There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Ever since its discovery, the Berry phase has permeated through all branches of physics. Over the last three decades, it was gradually realized that the Berry phase of the electronic wave function can have a profound effect on material properties and is responsible for a spectrum of phenomena, such as ferroelectricity, orbital magnetism, various (quantum/anomalous/spin) Hall effects, and quantum charge pumping. This progress is summarized in a pedagogical manner in this review. We start with a brief summary of necessary background, followed by a detailed discussion of the Berry phase effect in a variety of solid state applications. A common thread of the review is the semiclassical formulation of electron dynamics, which is a versatile tool in the study of electron dynamics in the presence of electromagnetic fields and more general perturbations. Finally, we demonstrate a re-quantization method that converts a semiclassical theory to an effective quantum theory. It is clear that the Berry phase should be added as a basic ingredient to our understanding of basic material properties.

/pdf/berry-phase-effects-on-electronic-properties-53dkbbi2er.pdf

Berry phase effects on electronic properties

Annals of physics

For a general quantum system driven by a slowly time-dependent Hamiltonian, transitions between instantaneous eigenstates are exponentially weak. But a nearby Hamiltonian exists for which the transition amplitudes between any eigenstates of the original Hamiltonian are exactly zero for all values of slowness. The general theory is illustrated by spins driven by changing magnetic fields, and implies that any spin expectation history, including those where the spin never precesses, can be generated by infinitely many driving fields, here displayed explicitly. Asymptotically, the absence of transitions is explained by continuation to complex time, where the complex degeneracies in the transitionless driving fields have a nongeneric structure for which there is no Stokes phenomenon; this is analogous to the explanation of reflectionless potentials.

https://iopscience.iop.org/article/10.1088/1751-8113/42/36/365303/pdf

Transitionless quantum driving

The simple act of slowly varying the parameters of a quantum system so that it remains always in its ground state is extremely rich from an information processing point of view. For an ideal, closed system, this adiabatic evolution is equivalent to full quantum computation, and it is convenient for establishing quantum algorithms for optimization. This review presents adiabatic quantum algorithms, proves the closed-system equivalence of the adiabatic and circuit models of quantum computation, reviews the placement of adiabatic quantum computation in the more general classification of computational complexity theory, and discusses the case of ``stoquastic'' quantum evolutions.

Adiabatic quantum computation

We consider a quantum mechanical system of N bosons with relativistic dispersion interacting through a mean field Coulomb potential (attractive or repulsive). We choose the initial wave function to describe a condensate where the N bosons are all in the same one-particle state. Starting from the N-body Schrodinger equation, we prove that, in the limit N ∞, the time evolution of the one-particle density is governed by the relativistic nonlinear Hartree equation. This equation is used to describe the dynamics of boson stars (Chandrasekhar theory). The corresponding static problem was rigorously solved in [10]. © 2005 Wiley Periodicals, Inc.

Mean Field Dynamics of Boson Stars

We prove the adiabatic theorem for quantum evolution without the traditional gap condition. All that this adiabatic theorem needs is a (piecewise) twice differentiable finite dimensional spectral projection. The result implies that the adiabatic theorem holds for the ground state of atoms in quantized radiation field. The general result we prove gives no information on the rate at which the adiabatic limit is approached. With additional spectral information one can also estimate this rate.

Adiabatic theorem without a gap condition

We study localization effects of disorder on the spectral and dynamical properties of Schrodinger operators with random potentials. The new results include exponentially decaying bounds on the transition amplitude and related projection kernels, including in the mean. These are derived through the analysis of fractional moments of the resolvent, which are finite due to the resonance-diffusing effects of the disorder. The main difficulty which has up to now prevented an extension of this method to the continuum can be traced to the lack of a uniform bound on the Lifshitz-Krein spectral shift associated with the local potential terms. The difficulty is avoided here through the use of a weak-L
 1 estimate concerning the boundary-value distribution of resolvents of maximally dissipative operators, combined with standard tools of relative compactness theory.

/pdf/moment-analysis-for-localization-in-random-schrodinger-vsmtxb1j8c.pdf

Moment analysis for localization in random Schrödinger operators

We consider a quantum mechanical system of N bosons with relativistic dispersion interacting through a mean field Coulomb potential (attractive or repulsive). We choose the initial wave function to describe a condensate, where the N bosons are all in the same one-particle state. Starting from the N-body Schroedinger equation, we prove that, in the limit N goes to infinity, the time evolution of the one-particle density is governed by the relativistic nonlinear Hartree equation. This equation is used to describe the dynamics of boson stars (Chandrasekhar theory). The corresponding static problem was rigorously solved by Lieb and Yau.

Alexander Elgart

Papers

Mean Field Dynamics of Boson Stars

Adiabatic theorem without a gap condition

Moment analysis for localization in random Schrödinger operators

Mean Field Dynamics of Boson Stars

The Adiabatic Theorem of Quantum Mechanics