Transverse-field $XY$ spin chain with the competing long-range interactions: Multi-criticality around the $XX$-symmetric point

The transverse-field $XY$ spin chain with competing antiferromagnetic long-range interactions, $J_r \propto 1/r^\alpha$ ($r$: distance between spins), and the exponent $\alpha$ was investigated numerically. The main concern is to clarify the character of the transverse-field-driven phase transition for the small-$\alpha$ regime around the $XX$-symmetric point, $\eta=0$ ($\eta$: $XY$-anisotropy parameter). To cope with the negative-sign problem, we employed the exact-diagonalization method, which enables us to evaluate the fidelity susceptibility $\chi_F$. Because the fidelity susceptibility does not assume any order parameter after a phase transition, it detects the multi-criticality around the $XX$-symmetric point in a systematic manner. As a preliminary survey, with $\eta=0.5$ and $\alpha=2$ fixed, we made the scaling analysis of $\chi_F$. The scaling behavior of $\chi_F$ shows that the transverse-field-driven phase transition belongs to the 2D-Ising universality class. Thereby, with the properly crossover-scaled $\eta$, the $\chi_F$ data is cast into the crossover scaling formula for the small-$\alpha$ regime. The multi-critical exponents fed into the scaling formula are argued through referring to related studies.