An extension of the Hellmann-Feynman theorem to one employing dynamical parameters that vary with time according to quantum dynamics is rigorously derived, avoiding any linear response or other approximations. The resulting theorem for the dynamics of observables, valid to all orders in external fields, is found to contain generalized Berry curvature type of quantities that incorporate the dynamics through explicit and nontrivial time-dependence; these Berry quantities resemble the so called anomalous terms (in semiclassical equations of motion in solids) but are both of magnetic and electric type, the former being associated with Quantum Hall Effect type of behaviors and the latter with Polarization type of behaviors. By way of application of the new theorem, the quantum equations of motion of a spinless and a spinfull electron in a solid are derived without any adiabatic or semiclassical approximation. The charge current formula for a many-body and interacting spinfull system is also derived and is found to consist of a longitudinal and a transverse part; phenomenological interpretations with respect to polarization and magnetization currents respectively then emerge in a natural way. In addition, a formula for the topological magnetoelectric effect for an interacting spinfull electron system is also provided. By carefully defining singlevaluedness in parameter space, in a nonstandard fashion and in higher rigor than usual, we are able to discuss in clarity the issue of possible obstruction of this singlevaluedness, the associated creation of Berry monopoles in parameter space and the quantization of the flux of Berry curvature (but with nontrivial dynamics included in its definition).
An extension of the Hellmann-Feynman theorem to one employing dynamical parameters that vary with time according to quantum dynamics is rigorously derived, avoiding any linear response or other approximations. The resulting theorem for the dynamics of observables, valid to all orders in external fields, is found to contain generalized Berry curvature type of quantities that incorporate the dynamics through explicit and nontrivial time-dependence; these Berry quantities resemble the so called anomalous terms (in semiclassical equations of motion in solids) but are both of magnetic and electric type, the former being associated with Quantum Hall Effect type of behaviors and the latter with Polarization type of behaviors. By way of application of the new theorem, the quantum equations of motion of a spinless and a spinfull electron in a solid are derived without any adiabatic or semiclassical approximation. The charge current formula for a many-body and interacting spinfull system is also derived and is found to consist of a longitudinal and a transverse part; phenomenological interpretations with respect to polarization and magnetization currents respectively then emerge in a natural way. In addition, a formula for the topological magnetoelectric effect for an interacting spinfull electron system is also provided. By carefully defining singlevaluedness in parameter space, in a nonstandard fashion and in higher rigor than usual, we are able to discuss in clarity the issue of possible obstruction of this singlevaluedness, the associated creation of Berry monopoles in parameter space and the quantization of the flux of Berry curvature (but with nontrivial dynamics included in its definition).
Geometric phases in quantum mechanics play an extraordinary role in broadening our understanding of fundamental significance of geometry in nature. One of the best known examples is the Berry phase [M.V. Berry (1984), Proc. Royal. Soc. London A, 392:45], which naturally emerges in quantum adiabatic evolution. So far the applicability and measurements of the Berry phase were mostly limited to systems of weakly interacting quasi-particles, where interference experiments are feasible. Here we show how one can go beyond this limitation and observe the Berry curvature, and hence the Berry phase, in generic systems as a nonadiabatic response of physical observables to the rate of change of an external parameter. These results can be interpreted as a dynamical quantum Hall effect in a parameter space. The conventional quantum Hall effect is a particular example of the general relation if one views the electric field as a rate of change of the vector potential. We illustrate our findings by analyzing the response of interacting spin chains to a rotating magnetic field. We observe the quantization of this response, which we term the rotational quantum Hall effect.
We point out that two of the most important theorems of Quantum Mechanics, the Ehrenfest theorem and the Hellmann-Feynman theorem, lack in their standard form important information: there are cases where non-Hermitian boundary contributions emerge. These contributions actually appear naturally, in order for the above theorems to be valid and applicable (i.e. in multiply-connected spaces), and this occurs for physical quantities that are not represented by well-defined self-adjoint operators (such as the position operator in a periodic potential, or in general Aharonov-Bohm configurations, either in real or in an arbitrary parameter-space, in the sense of Berry's adiabatic and cyclic procedures). In this short note, we report modifications of these two theorems when such non-Hermiticities appear, and we demonstrate how they resolve certain Quantum Mechanical paradoxes (most of them having been noticed in the past as violations of the so-called Hypervirial theorem in Quantum Chemistry). This resolution of paradoxes (essentially the re-establishment of applicability of the Ehrenfest theorem even in multiply-connected spaces) always proceeds through the appearance of certain generalized currents, in a theoretical picture with interesting structure (where a generalized continuity equation with a sink term shows up naturally).
