Non-Hermitian Structures in Soft Matter

Among the major advances in theoretical condensed matter physics in the past twenty years was to characterize topological insulators using the symmetry classes of Hermitian operators. These advancements were applied to various soft matter systems such as mechanical networks where they revealed the presence of topologically protected zero-frequency edge modes. They were also extended to Floquet operators (which occur in non-equilibrium settings) and non-Hermitian operators (occurring in systems with non-reciprocal couplings or subject to external gain/loss). In classical settings, such as in soft matter, non-Hermitian operators are ubiquitous and have revealed rich behavior such as odd elasticity/viscosity, skin effect, and nonreciprocal transitions across a variety of phenomenological systems. This dissertation deals with using non-Hermitian physics to understand collective behavior in soft matter systems. First, we consider a localization-to-delocalization phase transition when shear is applied to thermally fluctuating directed polymer chains. These chains cannot cross each other and are placed on a substrate consisting of a periodic arrangement of vertical grooves. We will characterize this phase transition using the properties of the diffusion operator governing the polymer configurations---this operator becomes non-Hermitian at nonzero shear. Second, we consider networks of classical mechanical oscillators with spring stiffnesses that are modulated in a time-periodic manner. We find the conditions for parametric resonance and one-way amplification to arise in these networks using the symmetries of the non-Hermitian Floquet operator governing the equations of motion. Specifically, we shall show how a clockwise moving wave in a ring of oscillators can be amplified while the counter-clockwise moving mode remains unamplified. In investigating these physical systems, we also developed some techniques which are widely applicable. Specifically, we developed a formulation to study systems that are invariant after a combined translation in both space and time. Compared to conventional Floquet techniques, this formulation involves integration of the system dynamics for shorter periods avoiding extraneous degeneracies of eigenvalues. We also characterized the real-to-complex eigenvalue transition in parametrized pseudo-Hermitian matrices which is typically accompanied by a drastic change in the behavior of the underlying system. This dissertation contains previously published as well as unpublished co-authored materials.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

doctoral

Discipline thesis:degree_discipline

Department of Physics

Grantor dc:publisher

University of Oregon

Year dc:date.issued

2024

Author dc:creator

• Melkani, Abhijeet

Advisor dc:contributor.advisor

• Paulose, Jayson

• Floquet
• non-Hermitian
• Non-reciprocity
• Soft matter
• Space-time symmetry
• Topological physics

Statement dc:rights

• All Rights Reserved.

Language dc:language.iso

en_US

Handle dc:identifier.uri

https://hdl.handle.net/1794/30281

OAI record oai:identifier

https://scholarsbank.uoregon.edu/server/oai/request?verb=GetRecord&metadataPrefix=dim&identifier=oai%3Ascholarsbank.uoregon.edu%3A1794%2F30281