Recent years have seen a renewed interest, both theoretically and experimentally, in the search for topological states of matter. On the theoretical side, while much progress has been achieved in providing a general classification of non-interacting topological states, the fate of these phases in the presence of strong interactions remains an open question. The purpose of this talk is to describe recent developments on this front. In the first part of the talk, we will consider, in a scenario with time-reversal symmetry breaking, dispersionless electronic Bloch bands (flatbands) with non-zero Chern number and show results of exact diagonalization in a small system at 1/3 filling that support the existence of a fractional quantum Hall state in the absence of an external magnetic field. In the second part of the talk, we will discuss strongly interacting electronic phases with time-reversal symmetry in two dimensions and propose a candidate topological field theory with fractionalized excitations that describes the low energy properties of a class of time-reversal symmetric states.

The description of non-Fermi liquid metals is one of the central problems in the theory of correlated electron systems. I present a holographic theory which builds on general features of the thermal entropy density and the entanglement entropy. Remarkable connections emerge between the holographic approach, and the postulated strong-coupling behavior of the field-theoretic approach.

Although the typical physical system achieves an ordered state at low temperatures, spin liquids stay disordered even in their ground state. In addition to an increasing number of experimental candidates for spin liquids, recent numerical work from Meng, et. al and Yan, et. al. has supplied strong numerical evidence for natural Hamiltonians having spin liquid ground states. Their featureless nature, though, makes learning about these states particularly difficult. In this talk, we explore what variational ansatz can teach us about them. Additionally, we examine whether the canonical theoretical framework makes sense in the context of the best wave-functions. Finally, we look at what other tools we have to make sense of spin liquids.

The entanglement spectrum denotes the eigenvalues of the reduced density matrix of a region in the ground state of a many-body system. Given these eigenvalues, one can compute the entanglement entropy of the region, but the full spectrum contains much more information. I will review geometric methods to extract this spectrum for special subregions in lorentz and conformally invariant field theories (and any theory whose universal low energy physics is captured by such a field theory). Using this technology, I give a new proof that the entanglement spectrum in quantum hall states is the same as that of a physical edge. I will discuss a number of other applications of the formalism, including, if there is time, a more complicated calculation of universal terms in the entanglement entropy at certain deconfined quantum critical points. The ultimate messages are that geometry is intimately bound up with entanglement and that the entanglement cut should be viewed as a physical cut.

Rare earth pyrochlores, with a chemical formula A2B2O7, exhibit many interesting features in A site spin system. Depending on A site rare earth elements, spin ice and magnetically ordered phases are shown in several experiments. Moreover, they have been also focused as possible candidates of U(1) spin liquid. In order to explore such versatile phases, we study the pseudospin-1/2 model, which is quite generic to describe rare earth pyrochlores with integer spins, in the presence of spin-orbit coupling and crystalline electric field. Using a new "gauge mean field theory", we show the possible ground states, corresponding to several phases listed above.

Novel phases can result from the interplay of electronic interactions and spin orbit coupling. In the first part, we discuss a simple Hubbard model for the pyrochlore iridates, whose phase diagram contains topological insulator (TI) and various magnetic phases. The latter host the novel topological Weyl semimetal, whose excitations behave like Weyl fermions. In the second part we study a novel spin liquid that was proposed to arise in the iridates, the 3D topological Mott insulator: a fractionalized TI where the neutral spinons acquire a topologically non-trivial band structure. The low-energy behavior is dominated by the 2D surface spinons strongly coupled to a bulk gauge field. This phase is characterized by the helical nature of the spinon surface states and the dimensional mismatch between the latter and the gauge bosons. We discuss experimental signatures as well as the possibility of dyonic excitations and a non-trivial magneto-electric response.

