I am a postdoctoral researcher working in the department for theoretical physics at the Leibniz University of Hannover. Since I started my diploma thesis in 2008 my main research has been centered around the investigation of methods for the solution and construction of 1D integrable quantum systems in condensed matter physics.

General context

Since the early 1930's it has been known that physical quantities of certain (strongly interacting) multi-particle quantum systems can be computed exactly. However, the question remained as to whether there was a common structure underlying these integrable models that distinguished them from other quantum systems. It was found that several such models can most conveniently be recast within the elegant and extremely powerful framework of the Quantum Inverse Scattering Method (QISM) which is intimately related to the celebrated Yang-Baxter algebra and ensures integrability of the respective model by construction.

Non-diagonal boundary conditions for graded quantum systems

In the course of time, several extensions to the original QISM scheme have been developed and refined. A most significant enhancement was provided by the possibility to equip integrable quantum chains with open boundary conditions while keeping their integrability intact. Likewise, the desire for treating fermionic systems within the QISM framework led to a graded generalization of the methods employed in the context of spin chains.

An interesting peculiarity arises when these two extensions get combined, i.e. when open boundary conditions are imposed upon integrable fermionic systems. In contrast to the ungraded case, integrability now requires some of the boundary parameters to be elements of a Grassmann algebra rather than ordinary c-numbers.

I have studied the possibility of incorporating non-diagonal (i.e. particle number changing) boundary conditions into one-dimensional quantum chains of interacting fermions as well as the applicability of Bethe ansatz methods for their exact solution. In the case of free fermions it turned out that a similarity transformation, diagonalizing one of the boundary super matrices on the auxiliary space led to a canonical transformation of the quantum space operators such that fermionic coherent states provided a suitable pseudo vacuum for a graded version of the algebraic Bethe ansatz. Unfortunately, this elegant procedure turned out to be inapt for non-trival cases of interacting models, which is why I shifted my attention to methods beyond Bethe ansatz techniques.

Fusion hierarchies

The QISM employs representations of the Yang-Baxter algebra to generate an infinite set of mutually commuting quantum space operators (transfer matrices) which in particular comprises the Hamiltonian of the associated quantum system. Given an R-matrix as solution to the Yang-Baxter equation, it is possible to systematically construct higher dimensional representations of the Yang-Baxter algebra by successively *fusing* copies of the original R-matrix.

The fusion process can mainly be performed in two distinct ways, by either increasing the dimensionality of the quantum space or of the auxiliary space. The first case is interesting as it can be used to include different particle types into an integrable model (e.g. a spin-S impurity in a spin-1/2 Heisenberg chain). The second case is of interest because it allows to construct a further infinite family of quantum space operators which not only commute among themselves but also with the original transfer matrices. These commuting families, obtained at respective fusion levels constitute a hierarchy of functional relations, which may in some cases be used to compute certain quantities of interest such as the energy spectrum of the model.

I have derived the fusion hierarchy for a particular model of interacting fermions, the *small polaron model*, which provides an effective description of the behavior of an additional electron in a polar crystal. For special values of one of its parameters, it was possible to show that the inherently infinite fusion hierarchy truncates after a finite amount of steps, thus yielding the desired TQ-Relations. Afterwards we managed to obtain the exact same relations *without* this constraint and derived the respective Bethe ansatz equations along with a closed expression for a particular eigenstate at arbitrary system size. Currently, we are studying transport properties of this system. I am also working to link our results to those (very recently) derived for anisotropic Heisenberg spin chains employing modified, so-called *inhomogeneous* TQ-Equations.

Hopf-algebras and the construction of integrable models

There is no simple way to see whether a given quantum model is integrable or not. But on the other hand it is quite straight forward to derive an integrable Hamiltonian from an R-matrix, satisfying the (parameter dependent) Yang-Baxter equation. This immediately raises the question of whether there is a systematic way for the construction of R-matrices.

In the context of Hopf-algebras, universal R-matrices appear as intertwiners between opposite co-products. In general, these quasi-triangular Hopf-algebras are difficult to find, especially those with a non-trivial R-matrix. However, by means of Drinfeld's quantum double construction it is possible to obtain quasi-triangular Hopf-algebras starting from any Hopf-algebra. In particular cases it is known how to Baxterize these universal R-matrices, i.e. how to obtain parameter dependent solutions to the Yang-Baxter equation from them.

Though not a distinct part of my actual research, I always attached great importance to a sound understanding of the deep algebraic structures underlying classical and in particular quantum integrability.

Computer Algebra and Numerics

Today, computers are indispensable research tools and in my opinion theoretical physicists should possess excellent general computer and programming skills, be it to support or verify their algebraic calculations, to produce numerical results or to simply present their work in an adequate and up-to-date manner.

Ever since my school days I have taken great pleasure in the design and optimization of software, ranging from x86 assembly code to modern high-level programming and script languages. During my research I designed very elaborate Mathematica packages that were custom-tailored for my special needs to perform non-commutative algebra on graded vector spaces and, for example, implemented various root finding procedures in Fortran to solve coupled sets of highly non-linear Bethe equations.

Conclusions

The study of integrable quantum systems facilitates an unprecedented insight into the behavior of strongly correlated many-particle systems at a non-perturbative level. It is my belief that the analysis of such systems is a challenging and exciting endeavor that will continue to reward us with a deeper understanding of the underlying mathematical structures.

But not only do integrable quantum models provide a rich playground for theoretical studies, they also have acquired substantial practical importance as quasi one-dimensional materials became experimentally available. Excellent realizations of spin chains have been observed in various compound materials such as CuPzN. Moreover, connections to research fields beyond the realms of condensed matter physics have been established; prominent examples being the AdS/CFT correspondence in high-energy physics or the consideration of anyonic chains in quantum information theory.

Apart from pursuing my previous research topics, I am always happy to learn about and get involved into new fields of research.