This series consists of talks in the area of Condensed Matter.
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.
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.
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.
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.
We consider the one dimensional, periodic spin chain with $N$ sites, similar to the one studied by Haldane \cite{hal}, however in the opposite limit of very large anisotropy and small nearest neighbour, anti-ferromagnetic exchange coupling between the spins, which are of large magnitude $s$. For a chain with an even number of sites we show that actually the ground state is non degenerate and given by a superposition of the two Néel states, due to quantum spin tunnelling. With an odd number of sites, the Néel state must necessarily contain a soliton.
I will present recent theoretical work on cluster Mott insulators (CMI) in which interesting physics such as emergent charge lattices, charge fractionalization and quantum spin liquids are proposed. For the anisotropic Kagome system like LiZn2Mo3O8, we find two distinct CMIs, type-I and type-II, can arise from the repulsive interactions. In type-I CMI, the electrons are localized in one half of the triangle clusters of the Kagome system while the electrons in the type-II CMI are localized in every triangle cluster.
One hallmark of topological phases with broken time reversal symmetry is the appearance of quantized non-dissipative transport coefficients, the archetypical example being the quantized Hall conductivity in quantum Hall states. Here I will talk about two other non-dissipative transport coefficients that appear in such systems - the Hall viscosity and the thermal Hall conductivity. In the first part of the talk, I will start by reviewing previous results concerning the Hall viscosity, including its relation to a topological invariant known as the shift.
Two-dimensional interacting electron gas in strong transverse magnetic field forms a collective state -- incompressible electron liquid, known as fractional quantum Hall (FQH) state. FQH states are genuinely new states of matter with long range topological order. Their primary observable characteristics are the absence of dissipation and quantization of the transverse electro-magnetic response known Hall conductance. In addition to quantized electromagnetic response FQH states are characterized by quantized geometric responses such as Hall viscosity and thermal Hall conductance.
In two spatial dimensions, it is well known that particle-like excitations can come with fractional statistics, beyond the usual dichotomy of Bose versus Fermi statistics. In this talk, I move one dimension higher to three spatial dimensions, and study loop-like objects instead of point-like particles. Just like particles in 2D, loops can exhibit interesting fractional braiding statistics in 3D. I will talk about loop braiding statistics in the context of symmetry protected topological phases, which is a generalization of topological insulators.
We present an analytic, gauge invariant tensor network ansatz for the ground state of lattice Yang-Mills theory for nonabelian gauge groups. It naturally takes the form of a MERA, where the top level is the strong coupling limit of the lattice theory. Each layer performs a fine-graining operation defined in a fixed way followed by an optional step of adiabatic evolution, resulting in the ground state at an intermediate coupling.