Papers, lectures, talks, and posters:

Foster group, Physics & Astronomy Department, Rice University

We present evidence that strongly suggests the equivalence between disordered surface states of topological superconductors (TSCs) and geometric percolation. Percolation is known to play a role in quantum Hall systems with magnetic fields. Our unexpected result implies that percolation applies to TSC surface states, in the absence of time-reversal symmetry breaking.

We numerically study the surface states of time-reversal TSCs in class CI with generic bulk winding numbers and quenched disorder. The low-energy states are predicted to be described by a Wess-Zumino-Novikov-Witten conformal field theory (WZNW-CFT), with universal wave function statistics that depend only on the bulk winding number. In this class, finite energy surface states were expected to be Anderson localized. We verify the WZNW-CFT results at low energy, but find critical delocalization at finite energy. The statistics at finite energy are universal and match those of the spin quantum Hall plateau transition in class C, which is equivalent to percolation.

Our result is surprising for a number of reasons. The most remarkable aspect is that percolation typically occurs only with fine-tuning, e.g. of the fluid level when water floods a landscape [Fig. (a)], or of the chemical potential in regular 2D quantum Hall physics. Here instead we find a stack of 2D critical states spanning the bulk gap of the TSC [Fig. (b)], where each state has statistics consistent with critical percolation. Our result is also surprising because all states EXCEPT the special one at zero energy were expected to be localized in this case.

Our finding of critical delocalization at the surface throughout the bulk gap is consistent with all other known topological edge or surface states (quantum Hall, 2D and 3D topological insulators), where these states are prevented from Anderson localization at all energies in the gap by the topology. Our work implies that delocalization throughout the gap may be a general principle for topological edge or surface states.

In recent years, many-body localization (MBL) has become a subject of intense study. Although much has been established about the MBL phase and the associated quantum transition between the ergodic and MBL phases in one dimension, fundamental questions regarding the nature (or even the existence) of these in higher dimensions remain unanswered. In this paper, we propose a new strategy to attack the 2D ergodic metal-to-MBL insulator transition. Our work explores the possibility that the seed of MBL resides within the diffusive metal (ergodic) phase. The key idea is to treat quantum conductance corrections and the dephasing thereof on equal footing, since dephasing is the only mechanism in an isolated, interacting quantum system that prevents Anderson localization at all energy densities (temperatures) in two dimensions.

For an isolated fermion system with short-range interactions, the self-generated thermal bath responsible for dephasing is itself diffusive. This should be contrasted with the conventional case of screened long-range Coulomb interactions, which leads to an effective Markovian bath. The Markovian dephasing problem was solved exactly long ago by Altshuler, Aronov, and Khmelnitsky (1982). The exact solution turns out to be the same as a standard approximation (self-consistent Born approximation, SCBA) in this case. We study the self-dephasing of the lowest order weak localization conductance correction due to the diffusive bath. The SCBA gives a finite dephasing rate for any finite temperature T, and this was conventionally accepted to imply that there is no true localization except at T = 0. However, in light of MBL this needs to be re-examined. The strongly non-Markovian character of the diffusive bath implies that long-range space and time correlations in the bath neglected in the SCBA can become important.

We cast the dephasing caused by a diffusive bath as a geometric statistical-mechanics problem of a polymer loop with self-interactions, and use the renormalization group (RG) approach to look for a critical point at nonzero temperature. In other words, we ask the question of whether dephasing can fail at sufficiently low temperature, owing to correlations in the bath. The answer turns out to be yes, at least within an epsilon-expansion about four spatial dimensions. We identify a renormalization group fixed point at which the dephasing length (the length scale of the curled-up polymer "blob") diverges. This occurs for T > 0 in d < 4 spatial dimensions. Our RG analysis suggests that this critical point is associated with a toy version of the ergodic-MBL phase transition, provided that it survives to 2D.

