(xii) Measuring space-group symmetry fractionalization in Z2 spin liquids (with Mike Zaletel and Yuan Ming Lu) arXiv:1501.01395
We identify physical observables that distinguish Z2 quantum spin liquids with different symmetry properties (SETs) and SU(2) spin rotation invariance. In the cylinder geometry we show that ground state quantum numbers for different topological sectors are robust invariants which can be used to identify the SET phase. More generally these invariants are related to 1D symmetry protected topological phases when viewing the cylinder geometry as a 1D spin chain. In particular we show that the Kagome spin liquid SET can be determined by measurements on one ground state, by wrapping the Kagome in a few different ways on the cylinder
Mott Insulators (quantum magnets with an odd number of spin 12 per unit cell) must either develop order that doubles the unit cell or display exotic behavior. For example, according to the Hastings-Oshikawa-Lieb-Schultz-Mattis theorem a fully gapped state of a 2D system that respects all symmetries must be topologically ordered i.e. contain anyon excitations. However, the theorem is silent on the nature of the topological order. We show that the double semion topological order is incompatible with time reversal and translation symmetry in Mott insulators. An application of our result is the Kagome lattice quantum antiferromagnet where recent numerical calculations of entanglement entropy indicate a ground state compatible with either toric code or double semion topological order. Our result rules out the latter possibility.
(x) Topological Phases with Strong Interactions (Xie Chen, Yuan Ming Lu, Lukasz Fidkowski):
Previous work in topological insulators and superconductors has been largely based on free fermions with topological `band' structures. We have focused on qualitatively new phenomena that arise with strong interactions, where no concept of a band structure is present. For example, we have discussed:
- New topological phases with protected edge modes that only appear in the presence of interactions. For example, bosons can form phases analogous to topological insulators in 2 and 3 dimensions, but these necessarily require interactions. 
- Previously, conventional wisdom held that 3D Topological Insulators and superconductors must be associated with gapless, metallic surface states if the symmetries are preserved. We found , while studying the simpler bosonic topological phases, that the 2D surface can in fact acquire a gap while remaining fully symmetric if it develops topological order, a possibility that was previously overlooked. Here, by topological order we mean a state that contains anionic excitations with fractional statistics, like in a fractional Quantum Hall state. These fractional excitations however realize the symmetries in a way that is impossible in a purely 2D system. The surface topological order for a specific bosonic topological phase was demonstrated in an exactly soluble model , and surface topological orders for the well known fermionic topological insulators and superconductors were constructed  . Interestingly, in some cases these are forced to be non-Abelian topological orders, that is, if one is guaranteed that the surface of a topological insulator is symmetric but gapped, then one must have realized a non-Abeilan topological state.
- Free fermion classification of topological phases can be modified on including interaction effects. Thus far only a few examples were known in 1D and 2D, where perturbative arguments based on weak interactions sufficed. Sometimes two phases that appear distinct at the free fermion level may be connected smoothly in the presence of strong interactions. We have very few theoretical tools to study such a problem, particularly in 3D. However, the surface topological order provides such a non-perturbative definition of a topological phase. Using this handle we were able to show that the integer classification of topological superconductors in 3D (class DIII, with time reversal symmetry) is actually reduced to a Z_16 classification.
 Theory and classification of interacting integer topological phases in two dimensions: A Chern-Simons approach. Yuan-Ming Lu and Ashvin Vishwanath. Phys. Rev. B 86, 125119 (2012)
 Physics of Three-Dimensional Bosonic Topological Insulators: Surface-Deconfined Criticality and Quantized Magnetoelectric Effect
Ashvin Vishwanath and T. Senthil, Phys. Rev. X 3, 011016 (2013).
 Exactly Soluble Model of a 3D Symmetry Protected Topological Phase of Bosons with Surface Topological Order,
F. J. Burnell, Xie Chen, Lukasz Fidkowski, Ashvin Vishwanath. arXiv:1302.7072
 Non-Abelian Topological Order on the Surface of a 3D Topological Superconductor from an Exactly Solved Model.
Lukasz Fidkowski, Xie Chen, and Ashvin Vishwanath, Phys. Rev. X 3, 041016 (2013).
 Symmetry Enforced Non-Abelian Topological Order at the Surface of a Topological Insulator,
Xie Chen, Lukasz Fidkowski, Ashvin Vishwanath. arXiv:1306.3250.
(ix) Criterion for non-Fermi liquid phases via interactions with Goldstone bosons. arXiv:1404.3728 (With Haruki Watanabe) PNAS 2014
Usually, Goldstone's theorem guarantees long lived gapless excitations when a continuous symmetry is spontaneously broken. We investigated if this remains true if the symmetry breaking occurs inside a metal - where gapless Fermi liquid excitations are also present. We found a simple criterion which determines if the coupling between these two kinds of excitations is relevant or irrelevant. In most cases the coupling is irrelevant - so both the Goldstone bosons and the Fermi liquid remain stable when coupled. But we found two cases where the coupling is relevant - which implies that something interesting must happen, eg. non-Fermi liquid physics. One case was previously known, but we also found an entirely new pattern of symmetry breaking that could be a new route to realizing non Fermi liquids.
(Featured in Journal Club for Condensed Matter Article by Jörg Schmalian: here )
(viii). Quantum Oscillations from Fermi arc surface states of Weyl and Dirac semimetals.
