Geometrodynamics of electrons in a crystal under position and time-dependent deformation

Liang Dong and Qian Niu

Phys. Rev. B 98, 115162 – Published 28 September 2018

Abstract

The semiclassical dynamics of Bloch electrons in a crystal under slowly varying deformation is developed in the geometric language of a lattice bundle. Berry curvatures and gradients of energy are introduced in terms of lattice covariant derivatives, with the corresponding connections given by the gradient and rate of strain. A number of physical effects are discussed: an effective post-Newtonian gravity at band bottom, polarization induced by spatial gradient of strain, orbital magnetization induced by strain rate, and the electron energy-stress tensor.

Received 16 February 2018
Revised 6 July 2018

DOI:https://doi.org/10.1103/PhysRevB.98.115162

Physics Subject Headings (PhySH)

Geometric & topological phases

Crystalline systems

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Liang Dong^1,2 and Qian Niu^2,1

¹International Center for Quantum Materials, Peking University, Beijing 100871, China
²Department of Physics, University of Texas, Austin, Texas 78712, USA

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Issue

Vol. 98, Iss. 11 — 15 September 2018

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Images

Figure 1
A schematic picture of how to identify the local lattices in a deforming crystal. Blue circles represent the lattice points of a deforming crystal. Dashed lines represent the crystalline lines of the fictitious periodic local lattice. As can be seen, the local distribution of lattice points is well described by the local lattice only with deviation far away from the local point. The right panel is the zooming-in picture. The lattice vector of a local lattice is given by the relative displacement between lattice points and located in the middle denoted by the black dots on the arrows. And a local lattice moves rigidly at velocity $W$ .
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Figure 2
(a) Elastic condition of lattice vectors. The dots on the four corners represent lattice points, which determine the local lattice vectors denoted by the four edges of the quadrilateral. The dot in the middle of each edge denotes where these lattice vectors reside. According to Eq. (1), it is straightforward to derive the condition for the four lattice vectors to form a closed quadrilateral. (b) Schematics of dislocation in a square lattice. Along the closed trajectory denoted by the bold line, if we count the change of lattice label, we always find a unit mismatch in the crystalline direction. (c) Schematics of a disclination in a square lattice. Along the closed trajectory denoted by the bold line, we find that the lattice vector continuously changes from $c$ to $\tilde{c}$ as shown in the picture. (d) A special case of disclination in a rectangular lattice, which is characterized by the $2 π$ rotation of lattice vectors along the defect center. In the region far enough from the center, locally we have a rectangular lattice, whose two lattice vectors $c_{1} = c_{1} θ, c_{2} = c_{2} r$ are along the angle direction and radius direction, respectively, in a polar coordinate. The disclination located at the origin is characterized by the rotation of lattice vectors by $2 π$ along the complete circle.
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Figure 3
The phase space of electrons in the one-dimensional case. The straight line at the bottom denotes the position space $x$ . Local Brillouin zones are denoted by circles at each position. The intersection between each circle and the straight line is the $k = 0$ point. The phase space is a “tube” with a varying radius. To compare two points in phase space, we show two alternatives. One is along the path denoted by $\partial_{x}$ where the value of $k$ is fixed, which then goes along the circle by the total change of wave vector $d k$ . Another path is to go along the correspondence line formed by parallel transport denoted by $\nabla_{x}$ , then go along the circle by the mechanical change of wave vector $D k$ .
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