Quantum Matter

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.

I will describe our recent work on a new topological phase of matter: topological Weyl semimetal. This phase arises in three-dimensional (3D) materials, which are close to a critical point between an ordinary and a topological insulator. Breaking time-reversal symmetry in such materials, for example by doping with sufficient amount of magnetic impurities, leads to the formation of a Weyl semimetal phase, with two (or more) 3D Dirac nodes, separated in momentum space. Such a topological Weyl semimetal possesses chiral edge states and a finite Hall conductivity, proportional to the momentum-space separation of the Dirac nodes, in the absence of any external magnetic field. Weyl semimetal demonstrates a qualitatively different type of topological protection: the protection is provided not by the bulk band gap, as in topological insulators, but by the separation of gapless 3D Dirac nodes in momentum space. I will describe a simple way to engineer such materials using superlattice heterostructures, made of thin films of topological insulators. References: arXiv:1110.1089; Phys. Rev. Lett. 107, 127205 (2011); Phys. Rev. B 83, 245428 (2011)."

Topology has many different manifestations in condensed matter physics. Real space examples include topological defects such as vortices, while momentum space ones include topological band structures and singularities in the electronic dispersion. In this talk, I will focus on two examples. The first is that of a vortex in a topological insulator that is doped into the superconducting state. This system, we find, has Majorana zero modes and thus, is a particularly simple way of obtaining these states. We derive general existence criteria for vortex Majorana modes and find that existing systems like Cu-doped Bi2Se3 fulfill them. In the process, we discover a rare example of a topological phase transition within a topological defect (the vortex) at the point when the criteria are violated. The second example is that of Weyl semimetals, which are three-dimensional analogs of graphene. Interestingly, the Dirac nodes here are topological objects in momentum space and are associated with peculiar Fermi-arc surface states. We discuss charge transport in these materials in the presence of interactions or disorder, and find encouraging agreement with existing experimental data.

Dualities in physics are well known for their conceptual depth and quantitative predictive power in contexts where perturbation theory is unreliable. They are also remarkable for the staggering arrange of physical problems that exploit them, ranging from the study of confinement and unconventional phases in statistical mechanics and field theory to the unification of the string theory landscape. In this talk I will present a new, completely general approach to dualities that affords a systematic theory of quantum dualities and incorporates classical dualities as well into one unified framework. This new algebraic approach is remarkably successful in extending powerful duality techniques to the context of topologically quantum ordered systems and quantum information processing, and affords a compelling foundation for a general theory of exact dimensional reduction or holographic correspondences. Many systems however display only approximate dimensional reduction, and so I will present general inequalities -some of them based on entanglement- linking quantum systems of different spatial dimensionality. These inequalities provide bounds on the expectation values of observables and correlators that can enforce an effective dimensional reduction. In closing I will discuss some implications for (topological) quantum memories.

Weak topological insulators have an even number of Dirac cones in their surface spectrum and are thought to be unstable to disorder, which leads to an insulating surface. Here we argue that the presence of disorder alone will not localize the surface states, rather, the presence of a time-reversal symmetric mass term is required for localization. Through numerical simulations, we show that in the absence of the mass term the surface always flow to a stable metallic phase and the conductivity obeys a one-parameter scaling relation, just as in the case of a strong topological insulator surface. With the inclusion of the mass, the transport properties of the surface of a weak topological insulator follow a two-parameter scaling form.

We report on our recent progress to investigate materials classes exhibiting d+id superconductivity, where topologically nontrivial pairing phases can emerge. Specifically, motivated by recent experimental progress, we show that graphene doped to the van Hove regime can give rise to a plethora of interesting ordering instabilities such as spin density wave and superconductivity. As a function of system parameters such as doping and range of Coulomb interaction, we explain which instability is favored by the system, and analyze the effect of long-range interactions on superconductivity giving rise to a competition between singlet d+id and triplet f wave. We also outline our work in progress for other materials classes which we believe are promising to stabilize such interesting topological superconducting states of matter.

It has been well-known that topological phenomena with fractional excitations, i.e., the fractional quantum Hall effect (FQHE) will emerge when electrons move in Landau levels. In this talk, I will show FQHE can emergy even in the absence of Landau levels in interacting fermion models and boson models. The non-interacting part of our Hamiltonian contains topologically nontrivial flat band.