We show how generic non-Hermitian tight-binding lattice models can be realized in an unconditional, quantum-mechanically consistent manner by constructing an appropriate open quantum system. We focus on the quantum steady states of such models for both fermionic and bosonic systems. Surprisingly, key features and spatial structures in the steady state cannot be simply understood from the non-Hermitian Hamiltonian alone. Using the 1D Hatano-Nelson model as a paradigmatic example, we show that the steady state has a marked sensitivity to boundary conditions. This dependence however is qualitatively and quantitatively distinct from the non-Hermitian skin effect, and has no simple connection to non-Hermitian topology. Further, particle statistics play an unexpected role: the steady-state density profile is dramatically different for fermions versus bosons. Our work highlights the key role of fluctuations in quantum realizations of non-Hermitian dynamics, and provides a starting point for future work on engineered steady states of open quantum systems.
Eigenstates exhibit localization at an open edge in a non-Hermitian lattice due to non-Hermitian skin effect. We here explore another interesting feature of non-Hermitian skin effect and predict quasi-stationary solutions, which are approximately time-independent. We show that the transition from such states to eigenstates is dramatically non-perturbative. We discuss that mathematically extending the boundary of a long non-Hermitian lattice to infinity can lead to nontrivial solution. We consider a non-Hermitian variant of the Su-Schrieffer-Heeger (SSH) model and predict non-topological but ro…
A reformulation of quantum mechanics called ``hybrid interaction picture'' (HIP) is proposed. In contrast to its current non-Hermitian-interaction-picture (NIP) predecessor the new formalism combines a simplification of the NIP Hilbert-space metric with an innovative preconditioning of the observables of interest. Via an illustrative example it is shown that the idea might open new ways towards optimal model-building strategies in quantum theory of unitary systems.
The interaction picture in a non-Hermitian realization is discussed in detail and considered for its practical use in many-body quantum physics. The resulting non-Hermitian interaction-picture (NHIP) description of dynamics, in which both the wave functions and operators belonging to physical observables cease to remain constant in time, is a non-Hermitian generalization of the traditional Dirac picture of standard quantum mechanics, which itself is widely used in quantum field theory calculations. Particular attention is paid here to the variational (or, better, bivariational) and dynamical (…
In study of pseudo(quasi)-hermitian operators, the key role is played by the positive-definite metric operator. It enables physical interpretation of the considered systems. In the article, we study the pseudo-hermitian systems with constant number of particles in equilibrium. We show that the explicit knowledge of the metric operator is not essential for study of thermodynamic properties of the system. We introduce a simple example where the physically relevant quantities are derived without explicit calculation of either metric operator or spectrum of the Hamiltonian.
Quasi-particles described by Green's functions of equilibrium systems exhibit non-Hermitian topological phenomena because of their finite lifetime. This non-Hermitian perspective on equilibrium systems provides new insights into correlated systems and attracts much interest because of its potential to solve open questions in correlated compounds. In this paper, we provide a concise review of the non-Hermitian topological band structures for quantum many-body systems in equilibrium as well as their classification.
In this paper we develop the Non-Equilibrium Green's Function (NEGF) formalism for a dissipative molecular junction that consists of a central molecular system with one-dimensional electronic transport coupled to a phonon environment and attached to multiple electronic leads. Our approach is partitionless - initial preparation of the system places the whole system in the correct canonical equilibrium state - and is valid for an external bias with arbitrary time dependence. Using path integrals as an intermediary tool, we apply a two-time Hubbard-Stratonovich transformation to the phonon influe…
We demonstrate a new type of non-Hermitian phase transition in open systems far from thermal equilibrium, which takes place in coupled systems interacting with reservoirs at different temperatures. The frequency of the maximum in the spectrum of energy flow through the system plays the role of the order parameter, and is determined by an analog of the -potential. The phase transition is exhibited in the frequency splitting of the spectrum at a critical point, the value of which is determined by the relaxation rates and the coupling strengths. Near the critical point, fluctuations of the order …
There are many formalisms to describe quantum decoherence. However, many of them give a non general and ad hoc definition of "pointer basis" or "moving preferred basis", and this fact is a problem for the decoherence program. In this paper we will consider quantum systems under a general theoretical framework for decoherence and we will present a tentative definition of the moving preferred basis. These ideas are implemented in a well-known open system model. The obtained decoherence and the relaxation times are defined and compared with those of the literature for the Lee- Friedrichs model.
