# Topological Phases in Cold Atom Systems

Europe/Stockholm
122:026 (Nordita, Stockholm)

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Description

### Venue

Nordita, Stockholm, Sweden

### Scope

Topological phases play an important role in condensed matter physics for understanding quantum effects like topological insulators and the quantum Hall effect. At the same time, cold atoms and ions have developed as testbeds to study complex quantum systems and are now used to generate, visualize and understand topological quantum states and phases. This program will bring together researchers from condensed matter and atomic physics, to share their knowledge how to describe and investigate topological quantum systems and to inspire collaborations between researchers of the two fields.

To facilitate the discussion between the cold atom and condensed matter communities, we plan to discuss different topics during the workshop. The timing will depend on the mix of participants, but the preliminary schedule of topics is

• Artificial gauge fields
• Driven and open quantum systems
• Topological phases and their applications

### Application

If you want to apply for participation in the program, please fill in the application form. You will be informed by the organizers shortly after the application deadline whether your application has been approved. Due to space restrictions, the total number of participants is strictly limited. (Invited participants are of course automatically approved, but need to register anyway.)

Application deadline: 31 March 2017

### Accommodation

Nordita provides a limited number of rooms in the Stockholm apartment hotel <href="https://www.nordita.org/guests/before/accommodation/index.php">BizApartments free of charge for accepted participants.

### Invited Participants

• M. Aidelsburger
• J. Budich
• N. Cooper
• N. Gemelke
• A. Hemmerich
• C. Monroe
• M. Müller
• C. de Morais Smith
• K. Sengstock
• I. Spielman
• Monday, July 31
• 10:00 AM 10:30 AM
Coffee
• 10:30 AM 11:30 AM
M. Burrello, One dimensional ladder limits of the quantum Hall states: the fermionic case at filling 1/2

Helical liquids have been experimentally detected in both
nanowires with strong spin-orbit coupling and
multi-component ultracold atomic chains with a
position-dependent coupling between the spin species. In
both cases the inner degree of freedom can be considered as
an additional space dimension, providing an interpretation
of these systems as synthetic ladders with artificial
magnetic fluxes determined by the spin-orbit terms.
In the presence of interactions, non-trivial helical states
appear for special ratios of the particle density and the
synthetic magnetic flux, like in the two-dimensional
fractional quantum Hall regime.
In this talk, I characterize the helical state which appears
in a ladder model of repulsive ultracold fermions at filling
ν = 1/2: this state cannot be explained in terms of a simple
fermionic Laughlin-like analog; it is generated by a gap
arising in the spin sector of the corresponding Luttinger
liquid and it corresponds to a Laughlin state at filling 1/8
of bosonic pairs. I will discuss the techniques required for
its analytical description as well as its main features,
derived from a DMRG study.

• 1:30 PM 2:30 PM
K. Schoutens, Many-body strategies for multi-qubit gates

The standard method for implementing algorithms for quantum
computation is
through quantum circuits. Such circuits typically contain
quantum gates
involving more than a
single or two qubits. Multi-qubit gates can be decomposed
into 1- and
2-qubit gates, but this is not necessarily the most
efficient strategy. We
present a framework for quantum control directly at the
level of multiple
qubits. A concrete proposal employs a so-called Krawtchouk
qubit chain,
which has previously been studied and experimentally
realised in the
context of perfect state transfer.

Based on arXiv:1707.05144 with Koen Greenland

• Tuesday, August 1
• 10:00 AM 10:30 AM
Coffee
• 10:30 AM 11:30 AM
G. Ortiz, Topological superfluidity with repulsive fermionic atoms

I will discuss a novel route to topological superfluidity in
repulsive fermionic systems.
The physical mechanism leading to pairing, and thus
superfluid behavior, is driven by
local kinetic-energy fluctuations and can be realized, for
instance, in multiband
systems with dissimilar localization properties.
Specifically, we propose to observe
this phenomenon in an optical superlattice with
alkaline-earth fermionic atoms, e.g.
Yb or Sr, carefully engineered to host itinerant and
spatially localized atoms whose
quantum fluctuations mediate an attractive interaction
between itinerant ones. This
mechanism gives rise to a topological $p$-wave superfluid
state in quasi-one-dimensional
wires, and a chiral p_x+ip_y superfluid in two dimensions.
Most importantly, we have
developed several experimental probes to characterize the
superfluid state, including
momentum-resolved RF spectroscopy and an analog of the
Edelstein magneto-electric
effect.

