Topological invariants and the ground state wavefunction of topological insulators

Yuan-Yao He, Han‐Qing Wu, Zi Yang Meng +1 more

Haoxiang Chen, Yubing Qian, Weiluo Ren +2 more

Many-body topological quantum states host exotic quantum phenomena and lie at the forefront of developing next-generation quantum technologies. Recently emerged neural network wavefunction methods have established themselves as a powerful computational framework for accessing these states, enabling the variational machine learning calculation of the system's ground state wavefunction. However, reliable computation of topological invariants remains an open challenge when the whole deterministic energy spectrum is not available. In this work, we introduce a robust approach to determining topological invariant based on simulating the charge pumping process, by monitoring the response of polarization upon flux insertion. By applying this method, we accurately extract the Chern numbers for Abelian fractional Chern insulators. Our approach also enables the first neural-network-wavefunction-based identification of anomalous composite Fermi liquid states. Our work resolves a key bottleneck in applying neural network wavefunctions to correlated topological matter, and the method proposed is also generally applicable to other many-body approaches, thereby opening up new avenues for future research in this field.

Zhong Wang, Shoucheng Zhang

Topological insulators are classified by ``topological invariants'' characterizing their electronic structures. Identifying and computing topological invariants for insulators in which electron-electron interactions are important is, however, difficult. Theorists now present a way to accomplish this task for a wide range of topological insulators.

Shoucheng Zhang

Xiao-Liang Qi

Charles-Edouard Bardyn

Meaningful topological invariants for mixed quantum states are challenging to identify as there is no unique way to define them, and most choices do not directly relate to physical observables. Here, we propose a simple pragmatic approach to construct topological invariants of mixed states while preserving a connection to physical observables, by continuously deforming known topological invariants for pure (ground) states. Our approach relies on expectation values of many-body operators, with no reference to single-particle (e.g., Bloch) wavefunctions. To illustrate it, we examine extensions to mixed states of $U(1)$ geometric (Berry) phases and their corresponding topological invariant (winding or Chern number). We discuss measurement schemes, and provide a detailed construction of invariants for thermal or more general mixed states of quantum systems with (at least) $U(1)$ charge-conservation symmetry, such as quantum Hall insulators.