Static vacancies as parametrized conformal defects in the critical $J_1$--$J_2$ transverse-field Ising chain

We revisit the problem of two static nonmagnetic vacancies in the transverse-field Ising chain with first- and second-neighbor couplings $J_1$ and $J_2$, now on the critical line, using density-matrix renormalization-group (DMRG) calculations in open chains of up to $N=300$ sites. In contrast to the gapped regime studied previously, where the vacancy-vacancy interaction decays exponentially, along the entire quantum critical line the interaction becomes algebraic, $|\Delta_b(r)|\sim r^{-\alpha}$, with $\alpha$ close to the universal Casimir value of unity and a weak but systematic dependence on the second-neighbor coupling, $\alpha_\infty \simeq 1.070 + 0.091\, J_2/J_1)$ across $J_2/J_1\in[0.1,1.0]$. The transmission ratio of the spin correlator across a vacancy approaches a $J_2$-dependent plateau $T_\infty(J_2)$ that grows from $0.11$ to $0.33$ over the same range, and the Affleck-Ludwig boundary entropy is small and approximately constant, $\log g_\infty \approx -0.073$, well above the Ising fixed-BC value $-\ln\sqrt{2}$ and close to the free-boundary value. The three observables vary smoothly and monotonically with $J_2$, consistent with a one-parameter family of partially transmissive conformal defects controlled by $J_2$. Throughout, the critical line is located using the bulk spin-correlator exponent $\eta=1/4$, the order-parameter exponent of the Ising universality class, which provides a robust criterion in this open geometry.