Equilibrium Thermodynamics of Non-Hermitian Dirac Fermions: Caloric and Magnetic Responses
Facultad de Ingeniería, Universidad San Sebastián（圣塞巴斯蒂安大学工程学院）；Departamento de Física, Universidad Técnica Federico Santa María（智利联邦理工大学物理系）；Instituto de Física, Pontificia Universidad Católica de Valparaíso（天主教瓦尔帕莱索大学物理研究所）
Topological phases are enriched in non-equilibrium open systems effectively described by non-Hermitian Hamiltonians. While several properties unique to non-Hermitian topological systems were uncovered, the fundamental role of symmetry in non-Hermitian physics has yet to be fully understood, and it has remained unclear how symmetry protects non-Hermitian topological phases. Here we show that two fundamental anti-unitary symmetries, time-reversal and particle-hole symmetries, are topologically equivalent in the complex energy plane and hence unified in non-Hermitian physics. A striking consequen…
In non-Hermitian scattering problems the behavior of the transmission probability is very different from its Hermitian counterpart; it can exceed unity or even be divergent, since the non-Hermiticity can add or remove the probability to and from the scattering system. In the present paper, we consider the scattering problem of a PT-symmetric potential and find a counter-intuitive behavior. In the usual PT-symmetric non-Hermitian system, we would typically find stationary semi-Hermitian dynamics in a regime of weak non-Hermiticity but observe instability once the non-Hermiticity goes beyond an …
The ordinary time-dependent perturbation theory of quantum mechanics, that describes the interaction of a stationary system with a time-dependent perturbation, predicts that the transition probabilities induced by the perturbation are symmetric with respect to the initial an final states. Here we extend time-dependent perturbation theory into the non-Hermitian realm and consider the transitions in a stationary Hermitian system, described by a self-adjoint Hamiltonian $\hat{H}_0$, induced by a time-dependent non-Hermitian interaction $f(t) \hat{P}$. In the weak interaction (perturbative) limit,…
A decomposition of the non-equilibrium stationary state of a quadratic Fermi system influenced by linear baths is obtained and used to establish a simulation protocol in terms of tensor states. The scheme is then applied to examine the occurrence of uncoupled Majorana fermions in Kitaev chains subject to baths on the ends. The resulting phase diagram is compared against the topological characterization of the equilibrium chain and the protocol efficiency is studied with respect to this model
In this work we study many-body 'steady states' that arise in the non-Hermitian generalisation of the non-interacting Su-Schrieffer-Heeger model at a finite density of fermions. We find that the hitherto known phase diagrams for this system, derived from the single-particle gap closings, in fact correspond to distinct non-equilibrium phases, which either carry finite currents or are dynamical insulators where particles are entrapped. Each of these have distinct quasi-particle excitations and steady state correlations and entanglement properties. Looking at finite-sized systems, we further modu…
In the context of traditional quantum-control considerations it is conjectured that one of the promising new strategies of the constructive model building could be sought in a non-stationary upgrade of the formalism of PT-symmetric quantum mechanics. This upgrade is briefly reviewed and characterized by the simultaneous participation of a triplet of Hilbert spaces in the representation of a single quantum system.
We consider a linear chain made of spins of one half in contact with a dissipative environment for which periodic delta-kicks are applied to the qubits of the linear chain in two different configurations: kicks applied to a single qubit and simultaneous kicks applied to two qubits of the linear chain. In both cases the system reaches a non-equilibrium stationary condition in the long time limit. We study the transient to the quasi stationary states and their properties as function of the kick parameters in the single kicked qubit case and report the emergence of stationary entanglement between…
We associate the stationary harmonic oscillator with time-dependent systems exhibiting non-Hermiticity by means of point transformations. The new systems are exactly solvable, with all-real spectrum, and transit to the Hermitian configuration for the appropriate values of the involved parameters. We provide a concrete generalization of the Swanson oscillator that includes the Caldirola-Kanai model as a particular case. Explicit solutions are given in both, the classical and quantum pictures.
