A central part of condensed matter physics is the classification and characterization of different states of matter. This is well-understood in many isolated, relatively simple systems. While it was believed for a long time that all phase transitions in materials can be understood as changes of the symmetry in the arrangement of atoms in the materials, it has been discovered in recent decades that symmetry arguments alone are not enough to find all phases. Namely, there exist phases that can only be understood using topological arguments.

An example of such a topological phase is the topological insulator, which is insulating in the bulk, but has a quantized electric current on its surfaces. The existence and properties of these surface states are determined by properties of the bulk and are described by integers – topological invariants. Therefore, it is important to find an easy way to study the bulk-physics. One way to do this is to look at the corresponding periodic or infinitely large system that does not have a surface. This system can be described by the so called Bloch Hamiltonian, which is easy to analyze. One expects this to be a good approximation, since the physics far away from the surface should depend very little on what happens on the actual surface. Remarkably, however, the topological invariant provides information also about boundary states of open system. The fact that we can use the Bloch Hamiltonian to calculate topological invariants in a system without boundaries and then use the result to accurately predict the number and character of the states on the boundary is called the bulk-boundary correspondence and is the foundation of the classification of topological materials.

Recently it has become popular to study so called non-Hermitian analogues of the above-mentioned systems, where we no longer require the Hamiltonian to be Hermitian. These are non-equilibrium systems interacting with the environment and they can be engineered in e.g. optical systems. One of the main problems with studying these systems theoretically is that the bulk-boundary correspondence does not hold in the sense that we cannot use the Bloch Hamiltonian to predict what happens in a system with surfaces. This means that the classification and description of these materials is much harder.

non-Hermitian Chern insulator
A schematic showing the band spectrum for a non-Hermitian Chern insulator, which is insulating in the bulk but supports chiral, electronic states on its edges. The gray spectrum corresponds to the eigenvalues of the Hamiltonian with periodic boundary conditions, while the blue corresponds to the eigenvalues of the Hamiltonian on a cylinder. The red corresponds to the chiral, edge state. Strikingly, the spectrum of the system with periodic boundary conditions is vastly different from the spectrum with open boundary conditions. Using the biorthogonal properties of the open system, however, the new theory can accurately predict when it attaches to the bulk spectrum, indicated by the orange and green dashed lines.


Now, in a publication highlighted as an Editors’ Suggestion in Physical Review Letters, scientists from Fysikum, Stockholm University, have solved this mystery by discovering a new more general principle that dictates the correspondence between the bulk and boundary that is also valid in non-Hermitian systems. More precisely, they have shown that while the usual tools to understand topological phases have become inadequate, these phases in the open system can be understood by making use of so-called biorthogonal quantum mechanics, where both left and right eigenvectors of the Hamiltonian enter the description, and by considering the biorthogonal properties of the electronic wave functions in any given microscopic model. This insight is of crucial importance for the basic understanding of topology in classical and quantum systems that are out of equilibrium, and is likely to provide a guiding principle in experimental applications ranging from mechanical systems and optical meta materials with loss and gain to heavy fermion materials with quasiparticles with a finite lifetime.

This work was performed by Flore Kunst, Elisabet Edvardsson (PhD students) and Emil Bergholtz (PI) at Fysikum, Stockholm University, in collaboration with Jan Budich, who is a previous postdoc from the department now holding a professorship in Dresden, Germany.

This work has been published as an Editors’ Suggestion in:
K. Kunst, E. Edvardsson, J. C. Budich, and E. J. Bergholtz, Biorthogonal Bulk-Boundary Correspondence in Non-Hermitian Systems, Phys. Rev. Lett. 121, 026808 (2018).