A fractional quantized Hall nematic (FQHN) is a novel phase in which a fractional quantum Hall conductance coexists with broken rotational symmetry characteristic of a nematic. Both the topological and symmetry-breaking order present are essential for the description of the state, e..g, in terms of transport properties. Remarkably, such a state has recently been observed by Xia et al. (cond-mat/1109.3219) in a quantum Hall sample at 7/3 filling fraction. As the strength of an applied in-plane magnetic field is increased, they find that the 7/3 state transitions from an isotropic FQH state to a FQHN. In this talk, I will provide a theoretical description of this transition and of the FQHN phase by deforming the usual Landau-Ginzburg/Chern-Simons (LG/CS) theory of the quantum Hall effect. The LG/CS theory allows for the computation of a candidate wave function for the FQHN phase and justifies, on more microscopic grounds, an alternative (particle-vortex) dual theory that I will describe. I will conclude by (qualitatively) comparing the results of our theory with the Xia et al. experiment.

Recent years have seen a renewed interest, both theoretically and experimentally, in the search for topological states of matter. On the theoretical side, while much progress has been achieved in providing a general classification of non-interacting topological states, the fate of these phases in the presence of strong interactions remains an open question. The purpose of this talk is to describe recent developments on this front. In the first part of the talk, we will consider, in a scenario with time-reversal symmetry breaking, dispersionless electronic Bloch bands (flatbands) with non-zero Chern number and show results of exact diagonalization in a small system at 1/3 filling that support the existence of a fractional quantum Hall state in the absence of an external magnetic field. In the second part of the talk, we will discuss strongly interacting electronic phases with time-reversal symmetry in two dimensions and propose a candidate topological field theory with fractionalized excitations that describes the low energy properties of a class of time-reversal symmetric states.

The scaling of entanglement entropy, and more recently the full entanglement spectrum, have become useful tools for characterizing certain universal features of quantum many-body systems.
Although entanglement entropy is difficult to measure experimentally, we show that for systems that can be mapped to non-interacting fermions both the von Neumann entanglement entropy and generalized Renyi entropies can be related exactly to the cumulants of number fluctuations, which are accessible experimentally. Such systems include free fermions in all dimensions, the integer quantum Hall states and topological insulators in two dimensions, strongly repulsive bosons in one-dimensional optical lattices, and the spin-1/2 XX chain, both pure and strongly disordered.
The same formalism can be used for analyzing entanglement entropy generation in quantum point contacts with non-interacting electron reservoirs. Beyond the non-interacting case, we show that the scaling of fluctuations in one-dimensional critical systems behaves quite similarly to the entanglement entropy, and in analogy to the full counting statistics used in mesoscopic transport, give important information about the system. The behavior of fluctuations, which are the essential feature of quantum systems, are explained in a general framework and analyzed in a variety of specific situations.

The multiscale entanglement renormalization ansatz can be reformulated in terms of a causality constraint on discrete quantum dynamics. This causal structure is that of de Sitter space with a flat spacelike boundary, where the volume of a spacetime region corresponds to the number of variational parameters it contains.

We consider the effect of an in-plane current on the magnetization dynamics of a quasi-two-dimensional spin-orbit coupled nanoscale itinerant ferromagnet. By solving the appropriate kinetic equation for an itinerant electron ferromagnet, we show that Rashba spin-orbit interaction provides transport currents with a switching action, as observed in a recent experiment (I. M.
Miron et al., Nature 476, 189 (2011)). The dependence of the effective switching field on the magnitude and direction of an external magnetic field in our theory agrees well with experiment.

Many-body entanglement, the special quantum correlation that exists among a large number of quantum particles, underlies interesting topics in both condensed matter and quantum information theory. On the one hand, many-body entanglement is essential for the existence of topological order in condensed matter systems and understanding many-body entanglement provides a promising approach to understand in general what topological orders exist. On the other hand, many-body entanglement is responsible for the power of quantum computation and finding it in experimentally stable systems is the key to building large scale quantum computers. In this talk, I am going to discuss how our understanding of possible many-body entanglement patterns in real physical systems contributes to the development on both topics. In particular, I am going to show that based on simple many-body entanglement patterns, we are able to (1) completely classify topological orders in one-dimensional gapped systems, (2) systematically construct new topological phases in two and higher dimensional systems, and also (3) find an experimentally more stable scheme for measurement-based quantum computation. The perspective from many-body entanglement not only leads to new results in both condensed matter and quantum information theory, but also establishes tight connection between the two fields and gives rise to exciting new ideas.