Fidelity Spectrum in Quantum Phase Transitions

A quantum phase transition (QPT) is incarnated by an abrupt change in the qualitative structure in the ground state wavefunction of a many-body system as the external driving parameter varies. The ground state fidelity, which is a measure of similarity between two states, is expected to show a sudden drop across the transition point and its possibility as a witness to QPTs has raised much interest in recent years. However, the ground state fidelity does not capture much information about the contribution of the low-lying excitations. In this presentation, we introduce the concept of fidelity spectrum, i.e. the matrix elements of $M=|\Psi(\lambda)\rangle\langle\Psi(\lambda+\delta\lambda)|$, where $\lambda$ is the external driving parameter and $\Psi(\lambda)$ is the wavefunction of the system at $\lambda$. By studying the fidelity spectrum, we hope to shed light on the role of excited states played in QPTs. We investigate the fidelity spectrum in two many-body systems, namely the one-dimensional transverse-field Ising model and the two-dimensional Kitaev model defined on a honeycomb lattice. We found that in different phases, as well as at the critical points, the fidelity spectrum shows significant different behaviors.