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Exploring the topologically nontrivial states in condensed matter, especially their electronic structure and transport property, is very important and timely. Most topics discussed in this paper are relevant to the topological properties of electronic wavefunctions. Besides a brief introduction, the paper contains five sections, including topological quantum phenomena, various topological phases, unusual transport properties of topological quasiparticles, topological collective excitations and coupled excitations, and ongoing research in topological quantum states. These presentations expound a variety of aspects in the studies of topological quantum states. With all these aspects together, a comprehensive understanding of the topological quantum states in condensed matter physics can be achieved. [ABSTRACT FROM AUTHOR]

A method based on the tracking of the peaks of interference fringes as a function of time-dependent analyzer setting is used to detect the Pancharatnam phase of light. The advantage of this method is demonstrated by observations of the nonlinearity of the Pancharatnam phase for certain paths on Poincare sphere where the fringe visibility reduces to almost zero. We also describe variations of this experiment with structured light beams, where the Pancharatnam phase leads to linear or nonlinear rotation of flower-pattern or spiral-shape interference fringes. [ABSTRACT FROM AUTHOR]

Motivated by J. Larson's work (Crespi et al., 2012), in which vacuum induced non-trivial Berry phases for Jaynes–Cummings (JC) model were considered to be the results from the application of rotating wave approximation (RWA) and become vanishing in the quantum Rabi model without RWA. Besides a counterpart relation between the trivialness or non-trivialness of Berry phases and the existence of conical intersections (CIs) of adiabatic potentials is illustrated by introducing semi-classical energy surface. We argue that there exist no definite connections between the trivialness or not of Berry phase and the CIs of adiabatic potentials because the CIs are fully controllable for the generalized JC and Rabi models discussed in this text by adjusting the dispersive non-linear interacting parameter. • We investigate Berry phases of the generalized Jaynes–Cummings model and Rabi model. • Berry phases has direct relation to the application of rotating wave approximation. • Berry phases has no necessary connection with conical intersections of adiabatic potentials. • It is analogous to tunable topologic phenomenon. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Skyrmions are topological solitons that emerge in many physical contexts. In magnetism, they appear as textures of the spin-density field stabilized by different competing interactions and characterized by a topological charge that counts the number of times the order parameter wraps the sphere. They behave as classical objects when the spin texture varies slowly on the scale of the microscopic lattice of the magnet. However, the fast development of experimental tools to create and stabilize skyrmions in thin magnetic films has led to a rich variety of textures, sometimes of atomistic sizes. In this paper, we discuss, in a pedagogical manner, how to introduce quantum interference in the translational dynamics of skyrmion textures, starting from the micromagnetic equations of motion for a classical soliton. We study how the nontrivial topology of the spin texture manifests in the semiclassical regime, when the microscopic lattice potential is treated quantum-mechanically, but the external driving forces are taken as smooth classical perturbations. We highlight close relations to the fields of noncommutative quantum mechanics, Chern–Simons theories, and the quantum Hall effect. [ABSTRACT FROM AUTHOR]

The relationship between atomic-scale and micromagnetic Dzyaloshinski–Moriya (DM) interactions has been investigated. By analyzing the Lifshitz invariants for different point groups, we have found that there is no unique link between the absence of inversion symmetry and DM interactions. The absence of inversion symmetry is a necessary condition for a net DM interaction in crystals, but several noncentrosymmetric point groups have zero DM interactions. In many cases, the key consideration is whether the crystals are polar and/or chiral. For example, MnSi-type spin spirals, which violate helical spin symmetry, are caused by the insertion of chiral atomic-scale building blocks into an achiral cubic lattice, and the scalar interaction parameter D used to describe the spirals is only loosely related to the DM vector D. It contains, in fact, magnetostatic and magnetocrystalline contributions of unknown magnitude. Finally, we discuss some aspects of the micromagnetism of the skyrmionics of nanoparticles and granular nanostructures. [ABSTRACT FROM AUTHOR]