Weyl semimetals have very unusual surface states that take the form of `Fermi arcs'. A standard probe of Fermi surfaces is quantum oscillations - and these surface states must display an unusual signatures given that Fermi arcs cannot on their own give rise to closed orbits. Predictions for quantum oscillations in thin films of both Weyl and the recently discovered Dirac semimetals are made.
(vii) Emergent Supersymmetry (with Tarun Grover and Donna Sheng): arXiv:1301.7449 Science vol. 344 no. 6181 pp. 280-283 (2014)
We show that space-time supersymmetry emerges naturally in topological superconductors that are well understood condensed matter systems. Specifically, we argue that the quantum phase transitions
at the boundary of topological superconductors in both two and three dimensions display super-
symmetry when probed at long distances and times.
Supersymmetry entails several experimental
consequences for these systems, such as, exact relations between quantities measured in disparate
experiments, and in some cases, exact knowledge of the universal critical exponents. The topological
surface states themselves may be interpreted as `Goldstino modes' arising from spontaneously broken supersymmetry,
indicating a deep relation between topological phases and SUSY.
We discuss prospects for experimental realizations in films of superfluid He3-B.
Emergentspace-time Supersymmetry at the Boundary of Topological Phases
(vi) Many Body Localization and topological phases (with Yasaman Bahri, Vosk, Altman) arXiv:1307.4092
Here, we extended the concept of a topological state of matter, usually sharply defined only in the quantum ground state, to highly excited states of a correlated quantum system, which is in a many-body localized phase. Specifically we considered a 1 dimensional system which realizes a symmetry protected topological phase. We demonstrate that in this case the bulk topology can give rise to a protected q-bit at the edge, surprisingly, even when the system is very 'hot' and strongly coupled to the degrees of freedom making up the q-bit.
Our work goes beyond parallel works establishing the existence of such phases, by demonstrating a quantum coherent spin-echo response, without the need to cool into the ground state of the system. Normally, achieving quantum coherence requires cooling to extremely low temperatures, which is a major obstruction toward practical realizations. The conceptual advance we make is to show that a class of systems exist where no cooling is needed to
attain quantum coherent responses. This result is enabled by a combination of topology and strong disorder, which preserve the topological structure, not only in the ground states, but in all of the exponentially many states of the spectrum. Here, the topological phase paradoxically appears to gain stability from disorder.
(v) Magnetic Frustration from Geometry and Spin-Orbit Coupling (Itamar Kimchi) :Many magnetic systems display "frustration", competing interactions that lead to multiple ground states at the classical level. The system is then poised to realize a variety of phases, including, it is believed, exotic spin liquid states that possess an unusual "topological order" arising from the expulsion of certain topological defect configurations. The different flavors of spin liquid states were studied in , for the triangular and Kagome magnets. Symmetries are realized in a subtle way for these states, which severely constrains them. Finding all possible phases consistent with these constraints led us to new spin-liquid states (see figure) with enhanced stability.
Thermal and quantum fluctuations, which are generally associated with their disordering effects, can paradoxically select ordered states in frustrated magnets ("order by disorder"). These have been studied in the context of two spin 1/2 frustrated magnets on a distorted kagome (volborthite ) lattice and on the hyper-kagome lattice  (NaIrO), a new three dimensional frustrated lattice.
Unified theory of spiral magnetism in the 3D harmonic honeycomb iridates Li2IrO3: arxiv:14083640 (With Itamar Kimchi, Radu Coldea, James Analytis) We show that the Kitaev interaction stabilizes the counter-rotating spiral. By studying a minimal model of zig zag chains. This is argued to be the key to understanding of the remarkably similar magnetic ordering found in two different structures of Li2IrO3.
(iv) Unconventional Quantum Criticality: In  we studied the physics of 2D quantum magnets when certain space-time defects (hedgehog defects - see figure) are absent. This lack of "topological disorder" completely changes the physics of the model, in particular, even when the spins are fluctuating, the system possesses a hidden order that manifests itself in the form of an emergent photon excitation (light!) and excitations with fractional quantum numbers. Also, continuous phase transitions in these models turn out to be very different in character; and in one situation even turns out to be self-dual. Interestingly, these deconfined critical points may actually occur quite naturally in certain quantum magnets  where quantum interference effects help to a suppress hedgehog defects. Moreover, they can control `Landau Forbidden' continuous transition - that is, transitions between two states of different symmetry that according to Landau's theory of phase transitions (and common sense) would be continuous without special fine tuning.
We are currently engaged in searching for microscopic models where such phenomena arise, and looking to generalize these results to three dimensions.
- TALK: Emerging Photons and Decaying Electrons
(iii) Skyrmion Crystals in Metallic Helimagnets (B. Binz, Haruki Watanabe, S. Parameswaran):Under pressure, MnSi a metal with a tendency to form helical magnetic structures, undergoes a transition into an unusual phase that shows "partial order" and non-Fermi liquid electrical transport. We have developed a theory of helical magnetic crystals - a coherent multi-spiral state (depicted in the figure), that agrees well with existing data on the magnetic structure. Moreover, it makes several predictions for unusual electrical transport arising from the presence of non-trivial magnetic structures .
(ii) Entanglement and the Quantum Phases of Matter: (under construction)