TY - JOUR

T1 - Viscoelastic response of topological tight-binding models in two and three dimensions

AU - Shapourian, Hassan

AU - Hughes, Taylor L.

AU - Ryu, Shinsei

N1 - Publisher Copyright:
© 2015 American Physical Society.

PY - 2015/10/29

Y1 - 2015/10/29

N2 - The topological response to external perturbations is an effective probe to characterize different topological phases of matter. Besides the Hall conductance, the Hall viscosity is another example of such a response that measures how electronic wave functions respond to changes in the underlying geometry. Topological (Chern) insulators are known to have a quantized Hall conductance. A natural question is how the Hall viscosity behaves for these materials. So far, most of studies on the Hall viscosity of Chern insulators have focused on the continuum limit. The presence of lattice breaks the continuous translational symmetry to a discrete group and this causes two complications: it introduces a new length scale associated with the lattice constant, and makes the momentum periodic. We develop two different methods of how to implement a lattice deformation: (1) a lattice distortion is encoded as a shift in the lattice momentum and (2) a lattice deformation is treated microscopically in the gradient expansion of the hopping matrix elements. After establishing the method of deformation, we can compute the Hall viscosity through a linear response (Kubo) formula. We examine these methods for three models: the Hofstadter model, the Chern insulator, and the surface of a 3D topological insulator. Our results in certain regimes of parameters, where the continuum limit is relevant, are in agreement with previous calculations. We also provide possible experimental signatures of the Hall viscosity by studying the phononic properties of a single-crystal 3D topological insulator.

AB - The topological response to external perturbations is an effective probe to characterize different topological phases of matter. Besides the Hall conductance, the Hall viscosity is another example of such a response that measures how electronic wave functions respond to changes in the underlying geometry. Topological (Chern) insulators are known to have a quantized Hall conductance. A natural question is how the Hall viscosity behaves for these materials. So far, most of studies on the Hall viscosity of Chern insulators have focused on the continuum limit. The presence of lattice breaks the continuous translational symmetry to a discrete group and this causes two complications: it introduces a new length scale associated with the lattice constant, and makes the momentum periodic. We develop two different methods of how to implement a lattice deformation: (1) a lattice distortion is encoded as a shift in the lattice momentum and (2) a lattice deformation is treated microscopically in the gradient expansion of the hopping matrix elements. After establishing the method of deformation, we can compute the Hall viscosity through a linear response (Kubo) formula. We examine these methods for three models: the Hofstadter model, the Chern insulator, and the surface of a 3D topological insulator. Our results in certain regimes of parameters, where the continuum limit is relevant, are in agreement with previous calculations. We also provide possible experimental signatures of the Hall viscosity by studying the phononic properties of a single-crystal 3D topological insulator.

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U2 - 10.1103/PhysRevB.92.165131

DO - 10.1103/PhysRevB.92.165131

M3 - Article

AN - SCOPUS:84946945301

SN - 1098-0121

VL - 92

JO - Physical Review B - Condensed Matter and Materials Physics

JF - Physical Review B - Condensed Matter and Materials Physics

IS - 16

M1 - 165131

ER -