Metasurfaces deployed for generating electromagnetic waves that carry orbital angular momentum (OAM) in the transmission mode are generally inefficient in operation. In this paper, we present a method for the design, fabrication, and characterization of an ultra-thin metasurface which could be used for generating OAM waves at microwave frequencies with high efficiency. We achieve this objective by proposing a novel bilayer subwavelength scatterer having $\pi $ retardation phase between two orthogonal polarizations of an incident circularly polarized (CP) wave that also have high amplitudes. When analyzed our setup using Jones matrices, we find that the scatterer inverts the spin direction of the incident CP wave with high efficiency. Furthermore, the setup provides full phase control with the spatial rotation of the scatterer as per Pancharatnam-Berry Phase Mechanism. To further illustrate the utility of the proposal, a metasurface was designed based on the proposed scatterer and the generation of OAM waves is validated through both simulations and experimental measurements. To further clarify the points, a rigorous analysis for the transmission and conversion efficiencies is presented. [ABSTRACT FROM AUTHOR]

In terms of quantum probability statistics, the Bell inequality (BI) and its violation are extended to spin- s entangled Schrödinger cat state (called the Bell cat state) with both parallel and antiparallel spin-polarizations. Except the spin-1/2, the BI is never ever violated by the Bell cat states with the measuring outcomes including entire Hilbert space. If, on the other hand, measuring outcomes are restricted in the subspace of spin coherent state (SCS), a universal Bell-type inequality (UBI), p s l c ≤ 0 , is formulated in terms of the local realistic model. We observe a spin parity effect that the UBI can be violated only by the Bell cat states of half-integer but not the integer spins. The violation of UBI is seen to be a direct result of nontrivial Berry phase between the SCSs of south- and north-pole gauges for half-integer spin, while the geometric phase is trivial for the integer spins. A maximum violation bound of UBI is found as p s max = 1 , which is valid for arbitrary half-integer spin- s states. [ABSTRACT FROM AUTHOR]

Abstract We investigate the physical properties of massive Dirac fermions in SrMnSb 2 using transport, specific heat, electronic structure calculations, and Shubnikov-de Haas (SdH) oscillations. SrMnSb 2 is a candidate Dirac antiferromagnet, consisting of the MnSb layers and the distorted square net of Sb atoms with a zigzag chain structure. This structural distortion leads to gap opening at the band crossing point found in the square lattice of the sister compound SrMnBi 2 but leaves another Dirac band crossing near the Brillouin zone boundary. The small 2D Fermi surface with a light electron mass and a small Fermi energy is confirmed by the large resistivity anisotropy, the large Seebeck coefficient, and also the angle and temperature dependent SdH oscillations. The Berry phase obtained from the SdH oscillations is trivial, in contrast to the case of SrMnBi 2. The relatively large spin orbit coupling gap and the small Fermi energy in SrMnSb 2 is found to be essential to understand this contrasting behavior of the massive Dirac fermions as compared to SrMnBi 2. Our observations demonstrate that the Berry phase of the mobile electrons in SrMnSb 2 is sensitive to the Fermi level change and can be tuned by doping or deficiency. Highlights • A small Fermi surface of SrMnSb 2 is confirmed by Shubnikov-de Haas oscillations. • The Berry phase from quantum oscillations is found to be nearly zero in SrMnSb 2. • SrMnSb 2 hosts massive Dirac fermions due to the spin-orbit coupling. [ABSTRACT FROM AUTHOR]

