The project concerns the investigation of one dimensional quantum systems and statistical quantum field theories.

The significance of low dimensional quantum systems stems from various aspects. On the one hand, low dimensionality enhances quantum fluctuations, so these systems are often strongly correlated. On the other hand, distinguished members of this class of models are the so-called integrable systems which allow for the non-perturbative or exact description of strongly interacting quantum systems. Apart from their theoretical significance, these systems can be studied experimentally both in condensed matter systems (spin chains, carbon nanotubes etc.) and with trapped ultra-cold atoms. Certain 1+1-dimensional quantum field theories, like the paradigmatic sine-Gordon model, provide the effective low energy description of 1D systems and transport through impurities [1].

The work is focused on the “method of fluctuating surface” [2] which is a nonperturbative tool for calculating a certain class of partition functions that appears in a variety of problems, including the Yuval–Anderson representation of the Kondo model, partition functions of boundary and bulk sine–Gordon models, or the dynamics of dissipative two-level systems. It works by mapping the partition function or correlation functions to the statistics of random surfaces subject to classical noise. The partition function of the boundary sine–Gordon model also appears in the calculation of the full distribution of contrast in the matter wave interference experiments [3]. Completely inhomogeneous or even disordered systems can also be studied.

We will use this approach to understand the properties of the sine–Gordon field theory at finite temperature. Due to the integrability of the model, there are analytic results either in terms of closed form expressions or in the form of non-linear integral equations for the thermal free energy and for one-point functions of vertex operators, giving an excellent opportunity to benchmark the method. However, much less is known about the two-point correlation functions which will be the main focus of the study. While the fluctuating surface method is essentially a numerical technique, it can also be used to obtain analytical results and to make controllable approximations.

As possible continuations of this work, applying the method to dynamic correlation functions, to the nonequilibrium dynamics, and to disordered systems provide exciting future directions of research. 
The work requires analytic calculations as well as numerical simulations and gives an opportunity to learn about quantum field theories and to master several analytic and numerical techniques.

1.     T. Giamarchi, Quantum Physics in One Dimension, Oxford University Press (2003)

2.     A. Imambekov, V. Gritsev, E. Demler, Mapping of Coulomb gases and sine-Gordon models to statistics of random surfaces, Phys. Rev. A 77, 063606 (2008); S. Hofferberth et al., Probing quantum and thermal noise in an interacting many-body system, Nature Physics 4, 489 (2008)

3.     M. Gring et al., Relaxation and Pre-thermalization in an Isolated Quantum System, Science 337, 1318 (2012); T. Langen et al., Experimental observation of a generalized Gibbs ensemble, Science 348, 207 (2015)