# Thesis projects

Here is a list of topics suitable for bachelor and master thesis projects:## Entanglement entropy in the random Heisenberg model

Basically all eigenstates of many-body Hamiltonians show
some degree of entanglement - in other words, product states are
rare exceptions. Now consider the one-dimensional Heisenberg model
and a division of the system in two parts (A and B), as shown
in Fig. a). The entanglement entropy *S _{e}* measures
the degree of entanglement between the subsystems A and B and its

*L*-dependence (

*L*is the number of sites in A) is shown in Fig. b) for the groundstate of the antiferromagnetic case. The question to be studied in this project is how the entanglement properties change in a

*random*Heisenberg model, with the randomness introduced, for example, by setting the nearest-neighbour couplings

*J*as random numbers in some interval

_{n}*J*<

_{min}*J*<

_{n}*J*. Averaging over many sets of {

_{max}*J*} gives a distribution

_{n}*P(S*.

_{e})## Iterative diagonalization of the Heisenberg chain

The Schrödinger equation for a cluster of interacting quantum particles can be solved by diagonalization of the corresponding Hamilton matrix, provided the numerical diagonalization of this matrix does not exceed the limits of computing time and memory. In this exact (or full) diagonalization approach, a single matrix is set up for the whole system. In the iterative diagonalization approach, however, one site of the system is added in each iteration - the enlarged cluster is then diagonalized using the information that has been kept from the previous iteration. Combined with a suitable truncation scheme, the iterative diagonalization allows to treat much larger system sizes. The aim of this project is to set up such an iterative diagonalization for the one-dimensional Heisenberg model. Various truncation schemes can be used here, based on the spectrum of eigenenergies or the entanglement spectrum.

## Engineering of harmonic chains

Consider a (semi-infinite) harmonic chain with the first body of
the chain displaced at time t=0, q_{o}(t=0)>0, and all other
bodies at rest (at their respective equilibrium positions). The
parameters of the chain, i.e. the values of the masses m_{i}
and spring constants k_{i}, determine the precise form of the
displacement q_{o}(t). The aim of this project is to
calculate the chain parameters m_{i} and k_{i} for a
given q_{o}(t), this means to engineer a harmonic chain to
produce a desired q_{o}(t). On a technical level this can be
achieved by a Laplace tranform of q_{o}(t), which results in
a function Q_{o}(s), and a subsequent continued fraction
decomposition of Q_{o}(s). This procedure clearly cannot work
for any given q_{o}(t), so one of the questions to look at is
which constraints can be assigned to q_{o}(t) such that it can
be realized as the displacement of a harmonic chain.

## Zero-point entropy of quantum impurity systems

The thermodynamic entropy of a quantum system in the limit of
temperature to zero is given by S(T to 0) = ln(d_{g}),
with d_{g} the degeneracy of the ground state. This
degeneracy can only have integer values, d_{g}=1,2,3,...
There are, however, quantum impurity systems for which d_{g}
seems to acquire non-integer values: as an example, the two-channel
Kondo model has a zero-point entropy of S_{imp} = 0.5 ln(2),
corresponding to d_{g}=sqrt(2). The idea of this project is
to derive this anomalous value of S_{imp} from the
specific fixed point structure of the two-channel
Kondo model.

## Entanglement in fermionic systems

A model of non-interacting fermions, such as the one-dimensional
tight-binding model, can be easily transformed between a real-space
representation (with the fermionic operators acting on individual
sites) and a representation in *k*-space. This transformation
can be used to express quantum impurity models (here: the
single-impurity Anderson model) in different representations: the
site- and *k*-representation. To investigate the entanglement
properties of a quantum impurity model (e.g.: "How strongly is
subsystem A entangled with subsystem B"), the representation of the
fermionic part has to be specified and the results will in fact
depend on this choice. The focus of this project is the
single-impurity Anderson model on a small cluster such that the
Hamilton matrix of the model can be diagonalized exactly. For this
model, the entanglement entropy should be calculated for different
representations and different divisions in subsystems A and B. The
results will be useful to decide which representation is best suited
to characterize the entanglement structure of quantum impurity
systems. (See also the paper by Chun Yang and Adrian E. Feiguin,
*Unveiling the internal entanglement structure of the Kondo
singlet*, Phys. Rev. B
95, 115106 (2017).)