effect of inhomogeneous non-Hermiticity then reveals itself via anomalous dispersion, instabilities, and superluminal group velocities at very high momenta, thus setting an upper bound on the particle momentum propagating through inhomogeneous backgrounds characterized by Lagrangians with non-Hermitian mass matrices. Such a non-Hermitian extension of the Standard Model of particle physics, encoded in a weak inhomogeneity of the non-Hermitian part of the fermion mass matrix, may nevertheless provide us with a low-energy particle spectrum consistent with experimentally observed properties. Published by the American Physical Society 2024

Non-Hermitian Moiré Valley Filter.

A valley filter capable of generating a valley-polarized current is a crucial element in valleytronics, yet its implementation remains challenging. Here, we propose a valley filter made of a graphene bilayer which exhibits a 1D moiré pattern in the overlapping region of the two layers controlled by heterostrain. In the presence of a lattice modulation between layers, electrons propagating in one layer can have valley-dependent dissipation due to valley asymmetric interlayer coupling, thus giving rise to a valley-polarized current. Such a process can be described by an effective non-Hermitian theory, in which the valley filter is driven by a valley-resolved non-Hermitian skin effect. Nearly 100% valley polarization can be achieved within a wide parameter range and the functionality of the valley filter is electrically tunable. The non-Hermitian topological scenario of the valley filter ensures high tolerance against imperfections such as disorder and edge defects. Our work opens a new route for efficient and robust valley filters while significantly relaxing the stringent implementation requirements.

Complex energy-based description of alpha-cluster lifetime in intense laser fields

The phenomenon of laser-assisted alpha decay is studied within the framework of non-Hermitian quantum theory integrated with the (t, t′)-formalism. The width and the lifetime of the alpha cluster of a specific isotope is extracted from the spectrum of the complex-scaled Hamiltonian operator of the nuclear system and estimated numerically. The laser-field-induced correction to the lifetime is computed by first-order (t, t′)-perturbation calculation with regard to different polarization states and control parameters of the external laser field, adjusting appropriate intensity and photon energy pairs by examining the limit of the non-relativistic approximation of the problem.

Non-Hermitian non-equipartition theory for trapped particles

The equipartition theorem is an elegant cornerstone theory of thermal and statistical physics. However, it fails to address some contemporary problems, such as those associated with optical and acoustic trapping, due to the non-Hermitian nature of the external wave-induced force. We use stochastic calculus to solve the Langevin equation and thereby analytically generalize the equipartition theorem to a theory that we denote the non-Hermitian non-equipartition theory. We use the non-Hermitian non-equipartition theory to calculate the relevant statistics, which reveal that the averaged kinetic and potential energies are no longer equal to kBT/2 and are not equipartitioned. As examples, we apply non-Hermitian non-equipartition theory to derive the connection between the non-Hermitian trapping force and particle statistics, whereby measurement of the latter can determine the former. Furthermore, we apply a non-Hermitian force to convert a saddle potential into a stable potential, leading to a different type of stable state.

Speeding Up Entanglement Generation by Proximity to Higher-Order Exceptional Points.

Entanglement is a key resource for quantum information technologies ranging from quantum sensing to quantum computing. Conventionally, the entanglement between two coupled qubits is established at the timescale of the inverse of the coupling strength. In this Letter, we study two weakly coupled non-Hermitian qubits and observe entanglement generation at a significantly shorter timescale by proximity to a higher-order exceptional point. We establish a non-Hermitian perturbation theory based on constructing a biorthogonal complete basis and further identify the optimal condition to obtain the maximally entangled state. Our study of speeding up entanglement generation in non-Hermitian quantum systems opens new avenues for harnessing coherent nonunitary dissipation for quantum technologies.

Non-Hermitian Waveguide Cavity QED with Tunable Atomic Mirrors.

Optical mirrors determine cavity properties by means of light reflection. Imperfect reflection gives rise to open cavities with photon loss. We study an open cavity made of atom-dimer mirrors with a tunable reflection spectrum. We find that the atomic cavity shows anti-PT symmetry. The anti-PT phase transition controlled by atomic couplings in mirrors indicates the emergence of two degenerate cavity supermodes. Interestingly, a threshold of mirror reflection is identified for realizing strong coherent cavity-atom coupling. This reflection threshold reveals the criterion of atomic mirrors to produce a good cavity. Moreover, cavity quantum electrodynamics with a probe atom shows mirror-tuned properties, including reflection-dependent polaritons formed by the cavity and probe atom. Our Letter presents a non-Hermitian theory of an anti-PT atomic cavity, which may have applications in quantum optics and quantum computation.

