Lectures, talks, and posters:

Foster group, Physics & Astronomy Department, Rice University

These are course notes for a class being taught in Fall 2016 at Rice on Lie algebras and their representations.
Most of the material will be standard, but the plan is for the presentation to be unabashedly applied, emphasizing visualization and
algorithms at the expense of rigor and generality. I am following roughly the presentation in Robert Cahn's excellent
*Semi-simple Lie Algebras and Their Representations,* available on the web
here.
The main difference is that I will work many examples and elaborate on various topics. All lecture notes will be posted here. These
are intended to be complete enough for self study, and I hope they will prove generally useful for physics students that wish to
learn this material.
As the current course is being taught at the graduate level,
I hope to cover some advanced topics by the end (some subset of
Riemannian symmetric spaces and random matrix theory;
classification of topological phases;
affine Lie algberas, WZNW models, quantum equivalence;
quantum groups, anyons, fusion and braiding rules).

- Rotations, so(3) and su(2). [v1.9]
- Lie groups as manifolds. SU(2) and the 3-sphere. [v1.1]
- Introduction to su(3). [v1.4]

1.
2016 SLT Workshop
Universal transport at the edge: Disorder, interactions, and topological protection

2.
2016 Colloquium

Topological insulators and superconductors provide condensed matter realizations of the holographic principle: a global property of the bulk translates into an anomalous time-reversal symmetry at the material surface. This symmetry underlies "topological protection" of the edge or surface states. Protection from disorder effects (Anderson localization) is particularly nontrivial, because surfaces are low dimensional. While this was previously understood for noninteracting models of edge and surface states, the more complicated problem of combined disorder and interaction effects had not been addressed until recently. Here I consider the edge states of 2D topological insulators with Rashba spin-orbit coupling (RSOC). With RSOC, disorder induces a backscattering term in the edge theory. We have shown that transport remains perfectly ballistic in a model that incorporates this term and interactions. The solution involves a mapping to a spin 1/2 moment that executes perfect adiabatic evolution in a random magnetic field. This work was published in Phys. Rev. Lett.,

The SLT link is a video for a talk given at the Landau Institute for Theoretical Physics in June 2016 combining this edge state physics and our topological superconductor work described below. The second link gives slides for the edge state part of a colloquium given at Texas A&M in March 2016.

1.
2015 MBL Program (KITP)
Transport and delocalization at the surface of a 3D topological superconductor

2.
2015 SPICE Junior Research Leaders

3.
2015 Colloquium

The KITP and SPICE workshop talks were given in Fall 2015. The colloquium is a condensed, less technical version that was given at the University of Houston in February 2015. These talks try to connect and contrast the attributes of 2D Majorana fluids (expected to form at the surface of a 3D topological superconductor) with the physics of the integer quantum Hall effect. Key results are presented from 4 of our papers,

- Foster and Yuzbashyan PRL 2012,
- Foster, Xie, and Chou PRB 2014,
- Chou and Foster PRB 2014,
- Xie, Chou, and Foster PRB 2015.

Our emphasis here is twofold: we ask (1) what are the robust physical characteristics of such a fluid, and (2) how stable is it to the combined effects of disorder and interactions, which are inevitable at the surface of a real superconductor. We show that Majorana surface fluids are not always stable, but when they are we predict universal heat and (if conserved) spin transport coefficients that encode the bulk winding number. The implication is that transport measurements could provide a smoking gun for the detection of a bulk topological superconductor, and this is important because spectroscopic measurements like ARPES have limited energy resolution that might preclude direct imaging for a small bulk gap. 2D Majorana surface fluids also turn out to be amenable to a wide variety of powerful analytical tools, including 2D conformal field theory (to treat the effects of disorder) and a large winding number expansion (to treat interactions).

