Geometric Phases and Wannier functions of Bloch electrons in one dimension (original) (raw)

Wannier Functions and the Phases of the Bloch Functions

Physical Review B, 1970

We discuss the problem of what phases should be attributed to the Bloch functions in order to obtain Wannier functions with minimal widths. An exact solution to this problem is foundby means of the k'p perturbation formalism. We also consider localized crystal functions of a more general kind. These functions are found to obey a Schrodinger equation in reciprocal space, and may have some important assets when compared to the Wannier functions.

Ewald-type formulas for Gaussian-basis Bloch states in one-dimensionally periodic systems

The Journal of Chemical Physics, 2010

Expressions for integrals involving general gaussian (s, p, d, . . . ) basis Bloch functions are presented. Applying Poisson transformation and Ewald-type partitioning scheme, all lattice sums appearing in these expressions lead to fast convergence in both direct and Fourier spaces. Numerical results produced for selected test cases show that a limited number of terms in the lattice sums are necessary to get convergence in the two spaces. 71.15.Dx,31.15.xr,31.15.xp

Bloch oscillations in two-dimensional lattices

New Journal of Physics, 2004

Bloch oscillations in a two-dimensional periodic potential under a (relatively weak) static force are studied for separable and non-separable potentials. The dynamics depends sensitively on the direction of the static field with respect to the lattice. Almost dispersionless periodic motion of the wavepackets is observed, as well as breathing modes. The origin of the weak dispersion is analysed. In addition, the coherent Bloch-Zener oscillation for a double-period potential inducing an additional Rowland ghost gap is discussed.

Generalization of Bloch's theorem for arbitrary boundary conditions: Theory

We present a generalization of Bloch's theorem to finite-range lattice systems of independent fermions, in which translation symmetry is broken solely due to arbitrary boundary conditions, by providing exact, analytic expressions for all energy eigenvalues and eigenstates. Starting with a reordering of the fermionic basis that transforms the single-particle Hamiltonian into a corner-modified banded block-Toeplitz matrix, a key step is a Hamiltonian-dependent bipartition of the lattice, which splits the eigenvalue problem into a system of bulk and boundary equations. The eigensystem inherits most of its solutions from an auxiliary, infinite translation-invariant Hamiltonian that allows for nonunitary representations of translation—hence complex values of crystal momenta with specific localization properties. A reformulation of the boundary equation in terms of a boundary matrix ensures compatibility with the boundary conditions, and determines the allowed energy eigenstates in the form of generalized Bloch states. We show how the boundary matrix quantitatively captures the interplay between bulk and boundary properties, leading to the construction of efficient indicators of bulk-boundary correspondence. Remarkable consequences of our generalized Bloch theorem are the engineering of Hamiltonians that host perfectly localized, robust zero-energy edge modes, and the predicted emergence, for instance, in Kitaev's Majorana chain, of localized excitations whose amplitudes decay in space exponentially with a power-law prefactor. We further show how the theorem may be used to construct numerical and algebraic diagonalization algorithms for the class of Hamiltonians under consideration, and use the proposed bulk-boundary indicator to characterize the topological response of a multiband time-reversal invariant s-wave topological superconductor under twisted boundary conditions, showing how a fractional Josephson effect can occur without entailing a fermionic parity switch. Finally, we establish connections to the transfer matrix method and demonstrate, using the paradigmatic Kitaev's chain example, that a defective (nondiagonalizable) transfer matrix signals the presence of solutions with a power-law prefactor.

Bloch oscillations in uniform magnetic and electric fields

Il Nuovo Cimento B, 1987

In the present paper, by using the method of the effective Hamiltonian, we study Bloch's electrons in the presence of a uniform electric and magnetic field. We show that all wave functions are periodic in time. This result is the fully quantum-mechanical analog of the well-known Bloch oscillations predicted by quasi-classical dynamics. Furthermore, using the tight-binding method for a single band of an electron in a simple cubic lattice in the presence of a uniform electric and magnetic field, we obtain an equation of Harper's form. From this equation we take exactly the points at which Bragg reflection occurs for the subbands of Landau.