The aim of the paper is to study the question whether or not equilibrium states exist in open quantum systems that are embedded in at least two environments and are described by a non-Hermitian Hamilton operator $\cal H$. The eigenfunctions of $\cal H$ contain the influence of exceptional points (EPs) as well as that of external mixing (EM) of the states via the environment. As a result, equilibrium states exist (far from EPs). They are different from those of the corresponding closed system. Their wavefunctions are orthogonal although the Hamiltonian is non-Hermitian.
Non-Hermitian systems exhibit many peculiar dynamic behaviors which never showed up in Hermitian systems. The existence of spectral singularity (SS) for a non-Hermitian scattering center provides a lasing mechanism in the context of quantum mechanics. In this paper we show exactly that a finite system at exceptional point (EP) can also provide an alternative laser solution. We investigate the dynamics of non-Hermitian Su-Schrieffer-Heeger (SSH) model around EP, based on the analysis of parity-time (PT ) and chiral-time (CT ) symmetries of the Hamiltonian. In contrast to the SS lasing mechanism…
We show that non-Hermitian biorthogonal many-body phase transitions can be characterized by the enhanced decay of Loschmidt echo. The quantum criticality is numerically investigated in a non-Hermitian transverse field Ising model by performing the finite-size dynamical scaling of Loschmidt echo. We determine the equilibrium correlation length critical exponents that are consistent with previous results from the exact diagonalization. More importantly, we introduce a simple method to detect quantum phase transitions with the short-time average of rate function motivated by the critically enhanc…
In contrast to classical systems, actual implementation of non-Hermitian Hamiltonian dynamics for quantum systems is a challenge because the processes of energy gain and dissipation are based on the underlying Hermitian system-environment dynamics that is trace preserving. Recently, a scheme for engineering non-Hermitian Hamiltonians as a result of repetitive measurements on an anicillary qubit has been proposed. The induced conditional dynamics of the main system is described by the effective non-Hermitian Hamiltonian arisng from the procedure. In this paper we demonstrate the effectiveness o…
Starting on the basis of the non-commutative q-differential calculus, we introduce a generalized q-deformed Schrödinger equation. It can be viewed as the quantum stochastic counterpart of a generalized classical kinetic equation, which reproduces at the equilibrium the well-known q-deformed exponential stationary distribution. In this framework, q-deformed adjoint of an operator and q-hermitian operator properties occur in a natural way in order to satisfy the basic quantum mechanics assumptions.
We derive the universal real time $U(1)$ topological gauge field action for mixed quantum states of weakly correlated fermions in all dimensions, and demonstrate its independence of the underlying equilibrium or non-equilibrium nature of dynamics stabilizing the state. The key prerequisites are charge quantization and charge conservation. The gauge action encodes non-quantized linear responses as expected for mixed states, but also quantized non-linear responses, associated to mixed state topology and accessible in experiment. Our construction furthermore demonstrates how the physical pictures…
We analyze a correlated system in equilibrium with special emphasis on non-Hermitian topology inducing a skin effect. The pseudo-spectrum, computed by the real-space dynamical mean-field theory, elucidates that additional pseudo-eigenstates emerge for the open boundary condition in contrast to the dependence of the density of states on the boundary condition. We further discuss how the line-gap topology, another type of non-Hermitian topology, affects the pseudo-spectrum. Our numerical simulation clarifies that while the damping of the quasi-particles induces the non-trivial point-gap topology…
Non-Hermitian Hamiltonians possessing a discrete real spectrum motivated a remarkable research activity in quantum physics and new insights have emerged. In this paper we formulate concepts of statistical thermodynamics for systems described by non-Hermitian Hamiltonians with real eigenvalues. We mainly focus on the case where the energy and another observable are the conserved quantities. The notion of entropy and entropy inequalities are central in our approach, which treats equilibrium thermodynamics.
