Narrow Results

This paper deals with the construction of a suitable topological $K$-theory capable of classifying topological phases of dynamically stable systems described by gapped $\eta$-self-adjoint operators on a Krein space with indefinite metric $\eta$.

We present homotopy theoretic and geometric interpretations of the Kane–Mele invariant for gapped fermionic quantum systems in three dimensions with time-reversal symmetry. We show that the invariant is related to a certain 4-equivalence which lends it an interpretation as an obstruction to a block decomposition of the sewing matrix up to non-equivariant homotopy. We prove a Mayer–Vietoris Theorem for manifolds with ${\mathbb{Z}}_2$-actions which intertwines Real and ${\mathbb{Z}}_2$-equivariant de Rham cohomology groups, and apply it to derive a new localization formula for the Kane–Mele invariant. This provides a unified cohomological explanation for the equivalence between the discrete Pfaffian formula and the known local geometric computations of the index for periodic lattice systems. We build on the relation between the Kane–Mele invariant and the theory of bundle gerbes with ${\mathbb{Z}}_2$-actions to obtain geometric refinements of this obstruction and localization technique. In the preliminary part we review the Freed–Moore theory of band insulators on Galilean spacetimes with emphasis on geometric constructions, and present a bottom-up approach to time-reversal symmetric topological phases.

We use the Cayley transform to provide an explicit isomorphism at the level of cycles from van Daele $K$-theory to $KK$-theory for graded $C^{\ast }$-algebras with a real structure. Isomorphisms between $KK$-theory and complex or real $K$-theory for ungraded $C^{\ast }$-algebras are a special case of this map. In all cases, our map is compatible with the computational techniques required in physical and geometrical applications, in particular, index pairings and Kasparov products. We provide applications to real $K$-theory and topological phases of matter.

I review the topological phases of the Aharonov–Bohm type associated with Maxwell and Yang–Mills fields.

Chiral edge states of $(2+1)$-dimensional Abelian and non-Abelian topological phases can be represented by chiral conformal field theories with integer and noninteger values of central charge, respectively. In this work we describe certain edge states in terms of constrained fermionic fields that realize chiral coset CFT structures. This construction arises naturally in the so-called quantum wires approach for topological phases and allows for representing fractionalized edge states directly in terms of fermionic degrees of freedom. At the same time, the constrained fermions description introduces some subtleties concerning gauge anomalies since it involves the coupling of chiral fermions to gauge fields. We describe in this paper how to handle these issues.

We propose a framework to describe topological phases with global internal symmetries. This involves embedding the phase of interest into an intrinsic topological-order. Such an embedding is related to the physics of anyon condensation, although no physical condensation actually occurs here. Crucial physical properties of the symmetric phase, such as edge states and global charges of anyons are easily recovered from the embedding. The framework naturally reproduces all phases that are currently known in 2+1 d, and potentially provides a complete classification of symmetric topological phases in 2+1 d. We illustrate our idea with simple examples.

This note discusses the residual decoherence for a class of topologically protected qubits realized as small Josephson junction arrays. For static noise, such circuits present a decoherence rate which is exponentially small in the system size. However, in the presence of a dynamical coupling to an external bath, this length has to remain small enough so that no irreversible real transition to excited states can occur. When this condition is not satisfied, the rate of such transitions is computed through a generalization of the Fermi golden rule formula, adapted to this particular class of systems, which is derived here.

Quantum many-body systems realize many different phases of matter characterized by their exotic emergent phenomena. While some simple versions of these properties can occur in systems of free fermions, their occurrence generally implies that the physics is dictated by an interacting Hamiltonian. The interaction distance has been successfully used to quantify the effect of interactions in a variety of states of matter via the entanglement spectrum [C. J. Turner, K. Meichanetzidis, Z. Papic and J. K. Pachos, Nat. Commun.8 (2017) 14926, Phys. Rev. B97 (2018) 125104]. The computation of the interaction distance reduces to a global optimization problem whose goal is to search for the free-fermion entanglement spectrum closest to the given entanglement spectrum. In this work, we employ techniques from machine learning in order to perform this same task. In a supervised learning setting, we use labeled data obtained by computing the interaction distance and predict its value via linear regression. Moving to a semi-supervised setting, we train an autoencoder to estimate an alternative measure to the interaction distance, and we show that it behaves in a similar manner.

