Schedule for: 19w5034 - Topological Phases of Interacting Quantum Systems

Although absence of the local order parameters is a fundamental feature of the topological phases, the Berry connection that represents a quantum interference of many body states successfully characterizes the bulk of the topological phases. The Chern number for the bulk of the quantum Hall states is a typical example. Also, with boundaries, appearance of local modes as the edge states characterizes the phase as the bulk-edge correspondence. As for the short range entangled state, the Berry phase that is quantized due to symmetry is used as a “quantum” local order parameter of the bulk. $Z_2$ Berry phase for the Haldane phase of the spin 1 quantum spin chain is a typical example. Here again, with boundaries, low energy local modes as the edge states appear associated with the nontrivial Berry phases as the bulk-edge correspondence. Focusing on the systems with interaction, we demonstrate the use of the symmetry protected Berry phases and the bulk-edge correspondence for various examples such as the generic valence bond solid (VBS) states and the corner states of the higher order topological phases.

We discuss both the conductance and charge transport on the maximal abelian cover Z of a compact Riemann surface in a strong magnetic field B. We consider a natural Landau Hamiltonian on Z modelling the FQHE and show that the low lying spectrum is discrete with infinite multiplicity whenever B is large. Interestingly, these spectral subspaces consist of holomorphic sections. We calculate the von Neumann dimension and degree of these spectral subspaces, and obtain fractional quantum numbers as the conductance. A refined analysis also gives the charge transport. This is joint work in progress with Graeme Wilkin.

Every (2+1) dimensional quantum many-body system with an energy gap is believed to be described by a topological quantum field theory (TQFT) in the low energy limit. What this means concretely is that every many-body system of this kind is associated with a collection of universal topological data. Some of this data, however, is missing a precise definition that would allow for its computation from a microscopic Hamiltonian. In this talk, I will address this issue by giving a microscopic definition of the “F-symbol” --- one of the most poorly understood pieces of data that characterize TQFTs. I will also discuss applications of this definition to the computation of anomalies at the boundaries of (2+1) dimensional symmetry-protected topological phases.

"Flux attachment” to form composite particles gives rise to an emergent geometry in the fractional quantum Hall effect. I will describe how the composite boson and composite fermion pictures complement each other as respective analogs of the unit cell and mobile charge carriers in crystalline electronic matter.

Kitaev's quantum double models provide a rich class of examples of two-dimensional lattice systems with topological order in the ground states and a spectrum described by anyonic elementary excitations. The infinite volume ground states of the abelian quantum double models come in a number of equivalence classes called superselection sectors. We prove that the superselection structure remains unchanged under uniformly small perturbations of the Hamiltonians. We introduce a Dynamical Toric Code Model and discuss some of its features. (joint work with Matthew Cha, Pieter Naaijkens, and Nicholas Sherman)

Weyl semimetals are a recently discovered class of materials, whose band structure at the Fermi level mimics massless relativistic fermions in 3+1 dimensions. As predicted by Nielsen and Ninomiya three decades before their discovery, when exposed to electromagnetic fields these materials are expected to give rise to the analogue of the axial anomaly in QED. In this talk I will present a theorem that provides a rigorous generalization of Nielsen-Ninomiya's prediction to the case of interacting lattice Weyl semimetals (joint work with A. Giuliani and V. Mastropietro). If time permits, I will also discuss the effect of weak disorder on the correlations on noninteracting Weyl semimetals, in the framework of an effective supersymmetric hierarchical model (joint work with G. Antinucci and L. Fresta).

I will present an index associated to a local unitary and a projection in the setting of many-body interacting particles on a lattice. Its values are in general rational, being integer multiples of the inverse of the rank of the projection. This is joint work with Alex Bols, Wojciech De Roeck and Martin Fraas.

As an application of the many-body index (see Sven Bachmann's talk), the Avron-Dana-Zak relation is shown to hold in the context of interacting quantum lattice systems, for the integer and fractional quantum Hall effects. A key formal property used to obtain this result is the additivity of the many-body index.

We show that phases of 1D quantum systems may enjoy symmetry protection even in the absence of standard symmetries. This observation leads to a broader definition of symmetry protected topological phases than currently used in the literature.