• 1:30 PM 2:30 PM
A. Trombettoni, Junctions of weakly-coupled strongly-interacting ultracold systems

After briefly reviewing the use of ultracold atoms for the
implementation of quantum devices, I discuss two examples of
junctions made by strongly interacting systems weakly
coupled between them. I present in the first part of the
talk recent results on the Josephson dynamics of two
ultracold fermionic gases at the unitary limit weakly linked
by a controllable barrier [1]. In the second part I discuss
properties of 1D Bose gases and then of junctions of
Tonks-Girardeau gases. When three Tonks-Girardeau gases are
coupled, one can exactly map their Hamiltonian by means of a
suitable Jordan-Wigner transformation into the Hamiltonian
of the multichannel Kondo model. I will also show recent
results on the experimental realization of Y-geometries with
holographic traps [2].

[1] G. Valtolina, A. Burchianti, A. Amico, E. Neri, K.
Xhani, J. A. Seman, A. Trombettoni, A. Smerzi, M. Zaccanti,
M. Inguscio, and G. Roati, "Josephson effect in fermionic
superfluids across the BEC-BCS crossover", Science 350, 1505
(2015).
[2] F. Buccheri, G. Bruce, A. Trombettoni, D. Cassettari, H.
Babujian, V. Korepin, and P. Sodano, "Holographic optical
traps for atom-based topological Kondo devices", New J.
Phys. 18, 075012 (2016).

• Wednesday, August 2
• 10:00 AM 10:30 AM
Coffee
• 10:30 AM 11:30 AM
J. Budich, Topological Insulators far from Equilibrium

TBA

• 1:30 PM 2:30 PM
G. Martone, Quantum phases and excitation spectrum of a spin-orbit-coupled Bose-Einstein condensate in a one-dimensional optical lattice

Spin-orbit-coupled quantum gases are characterized by a rich
phase diagram, with the appearance of novel quantum phases
having intriguing static and dynamic features.
In this talk I shall discuss the effect of a periodic
potential generated by a one-dimensional optical lattice on
the properties of a S=1/2 spin-orbit-coupled Bose gas.
Because of the interplay between the modified band structure
and the two-body interaction the ground state exhibits a
mixed regime, where the condensate wave function
is given by a superposition of multiple Bloch-wave
components, and an unmixed one, in which the atoms occupy a
single Bloch state. Magnetic phase transitions between
polarized and unpolarized states can be induced by tuning
the Raman coupling and the lattice strength, with the
emergence of a quantum tricritical point whose existence
is a consequence of the spin-dependent interaction. At the
dynamic level, I shall illustrate the behavior of the phonon
and roton modes of the excitation spectrum,
pointing out the instabilities occurring when a phase
transition is approached.

• 5:00 PM 11:45 PM
Reception outside NORDITA building
• Thursday, August 3
• 10:00 AM 10:30 AM
Coffee
• 10:30 AM 11:30 AM
N. Cooper, Superradiance-induced particle flow via dynamical gauge coupling

Superradiance-induced particle flow via dynamical gauge coupling

• Friday, August 4
• 10:00 AM 10:30 AM
Coffee
• 10:30 AM 11:30 AM
R. Kraus, Localization transition in presence of cavity backaction

We study the localization transition of an atom confined by
an external optical lattice in a high-finesse cavity. The
atom-cavity coupling yields an effective secondary lattice
potential, whose wavelength is incommensurate with the
periodicity of the optical lattice. The cavity lattice can
induce localization of the atomic wave function analogously
to the Aubry-André localization transition. Starting from
the master equation for the cavity and the atom we perform a
mapping of the system dynamics to a Hubbard Hamiltonian,
which can be reduced to the Harper's Hamiltonian in
appropriate limits. We evaluate the phase diagram for the
atom ground state and show that the transition between
extended and localized wavefunction is controlled by the
strength of the cavity nonlinearity, which determines the
size of the localized region and the behaviour of the
Lyapunov exponent. The Lyapunov exponent, in particular,
exhibits resonance-like behaviour in correspondence with the
optomechanical resonances. Finally we discuss the
experimental feasibility of these predictions.

