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

We present a unified theory for wave-packet dynamics of electrons in crystals subject to perturbations varying slowly in space and time. We derive the wave-packet energy up to the first-order gradient correction and obtain all kinds of Berry phase terms for the semiclassical dynamics and the quantization rule. For electromagnetic perturbations, we recover the orbital magnetization energy and the anomalous velocity purely within a single-band picture without invoking interband couplings. For deformations in crystals, besides a deformation potential, we obtain a Berry-phase term in the Lagrangian due to lattice tracking, which gives rise to new terms in the expressions for the wave-packet velocity and the semiclassical force. For multiple-valued displacement fields surrounding dislocations, this term manifests as a Berry phase, which we show to be proportional to the Burgers vector around each dislocation.

Wave-packet dynamics in slowly perturbed crystals: Gradient corrections and Berry-phase effects

The Bohr-Sommerfeld quantization rule lies at the heart of the semiclassical theory of a Bloch electron in a magnetic field. This rule is predictive of Landau levels and quantum oscillations for conventional metals as well as for a host of topological metals, which have emerged in the recent synergy between band theory, crystalline symmetries, and topology. Here, the authors systematically formulate quantization rules that apply to energy-degenerate bands and to orbits that intersect at saddle points and type-II Dirac points where quantum tunneling is unavoidable. The formulation extends previous work by incorporating the orbital magnetic moment and the geometric Berry phase. The latter is an indispensable property of topological metals.

https://link.aps.org/accepted/10.1103/PhysRevB.97.144422

Semiclassical theory of Landau levels and magnetic breakdown in topological metals

The simulation of complex quantum systems on a quantum computer is studied, taking the kicked Harper model as an example. This well-studied system has a rich variety of dynamical behavior depending on parameters, displays interesting phenomena such as fractal spectra, mixed phase space, dynamical localization, anomalous diffusion, or partial delocalization, and can describe electrons in a magnetic field. Three different quantum algorithms are presented and analyzed, enabling us to simulate efficiently the evolution operator of this system with different precision using different resources. Depending on the parameters chosen, the system is near integrable, localized, or partially delocalized. In each case we identify transport or spectral quantities which can be obtained more efficiently on a quantum computer than on a classical one. In most cases, a polynomial gain compared to classical algorithms is obtained, which can be quadratic or less depending on the parameter regime. We also present the effects of static imperfections on the quantities selected and show that depending on the regime of parameters, very different behaviors are observed. Some quantities can be obtained reliably with moderate levels of imperfection even for large number of qubits, whereas others are exponentially sensitive to the number of qubits. In particular, the imperfection threshold for delocalization becomes exponentially small in the partially delocalized regime. Our results show that interesting behavior can be observed with as little as $7--8\phantom{\rule{0.3em}{0ex}}\mathrm{qubits}$ and can be reliably measured in presence of moderate levels of internal imperfections.

https://hal.archives-ouvertes.fr/hal-00002791/document

Quantum computation of a complex system: The kicked Harper model

We find that a 2D periodic potential, with different modulation amplitudes in the x and y directions, and a perpendicular magnetic field may lead to a transition to electron transport along the direction of stronger modulation and to localization in the direction of weaker modulation. In the experimentally accessible regime we relate this new quantum transport phenomenon to avoided band crossings due to classical chaos.

https://arxiv.org/pdf/cond-mat/9810313.pdf

Bloch electrons in a magnetic field: why does chaos send electrons the hard Way?

We consider a set of localized Wannier functions which can be used to describe the Bloch bands of one-dimensional Hamiltonians whose associated Weyl function is periodic under a hexagonal lattice of translations in phase space. These Hamiltonians can be used to describe electrons on two-dimensional hexagonal lattices penetrated by a magnetic field. The effect on the Wannier functions of a rotation of the phase plane is considered. The resulting transformation is found to be surprisingly complicated when the quantized Hall conductance integer for the Bloch band is non-zero.

A six-fold rotation operator for the Wannier functions of the phase-space lattice Hamiltonian

We develop a renormalization group analysis for a Hamiltonian which is a periodic function of and ; this is a model for Bloch electrons in a magnetic field, and includes Harper's model as a special case. The eigenfunctions are Bloch waves when , the ratio of to the area of the unit cell, is a rational number p/q. The renormalization procedure produces an effective Hamiltonian for a subset of the spectrum, which can be expanded in . The zeroth-order term has been obtained in earlier papers; here we obtain the first-order term of this expansion in terms of the q-dimensional vectors which define the rational Bloch states (k and are Bloch wavevectors). The effective Hamiltonian is not invariant under gauge transformations multiplying the vector . We show that the infinitesimal gauge transformation is equivalent to evolving under a gauge Hamiltonian obtained (to lowest order in ) by quantizing .

