# Fermion-Vortex Duality & Interacting Topological Phase

### Particle-vortex duality of 2d Dirac fermion from electric-magnetic duality of 3d topological insulators

(with Max Metlitski)

The physics of a single Dirac cone, the surface of a Topological Insulator (TI), is proposed to be described by a dual theory, QED 3 of a gauge field coupled to a Dirac fermion which is the surface of a topological superconductor. The dual theory provides an explicit derivation of the T-Pfaffian state, a proposed surface topological order of the TI, which is the pair-condensed state of the dual fermions. arXiv:1505.05142

### 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.[1] [2].
• 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 [2], 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 [3], and surface topological orders for the well known fermionic topological insulators and superconductors were constructed [4] [5]. 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[4].

[1] 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)
[2] 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).
[3] 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.
[4]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).
[5] Symmetry Enforced Non-Abelian Topological Order at the Surface of a Topological Insulator, Xie Chen, Lukasz Fidkowski, Ashvin Vishwanath. arXiv:1306.3250.

Review Article (with Ari Turner): Beyond Band Insulators: Topology of Semi-metals and Interacting Phases

### Emergent Supersymmetry

(with Tarun Grover and Donna Sheng)
Science 344:6181 pp. 280-283 (2014). arXiv:1301.7449.

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.