Emergence and Entanglement - 2010

In the Ho2Ti2O7 and Dy2Ti2O7 magnetic pyrochlore oxides, the Ho and Dy Ising magnetic moments interact via geometrically frustrated effective ferromagnetic coupling. These systems possess and extensive zero entropy related to the extensive entropy of ice water -- hence the name spin ice. The classical ground states of spin ice obey a constraint on each individual tetrahedron of interacting spins -- the so-called "ice rules". At large distance, the ice-rules can be described by an effective divergent-free field and, therefore, by an emergent classical gauge theory. In contrast, while it would appear at first sight to relate to the spin ices, the Tb2Ti2O7 material displays properties that much differ from spin ices and the behaviour of that system has largely remained unexplained for over ten years. In this talk, I will review the key features of the (Ho,Dy)2Ti2O7 spin ice materials, discuss the recent experimental results that support the emergent gauge theory description of spin ices and discuss how Tb2Ti2O7 is perhaps a ''quantum melted'' spin ice.

I discuss a class of systems with a very special property: exact results for physical quantities can be found in the many-body limit in terms of the original (bare) parameters in the Hamiltonian. A classic result of this type is Onsager and Yang's formula for the magnetization in the Ising model. I show how analogous results occur in a fermion chain with strong interactions, closely related to the XXZ spin chain. This is done by exploiting a supersymmetry, and noting that certain quantites are independent of finite-size effects. I also discuss how these ideas are related to an interacting generalization of the Kitaev honeycomb model.

Many crystalline materials predicted by band theory to be metals are insulators due to strong electron interactions. Both experiment and theory suggest that such Mott-insulators can exhibit exotic gapless spin-liquid ground states, having no magnetic or any other order. Such “critical spin liquids” will possess power law spin correlations which oscillate at various wavevectors. In a sub-class dubbed “Spin Bose-Metals” the singularities reside along surfaces in momentum space, analogous to a Fermi surface but without long-lived quasiparticle excitations. I will describe recent theoretical progress in accessing such states via controlled numerical and analytical studies on quasi-1d model systems.

Adiabatic evolutions connect two gapped quantum states in the same phase. We argue that the adiabatic evolutions are closely related to local unitary transformations which define a equivalence relation. So the equivalence classes of the local unitary transformations are the universality classes that define the different phases of quantum system. Since local unitary transformations can remove local entanglements, the above equivalence/universality classes correspond to pattern of long range entanglement, which is the essence of topological order. The local unitary transformation also allows us to define wave function renormalization, where a wave function can flow to a simpler one within the same equivalence/universality class. Using such a setup, we find conditions on the possible fixed-point wave functions where the local unitary transformations have finite dimensions. The solutions of the conditions allow us to classify this type of topological orders, which include all the string-net states.

Anyons are a special kind of excitations which are allowed in two dimensional systems, along with fermions and bosons. The topological nature of braiding of non-abelian anyons may allow a realization of quantum computing gates which is immune to noise. While the insensitivity of the such systems to a localized noise source is a built-in feature, an issue of great importance is more subtle: the robustness to slight deformations of the amiltonian describing the phase by perturbations which are locally tiny but are spread over through the entire system. Such will always arise if the realization of the Hamiltonian in a particular system is not quite perfect. The subject of the talk will be a proof of such stability, and the cluster expansion representation of deformed topological states.

I will discuss the question of thermal stability of a passive quantum memory, or finite-temperature topological order, in two or three spatial dimensions. We will analyze the criteria for thermal stability. We will present new results on Majorana fermion codes and a new extension of the 2D surface code to three dimensions.

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.

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.

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.