Research

We are interested in a variety of problems in condensed matter theory. Our main focus lies on the study of phenomena which arise due to quantum mechanical effects in systems of correlated electrons. Areas of research include the study of topological phases of matter, fractional charges on geometrically frustrated lattices, and the applications of quantum information concepts to quantum many-body systems.

Below you can find short summaries of some of our recent research projects.

Topological phases in one dimension: An Entanglement Point of View

A topological phase is a phase of matter which cannot be characterized by a local order parameter, and thus falls beyond the Landau paradigm of condensed matter physics.

An example of a symmetry protected topological phase is the Haldane phase  of S = 1 chains. Here we found that the entire phase is characterized by a double degeneracy of the entanglement spectrum (see figure). The degeneracy is protected by a set of symmetries (either the dihedral group of π-rotations about two orthogonal axes, time-reversal symmetry, or bond centered inversion symmetry), and cannot be lifted unless either a phase boundary to another, “topologically trivial”, phase is crossed, or the symmetry is broken. More generally, these results offer a scheme to classify gapped phases of one dimensional systems.

In a second application, we focus on one-dimensional fermionic systems. The effect of interactions on topological insulators and superconductors remains, to a large extent, an open problem. Here, we developed a framework for classifying phases of one-dimensional interacting fermions, focusing on spinless fermions with time-reversal symmetry and particle number parity conservation. In agreement with an example presented by Fidkowski et al., we find that in the presence of interactions there are only eight distinct phases, which obey a Z8 group structure. This is in contrast to the Z classification in the non-interacting case. Each of these eight phases is characterized by a unique set of bulk invariants, related to the transformation laws of its entanglement (Schmidt) eigenstates under symmetry operations, and has a characteristic degeneracy of its entanglement levels.

Further reading:
F. Pollmann, A.M. Turner, E. Berg, and M. Oshikawa, Phys. Rev. B 81, 064439 (2010).
A.M. Turner, F. Pollmann, E. Berg, arXiv:1008.4346 (2010).

Charge degrees of freedom on frustrated lattices

After the discovery of the FQHE, the question was left open whether fractionally charged excitations could exist in three-dimensions. In 2002, it was suggested by Fulde, Penc and Shannon that excitations with charge e/2 could exist in a model of strongly correlated fermions on certain frustrated lattices, e.g, the three-dimensional pyrochlore lattice and its two-dimensional projection, the checkerboard lattice (see figure).

We used exact diagonalization methods to study the two-dimensional version of the pyrochlore lattice (i.e., the checkerboard lattice) and showed that fractional charges are linearly confined. The confinement results from a reduction of vacuum fluctuations and a polarization of the vacuum in the vicinity of a connecting string. The spectral functions show broad low-energy excitations which are due to the dynamics of fractionally charged excitations. Signatures for quasiparticles with large spatial extent are found (bound pairs of two fractionally charged particles). An effective low-energy Hamiltonian can be mapped to a U(1) lattice gauge theory which relates the considered low-energy model to the compact quantum electrodynamics in 2+1 dimensions.

For an effective low-energy model for the (three-dimensional) pyrochlore lattice, we showed that a U(1)-liquid phase with fractionalized excitations can be stabilized for a certain range of parameters  using large scale quantum Monte Carlo methods. The U(1) phase exhibits transverse excitations with a linear dispersion (“photons”) and the mutual interaction between electric charges falls of as in Coulomb’s law.

Further reading:
F. Pollmann, Joseph J. Betouras, Kirill Shtengel, Peter Fulde, arXiv:1101.0335 (2011).
O. Sikora, F. Pollmann, N. Shannon, K. Penc, and P. Fulde, Phys. Rev. Lett. 103, 247001 (2009).
F. Pollmann, E. Runge, and P. Fulde, Phys. Rev. B 73, 125121 (2006).
F. Pollmann, J. J. Betouras, K. Shtengel, and 
P. Fulde, Phys. Rev. Lett. 97, 170407 (2006).

Bound states and E8 symmetry effects in perturbed quantum Ising chains

In a recent experiment on CoNb2O6, Coldea et al. found for the first time experimental evidence of the exceptional Lie algebra E8. The emergence of this symmetry was theoretically predicted long ago for the transverse quantum Ising chain in the presence of a weak longitudinal field.

We consider an accurate microscopic model of CoNb2O6 incorporating additional couplings and calculate numerically the dynamical structure factor (see figure) using a recently developed Time evolving block decimation (TEBD) method. The TEBD method provides an efficient method to perform a time evolution of quantum states in one-dimensional systems. It can be seen as a descendant of the density matrix renormalization group method and is based on a matrix product state (MPS) representation of the wave- functions. Algorithms of this type are efficient because they exploit the fact that the ground-state wave functions are only slightly entangled, especially away from criticality. As the entanglement grows linearly as a function of time, the simulations of long time evolutions is numerically very difficult. The excitation spectra show bound states characteristic of the weakly broken E8 symmetry. We compare the observed bound state signatures in this model to those found in the transverse Ising chain in a longitudinal field and to experimental data.

Further reading:
J.A. Kjäll, F. Pollmann, J.E. Moore, arXiv:1008.3534 (2010).