Emergence & Entanglement II 2013
In this talk I will show how to obtain a detailed characterization of the emergent topological order starting from microscopic Hamiltonian on a two dimensional lattice. A key step is to obtain a tensor network representation for a complete set of ground states of the Hamiltonian, ﬁrst on an inﬁnite cylinder and then on a ﬁnite torus. As an application of the method I will study lattice Hamiltonians that give rise to selected anyon models, namely chiral semion, Ising as well as Z_3 and Z_5 models.
Given two lattice Hamiltonians H_1 and H_2 that are identical everywhere except on a local region R of the lattice, we propose a relationship between their ground states psi_1 and psi_2. Specifically, assuming the states can be represented as multi-scale entanglement renormalization ansatz (MERA), we propose a principle of directed influence which asserts that the tensors in the MERA’s that represent the ground states can be chosen to be identical everywhere except within a specific, localized region of the tensor network. The validity of this principle is justified by demonstrating
I will review recent work in our group using Density Matrix Renormalization Group (DMRG) to search for and study quantum spin liquid and topologically ordered states in two dimensional model Hamiltonians. This proves an efficient way to study these phases in semi-realistic situations. I will try to draw lessons from several studies and theoretical considerations.
We discuss the general features of charge transport of quantum critical points described by CFTs in 2+1D. Our main tool is the AdS/CFT correspondence, but we will make connections to standard field theory results and to recent quantum Monte Carlo data. We emphasize the importance of poles and zeros of the response functions. In the holographic setting, these are the discrete quasinormal modes of a black hole/brane; they map to the excitations of the CFT. We further describe the role of particle-vortex or S-duality on the conductivity, which is argued to obey two powerful sum rules.
In this talk, I will discuss about the notion of quantum renormalization group, and explain how (D+1)-dimensional gravitational theories naturally emerge as dual descriptions for D-dimensional quantum field theories. It will be argued that the dynamical gravitational field in the bulk encodes the entanglement between low energy modes and high energy modes of the corresponding quantum field theory.
We will point out that there is a universal thermodynamical property of entanglement entropy for excited states. We will derive this by using the AdS/CFT correspondence in any dimension. We will also directly confirm this property from direct field theoretic calculations in two dimensions. We will define a new quantity called entanglement density by taking derivatives of entanglement entropy with respect to the shape of subsystem.
In quantum systems with symmetry, the same topological phase can be enriched by symmetry in different ways, resulting in different symmetry transformations of the superselection sectors in the phase. However, not all symmetry transformations are allowed on the superselection sectors in topological phases in purely 2D systems. In this talk, I will discuss some examples of such symmetry enrichment of topological phases, which seem to be consistent with the fusion and braiding rules of the superselection sectors in the theory but are nonetheless impossible to realize in 2D.
I will discuss a family of solvable 3D lattice models that have a ``trivial" bulk, in which all excitations are confined, but exhibit topologically ordered surface states. I will discuss perturbations to these models that can drive a phase transition in which some of these excitations become deconfined, driving the system into a phase with bulk topological order.