Topological Order and Quantum Criticality

In this chapter we discuss aspects of the quantum critical behavior that occurs at a quantum phase transition separating a topological phase from a conventionally ordered one. We concentrate on a family of quantum lattice models, namely certain deformations of the toric code model, that exhibit continuous quantum phase transitions. One such deformation leads to a Lorentz-invariant transition in the 3D Ising universality class. An alternative deformation gives rise to a so-called conformal quantum critical point where equal-time correlations become conformally invariant and can be related to those of the 2D Ising model. We study the behavior of several physical observables, such as non-local operators and entanglement entropies, that can be used to characterize these quantum phase transitions. Finally, we briefly consider the role of thermal fluctuations and related phase transitions, before closing with a short overview of field theoretical descriptions of these quantum critical points.Comment: 24 pages, 7 figures, chapter of the book "Understanding Quantum Phase Transitions", edited by Lincoln D. Carr (CRC Press / Taylor and Francis, 2010); v2: updated reference

Magnetic Monopoles in Spin Ice

Electrically charged particles, such as the electron, are ubiquitous. By contrast, no elementary particles with a net magnetic charge have ever been observed, despite intensive and prolonged searches. We pursue an alternative strategy, namely that of realising them not as elementary but rather as emergent particles, i.e., as manifestations of the correlations present in a strongly interacting many-body system. The most prominent examples of emergent quasiparticles are the ones with fractional electric charge e/3 in quantum Hall physics. Here we show that magnetic monopoles do emerge in a class of exotic magnets known collectively as spin ice: the dipole moment of the underlying electronic degrees of freedom fractionalises into monopoles. This enables us to account for a mysterious phase transition observed experimentally in spin ice in a magnetic field, which is a liquid-gas transition of the magnetic monopoles. These monopoles can also be detected by other means, e.g., in an experiment modelled after the celebrated Stanford magnetic monopole search.Comment: (6 pages, 6 figures) v2: fig 3 replaced with colour version. For the high-definition version of the paper click http://www-thphys.physics.ox.ac.uk/user/ClaudioCastelnovo/Publications/papersub.pd

Toric-boson model: Toward a topological quantum memory at finite temperature

We discuss the existence of stable topological quantum memory at finite temperature. At stake here is the fundamental question of whether it is, in principle, possible to store quantum information for macroscopic times without the intervention from the external world, that is, without error correction. We study the toric code in two dimensions with an additional bosonic field that couples to the defects, in the presence of a generic environment at finite temperature: the toric-boson model. Although the coupling constants for the bare model are not finite in the thermodynamic limit, the model has a finite spectrum. We show that in the topological phase, there is a finite temperature below which open strings are confined and therefore the lifetime of the memory can be made arbitrarily (polynomially) long in system size. The interaction with the bosonic field yields a long-range attractive force between the end points of open strings but leaves closed strings and topological order intact.Comment: updated to published versio

Quantum oscillations and criticality in a fermionic and bosonic dimer model for the cuprates

We study quantum oscillations for a system of fermionic and bosonic dimers and compare the results to those experimentally observed in the cuprate superconductors in their underdoped regime. We argue that the charge carriers obey the Onsager quantization condition and quantum oscillations take on a Lifshitz-Kosevich form. We obtain the effective mass and find good qualitative agreement with experiments if we tune the model to the point where the observed mass divergence at optimum doping is associated to a van Hove singularity at which four free-dimer Fermi pockets touch pairwise in the interior of the Brillouin zone. The same van Hove singularity leads to a maximum in the d-wave superconducting pairing amplitude when antiferromagnetic interactions are included. Our combined results therefore suggest that a quantum critical point separating the underdoped and overdoped regimes is marked by the location of the van Hove saddle point in the fermionic dimer dispersion.This work was supported, in part, by the Engineering and Physical Sciences Research Council (EPSRC) Grant No. EP/M007065/1 (C.Ca. and G.G.), and by DOE Grant No. DE-FG02- 06ER46316 (C.Ch.)

Entanglement negativity and sudden death in the toric code at finite temperature

We study the fate of quantum correlations at finite temperature in the two dimensional toric code using the logarithmic entanglement negativity. We are able to obtain exact results that give us insight into how thermal excitations affect quantum entanglement. The toric code has two types of elementary excitations (defects) costing different energies. We show that an $\mathcal{O}(1)$ density of the lower energy defect is required to degrade the zero-temperature entanglement between two subsystems in contact with one another. However, one type of excitation alone is not sufficient to kill all quantum correlations, and an $\mathcal{O}(1)$ density of the higher energy defect is required to cause the so-called sudden death of the negativity. Interestingly, if the energy cost of one of the excitations is taken to infinity, quantum correlations survive up to arbitrarily high temperatures, a feature that is likely shared with other quantum spin liquids and frustrated systems in general, when projected down to their low energy states. We demonstrate this behaviour both for small subsystems, where we can prove that the negativity is a necessary and sufficient condition for separability, as well as for extended subsystems, where it is only a sufficient condition. We further observe that the negativity per boundary degree of freedom at a given temperature increases (parametrically) with the size of the boundary, and that quantum correlations between subsystems with extended boundaries are more robust to thermal fluctuations

Vison crystal in quantum spin ice on the breathing pyrochlore lattice

Recent excitement in the quantum spin ice community has come from the experimental discovery of pseudospin-$1/2$ breathing pyrochlores, including Ba$_3$Yb$_2$Zn$_5$O$_{11}$, in which inversion symmetry is broken by the `up' and `down' tetrahedra taking different physical sizes. We show here that the often-neglected $J_{z\pm}$ coupling between Kramers ions, in combination with the breathing nature of the lattice, can produce an imaginary ring flip term. This can lead to an unconventional '$U(1)_{\pi/2}$ phase', corresponding to a maximally dense packing of visons on the lattice. Coherent dynamics persists in all phases, together with its emergent QED description, in a manner reminiscent of fragmentation in spinon crystals. We characterize the enlarged QSI phase diagram and its excitations, showing that the imaginary ring flip acts both as a chemical potential for visons and as an effective three-photon vertex akin to strong light-matter coupling. The novel coupling causes a structured high-energy continuum to emerge above the photon dispersion, which is naturally interpreted as three photon up-conversion in a nonlinear optical crystal.Comment: 26 pages, 21 figure