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:

• Non-Abelian anyons and topological degeneracy splitting
• Frustrated magnetism
• Exotic order
• Optimized ensembles
• Ultracold atoms in optical lattices
• Open source codes for strongly correlated systems
• Collective excitations of quantum spin liquids

Non-Abelian anyons and topological degeneracy splitting

Topological quantum liquids, such as fractional quantum Hall liquids, rotating Bose-Einstein condensates or p_x + ip_y superconductors, harbor exotic quasiparticle excitations, which due to their unusual exchange statistics are referred to as anyons. A particularly intriguing species are non-Abelian anyons, which have attracted considerable interest in proposals for topological quantum computation. Their defining characteristic manifests itself when considering a set of non-Abelian anyons, which gives rise to a (degenerate) manifold of states even when pinning the anyons to fixed locations in space.

One line of my current research explores how this manifold of states for a set of non-Abelian anyons is split in the presence of interactions between the anyons. We could recently show that as a result of this splitting a single ground state is selected [1]. This new collective state corresponds to a gapped quantum liquid nucleated within the original parent liquid (of which the anyons are excitations) with characteristic (neutral) edge states forming at the spatial interface between the two liquids. These edge states are in precise correspondence with the gapless modes found for one-dimensional arrangements of anyons, such as the "golden chain" [2] and variations thereof [3]. This physics is at play when tuning the magnetic field off the center of the plateau for a non-Abelian quantum Hall state and will have direct experimental signatures, such as modifications to the thermal heat transport.

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

Frustrated magnetism

A familiar feature in many magnetic solids is the competition between different interactions that cannot be simultaneously satisfied. Such a situation, which is often referred to as frustration, typically leads to highly degenerate ground states and a suppression of ordering, which is signified by a large reduction of the ordering temperature with regard to the Curie-Weiss temperature and the emergence of an intermediate "spin liquid" regime. In many cases the nature of the ordered state and the spin liquid regime emerge from otherwise weak residual effects that prevail in splitting the degenerate ground-state manifold.

I have been studying a number of phenomena along these lines in collaboration with Leon Balents' group. For a broad class of spinel materials, which form A-site diamond-lattice antiferromagnets, we have identified an unusual spiral spin liquid regime [1,2]. This regime which exhibits a peculiar type of short-range spin spiral correlations but lacks long-range order, emerges from a massive ground-state degeneracy that develops amongst spirals whose propagation wave vectors reside on a continuous two-dimensional 'spiral surface' in momentum space. Our theoretical predictions agree well with experimental observations for the spinel MnSc₂S₄.

More recently, we have suggested that the unusual two-peak specific heat measurements for NiGa₂S₄, a highly two-dimensional spin S=1 antiferromagnet with a triangular lattice structure, can be explained by proximity to a quantum critical point between a quadru-polar (nematic) ground state and an antiferromagnetic one [3].

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
[3] E.M. Stoudenmire, S. Trebst, and Leon Balents, Phys. Rev. B 79, 214436 (2009).

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.

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 the spin correlations can exhibit singularities along surfaces in momentum space. When restricting these phases to a quasi-1D geometry, e.g. by placing the system onto an N-leg ladder, there should be distinctive signatures of this two-dimensional behavior with a precise pattern of one-dimensional gapless modes characteristic of the parent 2D quantum liquid [1]. 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 phase [1].

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 elementary examples of such topological ground states, in particular so-called quantum loop gases [2-4], which can harbor both Abelian and non-Abelian excitations. 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 [2]. 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) [3].

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

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).