The anomalous low temperature properties of glasses are commonly believed to originate from localized quantum-mechanical two-level tunneling systems, which couple to phonons and other elementary excitations, as well as to external fields. Although a broad range of experimental observations can be explained in terms of this idea, the microscopic nature of these two-level systems is generally unknown. Therefore, another class of substances is often considered, namely mixed crystals. Here, the microscopic nature of the tunneling units is clear. A substitutional defect (e.g. a Li defect in a KCl host crystal) is located in one of several off-center positions, and the defect can move from one potential well to the other by quantum tunneling. Mixed crystals thus allow an immediate investigation of the physics of tunneling units.
Our research activities deal with different aspects of tunneling systems, in particular their microscopic characterisation, their interactions, and their coupling to heat baths or external fields. Our main emphasis has been on the study of microscopic models of amorphous materials, in which tunneling systems with a broad range of barrier heights and asymmetries are generated through interactions as a collective effect.
Cooperations exist with the experimental group of C. Enss, A. Fleischmann and S. Hunklinger in Heidelberg, one of the leading experimental teams in this field, and with the theory groups of Annette Zippelius in Göttingen and Alois Würger at Bordeaux.
We present a quantum statistical analysis of a microscopic mean-field model of structural glasses at low temperatures. The model can be thought of as arising from a random Born von Karman expansion of the full interaction potential. The problem is reduced to a single-site theory formulated in terms of an imaginary-time path integral using replicas to deal with the disorder. We study the physical properties of the system in thermodynamic equilibrium and develop both perturbative and non-perturbative methods to solve the model. The perturbation theory is formulated as a loop expansion in terms of two-particle irreducible diagrams, and is carried to three-loop order in the effective action. The non-perturbative description is investigated in two ways, (i) using a static approximation, and (ii) via Quantum Monte Carlo simulations. Results for the Matsubara correlations at two-loop order perturbation theory are in good agreement with those of the Quantum Monte Carlo simulations. Characteristic low-temperature anomalies of the specific heat are reproduced, both in the non-perturbative static approximation, and from a three-loop perturbative evaluation of the free energy. In the latter case the result so far relies on using Matsubara correlations at two-loop order in the three-loop expressions for the free energy, as self-consistent Matsubara correlations at three-loop order are still unavailable. We propose to justify this by the good agreement of two-loop Matsubara correlations with those obtained non-perturbatively via Quantum Monte Carlo simulations.
We introduce models of heterogeneous systems with finite connectivity defined on random graphs to capture finite-coordination effects on the low-temperature behavior of finite dimensional systems. Our models use a description in terms of small deviations of particle coordinates from a set of reference positions, particularly appropriate for the description of low-temperature phenomena. A Born-von-Karman type expansion with random coefficients is used to model effects of frozen heterogeneities. The key quantity appearing in the theoretical description is a full distribution of effective single-site potentials which needs to be determined self-consistently. If microscopic interactions are harmonic, the effective single-site potentials turn out to be harmonic as well, and the distribution of these single-site potentials is equivalent to a distribution of localization lengths used earlier in the description of chemical gels. For structural glasses characterized by frustration and anharmonicities in the microscopic interactions, the distribution of single-site potentials involves anharmonicities of all orders, and both single-well and double well potentials are observed, the latter with a broad spectrum of barrier heights. The appearance of glassy phases at low temperatures is marked by the appearance of asymmetries in the distribution of single-site potentials, as previously observed for fully connected systems. Double-well potentials with a broad spectrum of barrier heights and asymmetries would give rise to the well known universal glassy low temperature anomalies when quantum effects are taken into account.
We propose a microscopic translationally invariant glass model
which exhibits two level tunneling systems, and shows the salient low-temperature
anomalies of glassy systems. Results so far obtained are in good accord with
experiment. Qualitative universality is due to the collective origin of the
glassy potential energy landscape. However, we obtain a simple explanation
also for the mysterious so-called quantitative universality that manifests
itself e.g. in the in the unusually weak dependence of values for the internal
friction plateau on substance or system parameters.
We investigate effects of interactions between substitutional
defects on the properties of defect crystals at low temperatures, where defect
motion is governed by quantum effects. Both, thermal and dynamical properties
are considered. The influence of interactions on defect motion is described
via a collective effect. Our treatment is semiclassical in the sense that
we analyze collective effects in a classical setting, and analyze the influence
on quantized defect motion only thereafter. Our theory describes a crossover
to glassy behavior at sufficiently high defect concentration. Our approach
is meant to be general. For the sake of definiteness, we evaluate most of
our results with parameters appropriate for Li-doped KCl crystals.
