Abstract
Inspired by recent experiments on graphene, we examine the nondissipative viscoelastic response of anisotropic two-dimensional quantum systems. We pay particular attention to electron fluids with point group symmetries and those with discrete translational symmetry. We start by extending the Kubo formalism for viscosity to systems with internal degrees of freedom and discrete translational symmetry, highlighting the importance of properly considering the role of internal angular momentum. We analyze the Hall components of the viscoelastic response tensor in systems with discrete point group symmetry, focusing on the hydrodynamic implications of the resulting forces. We show that though there are generally six Hall viscosities, there are only three independent contributions to the viscous force density in the bulk. To compute these coefficients, we develop a framework to consistently write down the long-wavelength stress tensor and viscosity for multicomponent lattice systems. We apply our formalism to lattice and continuum models, including a lattice Chern insulator and anisotropic superfluid.Received 27 October 2019Accepted 28 February 2020DOI:https://doi.org/10.1103/PhysRevX.10.021005Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasTopological phases of matterTechniquesHydrodynamicsLinear response theoryCondensed Matter, Materials & Applied Physics
Highlights
One of the most peculiar and fascinating manifestations of topology in condensed matter physics is the appearance of nondissipative transport coefficients in insulating systems
It has been shown that there is a unique Hall viscosity coefficient ηH, which for a gapped phase is proportional to the particle density nand a quantized invariant of the ground state known as the shift S [6,7], ηH
Our three main conceptual innovations are as follows: (1) a novel detailed analysis of the nondissipative viscosity tensor, revealing that there are generally six viscosity coefficients, three are redundant in the bulk; (2) a relationship between band topology and Hall viscosity for free-fermion systems, showing how the six viscosity coefficients are expressible in terms of quadrupole moments of the Berry curvature of occupied bands and a correction due to the internal angular momentum of bands; and (3) the first consistent framework for momentum transport and viscosity on a lattice or tight-binding system, derived only from conservation laws
Summary
One of the most peculiar and fascinating manifestations of topology in condensed matter physics is the appearance of nondissipative transport coefficients in insulating systems. Our three main conceptual innovations are as follows: (1) a novel detailed analysis of the nondissipative viscosity tensor, revealing that there are generally six viscosity coefficients, three are redundant in the bulk (we find a similar redundancy in the dissipative viscosity); (2) a relationship between band topology and Hall viscosity for free-fermion systems, showing how the six viscosity coefficients are expressible in terms of quadrupole moments of the Berry curvature of occupied bands and a correction due to the internal (pseudospin) angular momentum of bands; and (3) the first consistent framework for momentum transport and viscosity on a lattice or tight-binding system, derived only from conservation laws In formulating these results, we develop an extension of the Belinfante-Rosenfeld symmetrization procedure to anisotropic continuum and lattice systems, fixing the antisymmetric part of the stress tensor operator. We give some brief background and establish our notational conventions
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