Spectrum of 2D Bloch electrons in a periodic magnetic field : algebraic approach

Journal de Physique, 1990

Résume. 2014 Les méthodes algébriques que l'on a introduites récemment pour l'étude des électrons de Bloch sous champ magnétique uniforme sont généralisées au cas des champs periodiques. En utilisant une approche semi-classique, on étudie le cas où la maille magnetique est commensurable avec celle du réseau. En general et selon la valeur 03A6 du flux magnétique moyen a travers la cellule 616mentaire, deux cas distincts semblent se distinguer. Le premier cas est 0 =A 0, où la structure en niveaux de Landau est retrouvée (cas non commutatif). Dans le second cas 03A6 = 0, on obtient une structure de bandes non triviale (cas commutatif). Nos résultats sont illustrés avec des exemples simples. En particulier on montre, sous certaines conditions, que le mecanisme de stabilisation de la mer de Fermi, avec un quantum de flux par fermion, se généralise au cas d'un champ magnétique périodique.

Spectral properties of reduced Bloch Hamiltonians

Annals of Physics, 1977

Bloch Hamiltonians are defined, and the existence of bands is proven for a large class of periodic operators. The results are strong enough to include most of the reasonable physical models of a single electron in crystals. A notable exception is the Dirac Bloch Hamiltonian for a Coulombic crystal with high charge. Properties of the Bloch waves are briefly described and it is shown that "simple" Bloch Hamiltonians do not have Bloch waves with a finite number of Fourier coefficients. The asymptotic distribution of the bands is determined, and it is shown that for a large class of Hamiltonians, it is determined by the kinetic energy alone.

Effective Dynamics for Bloch Electrons: Peierls Substitution and Beyond

Communications in Mathematical Physics, 2003

We consider an electron moving in a periodic potential and subject to an additional slowly varying external electrostatic potential, φ(εx), and vector potential A(εx), with x ∈ R d and ε ≪ 1. We prove that associated to an isolated family of Bloch bands there exists an almost invariant subspace of L 2 (R d ) and an effective Hamiltonian governing the evolution inside this subspace to all orders in ε. To leading order the effective Hamiltonian is given through the Peierls substitution. We explicitly compute the first order correction. From a semiclassical analysis of this effective quantum Hamiltonian we establish the first order correction to the standard semiclassical model of solid state physics.

Bloch representation of x for a finite crystal

Physics Letters A, 1988

The eigenstates and eigenvalues ofan electron in a finite (L=Na), 1-D, periodic potential subjected to a constant electric field are shown to be solutions to an Nth-order secular equation which does not couple band indices, Implications for Wannier-Stark ladders and related phenomena are stressed,

Fermionic and Bosonic Partition Functions at Imaginary Chemical Potential as Bloch Functions

American journal of modern physics, 2024

In this work it is pointed out that the phase transitions of the d + 1 Gross-Neveu (fermionic) and CP N -1 (bosonic) models at finite temperature and imaginary chemical potential can be mapped to transformations of Hubbard-like regular hexagonal to square lattice with the intermediate steps to be specific surfaces (irregular hexagonal kind) with an ordered construction based on the even indexed Bloch-Wigner-Ramakrishnan polylogarithm function. The zeros and extrema of the Clausen Cl d (θ) function play an important role to the analysis since they allow us not only to study the fermionic and bosonic theories and their phase transitions but also the possibility to explore the existence of conductors arising from the correspondence between the partition functions of the two models and the Bloch and Wannier functions that play a crucial role in the tight-binding approximation in solid state physics. The main aim of this work is not only to unveil the relevance of the canonical partition functions of a fermionic and a bosonic model to Bloch states by using an imaginary chemical potential but also to examine the overlap between two Bloch wave-functions that differ by a lattice momentum that calculates the momentum transfer of a Bloch wave during the interaction with a lattice point of a hexagonal construction.