We report on a refined version of our spin-glass type approach to the
low-temperature physics of structural glasses. Its key idea is based on
a Born von Karman expansion of the interaction potential about a set of reference
positions in which glassy aspects are modeled by taking the harmonic contribution
within this expansion to be random. Within the present refined version the
expansion at the harmonic level is reorganized so as to respect the principle
of global translational invariance. By implementing this principle, we have
for the first time a mechanism that fixes the distribution of the parameters
characterizing the local potential energy configurations responsible for glassy
low-temperature anomalies solely in terms of assumptions about interactions
at a microscopic level.
We propose a new treatment of the dynamics of a periodically time-dependent
Liouvillian by mapping it onto a time-independent problem and give a systematic
expansion for its effective Liouvillian. In the case of a two-level system,
the lowest order contribution is equivalent to the well-known rotating wave
approximation. We extend the formalism to a pair of coupled two-level systems.
For this pair, we find two Rabi frequencies and we can give parameter regimes
where the leading order of the expansion is suppressed and higher orders become
important. These results might help to investigate the interaction of tunneling
systems in mixed crystals by providing a tool for the analysis of echo experiments.
We discuss a spin-glass type approach to the physics of structural glasses,
which leads to a class of models that exhibit both glassy low-temperature
phases and double- and single-well configurations in their potential energy
landscape. The low-temperature anomalies characteristic of amorphous systems
are reproduced, and within our model the universality issue can be illuminated.
We consider the interaction between localized excitations and phonons, and
we present a general expression for the dynamic susceptibility, from which
dynamic properties such as the internal friction can be calculated.
We review a model-based rather than phenomenological approach to low-temperature
anomalies in glasses. Specifically, we present a solvable model inspired
by spin-glass theory that exhibits both, a glassy low-temperature phase,
and a collection of double- and single-well configurations in its potential
energy landscape. The distribution of parameters characterizing the local
potential energy configurations can be computed , and is found to
differ from those assumed in the standard tunneling model and its variants.
Still, low temperature anomalies characteristic of amorphous materials are
reproduced. More importantly perhaps, we obtain a clue to the universality
issue. That is, we are able to distinguish between properties which can be
expected to be universal and those which cannot. Our theory also predicts
the existence, under suitable circumstances of amorphous phases without
low-energy tunneling excitations.
Substitutional defects in a cubic symmetry (such as a lithium defect
in a KCl host crystal) can be modeled appropriately by an eight-state-system.
Usually this tunneling degree of freedom is approximated by a two-level-system.
We investigate the observable differences between the two models in three
contexts. First we show that the two models predict different relations
between the temperature dependence of specific heat and static susceptibility.
Second we demonstrate that in the presence of external forces (pressure and
electric field) the eight-state-system shows features that cannot be understood
within the framework of the two-level-approximation. In this context we propose
an experiment for measuring the parameter for tunneling along the face diagonal.
Finally we discuss the differences between the models appearing for strongly
coupled pairs. Geometric selection rules and particular forms of asymmetry
lead to clear differences between the two models.
A random matrix approach to glassy physics is introduced. It leads to
a class of models which exhibit both, glassy low-temperature phases, and
double- and single-well configurations in their potential energy. The distribution
of parameters characterizing the local potential energy configurations can
be computed, and differ from those assumed in the standard tunneling
model and its variants. Still, the low-temperature anomalies characteristic
of amorphous systems are reproduced, and we are able to distinguish properties
which can be expected to be universal from those which cannot.
An analytically tractable model is introduced which exhibits both, a
glass-like freezing transition, and a collection of double-well configurations
in its zero-temperature potential energy landscape. The latter are generally
believed to be responsible for the anomalous low-temperature properties
of glass-like and amorphous systems via a tunneling mechanism that allows
particles to move back and forth between adjacent potential energy minima.
Using mean-field and replica methods, we are able to compute the distribution
of asymmetries and barrier-heights of the double-well configurations analytically,
and thereby check various assumptions of the standard tunneling model. We
find, in particular, strong correlations between asymmetries and barrier-heights
as well as a collection of single-well configurations in the potential energy
landscape of the glass-forming system - in contrast to the assumptions of
the standard model. Nevertheless, the specific heat scales linearly with temperature
over a wide range of low temperatures.
