Geometric response in topological phases and exotic classical fluids

Asking how a system responds to deformations is a longstanding question in science, and one that has given rise to concepts such as stiffness and viscosity that are indispensable in our thinking about the materials and liquids that surround us. In this thesis, we will revisit this question in an array of contemporary systems from topological materials to chiral active fluids, and open the door to new answers in the form of novel response phenomena and new perspectives. We begin by focusing our discussion on an unusual type of viscosity that can arise in systems with broken time reversal symmetry called the Hall (or odd) viscosity, which is a response to time-dependent spatial strains of the system that does not dissipate power. The Hall viscosity has been of theoretical and experimental interest in a wide variety of settings, from the quantum Hall effect, to the hydrodynamic regime of strongly correlated materials, to classical self-spinning chiral active fluids. Despite this wide interest, this viscosity was only well understood with the simplifying assumptions of full rotational and translational symmetry. The first part of this thesis is dedicated to broadening the scope of the non-dissipative viscosity. We start in Chapter 2 by focusing on the relationship between anisotropy, spin and viscosity in two- dimensional continuum systems. We first present a decomposition of the Hall viscosity tensor for anisotropic systems, and consider the physical implications of the viscosity coefficients. We find that while there are generally six Hall viscosity coefficients, only three contribute for the bulk force, a concept which we call the viscous redundancy. We show that there exists a similar redundancy for the dissipative viscosity as well. To treat the stress response of systems with spin, we develop a non-relativistic generalization of the Belinfante procedure for anisotropic systems. With these ingredients in place, we consider the non- dissipative viscosity for general free fermion systems, finding a relationship between viscosity, spin and band topology. In Chapter 3 we turn our attention to the lattice, and develop a systematic way to treat the momentum transport and stress response of systems with discrete translational symmetry. We also extend the Belin- fante procedure to consider lattice systems with internal rotational degrees of freedom. We revisit the case of free fermions, finding a similar but more nuanced version of the relationship between viscosity, spin and ii band topology in this case. We systematically treat the Hall viscosity of a Chern insulator as an example showing the importance of the lattice stress response framework and Belinfante procedure. We consider the physical implications of the viscosity coefficients from the lens of the bulk viscous forces. With the framework in place to consider viscosity in the presence of discrete symmetries, we consider a few extensions and applications. In Chapter 4, we consider the case of three spatial dimensions, where previous understanding of the Hall viscosity was limited to quasi-2D responses. We find that for systems with tetrahedral symmetries there is a manifestly three dimensional Hall viscosity, which we call the cubic Hall viscosity. We study this in an experimentally motivated model of a chiral magnetic metal. In Chapter 5, we revisit the viscous redundancy in two dimensions, and consider the implications the has on hydrodynamics, considering entropy production and stress boundary conditions in light of the redundancy. We then propose an experimentally accessible method to resolve the redundancy, showing that fluid flow in systems with a boundary can distinguish between otherwise redundant viscosity coefficients. Lastly, we chart a path from the non-dissipative viscosity and geometric response of continuum topo- logical phases to the geometric response of Higher-order topological insulators (HOTIs) to lattice defects. The common ingredient is the Wen-Zee (WZ) action, which is a mixed Chern-Simons term describing a coupling between the geometry of space to electromagnetism, and has emerged as a potential mechanism for the disclination response in HOTIs. In Chapter 6, we consider the role of the Gromov-Abanov-Jensen boundary (GJA) term, a key ingredient in the geometric response in the continuum, in the context of HOTIs. We find that this boundary term explains the filling anomaly (excess corner charge) and gives a valuable perspective on the bulk disclination response as well.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Physics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2023

Author dc:creator

• Rao, Pranav Valluru

Contributors dc:contributor

• Bradlyn, Barry
• Stone, Michael
• Mason, Nadya
• Ertekin, Elif

• Geometric Response
• Topological Phases Of Matter

Statement dc:rights

• Copyright 2023 Pranav Rao

Language dc:language

en, eng

Handle dc:identifier

https://hdl.handle.net/2142/120104

OAI record oai:identifier

https://www.ideals.illinois.edu/oai-pmh?verb=GetRecord&metadataPrefix=etdms&identifier=oai%3Awww.ideals.illinois.edu%3A2142%2F120104