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KITP Conference on Strong Correlations and Unconventional Superconductors
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Speaker: J. A. Sauls

Department of Physics & Astronomy, Northwestern University, Evanston, IL 60208

September 24, 2014

**Abstract:**
Broken symmetries in bulk condensed matter systems have implications for the spectrum of Fermionic
excitations bound to surfaces and topological defects. The A-phase of superfluid ^{3}He,
described by the Anderson-Morel (AM) state, is the realization of a two-dimensional broken
time-reversal topological superfluid. This state is also representative of spin-triplet, chiral
superconductors discussed as possible ground states for the superconductors
Sr_{2}RuO_{4} and UPt_{3}. I discuss the relationship between broken
symmetries of the ground state and the topological nature of the Fermionic states confined near
surfaces, domain walls and topological defects for the superfluid phases of ^{3}He. The
Fermionic spectrum of confined (2D) ^{3}He-A are chiral edge states (Weyl Fermions). The
negative energy states are related to the ground-state angular momentum, L_{z} = (N/2)
ℏ, of superfluid ^{3}He-A for N/2 Cooper pairs. The power law suppression of the
angular momentum, L_{z}(T) ≃ (N /2) ℏ [1 - β T^{2}] for 0 ≤ T
≪ T_{c}, in the fully gapped 2D chiral A-phase reflects the thermal excitation of the
Weyl branch of Fermions. I discuss the sensitivity of the edge current and ground-state angular
momentum to topology, geometry and symmetry of the confining boundaries, and show that the edge
current is protected by symmetry, not by topology of the bulk Hamiltonian. For local
translationally invariant boundaries and axially symmetry the edge current accounts for the
ground-state angular momentum, L_{z} = (N/2) ℏ, of a cylindrical disk of chiral
superfluid with N/2 fermion pairs. Non-specular scattering can dramatically suppress the edge
current. In the limit of perfect retro-reflection the edge states form a flat band of zero modes
that are non-chiral and generate no edge current. For a chiral superfluid film confined in a
cylindrical toroidal geometry the ground-state angular momentum can have a value ranging from
L_{z} > (N/2) ℏ to L_{z} < -(N/2) ℏ depending on the ratio of the
inner and outer radii and the degree of back scattering on the inner and outer surfaces. The
scaling of L_{z}, with (R_{2}/R_{1})^{2} and the reversal of the
ground-state angular momentum for a toroidal geometry, would provide a unique signature of broken
time-reversal symmetry of the ground state of superfluid ^{3}He-A, as well as direct
observation of chiral edge currents.

This research was supported by NSF Grant: DMR-1106315.

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