• 1:30 PM 2:30 PM
L. Mazza, Majorana zero modes in earth-alkaline(-like) one-dimensional quantum gases

In this seminar, I discuss the problem of creating Majorana
zero modes in a cold-atom system, which is naturally
number-conserving, and without assuming the presence of a
superconductor.
I begin by presenting a model of interacting fermions in a
two-leg ladder supporting zero-energy Majorana edge modes.
The model has a quasi-exactly solvable line for all possible
densities of fermions described by a topologically
non-trivial ground state wave-function which can be fully
characterized.
In the second part of the seminar, I argue that all the
necessary ingredients for the appearance of Majorana zero
modes in number-conserving ladders are naturally present in
one-dimensional Ytterbium gases (and more generally in
earth-alkaline(-like) atomic gases), which are currently
studied in several state-of-the-art experiments. I support
the claim with extensive numerical simulations.

References:
[1] Iemini, Mazza, Rossini, Fazio, and Diehl, Phys. Rev.
Lett. 115, 156402 (2015).
[2] Iemini, Mazza, Fallani, Zoller, Fazio, and Dalmonte,
Phys. Rev. Lett. 118, 200404 (2017).

• Monday, August 7
• 10:00 AM 10:30 AM
Coffee
• 10:30 AM 11:30 AM
N. Gemelke, TBA

TBA

• Tuesday, August 8
• 10:00 AM 10:30 AM
Coffee
• 10:30 AM 11:30 AM
F. Kunst, Anatomy of Topological Flat and Surface States: Exact Solutions from Destructive Interference on Frustrated Lattices

The main feature of topological phases is the presence of
robust boundary states, which appear
for example in the form of chiral edge states in Chern
insulators and open Fermi arcs on the
surfaces of Weyl semimetals. Even though, non-interacting,
topological systems can be
straightforwardly described in fully periodic systems, the
understanding of the corresponding
boundary states has almost exclusively relied on numerical
studies. In our work, we present a
general method on how to find exact, analytical solutions
for topological as well as trivial
boundary states using a generic tight-binding model on a
large class of geometrically frustrated
lattices without the necessity of having to fine-tune
hopping amplitudes. Our method is inspired
by a similar approach that has been used in the past to
construct, topologically-trivial, flat band
models from local constraints on ‘line graphs’, in which
case fine-tuning is required in the sense
that hopping is strictly local. We expand on this work by
considering a larger class of lattices,
finding solutions for both topologically trivial and
non-trivial bands, and going beyond the need
for fine-tuning. This class of lattices are highly suited to
be realized in cold atom systems. In this
sense, it is likely that our work will contribute to both
the research fields of flat-band physics and
that of topological matter, as well as advance the
cross-fertilization between them. In this talk, I
will present a number of examples to illustrate our
discoveries, some of which are
experimentally relevant such as the derivation of exact
solutions for Fermi arcs in the recently
synthesized slabs of pyrochlore iridates.

• Wednesday, August 9
• 10:00 AM 10:30 AM
Coffee
• 10:30 AM 11:30 AM
C. Repellin, Creating a bosonic fractional quantum Hall state by pairing fermions

We numerically study the behavior of spin-1/2 fermions on a
two-dimensional square lattice subject to a uniform magnetic
field, where opposite spins interact via an on-site
attractive interaction. The single-particle Hamiltonian is
the Harper-Hofstadter model which was recently realized in
several cold atomic gas experiments using artificial gauge
fields. In this context, a Feschbach resonance can be used
to implement a highly tunable on-site interaction. Starting
from the non-interacting case where each spin population is
prepared in a quantum Hall state with unity filling, we
follow the evolution of the system as the interaction
strength is increased. Above a critical value and for
sufficiently low flux density, we observe the emergence of a
twofold quasidegeneracy accompanied by the opening of an
energy gap to the third level. Analysis of the entanglement
spectra shows that the gapped ground state is the bosonic
1/2 Laughlin state. Our work therefore provides compelling
evidence of a topological phase transition from the
fermionic integer quantum Hall state to the bosonic Laughlin
state at a critical attraction strength. I will present the
numerical signatures of these two phases, and analyze the
equilibrium properties of the phase transition. Finally, I
will discuss some preliminary results concerning the
dynamics of the phase transition.

