Quantum colored strings in the hole-doped $t$-$J_z$ model

The stripe phase, an intertwined order observed in high-temperature superconductors, is regarded as playing a key role in elucidating the underlying mechanism of superconductivity, especially in cuprates. Following Jan Zaanen's early scenario, the filled charge stripe, with one hole per unit cell of the charge order, can be taken as the interactive elastic quantum strings of holes, stabilized by $\pi$-phase shifts between neighboring magnetic domains. However, this scenario is challenging to explain, particularly in terms of electron pairing, which necessitates hole pairs. In this work, we propose a new effective model for describing the stripe phase in the hole-doped $t$-$J_z$ model. With respect to the antiferromagnetic background, the model comprises three types of color-labeled point-defects coupling to an effective spin field, so named as ``colored string". Comparing with numerical results from large-scale density matrix renormalization group (DMRG) simulations, we find semi-quantitative agreement in local hole density, magnetic moment, and the newly proposed spectrum features of the effective spin field. By systematically analyzing the hole-density distribution and the scaling of groundstate energy at different system sizes, we determine the effective core radius and the effective hopping amplitude of the quantum string. Furthermore, the local pinning field can be finely adjusted to drag the quantum string, offering a potential method for detecting it in optical lattices. At last, we further demonstrate the partially-filled stripe with less than one hole per unit cell of the charge order can also be well described by the effective theory.