Topological spin liquids: Robustness under perturbations

We study the robustness of the paradigmatic kagome Resonating Valence Bond (RVB) spin liquid and its orthogonal version, the quantum dimer model. The non-orthogonality of singlets in the RVB model and the induced finite length scale not only makes it difficult to analyze, but can also significantly affect its physics, such as how much noise resilience it exhibits. Surprisingly, we find that this is not the case: The amount of perturbations which the RVB spin liquid can tolerate is not affected by the finite correlation length, making the dimer model a viable model for studying RVB physics under perturbations. Remarkably, we find that this is a universal phenomenon protected by symmetries: First, the dominant correlations in the RVB are spinon correlations, making the state robust against doping with visons. Second, reflection symmetry stabilizes the spin liquid against doping with spinons, by forbidding mixing of the initially dominant correlations with those which lead to the breakdown of topological order.Comment: v2: accepted versio

Matrix Product Density Operators: Renormalization Fixed Points and Boundary Theories

We consider the tensors generating matrix product states and density operators in a spin chain. For pure states, we revise the renormalization procedure introduced by F. Verstraete et al. in 2005 and characterize the tensors corresponding to the fixed points. We relate them to the states possessing zero correlation length, saturation of the area law, as well as to those which generate ground states of local and commuting Hamiltonians. For mixed states, we introduce the concept of renormalization fixed points and characterize the corresponding tensors. We also relate them to concepts like finite correlation length, saturation of the area law, as well as to those which generate Gibbs states of local and commuting Hamiltonians. One of the main result of this work is that the resulting fixed points can be associated to the boundary theories of two-dimensional topological states, through the bulk-boundary correspondence introduced by Cirac et al. in 2011.Comment: 63 pages, Annals of Physics (2016). Accepted versio

On tensor network representations of the (3+1)d toric code

We define two dual tensor network representations of the (3+1)d toric code ground state subspace. These two representations, which are obtained by initially imposing either family of stabilizer constraints, are characterized by different virtual symmetries generated by string-like and membrane-like operators, respectively. We discuss the topological properties of the model from the point of view of these virtual symmetries, emphasizing the differences between both representations. In particular, we argue that, depending on the representation, the phase diagram of boundary entanglement degrees of freedom is naturally associated with that of a (2+1)d Hamiltonian displaying either a global or a gauge $\mathbb Z_2$-symmetry

Projected entangled-pair states can describe chiral topological states

We show that Projected Entangled-Pair States (PEPS) in two spatial dimensions can describe chiral topological states by explicitly constructing a family of such states with a non-trivial Chern number. They are ground states of two different kinds of free-fermion Hamiltonians: (i) local and gapless; (ii) gapped, but with hopping amplitudes that decay according to a power law. We derive general conditions on topological free fermionic PEPS which show that they cannot correspond to exact ground states of gapped, local parent Hamiltonians, and provide numerical evidence demonstrating that they can nevertheless approximate well the physical properties of topological insulators with local Hamiltonians at arbitrary temperatures.Comment: v2: minor changes, references added. v3: accepted version, Journal-Ref adde

Nonlocal resources in the presence of Superselection Rules

Superselection rules severely alter the possible operations that can be implemented on a distributed quantum system. Whereas the restriction to local operations imposed by a bipartite setting gives rise to the notion of entanglement as a nonlocal resource, the superselection rule associated with particle number conservation leads to a new resource, the \emph{superselection induced variance} of local particle number. We show that, in the case of pure quantum states, one can quantify the nonlocal properties by only two additive measures, and that all states with the same measures can be asymptotically interconverted into each other by local operations and classical communication. Furthermore we discuss how superselection rules affect the concepts of majorization, teleportation and mixed state entanglement.Comment: 4 page

Symmetry Protected Topological Order in Open Quantum Systems

We systematically investigate the robustness of symmetry protected topological (SPT) order in open quantum systems by studying the evolution of string order parameters and other probes under noisy channels. We find that one-dimensional SPT order is robust against noisy couplings to the environment that satisfy a strong symmetry condition, while it is destabilized by noise that satisfies only a weak symmetry condition, which generalizes the notion of symmetry for closed systems. We also discuss "transmutation" of SPT phases into other SPT phases of equal or lesser complexity, under noisy channels that satisfy twisted versions of the strong symmetry condition

