Overview
The advent of fast and cost-effective computers as well as efficient
algorithms has made computational physics into a powerful third way,
besides experiment and theory, to do research. Disordered and complex
systems, my main area of research, have many applications to other
disciplines, such as biology, computer science, finance, etc. Due to
their complexity, computational physics is the tool of choice to study
these because, in general, analytic approaches are only fruitful for
the simplest problems.
In the following, I provide a short background to my research in
the physics of complex systems, and how I intend to improve our
understanding of complex systems such as spin, electron, and structural
glasses, as well as cold atomic gases and topologically-protected
quantum computing. A key component of my research is developing
innovative algorithms in order to achieve my research goals, as well as
to study realistic model systems that are not accessible analytically.
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Glassy systems
Glassy systems are characterized by disorder and frustration. The most
prominent representatives are spin glasses which exhibit high magnetic
frustration, as well as structural glasses which are characterized
by geometric frustration. Although spin glasses find few direct
applications in experimental systems, they are paradigmatic models that
deliver concepts relevant for a variety of systems such as optimization
problems, disordered magnets and economics applications. However,
despite ongoing research spanning several decades in the area of
glassy systems, there remain many fundamental open questions.
One of my main research foci over the past few years has been the
study of spin glasses where I have studied in detail the nature of
the low-temperature phase [1,8], the existence of a spin-glass state in
a field [2,3,9] (see figure for possible scenarios — for short-range
spin glasses there is no spin glass state in a field), chaos in
spin glasses [4], as well as if spin glasses obey universality
[5]. In these studies, the analysis of a one-dimensional model with
power-law interactions has proven to be very useful in elucidating
the properties of higher-dimensional systems because the tuning of
the power-law exponent corresponds to effectively changing the space
dimension of the system. In addition, I have studied models for
vortex glasses (gauge glass) [6] as well as hysteretic properties of
disordered magnetic systems [7]. More recently I have branched out
to the study of electron as well as structural glasses.
References:
- [1] H. G. Katzgraber,
M. Palassini, A. P. Young, Phys. Rev. B 63, 184422 (2001)
- [2] A. P. Young,
Helmut G. Katzgraber, Phys. Rev. Lett. 93, 207203 (2004)
- [3] Helmut G.
Katzgraber, A. P. Young, Phys. Rev. B 72, 184416 (2005)
- [4] Helmut G. Katzgraber,
Florent Krzakala, Phys. Rev. Lett. 98, 017201 (2007)
- [5] Helmut G. Katzgraber,
Mathias Koerner, A. P. Young, Phys. Rev. B, 73, 224432 (2006)
- [6] Helmut G. Katzgraber,
D. Wuertz, G. Blatter, cond-mat/0612511
- [7] H. G. Katzgraber et al., Phys. Rev. Lett. 89, 257202 (2002)
- [8] T. Joerg and H. G. Katzgraber, Phys. Rev. Lett., submitted
- [9] Thomas Jorg, Helmut G.
Katzgraber, Florent Krzakala, Phys. Rev. Lett., in press
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Topologically-protected quantum computing
Besides traditional implementations of quantum bits for quantum
computers, there have recently been novel approaches using
solid-state devices. In particular, Ioffe et al. have suggested
the use of quantum Josephson junction arrays for the construction of
a topologically-protected qubit (see figure). Based on the work of
Kitaev, this implementation shows good prospects for large decoherence
times and scalability; the latter being a crucial problem in common
qubit implementations. These implementations exploit topological order,
where the different states are characterized by topological properties
of the system instead of broken symmetries. Such states have been
observed in quantum Hall states, two-dimensional correlated Fermions,
as well as Bosonic systems. The main microscopic Hamiltonian with a
fractionalized quantum phase is the Rokhsar-Kivelson quantum hard-core
dimer model on the triangular lattice. In two dimensions it forms a
spin liquid with Z_2 symmetry relevant for quantum computing. Thus
it is of paramount importance to better understand the properties
of this model, study the robustness of the liquid phase towards
perturbations of the Hamiltonian, and search for new models exhibiting
these topologically-ordered phases.
Current research focuses on the feasibility of the implementation of
a topologically-protected qubit [1] as well as the general study of
exotic phases.
