Algebraic theory of crystal vibrations: Singularities and zeros in vibrations of one- and two-dimensional lattices

Abstract

A novel method for the calculation of the energy dispersion relation (EDR) and density of states (DOS) in one (1D) and two (2D) dimensions is introduced and applied to linear lattices (1D) and square and hexagonal lattices (2D). The (van Hove) singularities and (Dirac) zeros of the DOS are discussed. Results for the 2D hexagonal lattice (graphene-like materials) are compared with experimental data in microwave photonic crystals.

Algebraic Theory of Crystal Vibrations: Localization Properties of Wave Functions in Two-Dimensional Lattices

“…The symmetry adaptation of this lattice, Figure 1, was discussed in [12]. The basis vectors a i and b i generating the lattice and the reciprocal lattice, respectively, are given as Within a unit cell, with symmetry D 4h , we have two types of interactions, nearest-neighbor, λ (I) , and next-to-nearest neighbor, λ (I I) .…”

“…Here, k 1 and k 2 are the components of k in the reciprocal lattice basis, k = k 1 b 1 + k 2 b 2 , and the sign convention for λ (I) , λ (I I) is that of [12,41]. Accordingly, in terms of the components k x and k y , the EDR is given by [12,41,42] E k x , k y = ε − λ (I) 2 cos k x a + cos k y a − λ (I I) 4 cos k x a cos k y a,…”

“…One-dimensional vibrations have been studied both in molecules (0D) [14], and in polymers (1D) [15,35,36]. Applications to 2D lattices were introduced in [12], where both harmonic and anharmonic vibrations were studied.…”

“…Our sign convention for these coefficients is that of Reference [12], and they are assumed to be positive. In a previous publication [12], we have studied the energy eigenvalues of the Hamiltonian Equation (4) and, from those, we have obtained the EDR and the DOS for linear (1D), square and hexagonal (2D) lattices, with symmetry C ∞v , D 4h , and D 6h , respectively. In this article, we study the corresponding eigenfunctions.…”

Excited-state quantum phase transitions and their manifestations in an extended Dicke model

“…As indicated, the irregular part of the level density can be also considered for noninteger values of f . This is relevant in lattice systems (where the level density is calculated by integration of the dispersion relations over a space with an arbitrary integer dimension, hence f can be effectively half-integer [30]) or in systems explicitly dependent on time, where the 2 f dimensional phase space is supplemented by the time dimension, leading to the effective number of degrees of freedom equal to f + 1/2 [31].…”