Taming Dyson-Schwinger Equations with Null States.

In quantum field theory, the Dyson-Schwinger equations are an infinite set of coupled equations relating n-point Green's functions in a self-consistent manner. They have found important applications in nonperturbative studies, ranging from quantum chromodynamics and hadron physics to strongly correlated electron systems. However, they are notoriously formidable to solve. One of the main obstacles is that a finite truncation of the infinite system is underdetermined. Recently, Bender etal. [Phys. Rev. Lett. 130, 101602 (2023)PRLTAO0031-900710.1103/PhysRevLett.130.101602] proposed to make use of the large-n asymptotic behaviors and successfully obtained accurate results in D=0 spacetime. At higher D, it seems more difficult to deduce the large-n behaviors. In this Letter, we propose another avenue in light of the null bootstrap. The underdetermined system is solved by imposing the null state condition. This approach can be extended to D>0 more readily. As concrete examples, we show that the cases of D=0 and D=1 indeed converge to the exact results for several Hermitian and non-Hermitian theories of the gϕ^{n} type, including the complex solutions.

Q-Factor Optimization of Modes in Ordered and Disordered Photonic Systems Using Non-Hermitian Perturbation Theory.

The quality factor, Q, of photonic resonators permeates most figures of merit in applications that rely on cavity-enhanced light-matter interaction such as all-optical information processing, high-resolution sensing, or ultralow-threshold lasing. As a consequence, large-scale efforts have been devoted to understanding and efficiently computing and optimizing the Q of optical resonators in the design stage. This has generated large know-how on the relation between physical quantities of the cavity, e.g., Q, and controllable parameters, e.g., hole positions, for engineered cavities in gaped photonic crystals. However, such a correspondence is much less intuitive in the case of modes in disordered photonic media, e.g., Anderson-localized modes. Here, we demonstrate that the theoretical framework of quasinormal modes (QNMs), a non-Hermitian perturbation theory for shifting material boundaries, and a finite-element complex eigensolver provide an ideal toolbox for the automated shape optimization of Q of a single photonic mode in both ordered and disordered environments. We benchmark the non-Hermitian perturbation formula and employ it to optimize the Q-factor of a photonic mode relative to the position of vertically etched holes in a dielectric slab for two different settings: first, for the fundamental mode of L3 cavities with various footprints, demonstrating that the approach simultaneously takes in-plane and out-of-plane losses into account and leads to minor modal structure modifications; and second, for an Anderson-localized mode with an initial Q of 200, which evolves into a completely different mode, displaying a threefold reduction in the mode volume, a different overall spatial location, and, notably, a 3 order of magnitude increase in Q.

Extension of the Goldstone and the Englert-Brout-Higgs mechanisms to non-Hermitian theories

We discuss the extension of the Goldstone and Englert-Brout-Higgs mechanisms to non-Hermitian Hamiltonians that possess an antilinear PT symmetry. We study a model due to Alexandre, Ellis, Millington and Seynaeve and show that for the spontaneous breakdown of a continuous global symmetry we obtain a massless Goldstone boson in all three of the antilinear symmetry realizations: eigenvalues real, eigenvalues in complex conjugate pairs, and eigenvalues real but eigenvectors incomplete. In this last case we show that it is possible for the Goldstone boson mode to be a zero-norm state. For the breakdown of a continuous local symmetry the gauge boson acquires a non-zero 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.

Real energies and Berry phases in all PT-regimes in time-dependent non-Hermitian theories

We demonstrate that the existence of a Hermitian time-dependent intertwining operator that maps the non-Hermitian time-dependent energy operator to its Hermitian conjugate and its right to its left eigenstates guarantees the reality of the instantaneous energies. This property holds throughout all three -regimes, in the time-independent scenario referred to as the -symmetric regime, the exceptional point and the spontaneously broken -regime. We also propose a modified adiabatic approximation consisting of an expansion of the wavefunctions in terms the instantaneous eigenstates of the energy operator, instead of the usually used eigenfunctions of the Hamiltonian. We show that this proposal always leads to real Berry phases. We illustrate the working of our general proposals with two explicit examples for a time-dependent non-Hermitian spin model.