The dephasing problem considered here is the lowest order correction in the standard theory of the ergodic phase in d > 1 spatial dimensions (the interacting Finkel'stein sigma model). We recently recast this as finite-temperature Keldysh response theory suitable for calculations in any of the 10 symmetry classes. This allows the correct incorporation of real and virtual processes for transport, as well as Altshuler-Aronov corrections and interaction renormalization. See

A recent series of experiments in which an intense, ultrashort THz pump
pulse is injected into a superconductor have demonstrated fascinating far-from-equilibrium dynamics. These may indicate quantum coherent many-body evolution through a transient time window ("pre-thermalization plateau")
now accessible to ultrafast spectroscopy. In this work, we theoretically study
a particular THz pump-probe experiment [R. Matsunaga et al.]
wherein it was argued that dynamics of the Higgs amplitude
mode was observed, consistent with theoretical predictions originally
made in an entirely different physical context (interaction quantum quenches
in ultracold Fermi gases).
A crucial aspect of the experiment is that the majority of the pump pulse
spectral weight falls below the optical BCS gap, so that a weak pulse would
not be expected to couple strongly to the sample. Here we demonstrate
that an intense subgap pulse can couple * nonlinearly * to the Higgs mode, and
that the main effect of the pump is to induce a "twisted" initial state that
subsequently evolves according to the BCS Hamiltonian. We show that the
Lax reduction method (which exploits the integrability of the BCS Hamiltonian)
can be applied to the pump injection. The Lax method allows calculation
of the asymptotic gap from the state of the system obtained immediately
after the cessation of the pump.
We show that a phase of gapless (fluctuating) superconductivity ("phase I")
originally predicted in the context of cold atoms should be
accessible using more intense THz pump pulses. This opens the door to engineering
and observing dynamical phase transitions in coherent many body quantum evolution
within solid state systems, exploiting intense tabletop THz sources.

Slides for a guest lecture in undergraduate solid state physics being taught at Rice in Spring 2017. This is an introduction to topological materials, mainly by way of the dimerized chain (Su-Schrieffer-Heeger) model.

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).

** 1. Rotations, so(3) and su(2). [v2.0] **

1.
2018 Seminar

(I) Critical percolation without fine-tuning on the surface of a topological superconductor, and

(II) Dephasing catastrophe in 4 - ε: A possible instability of the ergodic phase

2.
2017 From MBL to black holes

Talks given at Caltech in Spring 2018 and at the Simons Center for Geometry and Physics in Fall 2017.
I discuss results relating to electrons in two spatial dimensions (2D), subject to the effects of quenched disorder (impurities) and quantum interference [Anderson (de)localization]. In both cases, the key physics is tied to classical geometric critical phenomena in 2D. I first present numerical evidence that most surface states of 3D topological superconductors exhibit universal critical statistics in the presence of nonmagnetic disorder. This is highly unusual, since critical (multifractal) statistics typically arise only by fine-tuning to a mobility edge. For a particular class, we show that (almost) every state within the surface energy band shows universal statistics consistent with the plateau transition of the spin quantum Hall effect ("equivalent" to critical percolation in 2D). Thus critical percolation can be robustly realized at the surface of a topological superconductor without fine tuning. Our results may facilitate new approaches to the logarithmic conformal field theories believed to describe plateau transitions, using perturbed WZNW models. Second, I discuss a "hole" in the standard theory of the ergodic phase in 2D. I demonstrate that the self-dephasing of weak localization for an isolated many-fermion system with short-ranged interactions is described by a strongly coupled auxiliary quantum field theory. The problem can be cast as a type of self-interacting walk. I present analytical results of a controlled expansion which suggest that dephasing can fail at sufficiently low temperatures, and speculate on the possible connection to many-body localization.

The material in these talks is derived from our papers,

1.
2016 SLT
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.

I also presented an overview of our work on 3D topological superconductors in a 2014 poster at the 26th Annual Kavli Frontiers of Science Symposium.

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.