# Topological order

For a long time it was believed that all ordered phases of matter, as well as all (continuous) phase transitions, can be understood in terms of Landau's theory of symmetry-breaking.
In this theory, Landau introduced the important notion of order as a broken symmetry.
Standard examples are solids, where the regular arrangement of atoms in a lattice breaks continuous translation symmetry, or magnets, where the spins align to give rise to a finite magnetization and spin-rotation symmetry is broken.
For each symmetry there is a corresponding local order parameter, which has a finite expectation value in the ordered phase and vanishes otherwise.
After the discovery of the quantum Hall effect it was soon realized
that Landau's theory of symmetry-breaking cannot be complete, as
different quantum Hall liquids obey the same symmetries and cannot
be distinguished by a local order parameter. Instead, one needs to
resort to non-local measurements, such as measuring the Hall
conductance, to tell them apart.

Quantum Hall liquids are the paradigmatic examples of so-called
'topological states'. Another example are the recently discovered
topological insulators and topological superconductors.
Common for such states is the existence of robust properties that do
not depend on microscopic details and are insensitive to local
perturbations. Prominent examples are the presence of protected
gapless edge states for topological insulators and the remarkably
precise quantization of the Hall conductance for quantum Hall liquids. The robustness of these
properties can be traced back to a topological invariant, such as
the Chern number for quantum Hall liquids, which is quantized to
integer values and can only be changed by closing the bulk
gap.
In the presence of strong correlations the system can exhibit even more fascinating properties, for instance particle excitations with fractional electric charge. States with such properties are commonly referred to as `topologically ordered', alluding to the fact that the emergent low-energy theory of these systems is a topological field theory.

# Entanglement spectrum

Determining topological order in strongly correlated systems is a hard task, as it cannot be characterized by any kind of local observable or correlation function.
One promising method is to measure the long-range entanglement via the
entanglement spectrum by dividing the system into two parts A and
B and tracing out the degrees of freedom in part B, which yields
the reduced density matrix of A, ρ_{A}. The
entanglement spectrum is the negative logarithm of the eigenvalues of
ρ_{A} and contains important information about the
properties of the state. For instance, a spatial partition of the
system can give valuable information about the edge physics while tracing
out particles yields information about quasihole
excitations. Entanglement spectra are, therefore, widely used as a
diagnostic tool to obtain information about the topological properties
of the system.

For non-interacting systems, the relationship between properties of
the ground state and its entanglement spectrum are well
understood. However, for strongly correlated systems most of the
conjectured properties are based on (numerical) observations. During
my postdoc in Bernevig's group we established some of the few exact
results known for such strongly correlated systems by relating the
universal properties of two distinct types of entanglement spectra to
each other [1] and thus proving a bulk-boundary
correspondence in the entanglement spectra. We also showed that
non-universal properties, such as finite size effects, can encode
valuable information about the ground state
[2].

**References**

[1] A. Chandran, M. Hermanns, N. Regnault, B.A. Bernevig, Phys. Rev. B

**84**, 205136 (2011).

[2] M. Hermanns, A. Chandran, N. Regnault, B.A. Bernevig, Phys. Rev. B

**84**, 121309(R) (2011).

# Quantum Hall hierarchy

Perhaps the most intriguing property of quantum Hall states
is that the emergent quasiparticle excitations can be
anyons. By exchanging two identical anyons, the wave
function acquires a phase factor, which is neither 0 (bosons) nor π
(fermions), but can take an arbitrary value in between. Some
quantum Hall states - the most prominent example being the Moore-Read
state - harbor even more exotic quasiparticles, namely
non-Abelian anyons.
In this case, a system with several localized quasiparticles is not
described by a single state, but rather a manifold of degenerate
states and the result of successive braiding operations of these
quasiparticles depends on the order of the braiding.

A useful framework to describe such particles and their properties is
via conformal field theory. This gives a natural operator description for
electrons and quasiholes, but not quasielectrons. The latter is more
complicated due to the Pauli principle that does not allow to
arbitrarily accumulate excess charge. A proper
description of the quasielectrons in terms of a conformal field theory
operator was first found in [1]. An important feature of this operator description is that it is
applicable for any quantum Hall state, even the non-Abelian
ones. Thus, it allows to generalize the Haldane-Halperin hierarchy to
non-Abelian states [2]. A systematic analysis of these
hierarchical states and their properties is currently an active
research topic in this group.

**References**

[1] T.H. Hansson, M. Hermanns, N. Regnault, S. Viefers, Phys. Rev. Lett.

**102**, 166805 (2009).

[2] M. Hermanns, Phys. Rev. Lett.

**104**, 056803 (2010).

# Quantum spin liquids

Spin liquids are exotic states, in which the local moments are
highly correlated, but do not order magnetically due to strong
quantum fluctuations down to zero temperature.
They can occur when different spin interactions cannot be simultaneously satisfied - a situation, which is often referred to as frustration.
One way to frustrate a system is via anisotropic spin
interactions that are incompatible with each other - also called exchange
frustration. The paradigmatic example of exchange frustration is the
Kitaev model on tri-coordinated lattices, where each spin interacts with its
three nearest neighbors via a different spin component. Even though
the spin system is strongly interacting, it can be mapped to a
system of non-interaction Majorana fermions, thus allowing us to
study such strongly correlated quantum states exactly.

Currently, there is a lot of interest in Kitaev models in three
spatial dimensions, mainly fueled by recent progress in the
synthesis of certain iridates. The latter are believed to exhibit
highly anisotropic spin-interactions of the Kitaev type. Depending on
the lattice, the properties of the resulting spin liquid phase and its stability
against various types of perturbations can differ widely.
The hyperoctagon lattice (illustrated in the figure) is of special
interest as it is the first example of an exactly solvable spin-1/2
model in three spatial dimensions that harbors a gapless spin liquid with a
two-dimensional spinon Fermi surface [1]. In contrast,
the Kitaev interaction on the (tri-coordinated) hyperhoneycomb lattice
yields a gapless spin liquid with a Fermi line. The line can be
destabilized by breaking time-reversal symmetry (e.g. by applying a
magnetic field) leading to a Weyl spin liquid [2],
where the gapless modes form Weyl points. The latter are topological
objects that imply various interesting properties of these novel spin
liquids, in particular the presence of protected, gapless surface
modes called Fermi arcs.

**References**

[1] M. Hermanns, S. Trebst, Phys. Rev. B 89, 235102 (2014).

[2] M. Hermanns, K. O'Brien, S. Trebst, Phys. Rev. Lett. 114, 157202 (2015);