GP2 - Birmingham Workshop Program: Local Topological Markers for Amorphous and Interacting Quantum Matter

Session Information

Location: Physics West 117 | School of Physics and Astronomy, University of Birmingham
Day: 1. Tuesday 21st April
Time: 10:40-11:00
Chairperson: Nur Unal

Presentation Details

Presentation Type: Oral | 
Title: Local Topological Markers for Amorphous and Interacting Quantum Matter
Abstract: Topological phases of matter have traditionally been studied in crystalline systems using momentum-space methods. Recent experiments have demonstrated topological edge states in amorphous materials, motivating the development of new tools for systems lacking translation invariance. In this work, we introduce real space local topological markers for odd dimensional quantum systems. Local topological markers are real space expressions of topological invariants, which are valuable for characterizing topological phases in materials lacking translation invariance. The Chern marker---the Chern number expressed in terms of the Fourier transformed Chern character---is an easily applicable local marker in even dimensions, but which until now lacked an analogue in odd dimensions.

The Chern character may be expressed in terms of the single-particle density matrix, which is a projector onto the filled bands. I will introduce a one-parameter family $P_{\vartheta}$ of single-particle density matrices interpolating between a trivial state and the state of interest in odd dimensions. By interpreting the parameter $\vartheta$ as an additional dimension, we can write down the Chern character in terms of the family $P_{\vartheta}$ and express the Chern marker in the combined space of $\vartheta$ and the odd dimensional Brillouin zone. Integrating over $\vartheta$ gives a closed form expression for local markers in odd dimensions.These include the chiral marker, a local $\mathbb Z$ marker which in the case of translation invariance is equivalent to the chiral winding number, and a Chern-Simons marker, a local $\mathbb Z_2$ marker characterizing all nonchiral phases in odd dimensions, including topological insulators in three dimensions.

The single-particle density matrix formalism has the advantage that the markers can be used to characterise the topology of a given quantum state, where no knowledge of the parent Hamiltonian is required. This formalism extends to interacting systems through the one-particle density matrix. While the one-particle density matrix of an interacting state is no longer a projector, as long as its spectrum remains gapped, it can be flattened to a topologically equivalent free fermion-like form. These results provide a practical and versatile approach to identifying topological phases in both disordered and interacting quantum systems.

Presenter

Dr Julia Hannukainen
University of Cambridge | United Kingdom

Authors

1. Hannukainen, Julia | T.C.M. Group, Cavendish Laboratory, J.J. Thomson Avenue, Cambridge CB3 0US, United Kingdom
2. Martinez, Miguel | Department of Physics, KTH Royal Institute of Technology, 106 91, Stockholm, Sweden
3. Bardarson, Jens | Department of Physics, KTH Royal Institute of Technology, 106 91, Stockholm, Sweden
4. Klein Kvorning, Thomas | Department of Physics, KTH Royal Institute of Technology, 106 91, Stockholm, Sweden