1.
2016 RCQM IQSDOoE
Quenched BCS superfluids: Topology and spectral probes

2.
2014 TOMAEQ14
Far from equilibrium topological p-wave superfluids

These talks provide an overview of our work on quenched p-wave BCS superfluids, given at the Rice Center for Quantum Materials in May 2016 and
the MPIPKS in Dresden in March of 2014.
In this work we study the dynamics of the amplitude ("Higgs") mode of a fermionic superfluid, following an instantaneous quench of the coupling strength. What is new relative to previous studies of quenched s-wave superfluids are the topological aspects relevant to 2D p+ip pairing.
The talks include a discussion of the dynamical phase diagram characterized by different steady-state behaviors of the Higgs mode, pseudospin versus "spectral"
(retarded GF) winding numbers far from equilibrium, quench-induced Floquet edge states, and bulk RF signatures of the topology following a quench. Timescales relevant to ultracold lithium are discussed, and a proposal for using a quench to overcome the problem of 3-body losses that plague adiabatic cooling schemes.
The backup slides at the end of the 2014 talk give a lightning introduction to the Lax spectral method used to solve the (classically integrable) dynamics.

One of the main takeaways is that far from equilibrium, one must distinguish two different notions of quantum topology. There is the topology of the state, i.e. the pseudospin winding number that can be (partly) obtained from an equal-time measurement such as time-of-flight. Then there is the topology of the * spectrum of excitations * on top of the non-equilibrium state. The latter encodes whether you have Majorana edge modes, for example, and can be probed through a non-equal-time measurement such as RF spectroscopy, wherein one drives transitions to "nearby" states. The reconciliation of these two notions (which are equivalent in equilibrium) leads to the prediction of quasiparticle population inversion in the non-equilibrium spectrum of excitations, whenever there is a mismatch.
Both talks contain results from my papers with Maxim Dzero, Victor Gurarie, and Emil Yuzbashyan,

The RCQM talk also features results obtained by Yunxiang Liao comparing different spectroscopic measures (RF, tunneling, ARPES) in the self-generated Floquet topological state. Here there is an interesting interplay between the Floquet states and the quench-induced population inversion of these states. This work was published in

1. 2009 March meeting Slow imbalance relaxation and thermoelectric transport in graphene

Older talk on slow imbalance relaxation and hydrodynamic ("interaction-limited") thermoelectric transport in graphene. Contains the main results from my paper with Igor Aleiner,

Recent experiments have confirmed the crossover into the hydrodynamic regime, including our collaboration with Fereshte Ghahari and Philip Kim on TEP in ultraclean graphene:

See also Hong-Yi Xie's poster (2016), described below.

1. 2009 MESO09 Termination of typical wavefunction multifractal spectra at the Anderson metal-insulator transition

Older talk on the "termination" of the typical multifractal spectrum of wavefunction probability amplitudes, at the Anderson metal-insulator transition. Results obtained via an extended non-linear sigma model and the operator product expansion, combined with a functional renormalization group that maps the problem to the propagation of a front in a 1D non-linear diffusion equation (KPP). Contains the main results of

1. 2012 Colloquium When Quantum Waves Crash upon Strange Shores: Splash Statistics in Disordered Media, and Correlated Fronts Far from Equilibrium

Research overview talk given at Rice in 2012.
The second half discusses our studies of quench dynamics in the quantum sine Gordon and XXZ models.
The main result is the observation of a "supersoliton" that is generated by the quench
from an initial density inhomogeneity, whenever there is * relative fractionalization *
between the (effective) free fermion descriptions of the
initial and final Hamiltonians. The former is used to define the initial state
(taken as the ground state of the initial Hamiltonian), while the latter generates the dynamics.
Contains results from 4 papers,

- Foster, Ryu, and Ludwig 2009,
- Foster, Yuzbashyan, and Altshuler PRL 2010,
- Foster, Berkelbach, Reichman, and Yuzbashyan PRB 2011,
- Foster PRB 2012.

