The well-known geometric phase present in the quantum adiabatic evolution discovered by Berry many years ago has its analogue, the Hannay phase, in the classical domain.We calculate the Berry phase with examples for quantum hermitian and non-hermitian $PT$-symmetric Hamiltonians and compare with the Hannay phase in their classical equivalents.We use the analogy to propose resonant electric circuits which reproduce the theoretical solutions in simulated laboratory experiments.
In this paper we have considered closed trajectories of a particle on a two-torus where the loops are noncontractible (poloidal and toroidal loops and knots embedded on a regular torus). We have calculated Hannay angle and Berry phase for particle traversing such loops and knots when the torus itself is adiabatically revolving. Since noncontractible loops do not enclose any area Stokes theorem has to be applied with caution. In our computational scheme we have worked with line integrals directly thus avoiding Stokes theorem.
The aim of this paper is to construct the Berry-Hannay model of quantum mechanics on a 2n-dimensional symplectic torus. We construct a simultaneous quantization of the algebra $\cal A$ of functions on the torus and the linear symplectic group $\G = \Sp(\mathrm{2n},\Z)$. In the construction we use the quantum torus $\A$, which is a deformation of $\cal A$, together with a $\G$-action on it. We obtain the quantization via the action of $\G$ on the set of equivalence classes of irreducible representations of $\A$. For $\h \in \Q$ this action has a unique fixed point. This gives a canonical projec…
We numerically investigate topological phases of periodic lattice systems in tight-binding description under the influence of dissipation. The effects of dissipation are effectively described by $\mathcal{PT}$-symmetric potentials. In this framework we develop a general numerical gauge smoothing procedure to calculate complex Berry phases from the biorthogonal basis of the system's non-Hermitian Hamiltonian. Further, we apply this method to a one-dimensional $\mathcal{PT}$-symmetric lattice system and verify our numerical results by an analytical calculation.
We in this paper demonstrate that the $PT$-symmetric non-Hermitian Hamiltonian for a periodically driven system can be generated from a kernel Hamiltonian by a generalized gauge transformation. The kernel Hamiltonian is Hermitian and static, while the time-dependent transformation operator has to be $PT$ symmetric and non-unitary in general. Biorthogonal sets of eigenstates appear necessarily as a consequence of non-Hermitian Hamiltonian. We obtain analytically the wave functions and associated non-adiabatic Berry phase $γ_{n}$ for the $n$th eigenstate. The classical version of the non-Hermiti…
We systematically study the topology of the exceptional point (EP) in the finite non-Hermitian system. Based on the concrete form of the Berry connection, we demonstrate that the exceptional line (EL), at which the eigenstates coalesce, can act as a vortex filament. The direction of the EL can be identified by the corresponding Berry curvature. In this context, such a correspondence makes the topology of the EL clear at a glance. As an example, we apply this finding to the non-Hermitian Rice-Mele (RM) model, the non-Hermiticity of which arises from the staggered on-site complex potential. The …
We study a non-Hermitian Rice-Mele model without breaking time-reversal symmetry, with the non-Hermiticity arising from imbalanced hopping rates. The Berry connection, Berry curvature and Chern number are introduced in the context of biorthonormal inner product. It is shown that for a bulk system, although the Berry connection can be complex numbers, the Chern number is still quantized, as topological invariant. For an opened chain system, the mid-gap edge modes are obtained exactly, obeying the bulk-edge correspondence. Furthermore, we also introduce a local current in the context of biorthon…
We propose a non-Hermitian generalization of the correspondence between the spectral flow and the topological charges of band crossing points (Berry-Chern monopoles). A class of non-Hermitian Hamiltonians that display a complex-valued spectral flow is built by deforming an Hermitian model while preserving its analytical index. We relate those spectral flows to a generalized Chern number that we show to be equal to that of the Hermitian case, provided a line gap exists. We demonstrate the homotopic invariance of both the non-Hermitian Chern number and the spectral flow index, making explicit th…
We propose a pair of the complex Berry curvatures associated with the non-Hermitian Hamiltonian and its Hermitian adjoint to reveal new physics in non-Hermitian systems. We give the complex Berry curvature and Berry phase for the two-dimensional non-Hermitian Dirac model. The imaginary part of the complex Berry phase induces the quantum Hall susceptance such that the quantum Hall conductance is generalized to quantum Hall admittance for non-Hermitian systems, which implies that the non-Hermiticity of systems could induce an intrinsic quantum capacitance or inductance of systems depending on th…
