Fidelity susceptibility and conductivity of the current in one-dimensional lattice models with open or periodic boundary conditions

We study, both numerically and analytically, the finite-size scaling of the fidelity susceptibility chi(J) with respect to the charge or spin current in one-dimensional lattice models and relate it to the low-frequency behavior of the corresponding conductivity. It is shown that in gapless systems with open boundary conditions the leading dependence on the system size L stems from the singular part of the conductivity and is quadratic, with a universal form chi(J) = [ 7(sic) (3)/2 pi(4)] KL2, where K is the Luttinger liquid parameter and. (x) is the Riemann. function. In contrast to that for periodic boundary conditions the leading system size dependence is directly connected to the regular part of the conductivity and is subquadratic, chi(J). L-gamma, where the K-dependent exponent. is equal to 1 in most situations (as a side effect, this relation provides an alternative way to study the low-frequency behavior of the regular part of the conductivity). For open boundary conditions, we also study another current-related quantity, the fidelity susceptibility to the lattice tilt.P, and show that it scales as the quartic power of the system size, chi P = [ 31(sic) (5)/8p6](KL4/pi(2)), where u is the sound velocity. Thus, the ratio L-2 chi J /chi P directly measures the sound velocity in open chains. The behavior of the current fidelity susceptibility in gapped phases is discussed, particularly in the topologically ordered Haldane state.