Conference

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.

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.

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.

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.

Many one dimensional random quantum systems exhibit infinite randomness phases, such as the random singlet phase of the spin-1/2 Heisenberg model. These phases are typically the result of destabilizing systems described by a conformal field theory with disorder. Interestingly, entanglement entropy in 1d infinite randomness phases also exhibits a universal log scaling with length. In my talk I will touch upon calculating the entanglement entropy for inifinite-randomness phases, as well as describe the exotic infinite randomness phases realized in chains of non-abelian anyon chains. It was speculated that the entanglement entropy of an infinite-randomness phase is associated with the direction of RG flow, just as the c-theorem dictates the direction of RG flows for CFT's. I will also show that the entanglement entropy in disordered non-abelian chains provide the only known counter example.

Many one dimensional random quantum systems exhibit infinite randomness phases, such as the random singlet phase of the spin-1/2 Heisenberg model. These phases are typically the result of destabilizing systems described by a conformal field theory with disorder. Interestingly, entanglement entropy in 1d infinite randomness phases also exhibits a universal log scaling with length. In my talk I will touch upon calculating the entanglement entropy for inifinite-randomness phases, as well as describe the exotic infinite randomness phases realized in chains of non-abelian anyon chains. It was speculated that the entanglement entropy of an infinite-randomness phase is associated with the direction of RG flow, just as the c-theorem dictates the direction of RG flows for CFT's. I will also show that the entanglement entropy in disordered non-abelian chains provide the only known counter example.

Condensed matter theorists have recently begun exploiting the properties of entanglement as a resource for studying quantum materials. At the forefront of current efforts is the question of how the entanglement of two subregions in a quantum many-body groundstate scales with the subregion size. The general belief is that typical groundstates obey the so-called "area law", with entanglement entropy scaling as the boundary between regions. This has lead theorists to propose that sub-leading corrections to the area law provide new universal quantities at quantum critical points and in exotic quantum phases (i.e. topological Mott insulators). However, away from one dimension, entanglement entropy is difficult or impossible to calculate exactly, leaving the community in the dark about scaling in all but the simplest non-interacting systems. In this talk, I will discuss recent breakthroughs in calculating entanglement entropy in two dimensions and higher using advanced quantum Monte Carlo simulation techniques. We show, for the first time, evidence of leading-order area law scaling in a prototypical model of strongly-interacting quantum spins. This paves the way for future work in calculating new universal quantities derived from entanglement, in the plethora of real condensed matter systems amenable to numerical simulation.

In recent years the characterization of many-body ground states via the entanglement of their wave-function has attracted a lot of attention. One useful measure of entanglement is provided by the entanglement entropy S. The interest in this quantity has been sparked, in part, by the result that at one dimensional quantum critical points (QCPs) S scales logarithmically with the subsystem size with a universal coefficient related to the central charge of the conformal field theory describing the QCP. On the other hand, in spatial dimension d > 1 the leading contribution to the entanglement entropy scales as the area of the boundary of the subsystem. The coefficient of this ''area law'' is non-universal. However, in the neighbourhood of a QCP, S is believed to possess subleading universal corrections. In this talk, I will present the first field-theoretic study of entanglement entropy in dimension d > 1 at a stable interacting QCP - the quantum O(N) model. Our results confirm the presence of universal corrections to the entanglement entropy and exhibit a number of surprises such as different epsilon -> 0 limits of the Wilson-Fisher and Gaussian fixed points, violation of large N counting and subtle dependence on replica index.

In this talk, I will present two schemes which would result in substantial entanglement between distant individual spins of a spin chain. One relies on a global quench of the couplings of a spin chain, while the other relies on a bond quenching at one end. Both of the schemes result in substantial entanglement between the ends of a chain so that such chains could be used as a quantum wire to connect quantum registers. I will also examine the resource of entanglement already existing between separated parts of a many-body system at criticality as the size of the parts and their separation is varied. This form of entanglement displays an interesting scale invariance.

Using the mapping of the Fokker-Planck description of classical stochastic dynamics onto a quantum Hamiltonian, we argue that a dynamical glass transition in the former must have a precise definition in terms of a quantum phase transition in the latter. At the dynamical level, the transition corresponds to a collapse of the excitation spectrum at a critical point. At the static level, the transition affects the ground state wavefunction: while in some cases it could be picked up by the expectation value of a local operator, in others the order may be non-local, and impossible to be determined with any local probe. Here we propose instead to use concepts from quantum information theory that are not centered around local order parameters, such as fidelity and entanglement measures. We show that for systems derived from the mapping of classical stochastic dynamics, singularities in the fidelity susceptibility translate directly into singularities in the heat capacity of the classical system. In classical glassy systems with an extensive number of metastable states, we find that the prefactor of the area law term in the entanglement entropy jumps across the transition. We also discuss how entanglement measures can be used to detect a growing correlation length that diverges at the transition. Finally, we illustrate how static order can be hidden in systems with a macroscopically large number of degenerate equilibrium states by constructing a three dimensional lattice gauge model with only short-range interactions but with a finite temperature continuous phase transition into a massively degenerate phase.

A lattice gauge theory is described by a redundantly large vector space that is subject to local constraints, and it can be regarded as the low energy limit of a lattice model with a local symmetry. I will describe a coarse-graining scheme capable of exactly preserving local symmetries. The approach results in a variational ansatz for the ground state(s) and low energy excitations of a lattice gauge theory. This ansatz has built-in local symmetries, which are exploited to significantly reduce simulation costs. I will describe benchmark results in the context of Kitaev’s toric code with a magnetic field or, equivalently, Z2 lattice gauge theory, for lattices with up to 16 x 16 sites (16^2 x 2 = 512 spins) on a torus.