We consider the out-of-equilibrium dynamics generated by joining two domains with arbitrary opposite magnetisations. We study the stationary state which emerges by the unitary evolution via the spin $1/2$ XXZ Hamiltonian, in the gapless regime, where the system develops a stationary spin current. Using the generalized hydrodynamic approach, we present a simple formula for the space-time profile of the spin current and the magnetisation exact in the limit of large times. As a remarkable effect, we show that the stationary state has a strongly discontinuous dependence on the strength of interact…
Discovery of an intrinsic non-Hermitian phase transition in a bulk condensed-matter system
ETH Zurich（苏黎世联邦理工学院）；University of Bonn（波恩大学）；The University of Osaka（大阪大学）；Tohoku University（东北大学）；Japan Science and Technology Agency (JST)（日本科学技术振兴机构）；Goethe-Universität Frankfurt（法兰克福歌德大学）；National Institute of Science Education and Research（国家科学教育与研究机构）；University of St. Andrews（圣安德鲁斯大学）
We show how the band structure and beam dynamics of non-Hermitian $PT$-symmetric sinusoidal optical lattices can be approached from the point of view of the equivalent Hermitian problem, obtained by an analytic continuation in the transverse spatial variable $x$. In this latter problem the eigenvalue equation reduces to the Mathieu equation, whose eigenfunctions and properties have been well studied. That being the case, the beam propagation, which parallels the time-development of the wave-function in quantum mechanics, can be calculated using the equivalent of the method of stationary states…
We present a simple non-hermitian model to describe the phenomenon of asymmetric tunneling between two energy-degenerate sites coupled by a non-reciprocal interaction without dissipation. The system was described using a biorthogonal family of energy eigenvectors, the dynamics of the system was determined by the Schrödinger equation, and unitarity was effectively restored by proper normalization of the state vectors. The results show that the tunneling rates are indeed asymmetrical in this model, leading to an equilibrium that displays unequal occupation of the degenerate systems even in the a…
We consider wave dynamics for a Schrödinger equation with a non-Hermitian Hamiltonian $\mathcal{H}$ satisfying the generalized (anyonic) parity-time symmetry $\mathcal{PT H}= \exp(2 i φ) \mathcal{HPT}$, where $\mathcal{P}$ and $ \mathcal{T}$ are the parity and time-reversal operators. For a stationary potential, the anyonic phase $φ$ just rotates the energy spectrum of $\mathcal{H}$ in complex plane, however for a drifting potential the energy spectrum is deformed and the scattering and localization properties of the potential show intriguing behaviors arising from the breakdown of the Galilea…
We investigate a model system for the injection of fermionic particles from filled source sites into an empty chain. We study the ensuing dynamics for Hermitian as well as for non-Hermitian time evolution where the particles cannot return to the bath sites (quantum ratchet). A non-homogeneous hybridization between bath and chain sites permits transient currents in the chain. Non-interacting particles show decoherence in the thermodynamic limit: the average particle number and the average current density in the chain become stationary for long times, whereas the single-particle density matrix d…
We demonstrate general classifications of Riemann surface topology generated by multiple arbitrary-order exceptional points of quasi-stationary states. Our studies reveal all possible product permutations of holonomy matrices that describe a stroboscopic encircling of 2nd order exceptional points. The permutations turn out to be categorized into a finite number of classes according to the topological structures of the Riemann surfaces. We further show that the permutation classes can be derived from combinations of cyclic building blocks associated with higher-order exceptional points. Our res…
We study an interacting one-dimensional gas of spin-1/2 fermions with two-body losses. The dynamical phase diagram that characterises the approach to the stationary state displays a wide quantum-Zeno region, identified by a peculiar behaviour of the lowest eigenvalues of the associated non-Hermitian Hamiltonian. We characterise the universal dynamics of this Zeno regime using an approximation scheme based on a effective decoupling of charge and spin degrees of freedom, where the latter effectively evolve in a non-Markovian way according to a non-Hermitian Heisenberg Hamiltonian. We present det…