Rarely noted paradoxes in applications of fundamental quantum relations are pointed out, with their resolution leading to emergent non-Hermitian behaviors due to boundary terms – even for closed systems and with real potentials. The role played by these non-Hermiticities on the consistency of quantum mechanical uncertainty relations is discussed, especially in multiply-connected spaces (more generally, for any system that satisfies the Bloch theorem of Solid State Physics). These subtleties – reflections of topological quantum anomalies – follow their own patterns (for any dimensionality, for both Schrödinger and Dirac/Weyl Hamiltonians and for either continuous or lattice (tight-binding) models): they can always be written as global fluxes of certain generalized current densities Jg. In continuous nonrelativistic models, these have the forms that had earlier been used by Chemists to describe atomic fragments of polyatomic molecules, while for Dirac/Weyl or other lattice models they have more interesting relativistic forms only recently worked out in graphene models. In spite of the deep mathematical origin as quantum anomalies examples are provided here, where such non-Hermiticities have a direct physical significance (for both conventional and topological materials). In all stationary state examples considered, these non-Hermitian boundary terms turn out to be quantized, this quantization being either of conventional or of a topological (Quantum Hall Effect (QHE)-type) origin. The latter claim is substantiated through direct application to a simple QHE arrangement (2D Landau system in an external in-plane electric field), where some particular Jg seems to be related to the well-known dissipationless edge currents. More generally, the non-Hermitian terms play a subtle role on Berry curvatures in solids and seem to be crucial for the consistent application of the so called Modern Theories of Polarization and Orbital Magnetization. It is emphasized that the above systems can be _closed_ (in multiply-connected space, so that the boundaries disappear, but the non-Hermiticity remains), a case in non-Hermitian physics that is not usually discussed in the literature; it is also stressed that a mapping between the above non-Hermiticity (for continuous systems) and the many recent available results in tight-binding solid state models (leading to the so-called exceptional points) is expected to promote enhanced understanding of quantum behavior at the most fundamental level.
For real potentials and for arbitrary boundary conditions we point to previously noted paradoxes in applications of the Ehrenfest and Hellmann-Feynman theorems, their resolution and the role played on the consistency of quantum mechanical uncertainty relations. The paradoxes originate from a hidden non-Hermitian character of the kinetic energy operator and are resolved if the proper boundary terms (almost always discarded in the literature) are taken seriously. These non-Hermitian contributions (reflections at a deeper level of topological anomalies) follow their own patterns (for any dimensionality, for both Schroedinger and Dirac/Weyl Hamiltonians and for either continuous or lattice (tight-binding) models): they can always be written as global fluxes of certain generalized current densities. In continuous nonrelativistic cases, these have the forms that had earlier been used by Chemists to describe atomic parts of polyatomic molecules while for Dirac/Weyl or lattice models they appear to have the corresponding relativistic forms. In spite of the fact that these boundary terms originate from a deep mathematical anomaly, we point out examples where such non-Hermiticities have physical significance in Quantum Condensed Matter Physics (for both conventional and topologically nontrivial materials). In all stationary state examples considered, these non-Hermitian boundary terms have turned out to be quantized, this quantization being either of conventional or of a topological (Quantum Hall Effect (QHE)-type) origin. The latter claim is substantiated through direct application to a simple QHE arrangement (2-D Landau system in an external electric field). Finally, the non-Hermitian terms are also shown to be crucial for the consistency of the standard uncertainty relations (of Kennard/Robertson type) in multiply-connected space or, generally, in any system that satisfies the Bloch theorem.
The quantum dynamics of a correlated quasi-twodimensional (q2d) electron system showing the Quantum Hall Effect has been investigated. Galilei transformations are discussed for the correct description of quantum mechanical states in crossed electric and magnetic fields. Our quantum liquid site model with quasibosons carrying electromagnetic flux quanta uses a spatio-temporary varying configuration of sites {R(t)}. Due to the nontrivial topology of the system and the multiple degeneracy of its ground state non Abelean Berry phases appear in the adiabatic time evolution. The calculation of the effective action functional has been carried out on the basis of a path integral approach. It is shown that the Berry phases entering the Greens function produce a Chern-Simons-like topological action term. Our theory yields a quantized conductivity tensor and, thus, exibita Quantum Hall behaviour. In calculating the local density of states of the inhomogeneous quantum liquid one obtains a remarkable broadening of peaks and a reduction of their heights.
The history of modern condensed matter physics may be regarded as the competition and reconciliation between Stoner’s and Anderson’s physical pictures, where the former is based on momentum–space descriptions focusing on long wave-length fluctuations while the latter is based on real-space physics emphasizing emergent localized excitations. In particular, these two view points compete with each other in various nonperturbative phenomena, which range from the problem of high [Formula: see text] superconductivity, quantum spin liquids in organic materials and frustrated spin systems, heavy-fermion quantum criticality, metal-insulator transitions in correlated electron systems such as doped silicons and two-dimensional electron systems, the fractional quantum Hall effect, to the recently discussed Fe-based superconductors. An approach to reconcile these competing frameworks is to introduce topologically nontrivial excitations into the Stoner’s description, which appear to be localized in either space or time and sometimes both, where scattering between itinerant electrons and topological excitations such as skyrmions, vortices, various forms of instantons, emergent magnetic monopoles, and etc. may catch nonperturbative local physics beyond the Stoner’s paradigm. In this review paper, we discuss nonperturbative effects of topological excitations on dynamics of correlated electrons. First, we focus on the problem of scattering between itinerant fermions and topological excitations in antiferromagnetic doped Mott insulators, expected to be relevant for the pseudogap phase of high [Formula: see text] cuprates. We propose that nonperturbative effects of topological excitations can be incorporated within the perturbative framework, where an enhanced global symmetry with a topological term plays an essential role. In the second part, we go on to discuss the subject of symmetry protected topological states in a largely similar light. While we do not introduce itinerant fermions here, the nonperturbative dynamics of topological excitations is again seen to be crucial in classifying topologically nontrivial gapped systems. We point to some hidden links between several effective field theories with topological terms, starting with one-dimensional physics, and subsequently finding natural generalizations to higher dimensions.