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 [Silva et al.[1]], where the vacancy-vacancy interaction decays exponentially, along the entire quantum critical line the interaction becomes algebraic, |∆ b (r)| ∼ r -α , with α close to the universal Casimir value of unity and a weak but systematic dependence on the second-neighbor coupling, α ∞ ≃ 1.070 + 0.091 (J 2 /J 1 ) across J 2 /J 1 ∈ [0.1, 1.0]. The transmission ratio of the spin correlator across a vacancy approaches a J 2 -dependent plateau T ∞ (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 ∞ ≈ -0.073, well above the Ising fixed-BC value -ln √ 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 η = 1/4, the order-parameter exponent of the Ising universality class, which provides a robust criterion in this open geometry.