/pdf/first-order-correction-to-the-renormalization-of-the-phase-2rlds86unq.pdf

First order correction to the renormalization of the phase space lattice Hamiltonian

The phase space lattice Hamiltonian is a realistic model for Bloch electrons in a magnetic field. It has a fractal spectrum when the lattice has centres of threefold or fourfold rotational symmetry. This fact has been explained using a renormalization group (RG) method, assuming that the RG transformation preserves the symmetry of the Hamiltonian. The symmetry preservation property has previously been demonstrated for fourfold rotation; the threefold case is considerably more difficult to analyse. In this paper we present a simplified form of the RG equations which clearly exhibits the threefold symmetry preservation. We also discuss the case of sixfold rotational symmetry, for which the symmetry of the Hamiltonian may be reduced to threefold under the action of the RG.

/pdf/conserved-and-broken-symmetries-in-the-renormalization-of-57dwtpwhka.pdf

Conserved and broken symmetries in the renormalization of the phase space lattice Hamiltonian

Heff describing a subset of the spectrum, for which the rational limit b i! p y q is also a semiclassical limit; we give the first two terms of the expansion of ˆ Heff in powers of b2 p y q. We derive a Bohr-Sommerfeld quantization condition, involving a Berry phase correction, and an equation for the bandwidth when b › p1yq1 with q1 large. We also discuss the dynamics of ˆ Heff under infinitesimal gauge transformations of the rational Bloch states. is a discrete Schrodinger equation which is widely used as a model for electrons in two-dimensional lattice structures penetrated by a magnetic field, or for electrons in incom- mensurate potentials. The parameter b is the ratio of the area of a flux quantum to that of the unit cell, or the com- mensurability of the superposed potentials: Derivations of the Harper equation in the context of Bloch electrons in a magnetic field are given in (1,2). The solution of (1) for the energy levels E and eigenstates hcnj is a difficult problem for which semiclassical methods have been very useful (3,4); the semiclassical limit is b i! 0 , and we de- fine a dimensionless effective Planck constant ¯ h › 2pb. For many purposes this limit is sufficient for physical ap- plications; for example, in ordinary solids the number of flux quanta per unit cell is always small. Experiments on artificial lattices make values of b of order unity achiev- able, and distinctive features of the spectrum of Harper's equation may soon be detectable in semiconductor super- lattices (5), and in superconducting grids (6). In this Let- ter we discuss an effective Hamiltonian method for which the semiclassical limit is b i! p y q ( p and q are coprime integers): Because the rationals are a dense set, this pro- vides a far-reaching extension of the semiclassical ap- proach. The idea that b i! p y q can be a semiclassical limit was originally proposed by Sokoloff (7), and later developed by others (8 - 10); we will give simple deriva- tions of results which cannot be obtained using these earlier approaches. When b › pyq, the eigenstates of (1) are Bloch waves, and the spectrum consists of q bands. We will describe an effective Hamiltonian for a subset of the spectrum which collapses onto a Bloch band as b i! p y q ; because the effective Hamiltonian is similar in form to the original one, this is a renormalization group (RG) transformation. The results are obtained from an algorithm for constructing an exact effective Hamiltonian derived in (11), following an approach introduced in (12). This Letter gives explicit expressions for the first two terms of a series expansion in Db › b2 p y q. The zeroth order term was deduced in the earlier papers, but this is not sufficient to determine the spectrum in the limit b i! p y q , because the first order correction is required for the Bohr-Sommerfeld quantization condition. With the addition of the first order correction, the RG method gives a satisfying understanding of the spectrum of Harper's equation. Our expression for the first order term is written in terms of the rational Bloch eigenstates: It consists of two components, one of which is not invariant under "gauge transformations" which alter the relative phases of the Bloch states at different points in the Brillouin zone. We write the Bohr-Sommerfeld quantization condition in gauge invariant form by incorporating the contribution from the gauge dependent term as a Berry phase correc- tion; a related expression was obtained by Chang and Niu (13), but their calculation misses the gauge independent first order contribution. We also apply the first order cor- rection to a calculation of the total width of the spectrum of the band when b › p1yq1, which is a high order ration- al approximant of pyq. We will derive a formula recently proposed by Tan (14) on the basis of numer- ical experiments; efforts to derive this result using the methods of (8 - 10) were not successful. We anticipate that our results will find other applications, for example, in problems involving the total energy of the Harper spectrum (15). The Hamiltonian of Harper's equation is a special case of

http://users.mct.open.ac.uk/mw987/Published%20papers/PAPER39.PDF

Semiclassical Limits of the Spectrum of Harper's Equation

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A six-fold rotation operator for the Wannier functions of the phase-space lattice Hamiltonian

First order correction to the renormalization of the phase space lattice Hamiltonian

Conserved and broken symmetries in the renormalization of the phase space lattice Hamiltonian

Semiclassical Limits of the Spectrum of Harper's Equation