Cyclic quantum evolution is accompanied by a systematic change in the phase of the initial state vector. This change only depends upon the path traced out by the system itself. Such effects are collectively known as geometric phase factors. The geometric foundations of these phase factors are most elegantly formulated in terms of fibre bundle theory and differential forms, both of which can represent a significant hurdle to master. We present a derivation of the abelian and non-abelian Berry phase in terms of embedded manifolds of linear vector spaces. Embedded manifolds offer the advantage of being less abstract than fibre bundles, and are well-suited for explicit calculations. Essential features of the derivation reduce to matrix–vector manipulations. We further discuss a numerical strategy for the calculation of abelian and non-abelian phase factors. Our approach is based upon Hungarian method and the polar decomposition, and is made freely available as a SpinDynamica addon. Additionally, all derivations and analytic calculations are supported by Mathematica notebooks. [Display omitted] • Discussion of geometric phase factors in terms of embedded manifolds. • Discussion of key geometric concepts such as horizontal lifts and parallel transport. • Several worked out examples supported by SpinDynamica notebooks. • Description of an algorithm for geometric phase factors based on the Hungarian method and the polar decomposition. • Implementation of the algorithm as a freely available SpinDynamica addon. [ABSTRACT FROM AUTHOR]

The artificial gauge field for electrically neutral exciton polaritons devoid from the polarization degree of freedom can be synthesized by means of applying crossed electric and magnetic fields. The appearance of the gauge potential can be ascribed to the motional (magneto-electric) Stark effect which is responsible for the presence of a linear-in-momentum contribution to the exciton kinetic energy. We study the interplay of this phenomenon with the competing effect which arises from the Rabi-splitting renormalization due the reduction of the electron–hole overlap for a moving exciton. Accounting for this mechanism is crucial in the structures with the high ratio of Rabi splitting and the exciton binding energy. Besides, we propose an approach which boosts the gauge field in the considered system. It takes advantage of the crossover from the hydrogen-like exciton to the strongly dipole-polarized exciton state at a specific choice of electric and magnetic fields. The strong sensitivity of the exciton energy to the momentum in this regime leads to the large values of the gauge field. We consider the specific example of a GaAs ring-shape polariton Berry phase interferometer and show that the flux of the effective magnetic field may approach the flux quantum value in the considered crossover regime.

Majorana stars, the antipodal directions associated with the coherent states that are orthogonal to a spin state, provide a visualization and a geometric understanding of the structures of general quantum states. For example, the Berry phase of a spin-1/2 is given by half the solid angle enclosed by the close path of its Majorana star. It is conceivable that the Berry phase of higher spins may also be related to the geometry of the Majorana constellation. We find that for a spin-1 state, besides the expected contributions from the solid angles enclosed by the close paths of the two Majorana stars, the Berry phase includes a term related to the twist of the relative position vector around the barycenter vector of the two Majorana stars, i.e., the self-rotation of the constellation. Interestingly, if the spin-1 state is taken as a symmetrized two-qubit state, the extra contribution to the Berry phase is given by the self-rotation of the Majorana constellation weighted by the quantum entanglement of the two qubits. This discovery alludes to the relevance of the Majorana stellar geometry in representing the deep structures of quantum states and of quantum entanglement.

In this paper, based on first given Lewis–Riesenfeld invariant depicted by a unit vector in parameter space, we inverse engineering the time-dependent Hamiltonian of a system with su(2) Lie algebraic structure. The introduced method is then applied to investigate nonadiabatic Abelian geometric quantum computation. We demonstrate that, by employing the nonadiabatic Berry phase generated through nonadiabatic periodic evolution, a driven two-level system which undergoes a single cyclic evolution along a loop path in Bloch space can realize a universal set of one-qubit gates. Subsequently, under consideration of the influence of the systematic error and dissipation on nonadiabatic process, the result reveals arbitrary one-qubit gate can be implemented with a high fidelity. Moreover, to complete the universal set, arbitrary controlled-U gate is designed by utilizing a driven system consisted of a pair of coupled spin subsystems.

