Research group Hermanns


Topological order

For a long time it was believed that all ordered phases of matter, as well as all (continuous) phase transitions, can be understood in terms of Landau's theory of symmetry-breaking. In this theory, Landau introduced the important notion of order as a broken symmetry. Standard examples are solids, where the regular arrangement of atoms in a lattice breaks continuous translation symmetry, or magnets, where the spins align to give rise to a finite magnetization and spin-rotation symmetry is broken. For each symmetry there is a corresponding local order parameter, which has a finite expectation value in the ordered phase and vanishes otherwise. After the discovery of the quantum Hall effect it was soon realized that Landau's theory of symmetry-breaking cannot be complete, as different quantum Hall liquids obey the same symmetries and cannot be distinguished by a local order parameter. Instead, one needs to resort to non-local measurements, such as measuring the Hall conductance, to tell them apart.

Quantum Hall liquids are the paradigmatic examples of so-called 'topological states'. Another example are the recently discovered topological insulators and topological superconductors. Common for such states is the existence of robust properties that do not depend on microscopic details and are insensitive to local perturbations. Prominent examples are the presence of protected gapless edge states for topological insulators and the remarkably precise quantization of the Hall conductance for quantum Hall liquids. The robustness of these properties can be traced back to a topological invariant, such as the Chern number for quantum Hall liquids, which is quantized to integer values and can only be changed by closing the bulk gap. In the presence of strong correlations the system can exhibit even more fascinating properties, for instance particle excitations with fractional electric charge. States with such properties are commonly referred to as `topologically ordered', alluding to the fact that the emergent low-energy theory of these systems is a topological field theory.


Entanglement spectrum

Partition of the system into two spatial regions A and B. Determining topological order in strongly correlated systems is a hard task, as it cannot be characterized by any kind of local observable or correlation function. One promising method is to measure the long-range entanglement via the entanglement spectrum by dividing the system into two parts A and B and tracing out the degrees of freedom in part B, which yields the reduced density matrix of A, ρA. The entanglement spectrum is the negative logarithm of the eigenvalues of ρA and contains important information about the properties of the state. For instance, a spatial partition of the system can give valuable information about the edge physics while tracing out particles yields information about quasihole excitations. Entanglement spectra are, therefore, widely used as a diagnostic tool to obtain information about the topological properties of the system.

For non-interacting systems, the relationship between properties of the ground state and its entanglement spectrum are well understood. However, for strongly correlated systems most of the conjectured properties are based on (numerical) observations. During my postdoc in Bernevig's group we established some of the few exact results known for such strongly correlated systems by relating the universal properties of two distinct types of entanglement spectra to each other [1] and thus proving a bulk-boundary correspondence in the entanglement spectra. We also showed that non-universal properties, such as finite size effects, can encode valuable information about the ground state [2].

References
[1] A. Chandran, M. Hermanns, N. Regnault, B.A. Bernevig, Phys. Rev. B 84, 205136 (2011).
[2] M. Hermanns, A. Chandran, N. Regnault, B.A. Bernevig, Phys. Rev. B 84, 121309(R) (2011).


Quantum Hall hierarchy

Density profile of two localized non-Abelian
quasielectrons in the Moore-Read Pfaffian state. Perhaps the most intriguing property of quantum Hall states is that the emergent quasiparticle excitations can be anyons. By exchanging two identical anyons, the wave function acquires a phase factor, which is neither 0 (bosons) nor π (fermions), but can take an arbitrary value in between. Some quantum Hall states - the most prominent example being the Moore-Read state - harbor even more exotic quasiparticles, namely non-Abelian anyons. In this case, a system with several localized quasiparticles is not described by a single state, but rather a manifold of degenerate states and the result of successive braiding operations of these quasiparticles depends on the order of the braiding.

A useful framework to describe such particles and their properties is via conformal field theory. This gives a natural operator description for electrons and quasiholes, but not quasielectrons. The latter is more complicated due to the Pauli principle that does not allow to arbitrarily accumulate excess charge. A proper description of the quasielectrons in terms of a conformal field theory operator was first found in [1]. An important feature of this operator description is that it is applicable for any quantum Hall state, even the non-Abelian ones. Thus, it allows to generalize the Haldane-Halperin hierarchy to non-Abelian states [2]. A systematic analysis of these hierarchical states and their properties is currently an active research topic in this group.

References
[1] T.H. Hansson, M. Hermanns, N. Regnault, S. Viefers, Phys. Rev. Lett. 102, 166805 (2009).
[2] M. Hermanns, Phys. Rev. Lett. 104, 056803 (2010).

Quantum spin liquids

3D print of the hyperoctagon lattice. Spin liquids are exotic states, in which the local moments are highly correlated, but do not order magnetically due to strong quantum fluctuations down to zero temperature. They can occur when different spin interactions cannot be simultaneously satisfied - a situation, which is often referred to as frustration. One way to frustrate a system is via anisotropic spin interactions that are incompatible with each other - also called exchange frustration. The paradigmatic example of exchange frustration is the Kitaev model on tri-coordinated lattices, where each spin interacts with its three nearest neighbors via a different spin component. Even though the spin system is strongly interacting, it can be mapped to a system of non-interaction Majorana fermions, thus allowing us to study such strongly correlated quantum states exactly.

Currently, there is a lot of interest in Kitaev models in three spatial dimensions, mainly fueled by recent progress in the synthesis of certain iridates. The latter are believed to exhibit highly anisotropic spin-interactions of the Kitaev type. Depending on the lattice, the properties of the resulting spin liquid phase and its stability against various types of perturbations can differ widely. The hyperoctagon lattice (illustrated in the figure) is of special interest as it is the first example of an exactly solvable spin-1/2 model in three spatial dimensions that harbors a gapless spin liquid with a two-dimensional spinon Fermi surface [1]. In contrast, the Kitaev interaction on the (tri-coordinated) hyperhoneycomb lattice yields a gapless spin liquid with a Fermi line. The line can be destabilized by breaking time-reversal symmetry (e.g. by applying a magnetic field) leading to a Weyl spin liquid [2], where the gapless modes form Weyl points. The latter are topological objects that imply various interesting properties of these novel spin liquids, in particular the presence of protected, gapless surface modes called Fermi arcs.

References
[1] M. Hermanns, S. Trebst, Phys. Rev. B 89, 235102 (2014).
[2] M. Hermanns, K. O'Brien, S. Trebst, Phys. Rev. Lett. 114, 157202 (2015);