Overview

My particular specialization is in the field of computational physics where my research is located at the interface between condensed matter physics and computer science. Many interesting problems in condensed matter theory are connected with strong coupling and/or multiple energy scales and are therefore very hard to handle analytically. To gain deeper insights, and eventually get quantitative understanding, computational approaches are needed. Despite all the enduring progress in computer hardware, the success of brute force use of computer power is very limited - instead sophisticated algorithms are called for which exploit the physics of the problem in much more detail.

A major part of my research is devoted to the development, implementation and application of numerical methods for physical systems. Over the last years I have comprehensively used and further improved various computational methods including high-order strong coupling expansions, quantum Monte Carlo simulations, classical Monte Carlo simulations and exact diagonalization techniques. I implemented all these methods exploiting modern programming techniques such as object-oriented programming in C++, generic algorithms and standard libraries. I strive to facilitate the applicability of numerical methods across disciplines such as condensed matter theory, materials research, chemical engineering and quantum information processing.

The following sections give an overview of the major lines of my research in the past 3 to 5 years:

• Optimized ensembles
• Topological quantum computation
• Ultracold atoms in optical lattices
• Open source codes for strongly correlated systems
• Collective excitations of quantum spin liquids

Optimized ensembles

Competing phases or interactions in complex many-particle systems can result in free energy barriers that strongly suppress thermal equilibration. Prominent examples of slowly equilibrating systems are frustrated magnets, glasses or proteins. To study the equilibrium behavior of such systems I have developed in collaboration with David Huse an adaptive Monte Carlo simulation technique that is capable to explore and overcome the entropic barriers which cause the slow-down [1]. The algorithm systematically optimizes the simulated statistical ensemble in broad-histogram Monte Carlo simulations by maximizing the round-trip rates between low and high entropy states based on measurements of the local diffusivity. In contrast to flat-histogram sampling techniques which recently have become very popular we demonstrated that these optimized histogram methods do not suffer from a critical slowing down [1,2].

For a number of applications we have recently shown that the simulation of an optimized ensemble can speed up equilibration by orders of magnitude in systems which have long relaxation times in conventional simulations such as low-energy configurations of frustrated systems [1], domain walls in ordered phases [1] or dense Lennard-Jones liquids [3].

In a recent project I have been studying the folding of small proteins [4]. It turns out that the state-of-the-art parallel tempering algorithm for these systems can be significantly improved by applying our novel approach to optimize the simulated temperature/ replica set [5]. The adaptive optimization thereby reveals the multiple temperature scales governing the folding process of a single protein and systematically reallocates computational resources to the bottlenecks in the transition.

Our new algorithms bear great potential to improve existing Monte Carlo simulations in a variety of fields which I want to explore in the future spanning from condensed matter systems such as quantum systems in the vicinity of a first-order phase transition to applications in chemical engineering such as the simulation of complex fluids and transition path sampling.

For a short introductory review of the ensemble optimization technique see references [6,7].

References
[1] Simon Trebst, David A. Huse, Matthias Troyer, Phys. Rev. E 70, 046701 (2004).
[2] P. Dayal, S. Trebst, S. Wessel, D. Würtz, M. Troyer, S. Sabhapandit, S. N. Coppersmith, Phys. Rev. Lett. 92, 097201 (2004).
[3] Simon Trebst, Emanuel Gull, Matthias Troyer, J. Chem. Phys. 123, 204501 (2005).
[4] Simon Trebst, Matthias Troyer, Ulrich H. E. Hansmann, J. Chem. Phys. 124, 174903 (2006).
[5] H.G. Katzgraber, S. Trebst, D.A. Huse, M. Troyer, J. Stat. Mech. P03018 (2006).
[6] Simon Trebst et al., "Computer Simulation Studies in Condensed Matter Physics XIX"; Springer Proceedings in Physics, Volume 115 (2007).
[7] Simon Trebst, Talk on Optimized statistical ensembles

Topological quantum computation

In recent years novel exotic states of matter have been proposed that exhibit a topological order. The topological stability of such phases can be exploited in physical implementations of qubits that are protected from decoherence caused by local fluctuations. In order to identify possible physical implementations of such qubits I am currently exploring quantum lattice models proposed by field theoreticians employing powerful quantum Monte Carlo simulation techniques. We aim at finding a minimal model system with a topological phase that could be realized by solid-state devices such as Josephson junction arrays or complex materials. Most of this work is currently under way and done in collaboration with Michael Freedman, Alexei Kitaev and Chetan Nayak at Microsoft's Station Q in Santa Barbara.

References
[1] Simon Trebst, Philipp Werner, Matthias Troyer, Kirill Shtengel, Chetan Nayak, Phys. Rev. Lett. 98, 070602 (2007).
[2] Simon Trebst, Talk on Breakdown of a topological phase: Quantum phase transition(s) in a loop gas with tension
[3] A. Feiguin, S. Trebst, A. Ludwig, M. Troyer, A. Kitaev, Z. Wang, and M. Freedman, Phys. Rev. Lett. 98, 160409 (2007).