Berry phase is a very general concept. It is applied here to families of solutions of the Dirac equation with different values of spin. The value of the Berry phase in the spin space is given by the same expression as was found before in the momentum space. [ABSTRACT FROM AUTHOR]

Winding number is a topologically significant quantity that has found valuable applications in various areas of mathematical physics. Here, topological qubits are shown capable of formation from winding number superpositions and so of being used in the communication of quantum information in linear optical systems, the most common realm for quantum communication. In particular, it is shown that winding number qubits appear in several aspects of such systems, including quantum electromagnetic states of spin, momentum, orbital angular momentum, polarization of beams of particles propagating in free-space, optical fiber, beam splitters, and optical multiports. [ABSTRACT FROM AUTHOR]

Abstract The orbital angular momentum of photon is one of the key source of quantum information. Polarized light passing through twisted special wave plates show entangled states of both spin angular momentum (SAM) and orbital angular momentum (OAM). The variable circular and linear retarders (VCR and VLR) are studied here from the view point of quantum processing of entangled states. In association with twist dependent gain of OAM, the geometric phase has been identified here both for VCR and VLR. For the application of VLR in an antenna array, we established a correlation between geometric phase and concurrence to explain the gradual flattened shape of the emergent wave by controlling the external twist of wave plates. [ABSTRACT FROM AUTHOR]

The connection between topology and nonreciprocity in photonic systems is reviewed. Topological properties such as Chern number, and momentum-space properties such as Berry phase and Berry connection, are used to explain backscattering immune edge states and their topological protection. We consider several examples to illustrate the role of momentum-space topology on wave propagation, and discuss recent magnet-less approaches. [ABSTRACT FROM AUTHOR]

The question whether a Berry phase should be included in the calculation of rovibronic states of the ozone molecule in its electronic ground state has been addressed in the present work. Since several conical intersections connect the three lowest singlet states, a phase of π generated by a symmetry-demanded conical intersection between states 2 and 3 might be present, or it might be compensated by further conical intersections. Hyperspherical coordinates were employed here as they provide a natural choice for the cyclic coordinate that is needed to investigate the presence or absence of such a topological phase. On a hyperspherical grid, we have computed the electronic energies of the three lowest singlet states using a multi-reference configuration interaction procedure. Non-adiabatic coupling terms (NACTs) along the cyclic variable were evaluated numerically. Investigation of the behaviour of the adiabatic-to-diabatic transformation angles, using the quantisation criterion of the NACTs [M. Baer and A. Alijah, Chem. Phys. Lett. 319, 489 (2000)], shows that a geometrical phase is retained. This phase has no practical effect on the well-established energy values of the rovibrational states located deeply in the three equivalent potential wells, but will come into action as highly excited states reach the dissociation region. [ABSTRACT FROM AUTHOR]

One of the characteristics of topological materials is their nontrivial Berry phase. Experimental determination of this phase largely relies on a phase analysis of quantum oscillations. We study the angular dependence of the oscillations in a Dirac material ZrTe5 and observe a striking spin-zero effect (i.e., vanishing oscillations accompanied with a phase inversion). This indicates that the Berry phase in ZrTe5 remains nontrivial for arbitrary field direction, in contrast with previous reports. The Zeeman splitting is found to be proportional to the magnetic field based on the condition for the spin-zero effect in a Dirac band. Moreover, it is suggested that the Dirac band in ZrTe5 is likely transformed into a line node other than Weyl points for the field directions at which the spin zero occurs. The results underline a largely overlooked spin factor when determining the Berry phase from quantum oscillations. [ABSTRACT FROM AUTHOR]

One of the characteristics of topological materials is their nontrivial Berry phase. Experimental determination of this phase largely relies on a phase analysis of quantum oscillations. We study the angular dependence of the oscillations in a Dirac material ZrTe5 and observe a striking spin-zero effect (i.e., vanishing oscillations accompanied with a phase inversion). This indicates that the Berry phase in ZrTe5 remains nontrivial for arbitrary field direction, in contrast with previous reports. The Zeeman splitting is found to be proportional to the magnetic field based on the condition for the spin-zero effect in a Dirac band. Moreover, it is suggested that the Dirac band in ZrTe5 is likely transformed into a line node other than Weyl points for the field directions at which the spin zero occurs. The results underline a largely overlooked spin factor when determining the Berry phase from quantum oscillations. [ABSTRACT FROM AUTHOR]

