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:

• Interacting anyons
• Exotic order
• Frustrated magnetism
• Optimized ensembles
• Ultracold atoms in optical lattices
• Open source codes for strongly correlated systems
• Collective excitations of quantum spin liquids

Interacting anyons

Two-dimensional topological quantum liquids harbor exotic quasiparticle excitations which due to their unusual exchange statistics are referred to as anyons. Interchanging two anyons may result in not only a fractional exchange phase, but may also give rise to a unitary rotation of the original wave function in a degenerate ground-state manifold. This latter case of non-Abelian statistics is being studied in a variety of contexts such as unconventional p_x + ip_y superconductors, fractional quantum Hall states, and proposals for topological quantum computation.

One line of my current research explores the rich physics that arises in systems of interacting non-Abelian anyons. Probably the simplest scenario one can study are one-dimensional systems such as the "golden chain" of interacting Fibonacci anyons [1], which can be solved exactly, and further modifications with competing interactions [2], which exhibit a wide variety of collective states. It turns out that these systems of interacting su(2)_k degrees of freedom are quite different from their conventional SU(2) counterparts with a topological symmetry stabilizing their gapless ground states [1-3].

References
[1] A. Feiguin, S. Trebst, A. Ludwig, M. Troyer, A. Kitaev, Z. Wang, and M. Freedman, Phys. Rev. Lett. 98, 160409 (2007).
[2] S. Trebst, E. Ardonne, A. Feiguin, D. A. Huse, A. W. W. Ludwig, M. Troyer, Phys. Rev. Lett. 101, 050401 (2008).
[3] C. Gils, E. Ardonne, S. Trebst, A. W. W. Ludwig, M. Troyer, Z. Wang, Phys. Rev. Lett. 103, 070401 (2009).
[4] Simon Trebst, Talk on Interacting Anyons in Topological Quantum Liquids: Things Golden

Exotic order

Quantum spin liquids in two spatial dimensions can support exotic quantum states of matter including gapped spin liquids with topological order and stable, gapless states with no topological structure often called "algebraic" or "critical" spin liquids.

The stability of the ground-state degeneracy in gapped, topological quantum spin liquids is at the heart of proposals to implement topological qubits that are protected from decoherence caused by local fluctuations. Recently, I have been studying two-dimensional quantum loop gases which are elementary examples of such topological ground states with Abelian or non-Abelian anyonic excitations [1-3]. Abelian loop gases appear as ground states of local, gapped Hamiltonians such as the toric code. While the origin of the ground-state degeneracy in a topological quantum liquid might be subtle, these ground states (and their degeneracy) are rather stable and can only be destroyed by local perturbations of the order of the microscopic exchange interactions — making them as robust as an ordinary antiferromagnet [1]. Stabilizing a gapped, non-Abelian loop gas turns out to be a more challenging task, and we could show that this will, in general, require non-local Hamiltonians (or the realization of non-trivial inner products) [2].

Gapless spin liquids generically exhibit spin correlations that decay as a power law in space and which can oscillate at particular wave vectors. One intriguing possibility is that for 2D phases the spin correlations can exhibit singularities along surfaces in momentum space. When restricting these phases to a quasi-one-dimensional geometry, e.g. by placing the system onto an N-leg ladder, there should be distinctive signatures of this two-dimensional behavior [4]. Characteristic of each parent 2D quantum liquid would be a precise pattern of one-dimensional gapless modes on the N-leg ladder. As a first step in this direction we have recently explored itinerant-boson models with a frustrating ring-exchange interaction on the two-leg ladder and found compelling evidence for the existence of an unusual strong-coupling phase, which can be understood as a descendant of a two-dimensional d-wave-correlated Bose liquid (DBL) phase [4].

References
[1] Simon Trebst, Philipp Werner, Matthias Troyer, Kirill Shtengel, Chetan Nayak, Phys. Rev. Lett. 98, 070602 (2007).
[2] Matthias Troyer, Simon Trebst, Kirill Shtengel, Chetan Nayak, Phys. Rev. Lett. 101, 230401 (2008).
[3] Simon Trebst, Talk on Breakdown of a topological phase: Quantum phase transition(s) in a loop gas with tension
[4] D. N. Sheng, Olexei I. Motrunich, Simon Trebst, Emanuel Gull, Matthew P.A. Fisher, Phys. Rev. B 78, 054520 (2008).

Frustrated magnetism

The competition between different interactions that cannot be simultaneously satisfied is often referred to as frustration — a familiar feature in many magnetic solids. Strong frustration leads to highly degenerate ground states and a large suppression of ordering.

In collaboration with Leon Balents' group I have investigated the physics of A-site diamond-lattice antiferromagnets appropriate for numerous spinel materials [1,2]. Our model exhibits, for sufficiently strong frustration, a massive ground-state degeneracy that develops amongst spirals whose propagation wavevectors reside on a continuous two-dimensional 'spiral surface' in momentum space. This gives rise to a broad fluctuating spin-liquid regime at higher temperatures, the so-called spiral spin-liquid which takes its name from magnetic correlations that reveal the underlying spiral surface despite the lack of any long-range order. Our theoretical predictions seem to agree well with experimental observations for the spinel MnSc₂S₄.

References
[1] Doron Bergman, Jason Alicea, Emanuel Gull, Simon Trebst, Leon Balents, Nature Phys. 3, 487 (2007) and cond-mat/0612001
[2] Simon Trebst, Talk on Order by disorder and spiral spin liquids in frustrated diamond lattice antiferromagnets

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], dense Lennard-Jones liquids [3] or quantum systems [4].

In an interdisciplinary project I have been studying the folding of small proteins [5]. 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 [6]. 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 have been met with some enthusiasm by the broader numerical community and are now employed in a variety of fields beyond condensed matter physics, including biological physics, chemical engineering, high-energy physics and probabilistic optimization.

For a short introductory review of the ensemble optimization techniques see references [7,8].

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] S. Wessel, N. Stoop, E. Gull, S. Trebst, M. Troyer, J. Stat. Mech. P12005 (2007).
[5] Simon Trebst, Matthias Troyer, Ulrich H. E. Hansmann, J. Chem. Phys. 124, 174903 (2006).
[6] H.G. Katzgraber, S. Trebst, D.A. Huse, M. Troyer, J. Stat. Mech. P03018 (2006).
[7] Simon Trebst et al., "Computer Simulation Studies in Condensed Matter Physics XIX"; Springer Proceedings in Physics, Volume 115 (2007).
[8] Simon Trebst, Talk on Optimized statistical ensembles

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).