Fidelity Approach to the Disordered Quantum $XY$ Model

We study the random $XY$ spin chain in a transverse field by analyzing the susceptibility of the ground state fidelity, numerically evaluated through a standard mapping of the model onto quasifree fermions. It is found that the fidelity susceptibility and its scaling properties provide useful information about the phase diagram. In particular it is possible to determine the Ising critical line and the Griffiths phase regions, in agreement with previous analytical and numerical results.

Figure 1

Ising transition at $\gamma =1$. (a) Average fidelity susceptibility, $[\chi {]}_{\mathrm{ave}}$, for $L=512$ and ${10}^4$ realizations, with varying disorder strengths $\sigma \in \{0,0.1,0.3\}$; (b) the associated finite-size scaling dimension ${\Delta }_{\chi }$ of $[\chi {]}_{\mathrm{ave}}$; (c) probability distribution of $\ln \chi$ at the Ising transition for system sizes $L=128$, 256, 512, and disorder $\sigma =0.1$; (d) distribution of $\ln \chi$ away from the Ising transition for the same disorder and range of system sizes.

Figure 2

Elimination of anisotropy transition at $\gamma =0$ due to disorder. (a) Average fidelity susceptibility $[\chi {]}_{\mathrm{ave}}$, near $\gamma =0$ for $L=500$, $\lambda =0.2$, $\sigma =0.1$, and ${10}^4$ realizations. (b) Finite-size scaling dimension of $[\chi {]}_{\mathrm{ave}}$ (in this case $\gamma =0.036$); (c) probability distribution of $\ln \chi$ at $\gamma =0$ and at the value of $\gamma$ corresponding to the maximum of $[\chi {]}_{\mathrm{ave}}$; (d) distribution of $\ln \chi$ far away from the anisotropy line, where the finite-size scaling is extensive.

Figure 3

Average fidelity susceptibility for various disorder strengths $\sigma \in \{0,0.1,0.2,0.3\}$, with fixed system size $L=400$. Here $\lambda =0.5$, and the derivative in Eq. (6) is taken along $\gamma$.