Interactions require a new theory of topology

Physical systems can do two things – store information and process it. A perfect memory should be impervious to any unintended change. Topological systems, theoretically, are just such a perfect memory. In our work we asked whether our currently understood paradigm of topology can be extended to the much richer class of systems that allow for interactions between the underlying components.

As a global property of the whole system, the topology cannot be changed under a local perturbation. Topological  states are thus immune to any local disturbance. We can see the consequences in the laboratory. In Quantum Anomalous Hall(QAH) systems, for example, dissipationless currents emerge perpendicular to an applied electric field. These dissipationless currents could  serve as building blocks for quantum computation as well as a new generation of electronic devices. In their 1982 groundbreaking work Thouless, Kohmoto, Nightingale, and den Nijs (TKNN), showed that we have access to the topological invariant of a non-interacting system using a quantity N3. Calculating N3  only requires information about isolated particles, captured in the single-particle Green function. N3 controls whether a non-interacting system can have the hallmarks of topology including an unchanging, exact conductance, and robustness to local perturbations that come from disorder, heat, or external influences.

Interactions between electrons, however, are crucial to understanding physics of real materials.  This is particular striking for a class of materials known as a Mott insulator.  While the non-interacting band theory requires a half-filled band to conduct, interactions between electrons force Mott insulators to conduct. Mott insulating physics is believed to be key in understanding cuprate and Moire superconductors - two of the most promising materials for ever higher temperature superconductors. The signature of a Mott insulator is an eigenvalue of zero in its single particle Green function across the entire system. The presence of such ``zeros'' in the Green function indicates that there is no single particle description and a many-body description is essential. 

The TKNN formula, however was derived for non-interacting systems. Prior to our work, it had been conjectured that the TKNN formula could still apply to systems with strong correlations. In our work, we showed that this is not the case. A fundamentally different description is required to capture the topology of interacting systems. We demonstrated that the TKNN formula fails to capture the low energy physics of a strongly correlated system. Moreover, we found the topological response in an exactly solvable Mott insulator and showed that the TKNN formula does not describe the topology. In other words - in the presence of zeros,  N3 no longer controls either dissipationless currents or robustness to local perturbations. This result indicates that multi-particle responses play crucial roles in strongly correlated systems.

The reason behind this failure is consistent with prior results on the breakdown of a key relationship between the particle density and the single particle Green function known as Luttinger’s theorem. Not surprisingly, it is also zeros that disconnect the Luttinger particle count, imaginatively enough called N1, from the physical particle density. For both N1 and N3, this discrepancy arises precisely when the single-particle Green function fails to capture properties of the many-body ground state accurately. The emergence of Green function zeros signifies the importance of irreducibly collective behaviour. Since topology itself is a global property, the proper physics should go beyond single particles. In subsequent work, we hope to better understand just how to think about topology in this regime.