Equivalence classes of gapped Hamiltonians compatible with given symmetry constraints, such as those underlying topological insulators, can be defined in many ways. For the non-chiral classes modeled by vector bundles over Brillouin tori, physically relevant equivalences include isomorphism, homotopy, and K-theory, which are inequivalent but closely related. We discuss an important subtlety which arises in the chiral Class AIII systems, where the winding number invariant is shown to be relative rather than absolute as is usually assumed. These issues are then analyzed and reconciled in the language of K-theory.

The remarkable properties of topological phases and topological insulators are analyzed in very simple quantum mechanical models. A characteristic effect of magnetic fields on quantum systems is the appearance of degenerate ground states. The level of degeneracy of the corresponding quantum phases depends on the strength of the magnetic field and the topology of the configuration space. In topological insulators, in absence of magnetic fields, the essential ingredients are time reversal symmetry, the existence of edge states and a non-trivial topological structure of the Jacobian space (Brillouin zone). In both cases the main features are encoded by topological indices. In topological phases induced by magnetic fields, the index is a topological invariant that depends on the Chern class of the magnetic fields and the topology of the space. In topological insulators the topological index is given by the number of edge levels crossing the Fermi surface that remains invariant under time reversal invariant perturbations. The robustness of the corresponding effects under perturbations follows from the topological nature of both phenomena.

The role of mixed states in topological quantum matter is less known than that of pure quantum states. Generalisations of topological phases appearing in pure states have received attention in the literature only quite recently. In particular, it is still unclear whether the generalisation of the Aharonov–Anandan phase for mixed states due to Uhlmann plays any physical role in the behaviour of the quantum systems. We analyse, from a general viewpoint, topological phases of mixed states and the robustness of their invariance. In particular, we analyse the role of these phases in the behaviour of systems with periodic symmetry and their evolution under the influence of an environment preserving its crystalline symmetries.

We review recent theoretical progress in our understanding of electron-driven novel magnetic phases on frustrated lattices. Our specific focus is on Kondo-lattice or double-exchange models assuming finite magnetic moments localized at the lattice sites. A salient feature of systems with SU(2) symmetric local moments is the emergence of noncoplanar magnetic ordering driven by the conduction electrons. The complex spin textures then endow the electrons a nontrivial Berry phase, often giving rise to a topologically nontrivial electronic state. The second part of the review is devoted to the discussion of metallic spin ice systems, which are essentially frustrated Ising magnets with local spin ordering governed by the so-called ice rules. These rules are similar to those that describe proton configurations in solid water ice, hence the name "spin ice". The nontrivial spin correlations in the ice phase give rise to unusual electron transport properties in metallic spin-ice systems.

Real and complex K-theory have found applications in the study of topological phases of matter in condensed matter physics. Furthermore, T-duality transformations inspired from string theory help to conceptually simplify the bulk-boundary correspondence there. This is an overview of recent progress on mathematical ideas based around this string theory-condensed matter analogy.

We review recent results on lattice models for spin-less fermions with strong repulsive interactions. A judicious tuning of kinetic and interaction terms leads to a model possessing supersymmetry. In the 1D case, this model displays critical behavior described by superconformal field theory. On 2D lattices we generically find superfrustration, characterized by an extensive ground state entropy. For certain 2D lattices analytical results on the ground state structure reveal yet another quantum phase, which we tentatively call 'supertopological'.

I review the topological phases of the Aharonov–Bohm type associated with Maxwell and Yang–Mills fields.