Non-Hermitian topology and exceptional-point geometries

Non-Hermitian theory is a theoretical framework used to describe open systems. It offers a powerful tool in the characterization of both the intrinsic degrees of freedom of a system and the interactions with the external environment. The non-Hermitian framework consists of mathematical structures that are fundamentally different from those of Hermitian theories. These structures not only underpin novel approaches for precisely tailoring non-Hermitian systems for applications but also give rise to topologies not found in Hermitian systems. In this Review, we provide an overview of non-Hermitian topology by establishing its relationship with the behaviours of complex eigenvalues and biorthogonal eigenvectors. Special attention is given to exceptional points — branch-point singularities on the complex eigenvalue manifolds that exhibit nontrivial topological properties. We also discuss recent developments in non-Hermitian band topology, such as the non-Hermitian skin effect and non-Hermitian topological classifications. Non-Hermitian theory consists of mathematical structures that are used to describe open systems, which can give rise to non-Hermitian topology not found in Hermitian systems. This Review provides an overview of non-Hermitian band topology and discusses recent developments, such as the non-Hermitian skin effect and non-Hermitian topological classifications.

Time-dependent [formula omitted]-operators as Lewis-Riesenfeld invariants in non-Hermitian theories

C-operators were introduced as involution operators in non-Hermitian theories that commute with the time-independent Hamiltonians and the parity/time-reversal operator. Here we propose a definition for time-dependent C(t)-operators and demonstrate that for a particular signature they may be expanded in terms of time-dependent biorthonormal left and right eigenvectors of Lewis-Riesenfeld invariants. The vanishing commutation relation between the C-operator and the Hamiltonian in the time-independent case is replaced by the Lewis-Riesenfeld equation in the time-dependent scenario. Thus, C(t)-operators are always Lewis-Riesenfeld invariants, whereas the inverse is only true in certain circumstances. We demonstrate the working of the generalities for a non-Hermitian two-level matrix Hamiltonian. We show that solutions for C(t) and the time-dependent metric operator may be found that hold in all three PT-regimes, i.e., the PT-regime, the spontaneously broken PT-regime and at the exceptional point.

Floquet topological properties in the non-Hermitian long-range system with complex hopping amplitudes

Non-equilibrium phases of matter have attracted much attention in recent years, among which the Floquet phase is a hot point. In this work, based on the periodic driving non-Hermitian model, we reveal that the winding number calculated in the framework of the Bloch band theory has a direct connection with the number of edge states even though the non-Hermiticity is present. Further, we find that the change of the phase of the hopping amplitude can induce the topological phase transitions. Precisely speaking, the increase in the value of the phase can bring the system into a topological phase with a large topological number. Moreover, it can be unveiled that the introduction of the purely imaginary hopping term brings an extremely rich phase diagram. In addition, we can select the even topological invariant exactly from the unlimited winding numbers if we only consider the next-nearest neighbor hopping term. Here, the results obtained may be useful for understanding the periodic driving non-Hermitian theory.

Non-Hermitian morphing of topological modes.

Topological modes (TMs) are usually localized at defects or boundaries of a much larger topological lattice1,2. Recent studies of non-Hermitian band theories unveiled the non-Hermitian skin effect (NHSE), by which the bulk states collapse to the boundary as skin modes3-6. Here we explore the NHSE to reshape the wavefunctions of TMs by delocalizing them from the boundary. At a critical non-Hermitian parameter, the in-gap TMs even become completely extended in the entire bulk lattice, forming an 'extended mode outside of a continuum'. These extended modes are still protected by bulk-band topology, making them robust against local disorders. The morphing of TM wavefunction is experimentally realized in active mechanical lattices in both one-dimensional and two-dimensional topological lattices, as well as in a higher-order topological lattice. Furthermore, by the judicious engineering of the non-Hermiticity distribution, the TMs can deform into a diversity of shapes. Our findings not only broaden and deepen the current understanding of the TMs and the NHSE but also open new grounds for topological applications.