In many classical and quantum systems described by an effective non-Hermitian Hamiltonian, spectral phase transitions, from an entirely real energy spectrum to a complex spectrum, can be observed as a non-Hermitian parameter in the system is increased above a critical value. A paradigmatic example is provided by systems possessing parity-time (PT) symmetry, where the energy spectrum remains entirely real in the unbroken PT phase while a transition to complex energies is observed in the unbroken PT phase. Such spectral phase transitions are universally sharp. However, when the system is slowly …
Covariant formulation of the Berry connection in non-Hermitian systems
Joint Laboratory of Optics of Palacký University and Institute of Physics of CAS, Faculty of Science, Palacký University（帕拉茨基大学光学联合实验室与捷克科学院物理研究所，理学院，帕拉茨基大学）
Berry phases strongly affect the properties of crystalline materials, giving rise to modifications of the semiclassical equations of motion that govern wave-packet dynamics. In non-Hermitian systems, generalizations of the Berry connection have been analyzed to characterize the topology of these systems. While the topological classification of non-Hermitian systems is being developed, little attention has been paid to the impact of the new geometric phases on dynamics and transport. In this work, we derive the full set of semiclassical equations of motion for wave-packet dynamics in a system g…
Among non-Hermitian systems, pseudo-Hermitian phases represent a special class of physical models characterized by real energy spectra and by the absence of non-Hermitian skin effects. Here, we show that several pseudo-Hermitian phases in two and three dimensions can be built by employing $q$-deformed matrices, which are related to the representation of deformed algebras. Through this algebraic approach we present and study the pseudo-Hermitian version of well known Hermitian topological phases, raging from two-dimensional Chern insulators and time-reversal-invariant topological insulators to …
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 $\cal{PT}$-regimes, in the time-independent scenario referred to as the $\cal{PT}$-symmetric regime, the exceptional point and the spontaneously broken $\cal{PT}$-regime. We also propose a modified adiabatic approximation consisting of an expansion of the wavefunctions in terms the instantaneou…
The geometrical Berry phase is key to understanding the behaviour of quantum states under cyclic adiabatic evolution. When generalised to non-Hermitian systems with gain and loss, the Berry phase can become complex, and should modify not only the phase but also the amplitude of the state. Here, we perform the first experimental measurements of the adiabatic non-Hermitian Berry phase, exploring a minimal two-site $\mathcal{PT}$-symmetric Hamiltonian that is inspired by the Hatano-Nelson model. We realise this non-Hermitian model experimentally by mapping its dynamics to that of a pair of classi…
Direct measurement of the quantum geometric tensor in pseudo-Hermitian systems
Nanjing University（南京大学）；National Laboratory of Solid State Microstructures and School of Physics, Nanjing University（南京大学固体微结构物理实验室和物理学院）；Collaborative Innovation Center of Advanced Microstructures, Nanjing University（南京大学先进微结构协同创新中心）；Centre for Quantum Technologies, National University of Singapore（新加坡国立大学量子技术中心）；MajuLab, CNRS-UNS-NUS-NTU International Joint Research Unit, Singapore UMI 3654, Singapore（新加坡UMI 3654 MajuLab（CNRS-UNS-NUS-NTU国际联合研究单元））；Guangdong Basic Research Center of Excellence for Structure and Fundamental Interactions of Matter, Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials, School of Physics, South China Normal University（华南师范大学物理学院广东省物质结构与基本相互作用卓越基础研究中心及广东省量子工程与量子材料重点实验室）；Guangdong-Hong Kong Joint Laboratory of Quantum Matter, Frontier Research Institute for Physics, South China Normal University（华南师范大学前沿物理研究院粤港量子物质联合实验室）
Recently developed parity ($\mathcal{P}$) and time-reversal ($\mathcal{T}$) symmetric non-Hermitian quantum theory is envisioned to have far-reaching implications in basic science and applications. It is known that the $PT$-inner product is defined with respect to a non-canonical, system-generated dynamical symmetry, namely the $C$ symmetry. Here, we show that the $PT$ invariant equation of motion is defined by the simultaneous time evolution of the state $ψ(t)$ and the operator $C(t)$ to manifest unitarity. The dynamical $C$ operator lends itself to a new term in the Berry phase. The $PT$ sym…
Non-Hermitian wave-packet dynamics and its realization within a non-Hermitian chiral cavity