The book is a review of ten years of progress in our understanding of
interacting tunneling systems. The dielectric and elastic response of [111}
off-center impurities in alkali halides is investigated in detail and the
coherent motion of an impurity pair is discussed in view of recently observed
rotary echos. The low temperature properties of defect crystals with an
impurity concentration higher than 100ppm are reviewed and the dipolar interaction
is shown to drive the cross-over to relaxational dynamics. This comprehensive
presentation of the theory is illustrated by experimental data for lithium,
cyanide and hydroxyl impurities in various host crystals.
Paraelectric impurities contribute significantly to the low-temperature
properties of alkali halide crystals. Even at very low density the dipolar
interaction of adjacent defect ions may lead to deviations from the behaviour
expected for isolated impurities; in a certain range of concentration it
is sufficient to consider pairs of coupled defects. Applying a projection
operator method, previous work on this pair model is extended to the case
of finite asymmetry and weak coupling to acoustic phonons. After performing
the ensemble average the specific heat and the zero-frequency susceptibility
are calculated and compared with experimental data on KCl:Li and KCl:CN.
The isotope effect on Rabi frequency and relaxation rate is discussed.
Relaxation due to incoherent tunnelling is shown to be responsible for
the anomalous temperature dependence of the homogeneous linewidth of optical
transitions in glasses. The distribution function of the tunnelling model
for the parameters of the tunnelling systems (TS) is used , and dipole-dipole
coupling between the chromophore and the TS is assumed. Within this, a bending
from a linear to a quadratic temperature dependence of the homogeneous linewidth
at $T\approx 5K$ is derived. Measurements on SiO$_2$, GeO$_2$ and BeF$_2$
doped with Eu$^{3+}$ and $Pr$$^{3+}$ are quantitatively accounted for by our
theory. This interpretation is supported by acoustic measurements on oxide
glasses.
An exact solution for the dynamics of a coupled pair of symmetric two-level-systems
is given by calculating the resolvent of the Liouvillian and the statistical
operator of the problem. At low temperatures interference effects between
the two systems turn out to be of major importance. Depending on the value
and the sign of the interaction parameter the susceptibility of the pair
increases or decreases compared to the situation of isolated systems. For
high temperatures the interference contributions disappear.
A mode-coupling theory (MCT) is presented for the spin-boson model with
a spectral density which accounts for a heat bath made up of lattice vibrations
of a dielectric solid (superohmic dissipation). A usual decoupling approximation
provides a set of non-linear integral equations which are solved both numerically
by iteration on a computer and analytically by means of a frequency dependent
ansatz for the memory functions. There is a transition to incoherent motion
at a temperature $T^*$ where a bare two-level energy is equal to the damping
rate, in contradiction to results obtained previously from a path integral
formulation. The discrepancy arises since in the MCT the relevant self-energy
function does not exhibit a $1/z$-pole at $z=0$. For tunnelling systems
in dielectrics this yields a new relaxation mechanism due to incoherent tunnelling:
the present results might require to modify some of the basic assumptions
of the standard tunnelling model for dielectric glasses.
A new relaxation mechanism is shown to arise from 1< overdamped two-level
systems above a critical temperature $T^*\approx 5K$, thus yielding an explanation
for experimental observations in dielectric glasses in the temperature range
below the relaxation peak. Using the distribution function of the tunnelling
model for the parameters of the two-level systems, both the linear decrease
of the sound velocity and the linear increase of the absorption up to the
relaxation maximum, are quantitatively accounted for by our theory.
A Mori approach to the dynamics of coupled tunneling systems in defect crystals is presented. Transverse and longitudinal correlation functions of two-level systems are given as continued fractions; the memory kernels are evaluated in a usual decoupling approximation. Due to the random configuration of the defects on the host lattice both two-level splitting and relaxation rates show a broad distribution; the corresponding distribution function is derived for the case of a dipolar interaction of the defects. The theory covers both weak and strong coupling; the average interaction energy turns out to be the essential parameter. Dependence on frequency, temperature and concentration of the dynamical susceptibility is discussed. When passing from weak to strong coupling, the zero-temperature susceptibility shows a crossover from a constant value to a decrease with the third power of inverse defect concentration; there is quite a strong relaxational peak in the susceptibility. The theory accounts for several features observed in a recent low frequency experiment on KCL:LI.