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.
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.
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.
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)].
In strongly correlated quantum materials, the behavior of charge carriers is dominated by strong electron-electron interactions. These can lead to insulating states with spin order, and upon doping to competing ordered states including unconventional superconductivity. The underlying pairing mechanism remains poorly understood however, even in strongly simplified theoretical models. Recent advances in quantum simulation allow to study pairing in paradigmatic settings, e.g. in the t-J t − J and t-J_z t − J z Hamiltonians. Even there, the most basic properties of paired states of only two dopants, such as their dispersion relation and excitation spectra, remain poorly studied in many cases. Here we provide new analytical insights into a possible string-based pairing mechanism of mobile holes in an antiferromagnet. We analyze an effective model of partons connected by a confining string and calculate the spectral properties of bound states. Our model is equally relevant for understanding Hubbard-Mott excitons consisting of a bound doublon-hole pair or confined states of dynamical matter in lattice gauge theories, which motivates our study of different parton statistics. Although an accurate semi-analytic estimation of binding energies is challenging, our theory provides a detailed understanding of the internal structure of pairs. For example, in a range of settings we predict heavy states of immobile pairs with flat-band dispersions - including for the lowest-energy d d -wave pair of fermions. Our findings shed new light on the long-standing question about the origin of pairing and competing orders in high-temperature superconductors.
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.
Understanding the doped Mott insulator is a central challenge in condensed matter physics. In this work, we first explicitly identify a new sign structure in the $t\text{\ensuremath{-}}{t}^{\ensuremath{'}}\text{\ensuremath{-}}J$ model on the square lattice that replaces the conventional Fermi statistics for weakly interacting electrons. Then we show that the singular, i.e., the phase-string part of the sign structure in the partition function can be precisely turned off in a modified model. The density matrix renormalization group method is then employed to study these two models comparatively on finite-size systems, which is designed to unveil the consequences of the phase-string component. We find that the hole pairing is present not only in the quasi-long-range superconducting phase but also in the stripe phase of the $t\text{\ensuremath{-}}{t}^{\ensuremath{'}}\text{\ensuremath{-}}J$ model. However, once the phase-string is switched off, both the superconducting and stripe orders together with the underlying hole pairing disappear. The corresponding ground state reduces to a trivial Fermi-liquid-like state with small hole Fermi pockets that is decoupled from the antiferromagnetic spin background. It is in sharp contrast to the original $t\text{\ensuremath{-}}{t}^{\ensuremath{'}}\text{\ensuremath{-}}J$ model where large Fermi surfaces can be restored in the stripe phase found at ${t}^{\ensuremath{'}}/t<0$ or the superconducting phase at ${t}^{\ensuremath{'}}/t>0$ in the six-leg ladder calculation. Our study clearly demonstrates that the strong correlation effect in doped Mott insulator should be mainly attributed to the long-range quantum entanglement between the spin and charge, which is, nonperturbatively, beyond a simple spin-charge separation under the no double occupancy constraint.