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

Abstract

We study, both numerically and analytically, the finite-size scaling of the fidelity susceptibility ${\ensuremath{\chi}}_{J}$ 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 ${\ensuremath{\chi}}_{J}=[7\ensuremath{\zeta}(3)/2{\ensuremath{\pi}}^{4}]K{L}^{2}$, where $K$ is the Luttinger liquid parameter and $\ensuremath{\zeta}(x)$ is the Riemann $\ensuremath{\zeta}$ 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, ${\ensuremath{\chi}}_{J}\ensuremath{\propto}{L}^{\ensuremath{\gamma}}$, where the $K$-dependent exponent $\ensuremath{\gamma}$ 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 ${\ensuremath{\chi}}_{\mathcal{P}}$, and show that it scales as the quartic power of the system size, ${\ensuremath{\chi}}_{\mathcal{P}}=[31\ensuremath{\zeta}(5)/8{\ensuremath{\pi}}^{6}](K{L}^{4}/{u}^{2})$, where $u$ is the sound velocity. Thus, the ratio ${L}^{2}{\ensuremath{\chi}}_{J}/{\ensuremath{\chi}}_{\mathcal{P}}$ 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.