Edge theories in Projected Entangled Pair State models

We study the edge physics of gapped quantum systems in the framework of Projected Entangled Pair State (PEPS) models. We show that the effective low-energy model for any region acts on the entanglement degrees of freedom at the boundary, corresponding to physical excitations located at the edge. This allows us to determine the edge Hamiltonian in the vicinity of PEPS models, and we demonstrate that by choosing the appropriate bulk perturbation, the edge Hamiltonian can exhibit a rich phase diagram and phase transitions. While for models in the trivial phase any Hamiltonian can be realized at the edge, we show that for topological models, the edge Hamiltonian is constrained by the topological order in the bulk which can e.g. protect a ferromagnetic Ising chain at the edge against spontaneous symmetry breaking.Comment: 5 pages, 4 figure

Transfer Matrices and Excitations with Matrix Product States

We investigate the relation between static correlation functions in the ground state of local quantum many-body Hamiltonians and the dispersion relations of the corresponding low energy excitations using the formalism of tensor network states. In particular, we show that the Matrix Product State Transfer Matrix (MPS-TM) - a central object in the computation of static correlation functions - provides important information about the location and magnitude of the minima of the low energy dispersion relation(s) and present supporting numerical data for one-dimensional lattice and continuum models as well as two-dimensional lattice models on a cylinder. We elaborate on the peculiar structure of the MPS-TM's eigenspectrum and give several arguments for the close relation between the structure of the low energy spectrum of the system and the form of static correlation functions. Finally, we discuss how the MPS-TM connects to the exact Quantum Transfer Matrix (QTM) of the model at zero temperature. We present a renormalization group argument for obtaining finite bond dimension approximations of MPS, which allows to reinterpret variational MPS techniques (such as the Density Matrix Renormalization Group) as an application of Wilson's Numerical Renormalization Group along the virtual (imaginary time) dimension of the system.Comment: 39 pages (+8 pages appendix), 14 figure

Single copy/knock-in models of ALS SOD1 in C. elegans suggest loss and gain of function have different contributions to cholinergic and glutamatergic neurodegeneration

Mutations in Cu/Zn superoxide dismutase 1 (SOD1) lead to Amyotrophic Lateral Sclerosis (ALS), a neurodegenerative disease that disproportionately affects glutamatergic and cholinergic motor neurons. Previous work with SOD1 overexpression models supports a role for SOD1 toxic gain of function in ALS pathogenesis. However, the impact of SOD1 loss of function in ALS cannot be directly examined in overexpression models. In addition, overexpression may obscure the contribution of SOD1 loss of function in the degeneration of different neuronal populations. Here, we report the first single-copy, ALS knock-in models in C. elegans generated by transposon- or CRISPR/Cas9- mediated genome editing of the endogenous sod-1 gene. Introduction of ALS patient amino acid changes A4V, H71Y, L84V, G85R or G93A into the C. elegans sod-1 gene yielded single-copy/knock-in ALS SOD1 models. These differ from previously reported overexpression models in multiple assays. In single-copy/knock-in models, we observed differential impact of sod-1 ALS alleles on glutamatergic and cholinergic neurodegeneration. A4V, H71Y, G85R, and G93A animals showed increased SOD1 protein accumulation and oxidative stress induced degeneration, consistent with a toxic gain of function in cholinergic motor neurons. By contrast, H71Y, L84V, and G85R lead to glutamatergic neuron degeneration due to sod-1 loss of function after oxidative stress. However, dopaminergic and serotonergic neuronal populations were spared in single-copy ALS models, suggesting a neuronal-subtype specificity previously not reported in invertebrate ALS SOD1 models. Combined, these results suggest that knock-in models may reproduce the neurotransmitter-type specificity of ALS and that both SOD1 loss and gain of toxic function differentially contribute to ALS pathogenesis in different neuronal populations.Peer reviewe