References:
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Cold atomic gases
My recent projects in the field of ultracold atomic gases are mainly
driven by a productive collaboration with the
experimental quantum
optics group
of T. Esslinger at ETH Zurich. Superimposing optical lattices to
Bose-Einstein condensates has enabled the realization of complex
quantum phases. Whereas theoretically and numerically exotic
quantum phases are being predicted, experimentally they are far from
being observed due to the complexity of the experimental setups. In
particular, experiments require optical traps that can drastically
change the critical behavior of the cold gas being studied. For
example, while in a system with no trapping potential criticality has
been observed numerically, recent simulations of systems in quadratic
confining potentials have shown that the quantum criticality of the
superfluid—Mott insulator transition observed in uniform systems is
destroyed by the finite gradient of the confining potential. We have
performed large-scale quantum Monte Carlo simulations of the two-
and three-dimensional Bose-Hubbard model [1] where we show that
quartic traps are better suited for experiments than quadratic ones.
The figure shows the local density n for quadratic and quartic trap
shapes. The central Mott plateau is considerably larger for the quartic trap.
In contrast to bosonic systems, the physics of strongly interacting
fermions in 2D and 3D lattices is not yet properly understood. Indeed,
accurate numerical simulations on interacting fermionic systems in
2D and 3D lattice models are NP (nondeterministic polynomial) hard,
and no general numerical or analytical solution exists. Experiments
on ultracold fermionic gases in optical lattices could thus be very
useful in elucidating the properties of the fermionic Hubbard model
and to probe exotic quantum phases. Recently, we have attempted to
understand experimental results by the Esslinger group at ETH
Zurich where ultracold fermionic atoms are ramped across a Feshbach
resonance. Our results explain the molecule formation rates found
and show that current experiments are performed
at temperatures considerably higher than expected: lower temperatures
are required for fermionic systems to be used as quantum simulators [2].
References:
- [1] Olivier Gygi,
Helmut G. Katzgraber, Matthias Troyer, Stefan Wessel, G. George Batrouni,
Phys. Rev. A 73, 063606 (2006)
- [2] Helmut G. Katzgraber,
Aniello Esposito, Matthias Troyer, Phys. Rev. A 74, 043602 (2006)
- [3] S. Morrison, A. Kantian, A.
J. Daley, H.G. Katzgraber, M. Lewenstein, H. P. Büchler, P. Zoller, New J.
Phys., submitted
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Algorithms
Algorithm development plays a key role in computational physics
research. Even with ever-increasing computer power, the study of
complex systems, such as glasses in particular, requires efficient
algorithms.
My recent research has focused on improving the parallel tempering
Monte Carlo method [1] using an optimized ensemble method, simple
heuristic ground-state search algorithms for spin glasses [2,3,5],
and computing energy distributions of disordered systems to high
precision with a novel method using sampling in the disorder with
a guiding function [4]. The figure shows a ground-state energy
distribution for the mean-field Sherrington Kirkpatrick spin-glass
model. While conventional approaches deliver 5 - 6 orders of magnitude,
our algorithms extends this range to 16 - 18 orders of magnitude with
the same numerical effort.
References:
- [1] Helmut G.
Katzgraber, Simon Trebst, David A. Huse, Matthias Troyer, J. Stat. Mech.
P03018 (2006)
- [2] Helmut G.
Katzgraber, Mathias Koerner, Frauke Liers, Michael Juenger, A. K. Hartmann,
Phys. Rev. B 72, 094421 (2005)
- [3] J. J. Moreno, H.
G. Katzgraber, Alexander K. Hartmann, Int. J. of Mod. Phys. C 14(3), 285
(2003)
- [4] Mathias Koerner,
Helmut G. Katzgraber, Alexander K. Hartmann, J. Stat. Mech.
P04005 (2006)
- [5] Martin Pelikan, Helmut G.
Katzgraber, Sigismund Kobe, Technical Report
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Statistical mechanics data initiative
Hosted at www.stat-phys.org
and initiated by myself, the goal of the statistical mechanics data
initiative is to create a data repository along the lines of the
arXiv where research groups can upload and publish their raw data
sets. Furthermore, test instances for algorithm development will
be shared. Stay tuned since the site is under construction!
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last update: 06/2007
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