Quantum Matter

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.

Recent progress on the construction of holographic lattices and its applications to AdS/CMT correspondence will be briefly reviewed. Our special interests will focus on the building of bulk geometry of gravity whose holographic duals exhibit metal-insulator transitions (MIT).  In particular, the Peierls phase transition induced by charge density waves is implemented in a holographic manner. The holographic entanglement entropy close to quantum critical points will be discussed as well.

Symmetry-protected topological (SPT) phases can be thought of as generalizations of topological insulators. Just as topological insulators have robust boundary modes protected by time reversal and charge conservation symmetry, SPT phases have boundary modes protected by more general symmetries. In this talk, I will describe a method for analyzing 2D and 3D SPT phases using braiding statistics. More specifically, I will show that 2D and 3D SPT phases can be characterized by gauging their symmetries and studying the braiding statistics of their gauge flux excitations. The 3D case is of particular interest as it involves a generalization of quasiparticle braiding statistics to three dimensions.

I will show how hydrodynamics is modified if the underlying fluid constituents are massless Weyl fermions, which are anomalous at the quantum level. Because of the nondissipative nature of the modification I will construct a partition function which compactly describes the transport properties of the system and I will explain how the anomalous properties can be understood in terms of kinetic theory and heat kernels. Finally I will comment on possible applications to heavy-ion collisions, condensed matter systems such as Weyl-semi metals and potential future questions such as anomalous corrections to entanglement entropy.

We examine the interplay of symmetry and topological order in 2+1D topological phases of matter. We define the topological symmetry group, characterizing symmetry of the emergent topological quantum numbers, and describe its relation with the microscopic symmetry of the physical system.

We then derive a general classification of symmetry fractionalization in topological phases, including phases that are non-Abelian and symmetries that permute the quasiparticle types and/or are anti-unitary. We develop a general algebraic theory of extrinsic defects (fluxes) associated with elements of the symmetry group, which provides a general classification of symmetry-enriched topological phases derived from a topological phase of matter with symmetry. We also examine the promotion of the global symmetry to a local gauge invariance, wherein the extrinsic defects are turned into deconfined quasiparticle excitations, which results in a different topological phase.

Does a generic quantum system necessarily thermalize? Recent developments in disordered many-body quantum systems have provided crucial insights into this long-standing question. It has been found that sufficiently disordered systems may fail to thermalize leading to a 'many-body localized' phase. In this phase, the fundamental assumption underlying equilibrium statistical mechanics, namely, the equal likelihood for all states at the same energy, breaks down. A fundamental question is: what happens as the disorder becomes weaker so that one approaches the localization-delocalization transition? For example, does the system thermalize *at* the transition? 

In this talk, I will show that very general considerations on the scaling of entanglement entropy close to the transition imply that at a continuous many-body localization transition, the system is necessarily ergodic.

Finally, I will present recent results on "eigenstate thermalization", a long standing hypothesis which posits that a single eigenstate hides within itself a thermal ensemble. In particular, I will discuss which class of operators do or do not satisfy eigenstate thermalization.

Holographic duality is a duality between gravitational systems and non-gravitational systems. In this talk, I will propose a different approach for understanding holographic duality named as the exact holographic mapping. The key idea of this approach can be summarized by two points: 1) The bulk theory and boundary theory are related by a unitary mapping in the Hilbert space. 2) Space-time geometry is determined by the structure of correlations and quantum entanglement in a quantum state. When applied to lattice systems, the holographic mapping is defined by a unitary tensor network. For free fermion boundary theories, I will discuss how different bulk geometries are obtained as dual theories of different boundary states. A particularly interesting case is the AdS black hole geometry and the interpretation of the interior of a black hole. We will also discuss dual geometries of topological states of matter.

The entanglement spectrum, i.e. the logarithm of the eigenvalues of reduced density matrices of
quantum many body wave functions, has been the focus of a rapidly expanding research endeavor recently.
Initially introduced by Li & Haldane in the context of the fractional quantum Hall effect, its usefulness has been
shown to extend to many more fields, such as topological insulators, fractional Chern insulators, spin liquids,
continuous symmetry breaking states, etc.
After a general introduction to the field we review some of our own contributions to the field, in particular the
perturbative structure of the entanglement spectrum in gapped phases, the entanglement spectrum across the
Mott-insulator transition in the Bose-Hubbard model, and the relation of the entanglement spectrum of
(1+1) dimensional quantum critical systems to the operator content of their underlying CFT.