Ref: Creating a bosonic fractional quantum Hall state by
pairing fermions, C. Repellin, T. Yefsah, A. Sterdyniak,
arXiv:1612.09184

• 1:30 PM 2:30 PM
G. Brunn, Topological superfluid in a Fermi-Bose mixture with a high critical temperature

We show that a two-dimensional (2D) spin-polarised Fermi gas
immersed in a 3D Bose-Einstein condensate (BEC) constitutes
a very promising system to realise a topological superfluid.
The fermions attract each other via an induced interaction
mediated by the bosons, and the resulting pairing is
analysed with retardation effects fully taken into account.
This is further combined with
Berezinskii-Kosterlitz-Thouless (BKT) theory to obtain
reliable results for the superfluid critical temperature. We
show that both the strength and the range of the induced
interaction can be tuned experimentally, which can be used
to make the critical temperature approach the maximum value
allowed by general BKT theory. Moreover, this is achieved
while keeping the Fermi-Bose interaction weak so that
three-body losses are small. We furthermore show that if the
fermions are confined in two layers, they can realise a
so-called Z_2 topological superfluid with time-reversal
symmetry and chiral edge states in analogy with the quantum
spin Hall state.

• Thursday, August 10
• 10:00 AM 10:30 AM
Coffee
• 1:30 PM 2:30 PM
K. Sengstock, Topology and dynamics in Floquet engineered optical lattices

TBA

• 5:00 PM 11:30 PM
Reception outside (when weather permits) the Nordita building

Eating, drinking, enjoying the sun,...

• Friday, August 11
• 10:00 AM 10:30 AM
Coffee
• 10:30 AM 11:30 AM
M. Müller, Topological Quantum Computation: From Concepts to Experiments

Quantum computers hold the promise to allow one to solve
problems that cannot be efficiently treated on classical
computers. To date, the construction of a fault-tolerant
quantum computer remains a fundamental scientific and
technological challenge, due the influence of unavoidable
noise which affects the fragile quantum states. In our talk,
we will first introduce basic concepts of quantum error
correction based on topological quantum error-correcting
codes, which allow one to protect quantum information during
storage and processing by distributing logical quantum
information over quantum many-body spin systems. We
then discuss progress on experimental quantum error
correction, in particular the realisation of a minimal
topological color code with trapped ions, which for the
first time demonstrated basic quantum computations on an
encoded qubit. In the second part, I will present recent
theory work of our group and experimental progress towards
fault-tolerant quantum error correction and control of
coupled logical qubits of increasing size and robustness in
scalable trapped-ion architectures.

• 1:30 PM 2:30 PM
J. Slingerland, Spin chains with stable topological zero modes.

Ground state degeneracy is an important characteristic of
topological
order. It is a natural question under what conditions such
topological
degeneracy extends to higher energy states or even to the
full energy
spectrum of a model, in such a way that the degeneracy is
preserved
when the Hamiltonian of the system is perturbed. It appears that
Ising/Majorana wires have this property due to the presence
of robust
edge zero modes. Generalized wire models based on
parafermions also
have edge zero modes and topological degeneracy at special
points in
parameter space, but the stability of these modes is a much
more
intricate question, and generically the zero modes are
spoilt by the
presence of resonances between bands in the unpertrbed
model. Both the
Majorana and parafermionic models are part of a large class
of chain
models with edge modes, based on tensor categories. It is
natural to
ask which of these have stable edge zero modes and/or full
spectrum
degeneracy.

• Monday, August 14
• 10:00 AM 10:30 AM
Coffee
• 10:30 AM 11:30 AM
P. Laurell, Prethermal states of matter in interacting Floquet systems

Periodically driven quantum systems are able to host novel
phases not
found in equilibrium, such as topological Floquet states or time
crystals. Alternatively, one can use the periodic drive to
engineer
new band structures or interactions. In both cases integrability
breaking terms are generally present, and the absence of
conserved
quantities tend to cause the system to heat up to an effectively
infinite temperature. Thus, at first sight, it appears as if
interacting Floquet systems have a rather limited phase
structure.
However, it is possible to use time-dependent transformations to
remove almost all time-dependence from the Hamiltonian,
approximately
mapping it onto a system in thermal equilibrium. This
approach can be
valid up to exponentially long times, after which thermalization
occurs. I will discuss our recent numerical results on symmetry
breaking prethermal states in spin systems, and the
connections to
experimental realizations in cold atoms or solid state systems.