For the classification of SPT phases, defining an index is a central problem. In the famous paper, Pollmann, Tuner, Berg, and Oshikawa introduced ${\mathbb Z}_2$-indices for injective matrix products states (MPS) which have either ${\mathbb Z}_2\times {\mathbb Z}_2$ dihedral group (of $\pi$- rotations about $x$, $y$, and $z$-axes) symmetry, time-reversal symmetry, or reflection symmetry. we introduce an index which generalizes the index by Pollmann et.al. The index is an invariant of the $C^1$-classification of SPT phases.

We review the ${\mathbb Z}_2$-valued index map for quasifree states of the CAR algebra studied by Araki et al. and its connection to the ${\mathbb Z}_2$-valued spectral flow recently studied by Carey-Phillips-Schulz-Baldes. We then define a ${\mathbb Z}_2$-phase label for pure states of the CAR algebra over Z satisfying additional properties (pure states satisfying the split property being a key example). Basic properties of this $Z_2$-phase label are then shown. This is joint work with Hermann Schulz-Baldes.

For systems where periodic boundary conditions are not an option, the spectral localizer is a tool that can separate edge and bulk effects. This is the tool introduced by Kisil in defining the Clifford spectrum. The localizer can been thought of as an augmented Hamiltonian modeling a finite system and a probe, which picks up the symmetries from the underlying system. In all ten symmetry classes and all dimensions, one can use the localizer to create a K-theory class in either the Trout or Van Daele pictures of K-theory that is expected to correspond to a bulk invariant. As new methods for computing the boundary maps in K-theory are discovered, more and more instances of the index can be rigorously proven to equal a bulk invariant. Specific applications discussed will include numerical studies (by various investigators) of nanowires in class BDI and a Chern insulators built on the vertices of an aperiodic tiling of the plane.

We propose the Roe C*-algebra from coarse geometry as a model for topological phases of disordered materials. We explain the robustness of this C*-algebra and formulate the bulk-edge correspondence in this framework. We describe the map from the K-theory of the group C*-algebra of $Z^d$ to the K-theory of the Roe C*-algebra, both for real and complex K-theory. (Article with Elke Ewert)

It is known that a momentum space Chern class obstructs existence of good tight-binding models in position space. This is a T-duality, and holds in more general geometric contexts, e.g. crystallographic, defective, and non-Euclidean effective interacting topological phases. Namely, good Wannier bases correspond to free modules over pre-C*-algebras of the symmetry group. Thus K-theory and T-duality give precise tools to find topological insulators and understand their boundary gapless modes; explicit new examples will be given. Joint with M. Ludewig.

In the context of quantum spin liquids, it is long known that the condensation of fractionalized excitations can inevitably break certain physical symmetries. For example, condensing spinons will usually break spin rotation and time reversal symmetries. We generalize these phenomena to the context of a generic continuous quantum phase transition between symmetry enriched topological orders, driven by anyon condensation. We provide two rules to determine whether a symmetry is enforced to break across an anyon condensation transition or not. Using a dimensional reduction scheme, we establish a mapping between these symmetry-breaking anyon-condensation transitions in two spatial dimensions, and deconfined quantum criticality in one spatial dimension.

The one-dimensional AKLT spin chain is the prototypical example of a frustration-free quantum spin system with a spectral gap above its ground state. Affleck, Kennedy, Lieb, and Tasaki conjectured that the two-dimensional version of their model on the hexagonal lattice also exhibits a spectral gap. In this talk, we introduce a family of variants of the hexagonal AKLT model, defined by decorating each edge of the lattice with an AKLT chain of length $ n$, and prove that these decorated models are gapped for all $n \geq 3$.

As realized by TKNN in 1982, a relevant Transport-Topology Correspondence holds true for gapped periodic 2D systems, in the sense that a non-vanishing Hall conductivity corresponds to a non-trivial topology of the space of occupied states, decomposed with respect to the crystal momentum (the Bloch bundle). More recently, a related Localization-Topology Correspondence has been noticed and mathematically proved for 2D and 3D gapped periodic quantum system. The result states that the Bloch bundle is (Chern) trivial if and only if there exists a system of composite Wannier functions on which the expectation value of the squared position operator is finite. In other words, whenever the system is in a Chern-non-trivial phase, the composite Wannier functions are very delocalized, while in the Chern trivial phase they can be chosen exponentially localized (joint work with D. Monaco, A. Pisante and S. Teufel). During my talk, I will report on this result and the essential ideas of its proof, as well as on the ongoing attempt to generalize this correspondence to non-periodic gapped quantum systems (work in progress with G. Marcelli and M. Moscolari).

