Krylov complexity and fidelity susceptibility in two-band Hamiltonians

TL;DR

This paper explores Krylov spread complexity in two-band Hamiltonians, revealing its divergence at topological transitions and its relation to fidelity susceptibility, with applications to the SSH model.

Contribution

It introduces a geometric formulation of spread complexity, links it to fidelity susceptibility, and uncovers a non-unitary duality in the SSH model.

Findings

Derivative of spread complexity diverges at topological phase transition in SSH model.

Spread complexity is bounded by fidelity susceptibility in two-band models.

A non-unitary duality between phases affects spread complexity and fidelity susceptibility.

Abstract

We investigate Krylov spread complexity for the ground state of two-band Hamiltonians, where the reference state is a generic state on the Bloch sphere. The spread complexity is obtained by using a purely geometric formulation in terms of Bloch sphere data without constructing the circuit Hamiltonian. For generic reference states, the derivative of the spread complexity is logarithmically divergent at the topological phase transition in the Su-Schrieffer-Heeger (SSH) model. We demonstrate that the derivative of the spread complexity is bounded by fidelity susceptibility for general two-band models, indicating the sensitivity of the spread complexity to any gap closing (topological or trivial). This is illustrated in the massive Dirac Hamiltonian with a trivial gap closing. Finally, we introduce a non-unitary duality in the SSH model between the topological and trivial phases, which manifests itself in the spread complexity and fidelity susceptibility.