• 1:30 PM 2:30 PM
M. Aidelsburger Floquet engineering with interacting ultracold atoms: Floquet engineering with interacting ultracold atoms

Floquet engineering is an important tool for the engineering
of novel band structures with interesting properties that go
beyond those offered by static systems. Recently, Floquet
systems have enabled the generation of Bloch bands with
non-trivial topological properties, such as the Hofstadter
and Haldane model. This led to the observation of chiral
Meissner currents and the first Chern-number measurements
with charge-neutral atoms.
Besides this success studies of many-body phases in driven
systems remains experimentally challenging in particular due
to the interplay between periodic driving an interactions.
In a driven system energy is not conserved which can lead to
severe heating. In order to find stable parameter regimes
for the generation of driven many-body phases it is
essential to develop a deeper understanding of the
underlying processes.
Here, I briefly review recent experimental advances in the
generation of topological band structures in the
non-interacting regime using Floquet engineering and present
first studies of interacting atoms in driven 1D lattices.

• Tuesday, August 15
• 10:00 AM 10:30 AM
Coffee
• 10:30 AM 11:30 AM
M. Tylutki, Coherent Oscillations of a Small Fermi Polaron

We study a trapped polarized Fermi gas interacting with a single
impurity. First, we investigate the tunnelling dynamics of the
impurity through a potential barrier, such as one created by a
double-well trap. We perform an exact diagonalization of the
full
many-body Hamiltonian and analyze the results in a Local Density
Approximation. Second, we consider a radio-frequency
spectroscopy of
our system and the resulting spectral function. These
calculations can
motivate future experiments, which can provide us with a further
insight into the physics of a Fermi polaron.

• Wednesday, August 16
• 10:00 AM 10:30 AM
Coffee
• 10:30 AM 11:30 AM
A. Quelle, TBA

TBA

• 1:30 PM 2:30 PM
A. Hemmerich, Topological bosons in optical lattices

I will discuss possible schemes to implement topological
matter with bosons condensed in metastable higher bands of
an optical lattice.

• Thursday, August 17
• 10:00 AM 10:30 AM
Coffee
• 10:30 AM 11:30 AM
M. Hasan, Pushing Topologicity with Light: A Route towards Topotronics

I will outline how strong light-matter interaction can be
used to induce quantum phase transition between normal and
topological phases in two-dimensional topological
insulators. The example case is a HgTe quantum well, in
which band inversion occurs above a critical value of the
well-thickness, and it will be delineated that coupling
between electron states and the E-field from an off-resonant
laser provides a powerful tool to control topological
transitions, even for a thickness of the quantum well that
is below the critical value where normally topological edge
states occur. The dependence of topological phase properties
of the edge states, including their group velocity, will be
discussed from an analytic perspective. I will then hint
towards possible new experimental means with which to
investigate topological insulators, that are discussed in
the talk. Importantly, all the topological effects that will
be discussed here could be realized in a controllable and
reversible manner, simply by changing the intensity of the
electromagnetic radiation. Technology that relies on these
topological states will be extrapolated, hinting towards a
possible new field of topotronics.

I believe this will be interesting for a broad audience. For
a detail of the talk please have a look at the arXiv version
(https://arxiv.org/pdf/1701.06756.pdf). The paper is now
being reviewed in Physical Review Letters.

• 1:30 PM 2:30 PM
E. Bergholtz, Fractional Chern insulators and non-Abelian twist defects

In this talk I will discuss fractional Chern insulators with
emphasis on the analogy with more conventional continuum
Landau level physics — and on aspects that are qualitatively
new in the lattice setting such as Berry curvature
fluctuations, competing instabilities and novel collective
states of matter emerging in bands with higher Chern number.
I will also explain how the lattice setting naturally allows
for exotic extrinsic wormhole-like twist defects (aka
“genons”) that effectively increase the genus of space.

• 5:00 PM 11:30 PM
Reception ourside Nordita building

Sun, snacks, socializing, ...

• Friday, August 18
• 10:00 AM 10:30 AM
Coffee
• 10:30 AM 11:30 AM
J. Zheng, Effective field theory for a strong magnetic scatterer near the edge of a quantum spin Hall insulator

We derive the effective field theory describing a classical
magnetic
impurity located near the edge of a two-dimensional quantum
spin Hall
insulator focusing on the regime where the impurity
potential is larger
than the band gap. We show that, due to the new features