Guided by the many-particle quantum theory of interacting systems, we develop a uniform classification scheme for topological phases of disordered gapped free fermions, encompassing all symmetry classes of the Tenfold Way. We apply this scheme to give a mathematically rigorous proof of bulk-boundary correspondence. To that end, we construct real C$^\ast$-algebras harbouring the bulk and boundary data of disordered free-fermion ground states. These we connect by a natural bulk-to-boundary short exact sequence, realising the bulk system as a quotient of the half-space theory modulo boundary contributions. To every ground state, we attach two classes in different pictures of real operator $K$-theory (or $KR$-theory): a bulk class, using Van Daele's picture, along with a boundary class in Kasparov's Fredholm picture. We then show that the connecting map for the bulk-to-boundary sequence maps these $KR$-theory classes to each other.

As a general framework to study the bulk-boundary correspondence, Prodan and Shmalo proposed to study resonators on point patterns. A typical Hamiltonian in the dynamical system of resonators on a point pattern has the same form as the zeta matrix of a metric space used in the definition of its magnitude. The magnitude of a metric space is an invariant which counts an effective number of points, and is categorified to a notion of magnitude homology. To help a discovery of meaningful relationship with physics of resonators, I will talk about the basics of the magnitude (homology).

In condensed matter physics, topologically protected (codimension-one) edge states are known to appear on the surface (edges) of some insulators reflecting some topology of bulk. This correspondence is called the bulk-edge correspondence and was first proved by Hatsugai. Kellendonk-Richter-Schulz-Baldes used index theory for Toeplitz operators for its proof. In this talk, we consider systems with a codimension-two corner which appears as an intersection or a union of two half-planes. By using index theory for quarter-plane Toeplitz operators (or its variants), we show that topologically protected corner states appear reflecting some topology of gapped bulk and two edges. Further, a construction of explicit examples will be introduced. Recently, such systems with topologically protected corner states are studied actively in condensed matter physics under the name of higher-order topological insulators (HOTIs). If time permits, I will discuss one model of HOTIs by using the theory developed in this talk.

The scope of the index-theoretic approach to the bulk-boundary correspondence is extended to a pseudo-gap regime. For the case of a half-space graphene model with an edge of arbitrary cutting angle, this allows to express the density of surface as a linear combination of the winding numbers of the bulk. The new technical element is an index theorem for Toeplitz operators with non-commutative symbols from a Besov space for operators in a finite von Neumann algebra equipped with an R-action. For such operators a type II1 analogue of Peller's traceclass characterization for Toeplitz operators is proved. Joint work with H. Schulz-Baldes.

We will present a new formulation of bulk and edge indices for disordered Floquet systems. A byproduct is a space-time duality stating the equivalence of two settings: two systems may be placed next to one another in space or operate one after the other in time. A different type of systems to be addressed are disordered chiral chains, which may be viewed as Su-Schrieffer-Heeger models with random hopping. There localization occurs at all but possibly one energy, which is enough to endow the model with topological features. Different formulations of the index will be introduced and related to the Lyapunov spectrum of the chain.

A Delone set is a uniformly discrete and relatively dense subset of Euclidean space $\mathbb{R}^{d}$. As such they constitute a mathematical model for a general solid material. By choosing an abstract transversal for the translation action on the orbit space of the Delone set, one obtains an etale groupoid. In the absence of a $\mathbb{Z}^d$-labelling, the associated groupoid C*-algebra replaces the crossed product algebra as the natural algebra of observables. The K-theory of the groupoid C*-algebra is a natural home for the formulation of the bulk-boundary correspondence for topological insulators as well as a source for numerical invariants of (weak) topological phases. This is joint work with Chris Bourne

With the possibility to manufacture almost any structure which might exhibit topological phenomena and to control its boundary there is a need to generalise the standard approach to the bulk boundary correspondence which is based on crossed products to more general settings. For the bulk algebra this can be done using the pattern groupoid algebra. What we will focus on is an approach which allows to include boundaries and to construct more flexible half space algebras than in the cases considered so far. Applications to topological pumping will be given.