1. Thermoelectric transport in hydrodynamic graphene: Impurities, interactions, and optical phonons (2016)

Poster by Hong-Yi Xie on thermoelectric power (TEP) and thermal conductivity (TC) in graphene,
in the __interaction-dominated__ hydrodynamic regime. Motivated by recent thermopower measurements
performed by Fereshte Ghahari and Philip Kim, we consider thermoelectric transport in
ultraclean (boron-nitride encapsulated) graphene. In monolayer graphene, the TEP and TC
are predicted to be strongly enhanced relative to the Mott and Wiedemann-Franz laws, respectively.
These "classic" relations are valid when elastic impurity scattering dominates transport, but
fail when inelastic carrier-carrier scattering becomes more efficient. The latter should occur for
temperatures greater than 100 K in relatively clean samples.
By numerically solving the Boltzmann transport equation incorporating collision integrals for
quenched disorder, Coulomb interactions, and electron-optical-phonon scattering,
we demonstrate the crossover of the TEP and TC from the Fermi liquid to hydrodynamic regimes
as a function of density or interaction strength. On the other hand, we find that
__optical phonons__
become non-negligible at relatively low temperatures and prevent saturation of the TEP to the upper hydrodynamic bound.
Combining all of these scattering mechanisms, we obtain the thermopower that quantitatively coincides
with the experimental data.

2. Helical Quantum Edge Gears in 2D Topological Insulators (2015)

Poster by Yang-Zhi Chou on Coulomb drag in helical edge states of 2D topological insulators (TIs) with Rashba spin-orbit coupling. A remarkable and as-yet-unexploited aspect of topological insulator physics is the topology of the edge states, i.e. the fact that the edge liquid of a 2D TI forms a closed, unbreakable loop in the absence of electrical contacts or magnetic fields. We propose a novel experimental setup in which edge loops rotate as interlocking "gears" through Coulomb drag, in topological insulators with Rashba spin-orbit coupling. We show that this allows a simple two-terminal measurement of the Luttinger parameter that encodes electron correlations, a quantity that is otherwise notoriously difficult to measure. Our results should trigger new experiments and may open a new venue for edge gear-based electronic devices.

3. Distribution functions and probes of far-from-equilibrium topological matter (2015)

Poster by Yunxiang Liao on spectroscopic probes of isolated non-equilibrium quantum matter. Using our quench-induced Floquet topological p+ip superfluid as an example, we consider probes that are sensitive (radio-frequency spectroscopy, rf) and not (tunneling) to the distribution function that determines the occupation of the Floquet states. The post-quench Cooper pairs occupy a linear combination of "ground" and "excited" Floquet states, with coefficients determined by the distribution function. For a realization in ultracold atoms, the rf signal is well-captured by a quasi-equilibrium approximation, and shows a robust gap. The tunneling signal in a solid state realization is ignorant of the distribution function, being determined only by the Floquet (retarded) Green's function. It does not show a gap for deep quenches. Results are obtained from the exact analytical solution to the quench-induced Floquet topological superfluid in terms of elliptic functions.

4. Universal Surface Transport Coefficients of 3D Topological Superconductors (2014)

Poster by Hong-Yi Xie on the surface transport coefficients of bulk topological superconductors. Altshuler-Aronov quantum conductance corrections are shown to vanish in two schemes: (i) perturbative in the interactions, and (ii) in a large winding number expansion. Both schemes treat the effects of disorder exactly.

5. Quantum Critical Phenomena in Disordered Topological Superconductors (2014)

Poster by Yang-Zhi Chou on wavefunction and energy level statistics for disordred
Dirac fermions in 2D. The model can be realized on the surface of a dirty topological
superconductor, or in artificial graphene with a textured pattern of bonds.
The strong randomness limit exhibits a frozen regime wherein individual wavefunctions appear
* almost * localized. Despite this, our numerics confirm the analytical prediction of
power-law Chalker scaling with energy between different wavefunction profiles, and
Wigner-Dyson energy level statistics.

6. Quench spectroscopy of a Luttinger liquid: Fractionalized density waves in the XXZ chain (2011)

Lattice version of the "supersoliton" induced by quenching a gapless Luttinger liquid to a gapped insulator, in the presence of fractionalization. DMRG and analytical results from bosonization.

7. Termination of Typical Wavefunction Multifractal Spectra at the Anderson MIT (2009)

Functional renormalization group method used to obtain the "termination" of the typical multifractal spectrum of wavefunction probability amplitudes, at the Anderson metal-insulator transition. Results obtained via an extended non-linear sigma model and the operator product expansion, in the epsilon expansion.