Although both experimental observations and numerical simulations have reached a consensus that the stripe phase is intertwined with superconductivity in cuprates, the microscopic mechanism behind $d$-wave pairing in the presence of stripes remains unclear. Using the effective theory of quantum colored strings, we derive the wavefunction in Fock space. Our results show that two spinons with opposite chiralities tend to pair into a spinon singlet, which in turn facilitates the formation of negative pair-pair correlations between distant $x$-bonds and $y$-bonds, a hallmark of the $d$-wave pairing pattern. The same pair-pair correlation pattern is observed across various models, as confirmed by large-scale density matrix renormalization group calculations. Based on these results, we conclude that the spinon singlet is the origin of $d$-wave superconductivity in a fluctuating, partially-filled stripe, and this mechanism may also extend to multi-stripe configurations.
Spinon Singlet in Quantum Colored String: Origin of d-Wave Pairing in a Partially-Filled Stripe
Although both experimental observations and numerical simulations have reached a consensus that the stripe phase is intertwined with superconductivity in cuprates, the microscopic mechanism behind d-wave pairing in the presence of stripes remains unclear. Using the effective theory of quantum colored strings, we derive the wavefunction in Fock space. Our results show that two spinons with opposite chiralities tend to pair into a spinon singlet, which in turn facilitates the formation of negative pair-pair correlations between distant x-bonds and y-bonds, a hallmark of the d-wave pairing pattern. The same pair-pair correlation pattern is observed across various models, as confirmed by large-scale density matrix renormalization group calculations. Based on these results, we conclude that the spinon singlet is the origin of d-wave superconductivity in a fluctuating, partially-filled stripe, and this mechanism may also extend to multi-stripe configurations.
Significant research advances have led to a consensus that the Fermi-Hubbard model and its extended variants are archetypical frameworks for elucidating the intertwined relationship between stripe orders and superconductivity in hole-doped high-T_{c} materials. Notably, the Hubbard quantum simulator has recently achieved several remarkable breakthroughs, e.g., being cooled down to the cryogenic regime and enabling the observation of stable fluctuating stripes. However, the microscopic mechanism underlying the d-wave pairing of electrons in the presence of stripes at low temperatures remains poorly understood due to the intricate interplay between strongly correlated effects and non-negligible thermal fluctuations. Here, we conduct a close investigation of a partially filled stripe in the representative t-J and t-t^{'}-U models with both numerical and analytical methods. Analogous to quantum gas microscopy, the perfect sampling technique allows us to obtain the high-confidence-level statistics of the Fock basis states appearing in the ground-state wave function. In a novel physical paradigm, these data demonstrate that two spinons with opposite chiralities tend to pair spontaneously into a singlet state, thereby naturally giving rise to the d-wave pairing pattern. Then, using the updated effective theory of quantum colored string, we have reconstructed the wave function and have determined the nature of spinon pairing and its connection to the d-wave sign structure of pair-pair correlation. Furthermore, spinon singlet pairs enable the establishment of a long-range pair-pair correlation between the two stripes. Our Letter offers new insights into the microscopic physics of stripes and paves the way for further exploration of multistripe-mediated pairing mechanisms in the Fermi-Hubbard model.
Scanning tunneling microscopy (STM) is a powerful tool for visualizing the local density of states (LDOS) of individual stripes in cuprates. However, the microscopic nature of the observed exotic LDOS patterns and their connection to high-Tc superconductivity remain open questions. Within the framework of the quantum colored string model, we reveal that the ubiquitous 4a0×4a0 plaquettes originate from either the breaking of local spinon singlet pairs through hole insertion, or the formation of an unpaired spinon upon electron addition in a stripe. Moreover, by comparing our data with LDOS of cuprates, we identify an effect of particle-hole symmetry breaking (PHSB): a 2a0 shift, which is predicted and confirmed in a longer stripe (L=18). At last, we establish and verify a general relation between hole density and plaquette size across multiple fillings. Our work offers a fresh wavefunction-based perspective on interpreting STM signals in cuprate experiments and demonstrates that their origin may arise from spinon singlet pairing in the ground state of fluctuating stripes, the same mechanism underlying the d-wave sign structure [Phys. Rev. Lett. \textbf{137}, 086702 (2026)].
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 Zaanen's early scenario [J. Zaanen, Phys. Rev. Lett. 84, 753 (2000); J. Zaanen et al., Philos. Mag. B 81, 1485 (2001)], 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 $\ensuremath{\pi}$-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 paper, we propose an effective model for describing the stripe phase in the hole-doped $t\ensuremath{-}{J}_{z}$ model. With respect to the antiferromagnetic background, the model comprises three types of color-labeled point defects coupling to an effective spin field, so-called colored string. Comparing with numerical results from large-scale density matrix renormalization group simulations, we find semiquantitative agreement in local hole density, magnetic moment, and the proposed spectrum features of the effective spin field. By systematically analyzing the hole-density distribution and the scaling of ground-state 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. Lastly, 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.
Recent experimental results of angle-resolved photoemission spectroscopy suggested that an additional strong nearest neighbor attraction in the Hubbard model might be significant for the doped cuprates. The stripe-ordered patterns, which are formed by the inhomogeneous distribution of spin, charge, and pairing correlations in CuO2 planes, are a well-known feature of the doped cuprates. By employing constrained path quantum Monte Carlo, we examine the effect of the nearest neighbor attraction on a two-dimensional repulsive Hubbard model. Within and across the stripe regions, the ground-state spin correlations and the d-wave pairing correlation are calculated. We found that the spin-spin correlation is the largest when the interstripe region is nearly half filled, and we also found the d-wave superconducting correlation of the neighboring sites is enhanced in the presence of the stripe pattern. This reveals the crucial effects of the strong nearest neighbor attraction on superconductivity in doped cuprates.
Charge stripes have been widely observed in many different types of unconventional superconductors, holding varying periods ( P ) and intensities. However, a general understanding on the interplay between charge stripes and superconducting properties is still incomplete. Here, using large-scale unbiased numerical simulations on a general inhomogeneous Hubbard model, we discover that the charge-stripe period P , which is variable in different real material systems, could dictate the pairing symmetries—d wave for P≥4,s and d waves for P≤3 . In the latter, tuning hole doping and charge-stripe amplitude can trigger a d-s wave transition and magnetic-correlation shift, where the d-wave state converts to a pairing-density wave state, competing with the s wave. These interesting phenomena arise from an unusual stripe-induced selection rule of pairing symmetries around on-stripe region and within inter-stripe region, giving rise to a critical point of P=3 for the phase transition. In general, our findings offer important insights into the differences in the superconducting pairing mechanisms across many P -dependent superconducting systems, highlighting the decisive role of charge stripe. Charge stripes have been observed in various unconventional superconductors, but their interplay with the superconducting properties is not fully understood. Here, the authors show, via large-scale numerical simulations based on an inhomogeneous Hubbard model, that the charge stripe period can dictate the superconducting pairing symmetry.