Fidelity, dynamic structure factor, and susceptibility in critical phenomena

Motivated by the growing importance of fidelity in quantum critical phenomena, we establish a general relation between the fidelity and structure factor of the driving term in a Hamiltonian through the concept of fidelity susceptibility. Our discovery, as shown by some examples, facilitates the evaluation of fidelity in terms of susceptibility using well-developed techniques, such as density matrix renormalization group for the ground state, or Monte Carlo simulations for the states in thermal equilibrium.

Figure 1

(Color online) Left: Fidelity between two states separated by different $\delta U=0.04,0.05,0.06$, versus $U∕t$ of the half-filled Hubbard model with $N=L=10$ and periodic boundary conditions. Right: Fidelity susceptibility ${\chi }_F$ as a function of $U∕t$, obtained from the data of the left picture. All lines in the left picture collapse to a single line.

Figure 2

(Color online) Fidelity susceptibility as a function of $U∕t$ for various system sizes. In order to avoid degeneracy in the ground state, the systems of $L=6$ and 10 are diagonalized with periodic boundary conditions, and the systems of $L=8$ and 12 with antiperiodic boundary conditions.

Figure 3

Scaling behavior of the fidelity susceptibility up to 1906 sites for the half-filled Hubbard model at $U=0$. Clearly, $\chi ∕N$ is a constant if $N$ is very large.