Emergence and Entanglement - 2010

I describe a novel abelian gauge theory in 2+1 dimensions which has surprising theoretical and phenomenological features. The theory has a vanishing coefficient for the square of the electric field $e_i^2$, characteristic of a quantum critical point with dynamical critical exponent $z=2$, and a level-$k$ Chern-Simons coupling, which is marginal at this critical point. For $k=0$, this theory is dual to a free $z=2$ scalar field theory describing a quantum Lifshitz transition, but $k \neq 0$ renders the scalar description non-local. The $k \neq 0$ theory exhibits properties intermediate between the (topological) pure Chern-Simons theory and the scalar theory. For instance, the Chern-Simons term does not make the gauge field massive. Nevertheless, there are chiral edge modes when the theory is placed on a space with boundary, and a non-trivial ground state degeneracy $k^g$ when it is placed on a finite-size Riemann surface of genus $g$. The coefficient of $e_i^2$ is the only relevant coupling; it tunes the system through a quantum phase transition between an isotropic fractional quantum Hall state and an anisotropic fractional quantum Hall state. I describe zero-temperature transport coefficients in both phases and at the critical point, and comment briefly on the relevance of the results to recent experiments.

The AdS/CFT correspondence has opened the door to understand a class of strongly coupled quantum field theories. Although the original correspondence has been conjectured based on string theory, it is possible that the underlying principle is more general, and a wider class of quantum field theories can be understood through holographic descriptions. In this talk, I will discuss about a prescription to construct holographic theories for general quantum field theories. As an example, I will present a holographic dual theory for the D-dimensional O(N) vector model. The phase transition and critical behaviors of the model are reproduced through the holographic theory.

The entanglement entropy of a pure quantum state of a bipartite system is defined as the von Neumann entropy of the reduced density matrix obtained by tracing over one of the two parts. Critical ground states of local Hamiltonians in one dimension have entanglement that diverges logarithmically in the subsystem size, with a universal coefficient that is is related to the central charge of the associated conformal field theory. In this talk I will discuss the extension of these ideas to two dimensional systems, either at a special quantum critical point or in a topological phase. We find the entanglement entropy for a standard class of z=2 quantum critical points in two spatial dimensions with scale invariant ground state wave functions: in addition to a nonuniversal ‘‘area law’’ contribution proportional to the size of the boundary of the region under observation, there is generically a universal logarithmically divergent correction, and in its absence a universal finite piece is found. This logarithmic term is completely determined by the geometry of the partition into subsystems and the central charge of the field theory that describes the equal-time correlations of the critical wavefunction. On the other hand, in a topological phase there is no such logarithmic term but instead a universal constant term. We will discuss the connection between this universal entanglement entropy and the nature of the topological phase.

Recently several proposals are made for possible spin liquid and topological insulator phases in frustrated magnets. I will review some of these efforts and present some new results. Implications to real materials will also be made.

The entanglement entropy in conformal field theory was predicted to include a boundary term which depends on the choice of conformally invariant boundary condition. We have studied this effect in the Kondo model of a magnetic impurity in a metal, which exhibits a renormalization group flow between conformally invariant fixed points.

Recent work has explored some aspects of entanglement in topological insulators. Notably, the entanglement spectrum has been shown to mimic certain properties of the low-energy fermionic modes found on real spatial boundaries. I will discuss the many-body entanglement spectrum of topological insulators and show that it matches the expected CFT character structure that has been previously shown to hold in fractional quantum Hall effect ground states. I also present the analysis of a disorder-driven Anderson localization transition in a Chern-Insulator from an analysis of the entanglement spectrum. Interestingly, the disorder-averaged level-spacing statistics of the entanglement spectrum characterizes the system just as well as the statistics of the real energy spectrum, but with the advantage that only the ground state, and not the entire spectrum of excited states, needs to be calculated.

We study a superconductor-ferromagnet-superconductor (SC-FM-SC) Josephson junction array deposited on top of a two-dimensional quantum spin Hall (QSH) insulator. The existence of Majorana bound states at the interface between SC and FM gives rise to charge-e tunneling, in addition to the usual charge-2e Cooper pair tunneling, between neighboring superconductor islands. Moreover, because Majorana fermions encode the information of charge number parity, an exact Z_2 gauge structure naturally emerges and leads to many new insulating phases, including a deconfined phase where electrons fractionalize into charge-e bosons and topological defects. We will also discuss the ultracold alkaline earth atoms trapped in optical lattice, which naturally has a SU(N) spin symmetry with N as large as 10. An SU(2)xU(1) gauge theory emerges in a large part of the phase diagram of this system.