In the framework of the single-field slow-roll inflation, we derive the Hamiltonian of the linear primordial scalar and tensor perturbations in the form of time-dependent harmonic oscillator Hamiltonians. We find the invariant operators of the resulting Hamiltonians and use their eigenstates to calculate the adiabatic Berry phase for sub-horizon modes in terms of the Lewis–Riesenfeld phase. We conclude by discussing the discrepancy in the results of Pal et al. (2013) [21] for these Berry phases, which is resolved to yield agreement with our results. [ABSTRACT FROM AUTHOR]

Graphical abstract Abstract Dirac nodal-line semimetals with the linear bands crossing along a line or loop, represent a new topological state of matter. Here, by carrying out magnetotransport measurements and performing first-principle calculations, we demonstrate that such a state has been realized in high-quality single crystals of SrAs 3. We obtain the nontrivial π Berry phase by analysing the Shubnikov-de Haas quantum oscillations. We also observe a robust negative longitudinal magnetoresistance induced by the chiral anomaly. Accompanying first-principles calculations identifies that a single hole pocket enclosing the loop nodes is responsible for these observations. [ABSTRACT FROM AUTHOR]

We report Shubnikov-de Haas (SdH) and Hall oscillations in Cu-doped high quality bismuth selenide single crystals. To increase the accuracy of Berry phase determination by means of the of the SdH oscillations phase analysis we present a study of n-type samples with bulk carrier density n ∼ 10 19 − 10 20 cm − 3 at high magnetic field up to 60 Tesla. In particular, Landau level fan diagram starting from the value of the Landau index N = 4 was plotted. Thus, from our data we found π -Berry phase that directly indicates the Dirac nature of the carriers in three-dimensional topological insulator (3D TI) based on Cu-doped bismuth selenide. We argued that in our samples the magnetotransport is determined by a general group of carriers that exhibit quasi-two-dimensional (2D) behaviour and are characterized by topological π -Berry phase. Along with the main contribution to the conductivity the presence of a small group of bulk carriers was registered. For 3D-pocket Berry phase was identified as zero, which is a characteristic of trivial metallic states. [ABSTRACT FROM AUTHOR]

The Berry-phase treatment of the spontaneous polarization defines this quantity as the polarization difference between the investigated system and an appropriate reference state. In the case of wurtzite crystals, the spontaneous polarization is usually referenced to the zinc blende structure. Here, we demonstrate that the closely-related trigonal structure can also be used as a reference to determine the spontaneous polarization of wurtzite materials. We compare the trigonal-based values for wurtzite nitrides XN (X = Al, Ga, In) and oxides YO (Y Be, Zn) with the widely-known zinc blende results reported in the literature and find an excellent agreement. It is shown that the electronic contribution of polarization is greater than the ionic one. Furthermore, we reproduce the experimental value of the spontaneous polarization of wurtzite GaN reported by Lähnemann et al. [Phys. Rev. B 86, 081302(R) (2012)]. In order to do so, we calculate the spontaneous polarization for each type of basal stacking faults using periodic hexagonal supercells. This theoretical analysis leads to a value nearly identical to the experimental measurement. [ABSTRACT FROM AUTHOR]

We study the scaling behavior of the Berry phase in the Yang-Lee edge singularity (YLES) of the non-Hermitian quantum system. A representative model, the one-dimensional quantum Ising model in an imaginary longitudinal field, is selected. For this model, the dissipative phase transition (DPT), accompanying a parity-time (PT) symmetry-breaking phase transition, occurs when the imaginary field changes through the YLES. We find that the real and imaginary parts of the complex Berry phase show anomalies around the critical points of YLES. In the overlapping critical regions constituted by the (0 + 1)D YLES and (1 + 1)D ferromagnetic-paramagnetic phase transition (FPPT), we find that the real and imaginary parts of the Berry phase can be described by both the (0 + 1)D YLES and (1 + 1)D FPPT scaling theory. Our results demonstrate that the complex Berry phase can be used as a universal order parameter for the description of the critical behavior and the phase transition in the non-Hermitian systems.