Semiclassics for matrix Hamiltonians: The Gutzwiller trace formula with applications to graphene-type systems

M. Vogl, O. Pankratov, and S. Shallcross
Phys. Rev. B 96, 035442 – Published 27 July 2017

Abstract

We present a tractable and physically transparent semiclassical theory of matrix-valued Hamiltonians, i.e., those that describe quantum systems with internal degrees of freedoms, based on a generalization of the Gutzwiller trace formula for a n×n dimensional Hamiltonian H(p̂,q̂). The classical dynamics is governed by n Hamilton-Jacobi (HJ) equations that act in a phase space endowed with a classical Berry curvature encoding anholonomy in the parallel transport of the eigenvectors of H(p,q); these vectors describe the internal structure of the semiclassical particles. At the O(1) level and for nondegenerate HJ systems, this curvature results in an additional semiclassical phase composed of (i) a Berry phase and (ii) a dynamical phase resulting from the classical particles “moving through the Berry curvature”. We show that the dynamical part of this semiclassical phase will, generally, be zero only for the case in which the Berry phase is topological (i.e., depends only on the winding number). We illustrate the method by calculating the Landau spectrum for monolayer graphene, the four-band model of AB bilayer graphene, and for a more complicated matrix Hamiltonian describing the silicene band structure. Finally, we apply our method to an inhomogeneous system consisting of a strain engineered one-dimensional moiré in bilayer graphene, finding localized states near the Dirac point that arise from electron trapping in a semiclassical moiré potential. The semiclassical density of states of these localized states we show to be in perfect agreement with an exact quantum mechanical calculation of the density of states.

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  • Received 28 November 2016
  • Revised 1 June 2017

DOI:https://doi.org/10.1103/PhysRevB.96.035442

©2017 American Physical Society

Physics Subject Headings (PhySH)

Condensed Matter, Materials & Applied PhysicsGeneral Physics

Authors & Affiliations

M. Vogl,*, O. Pankratov, and S. Shallcross

  • Lehrstuhl für Theoretische Festkörperphysik, Staudtstrasse 7-B2, 91058 Erlangen, Germany

  • *Present address: Department of Physics, The University of Texas at Austin, Austin, TX 78712, USA.

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Vol. 96, Iss. 3 — 15 July 2017

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