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

Beginning on Sunday, June 2 and ending Friday June 7, 2019

All times in Oaxaca, Mexico time, CDT (UTC-5).

Sunday, June 2 | |
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14:00 - 23:59 | Check-in begins (Front desk at your assigned hotel) |

19:30 - 22:00 | Dinner (Restaurant Hotel Hacienda Los Laureles) |

20:30 - 21:30 | Informal gathering (Hotel Hacienda Los Laureles) |

Monday, June 3 | |
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07:30 - 08:45 | Breakfast (Restaurant at your assigned hotel) |

08:45 - 09:00 | Introduction and Welcome (Conference Room San Felipe) |

09:00 - 10:00 |
Yasuhiro Hatsugai: Symmetry protected Berry phases and edge states with interaction ↓ 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. (Conference Room San Felipe) |

10:00 - 11:00 |
Mathai Varghese: Fractional quantum numbers on non-compact Riemann surfaces ↓ 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. (Conference Room San Felipe) |

11:00 - 11:30 | Coffee Break (Conference Room San Felipe) |

11:30 - 12:30 |
Michael Levin: Bridging the gap between lattice models and TQFTs ↓ 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. (Conference Room San Felipe) |

12:30 - 13:30 |
F. Duncan M. Haldane: Geometry of Flux Attachment in the Fractional Quantum Hall Effect ↓ "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. (Conference Room San Felipe) |

13:30 - 13:35 | Group Photo (Hotel Hacienda Los Laureles) |

13:35 - 15:00 | Lunch (Restaurant Hotel Hacienda Los Laureles) |

16:00 - 16:30 | Coffee Break (Conference Room San Felipe) |

16:30 - 17:30 |
Bruno Nachtergaele: A Dynamical Toric Code model and Stability of the superselection sectors of two-dimensional quantum lattice models ↓ 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) (Conference Room San Felipe) |

17:30 - 18:30 |
Marcello Porta: Transport in interacting Weyl semimetals ↓ 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). (Conference Room San Felipe) |

19:00 - 21:00 | Dinner (Restaurant Hotel Hacienda Los Laureles) |

Tuesday, June 4 | |
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07:30 - 09:00 | Breakfast (Restaurant at your assigned hotel) |

09:00 - 10:00 | Yosi Avron: On the work of Hastings and Michalakis: why the adiabatic curvature of large gapped systems is constant (Conference Room San Felipe) |

10:00 - 11:00 |
Sven Bachmann: Fractional quantum transport ↓ 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. (Conference Room San Felipe) |

11:00 - 11:30 | Coffee Break (Conference Room San Felipe) |

11:30 - 12:30 |
Alexander Bols: The Avron-Dana-Zak relation constrains the allowed values of the Hall conductivity and the charge density in the ground state of a spatially periodic Hall fluid with rational flux per unit cell ↓ 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. (Conference Room San Felipe) |

12:30 - 13:30 |
Thomas Quella: Symmetry protected topological phases in one dimension: Beyond groups ↓ 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. (Conference Room San Felipe) |

13:30 - 15:00 | Lunch (Restaurant Hotel Hacienda Los Laureles) |

16:00 - 16:30 | Coffee Break (Conference Room San Felipe) |

16:30 - 17:30 |
Yoshiko Ogata: Classification of symmetry protected topological phases in quantum spin chains ↓ 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. (Conference Room San Felipe) |

17:30 - 18:30 |
Chris Bourne: On ${\mathbb Z}_2$-indices for ground states of fermionic chains ↓ 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. (Conference Room San Felipe) |

19:00 - 21:00 | Dinner (Restaurant Hotel Hacienda Los Laureles) |

Wednesday, June 5 | |
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07:30 - 09:00 | Breakfast (Restaurant at your assigned hotel) |

09:00 - 10:00 |
Terry Loring: The spectral localizer for estimating bulk gaps and calculating K-theory ↓ 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. (Conference Room San Felipe) |

10:00 - 11:00 |
Ralf Meyer: Coarse geometry and topological phases ↓ 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) (Conference Room San Felipe) |

11:00 - 11:30 | Coffee Break (Conference Room San Felipe) |

11:30 - 12:30 |
Guo Chuan Thiang: (Non)-existent atomic limits: geometric meaning of K-theory in the solid-state ↓ 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. (Conference Room San Felipe) |

12:30 - 13:30 | Lunch (Restaurant Hotel Hacienda Los Laureles) |

13:30 - 19:00 | Free Afternoon (Oaxaca) |

19:00 - 21:00 | Dinner (Restaurant Hotel Hacienda Los Laureles) |

Thursday, June 6 | |
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07:30 - 09:00 | Breakfast (Restaurant at your assigned hotel) |

09:00 - 10:00 |
Yuan-Ming Lu: Spontaneous symmetry breaking from anyon condensation ↓ 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. (Conference Room San Felipe) |

10:00 - 11:00 |
Amanda Young: A gapped family of two-dimensional AKLT models ↓ 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 $n \geq 3$. (Conference Room San Felipe) |

11:00 - 11:30 | Coffee Break (Conference Room San Felipe) |

11:30 - 12:30 |
Gianluca Panati: The Localization-Topology Correspondence: periodic systems and beyond ↓ 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). (Conference Room San Felipe) |

12:30 - 13:30 |
Christopher Max: Bulk-boundary correspondence for disordered free-fermion topological phases ↓ 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. (Conference Room San Felipe) |

13:30 - 14:45 | Lunch (Restaurant Hotel Hacienda Los Laureles) |

14:45 - 15:45 | Roger Mong: Topological Quantum Computing (Conference Room San Felipe) |

16:00 - 16:30 | Coffee Break (Conference Room San Felipe) |

16:30 - 17:00 |
Kiyonori Gomi: Is magnitude related to physics of patterned resonators? ↓ 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). (Conference Room San Felipe) |

17:00 - 17:30 |
Shin Hayashi: Topological invariants and corner states ↓ 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. (Conference Room San Felipe) |

17:30 - 18:00 |
Tom Stoiber: Flat bands of surface states via index theory of Toeplitz operators with Besov symbols ↓ 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. (Conference Room San Felipe) |

18:00 - 19:00 | Discussion (Conference Room San Felipe) |

19:00 - 21:00 | Dinner (Restaurant Hotel Hacienda Los Laureles) |

Friday, June 7 | |
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07:30 - 09:00 | Breakfast (Restaurant at your assigned hotel) |

09:00 - 10:00 |
Graf Gian Michele: Disorder and topology. The cases of Floquet and of chiral systems ↓ 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. (Conference Room San Felipe) |

10:00 - 11:00 |
Bram Mesland: Index theory and topological phases of aperiodic lattices ↓ 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 (Conference Room San Felipe) |

11:00 - 11:30 | Coffee break (Conference Room San Felipe) |

11:30 - 12:30 |
Johannes Kellendonk: Bulk boundary correspondance for quasiperiodic chains ↓ 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. (Conference Room San Felipe) |

12:30 - 14:30 | Lunch (Conference Room San Felipe) |