Ultracold atoms in optical lattices

In 1982 Richard Feynman formulated the pioneering idea that one quantum system could be simulated by another quantum system. A well-controlled implementation of such a quantum simulator would constitute a first milestone towards quantum computation. The first physical realization of a quantum simulator has recently been demonstrated by experiments that confine ultracold atoms in optical lattices. These experiments allow to widely tune the quantum mechanical interactions between individual atoms thereby allowing an unprecedented control of a quantum mechanical many-body system.

In a current line of research I use computer simulations to guide experiments on how to prepare and manipulate these quantum simulators. While the physics of interacting bosonic atoms is well understood theoretically and validated both by numerical and experimental simulations, the simulation of ultracold fermionic atoms holds promise to shed light on intriguing quantum phenomena occurring in electron systems which still elude a theoretical description such as high-temperature superconductivity. In collaboration with Peter Zoller I have studied how one can adiabatically prepare d-wave resonating valence bond states of fermionic atoms in two-dimensional optical lattices [1]. Ultimately, we believe that such an experimental setup will answer the open question whether the Hubbard model is sufficient to model d-wave superconductivity in cuprate superconductors.

References
[1] Simon Trebst, Ulrich Schollwöck, Matthias Troyer, Peter Zoller, Phys. Rev. Lett. 96, 250402 (2006).
[2] Andreas Läuchli, Guido Schmid, Simon Trebst, Phys. Rev. B 74, 144426 (2006).

Open source codes for strongly correlated systems

Unlike in other physics communities, there have been no high-performance "community codes" available to study strongly correlated quantum systems. I strongly believe that implementations of numerical methods should be publicly available to the physics community as open source codes. As a common framework to integrate and publish codes for numerical simulations of strongly correlated systems we have launched the ALPS project (Algorithms and Libraries for Physics Simulations) which is currently maintained by an international collaboration of researchers [1-3]. Besides contributing implementations of several applications and basic libraries - most notably the worm algorithm for continuous-time quantum Monte Carlo simulations [4] - I co-organized a series of workshops where the ALPS project was founded. On the ALPS webpages you can find a description of my ongoing and past projects.

References
[1] F. Alet et al. (ALPS collaboration), J. Phys. Soc. Jpn. Suppl. 74, 30 (2005).
[2] A. F. Albuquerque et al. (ALPS collaboration), J. Magn. Mag. Mat. 310, 1187 (2007).
[3] Simon Trebst, Talk on The ALPS Project: Open Source Software for Quantum Lattice Models
[4] Simon Trebst, Talk on The worm algorithm
[5] Simon Trebst, Talk on Series expansions for Quantum Lattice Models

Collective excitations of quantum spin liquids

Since my PhD with Hartmut Monien I have been working with strong coupling cluster expansions [1,2]. At the time, we expanded the technique to study multiparticle scattering of dressed excitations such as triplons in spin-1/2 quantum antiferromagnets [3]. These perturbative expansions allow for the first time to quantitatively study the appearance of bound states and continua in strongly correlated systems [4,5]. These collective states exhibit clear experimental signatures in neutron scattering experiments or optical spectroscopy such as Raman scattering and infrared absorption. The theoretical predictions work best for materials that form quantum spin liquids such as spin ladder compounds for which we actually predicted a bound state that was later observed experimentally. From the computational perspective, I provided a major improvement over existing codes by developing a C++ library to efficiently handle, enumerate and topologically classify the underlying clusters.

I collaborated with Anirvan Sengupta and Girsh Blumberg to describe the rich Raman spectrum of the transition metal oxid alpha-NaV₂O₅ which undergoes a phase transition at 34 Kelvin to a charge ordered spin liquid phase. We have used strong coupling series expansions to classify a variety of possible charge orderings on the underlying quarter-filled trellis lattice [6]. Thereby we could identify the actual zig-zag charge ordering and assign all magnetic excitations in the low temperature phase as single triplon or two-triplon bound states.

References
[1] Simon Trebst, Bound states in strongly correlated magnetic and electronic systems PhD thesis, Bonn University (2002)
[2] Simon Trebst, Talk on Series expansions for Quantum Lattice Models
[3] S. Trebst, H. Monien, C.J. Hamer, Z. Weihong, R.R.P. Singh, Phys. Rev. Lett. 85, 4373 (2000).
[4] Z. Weihong, C.J. Hamer, R.R.P. Singh, S. Trebst, H. Monien, Phys. Rev. B 63, 144410 (2001).
[5] Z. Weihong, C.J. Hamer, R.R.P. Singh, S. Trebst, H. Monien, Phys. Rev. B 63, 144411 (2001).
[6] Simon Trebst and Anirvan Sengupta, Phys. Rev. B 62, R14613 (2000).