Quantum Matter

Fractional quantum Hall effect in the sequence of filling factors n/(2np +- 1) is well understood by the integer quantum Hall effect of the composite fermions at the filling factor n. A composite fermion (CF) is a bound state of an electron and 2p number of quantized vortices. However, the experimentally observed states such as 4/11, 5/13, and 3/8 which are between 1/3 and 2/5 cannot be accommodated in the conventional noninteracting theory of composite fermions. The interaction between CFs in partially filled second effective Landau levels of CFs is important for these states. However, conventional fractional quantum Hall effect of these interacting composite fermions do not reproduce these incompressible states. By diagonalizing Coulomb Hamiltonian in a restricted Hilbert space of the CFs, we find that these states are incompressible for flux-particle relationships that are different from the conventional considerations. We interpret that 4/11 and 5/13 states occur due to fractional quamtum Hall effect of the composite fermions for Haldane Pseudopotential V_3 rather than V_1. And the panti-Pfaffian pair correlation between CFs in the half-filled second effective Landau level creates incompressible 3/8 state. The unconventional topologies of these states are reflected in anomalously low magneto-roton energies for these states.

The properties of a strange metal fermion model with infinite-range

interactions turn out to be closely related to those of charged black holes

with AdS2 horizons. I show that a microscopic computation of the ground

state entropy density of the fermion model yields precisely the Bekenstein-Hawking

entropy density of the black hole. The fermion model is UV finite and has no supersymmetry

Parafermions are the simplest generalizations of Majorana fermions that realize topological order. We propose a less restrictive notion of topological order in 1D open chains, which generalizes the seminal work by Fendley [J. Stat. Mech., P11020 (2012)]. The first essential property is that the groundstates are mutually indistinguishable by local, symmetric probes, and the second is a generalized notion of zero edge modes which cyclically permute the groundstates. These two properties are shown to be topologically robust, and applicable to a wider family of topologically-ordered Hamiltonians than has been previously considered. As an application of these edge modes, we formulate a new notion of twisted boundary conditions on a closed chain, which guarantees that the closed-chain groundstate is topological, i.e., it originates from the topological manifold of the open chain. Finally, we generalize these ideas to describe symmetry-breaking phases with a parafermionic order parameter.

These exotic phases are condensates of parafermion multiplets, which generalizes Cooper pairing in superconductors. The stability of these condensates are investigated on both open and closed chains.

We analyse a class of 1D lattice models, known as M_k models, which are characterised by an order-k clustering of spin-less fermions and by N=2 lattice supersymmetry. We show a fundamental connection to Z_k parafermions and identify a class of (bulk or edge) defects, which are in one-to-one correspondence with so-called spin fields in the parafermion CFT. In the gapped regime, injecting such defects leads to ground state degeneracies that are protected by the supersymmetry. The defects, which are closely analogous to quasi-holes over the fermonic Read-Rezayi quantum Hall states, display characteristic fusion rules, which are of Ising type for k=2 and of Fibonacci type for k=3.

The standard theory of topological insulators and superfluids (or superconductors) assumes that the fermionic elementary excitations in these systems – electrons in the insulator and Bogoliubov quasiparticles in the superfluid – do not interact with one another. In this talk I will discuss extensions of this theory to include the effects of interparticle interactions on the topological surface states of 3D topological insulators and superfluids. First, I will argue that bulk quantum fluctuations of the superfluid order parameter in the only 3D topological superfluid known to date, 3He-B, can generate an effective two-body interaction between the surface Majorana fermions.

Possible consequences of this interaction will be discussed. Second, I will propose an extension of the phenomenological Landau theory of Fermi liquids to the surface states of 3D topological insulators, leading to the concept of helical Fermi liquid.