Transport and real-time dynamics in one-dimensional quantum magnets and ultra-cold atomic gases

The goal of this thesis is to study the transport properties and
real-time dynamics of quantum magnets and ultra-cold atomic gases
in one spatial dimension using numerical methods. The focus will be
on the discussion of diffusive versus ballistic dynamics along with
a detailed analysis of characteristic velocities in ballistic
regimes. For the simulation of time-dependent density profiles we
use the adaptive time-dependent density matrix renormalization
group (DMRG). This numerical method allows for the simulation of
time-dependent wave functions close to as well as far from
equilibrium in a controlled manner. The studies of one-dimensional
quantum magnets are partially motivated by the experimental
evidence for a highly anisotropic and for insulators comparably
high thermal conductivity of certain cuprates. We use linear
response theory to study transport coefficients at arbitrary
temperatures by diagonalizing small systems exactly and then
calculating the current-current correlation functions. As first
application we discuss the spin transport in the spin-$1/2$
Heisenberg chain with anisotropic exchange interactions
(XXZ-chain). The second application of exact diagonalization, here
in combination with time-dependent DMRG, is a discussion of the
transverse components of the current-current correlation function.
While usually only a Zeeman field is considered in the theory of
transport coefficients, we here investigate the dynamic induced by
an additional transverse magnetic field. We find that in this
scenario the current-current correlation function exhibits coherent
oscillations. In addition a second non-trivial frequency, different
from the one expected from the usual Larmor precession, emerges and
is studied varying temperature and field. Finally we calculate the
frequency-dependent spin and heat conductivity of dimerized spin
chains in a magnetic field. Motivated by the recent experimental
studies of the phase diagram of C$_5$H$_{12}$N$_2$CuBr$_4$ we take
the dimerized chain as a minimal model that exhibits features of
the low-temperature region of the observed phase diagram. As a main
result, the spin and heat conductivity obtained from linear
response theory are enhanced in the field-induced gapless phase.
The last application in the field of one-dimensional quantum
magnets is the simulation of time-dependent energy-density
wave-packets close to as well as far from equilibrium using the
time-dependent density renormalization group. The main results are
ballistic energy dynamics independently of how far
out-of-equilibrium the initial state is and a detailed
understanding of the average expansion velocity. The applications
in the field of ultra-cold atomic gases focus on the sudden
expansion of an initially trapped gas into an empty optical
lattice. This setup was recently realized in an experiment
performed by U. Schneider {\it et al.} and discussed in the context
of electronic transport in the two-dimensional and the
three-dimensional Fermi-Hubbard model. Here we investigate the
sudden expansion of three different setups: For the expansion of a
spin-balanced cloud of fermions, we identify the ballistic regime,
and therein investigate the average expansion velocity of the
cloud. As a main result the expansion velocity is determined by a
small subset of the initial condition over a wide range of
parameters. For instance, the Mott-insulating phase of the Hubbard
model is characterized by a constant expansion velocity
independently of the strength of the interaction. In the case of
spinless bosons, we study the expansion from initial states that
have a fixed particle number per lattice site and a certain
concentration of defects. We study the expansion velocity as a
function of interaction strength and investigate whether the
time-dependent momentum distribution functions indicate a dynamical
quasi-condensation. The last example is the sudden expansion of a
spin-polarized gas of fermions in the presence of attractive
interactions. This study is motivated by current effort to
experimentally detect the Fulde-Ferrell-Larkin-Ovchinnikov state.
Our results for the time-dependent momentum distribution functions
and the wave-function of the pair condensate suggest that the
signatures of the FFLO state vanish quickly, yet a stationary form
of the momentum distribution also emerges fast. The latter is shown
to be determined by the initial conditions, which might eventually
allow for an indirect detection of the FFLO phase.