Topological Invariants for Quantum Quench Dynamics from Unitary Evolution.

Area of Science:

• Quantum physics
• Condensed matter physics
• Topological phases of matter

Background:

• Exploring topological properties of quantum systems after sudden Hamiltonian changes (quench dynamics) is an active research area.
• Existing theories for defining topological invariants are well-established for static or periodically driven systems, but not for general quenches in band insulators.
• Previous methods, like Hopf mapping, were limited to specific cases (trivial initial states, 2D two-band systems) and difficult to generalize.

Purpose of the Study:

• To develop a general method for defining and characterizing topological invariants in quantum quench dynamics of band insulators.
• To overcome limitations of previous approaches regarding initial states, system dimensions, and band structures.
• To establish a systematic framework for classifying the dynamical topology of quantum systems.

Main Methods:

• Introduction of the 'loop unitary' concept, derived from the unitary time-evolution operator.
• Utilizing homotopy invariants of the loop unitary to characterize dynamical topology.
• Proving the invariant's equivalence to the change in Chern number for 2D two-band systems.
• Analyzing the manifestation of nontrivial dynamical topology through defects and eigenvector behavior.

Main Results:

• The loop unitary's homotopy invariant provides a full characterization of dynamical topology.
• For 2D two-band systems, this invariant precisely equals the quench-induced change in the Chern number, irrespective of the initial state.
• Nontrivial dynamical topology is identified by hedgehog defects in the loop unitary.
• Winding and linking of eigenvectors along dynamical quantum phase transition curves indicate nontrivial topology.

Conclusions:

• The loop unitary offers a powerful and generalizable tool for studying topological properties during quantum quenches.
• This framework successfully classifies dynamical topology, extending beyond previous limitations.
• The findings provide a systematic route for understanding and characterizing complex quantum dynamics and phase transitions.