School of Physics and Astronomy, Shanghai Jiao Tong University（上海交通大学物理与天文学院）；Wilczek Quantum Center, School of Physics and Astronomy, Shanghai Jiao Tong University（上海交通大学物理与天文学院吴文裕量子中心）；Shanghai Research Center for Quantum Sciences（上海量子科学研究中心）；Tsung-Dao Lee Institute, Shanghai Jiao Tong University（上海交通大学李政道研究所）；Shanghai Branch, Hefei National Laboratory（合肥国家实验室上海分部）
Non-Hermitian systems can produce branch singularities known as exceptional points (EPs). Different from singularities in Hermitian systems, the topological properties of an EP can involve either the winding of eigenvalues that produces a discriminant number (DN) or the eigenvector holonomy that generates a Berry phase. The multiplicity of topological invariants also makes non-Hermitian topology richer than its Hermitian counterpart. Here, we study a parabola-shaped trajectory formed by EPs with both theory and acoustic experiments. By obtaining both the DNs and Berry phases through the measur…
Geometric modulation of transition and survival intensities in non-Hermitian systems
Hokkaido University（北海道大学）；The Institute for Solid State Physics, The University of Tokyo（东京大学固体物理研究所）；Institute of Industrial Science, The University of Tokyo（东京大学工业科学研究所）；Institute for Frontier Education and Research on Semiconductors, Hokkaido University（北海道大学半导体前沿教育与研究 institute）
We study in this paper the time evolution of $PT$-symmetric non-Hermitian Hamiltonian consisting of periodically driven $SU(1,1)$ generators. A non-Hermitian invariant operator is adopted to solve the Schrödinger equation, since the time-dependent Hamiltonian is no longer a conserved quantity. We propose a scheme to construct the non-Hermitian invariant with a $PT$-symmetric but non-unitary transformation operator. The eigenstates of invariant and its complex conjugate form a bi-orthogonal basis to formulate the exact solution. We obtain the non-adiabatic Berry phase, which reduces to the adia…
We develop a theoretical description of non-Hermitian time evolution that accounts for the break- down of the adiabatic theorem. We obtain closed-form expressions for the time-dependent state amplitudes, involving the complex eigen-energies as well as inter-band Berry connections calculated using basis sets from appropriately-chosen Schur decompositions. Using a two-level system as an example, we show that our theory accurately captures the phenomenon of "sudden transitions", where the system state abruptly jumps from one eigenstate to another.
In condensed matter physics, non-Abelian statistics for Majorana zero modes (or Majorana Fermions) is very important, really exotic, and completely robust. The race for searching Majorana zero modes and verifying the corresponding non-Abelian statistics becomes an important frontier in condensed matter physics. In this letter, we generalize the Majorana zero modes to non-Hermitian (NH) topological systems that show universal but quite different properties from their Hermitian counterparts. Based on the NH Majorana zero modes, the orthogonal and nonlocal Majorana qubits are well defined. In par…
The time evolution of periodically driven non-Hermitian systems is in general non-unitary but can be stable. It is hence of considerable interest to examine the adiabatic following dynamics in periodically driven non-Hermitian systems. We show in this work the possibility of piecewise adiabatic following interrupted by hopping between instantaneous system eigenstates. This phenomenon is first observed in a computational model and then theoretically explained, using an exactly solvable model, in terms of the Stokes phenomenon. In the latter case, the piecewise adiabatic following is shown to be…
This thesis explores important concepts in the area of quantum information geometry and their relationships. We highlight the unique characteristics of these concepts that arise from their quantum mechanical foundations and emphasize the differences from their classical counterparts. However, we also demonstrate that for Gaussian states, classical analogs can be used to obtain the same mathematical results, providing a valuable tool for simplifying calculations. To establish the groundwork for the subsequent analysis, we introduce some fundamental ideas from quantum field theory. We then explo…
The usual concepts of topological physics, such as the Berry curvature, cannot be applied directly to non-Hermitian systems. We show that another object, the quantum metric, which often plays a secondary role in Hermitian systems, becomes a crucial quantity near exceptional points in non-Hermitian systems, where it diverges in a way that fully controls the description of wavepacket trajectories. The quantum metric behaviour is responsible for a constant acceleration with a fixed direction, and for a non-vanishing constant velocity with a controllable direction. Both contributions are independe…
