Evolution of superconductivity from charge clusters to stripes in the $t$-$t'$-$J$ model

Competition and coexistence of charge orders and superconductivity are hallmarks in many strongly correlated electron systems. Here, we unravel the precise role of charge fluctuations on the superconducting state in the $t$-$t’$-$J$ model of the high-temperature cuprate superconductors. Using finite-temperature tensor network simulations, we investigate thermal snapshots in the underdoped regime where the ground state features a superconducting stripe phase. At intermediate temperatures, where stripes have melted and hole clustering is observed, we find that pairing correlations are tightly localized on the hole clusters. Upon entering the stripe regime at lower temperatures, pairing increasingly delocalizes across different hole clusters to ultimately become coherent across the full system in the ground state. This pair-charge locking gives rise to an intuitive picture of the parent state of the superconducting stripe phase: pairing is localized on hole clusters formed via hole attraction due to the onset of magnetic correlations at intermediate temperature. We discuss how this microscopic picture is consistent with a broad range of experimental observations in cuprate superconductors, including scanning tunneling microscopy (STM) evidence for local pairing above $T_c$ and nuclear magnetic resonance (NMR) signatures of charge clustering in the underdoped regime.

💡 Research Summary

In this work the authors investigate how charge fluctuations influence superconductivity in the t‑t′‑J model, a minimal model believed to capture the essential physics of cuprate high‑temperature superconductors. They focus on a parameter set (J/t = 0.4, t′/t = 0.2, hole doping p = 1/16) that previous zero‑temperature DMRG studies identified as hosting intertwined stripe order and d‑wave superconductivity. Using the finite‑temperature tensor‑network technique known as minimally entangled typical thermal states (METTS), they generate ensembles of pure states that faithfully represent the thermal density matrix at various temperatures.

For each METTS snapshot they compute the singlet‑pair two‑particle reduced density matrix (2RDM) on nearest‑neighbour bonds. By diagonalising the 2RDM they obtain eigenvalues εₙ (condensate fractions) and eigenvectors χₙ (pair wavefunctions). According to the Penrose‑Onsager criterion, a leading eigenvalue that scales with system size signals the emergence of a pairing condensate, while the spatial structure of χₙ reveals where the pairs are located.

The temperature evolution can be divided into three regimes. At high temperatures (T/t ≳ 0.2) the eigenvalue spectrum is featureless, indicating no significant pairing and a nearly uniform charge distribution. In the intermediate regime (T/t ≈ 0.04) the holes self‑organise into mesoscopic clusters of a few sites. Three eigenvalues (ε₁, ε₂, ε₃) separate from the bulk of the spectrum, and their corresponding χₙ are strongly localised on these hole‑rich clusters, displaying a d‑wave sign structure within each cluster. This demonstrates that pairing first nucleates on charge clusters rather than on the antiferromagnetic background.

When the temperature is lowered further (T/t ≲ 0.02) the system develops stripe order. The charge clusters merge into linear, stripe‑like patterns, and the previously isolated pair wavefunctions begin to hybridise. The splitting w = ε₁ − ε₃, which measures the Josephson coupling between condensates on different stripes, grows sharply, signalling the onset of inter‑stripe phase coherence. The dominant χ₁ evolves into a system‑wide d‑wave wavefunction, while χ₂ and χ₃ acquire finite momentum components (kₓ = π) that reflect the underlying charge‑density‑wave periodicity. In the ground state the three leading eigenvalues remain distinct, indicating a fragmented condensate tied to the stripe superlattice, but the overall pairing becomes coherent across the entire lattice.

To quantify charge clustering the authors adopt a density‑weighted cluster‑size distribution p(m) based on an adaptive hole‑density threshold. At high temperature the distribution is peaked at m = 1 (isolated holes). Cooling transfers weight to larger m, and the mean cluster size ⟨m⟩ grows, evidencing a “forest‑blocked” phase‑separation tendency that is arrested by the long‑range Coulomb interaction and lattice geometry. At the lowest temperatures the distribution concentrates around m ≈ 12, corresponding to the stripe wavelength observed in the simulations.

The numerical findings dovetail with a broad set of experiments. Scanning tunneling microscopy (STM) on underdoped cuprates reports nanoscale regions with a superconducting gap persisting above the bulk Tc, precisely the “cluster‑local pairing” identified here. Nuclear magnetic resonance (NMR) detects inhomogeneous charge environments in the pseudogap regime, consistent with the intermediate‑temperature hole clusters. Thus the authors propose a coherent microscopic picture: (i) at temperatures where magnetic correlations begin to develop, holes attract each other and form mesoscopic clusters; (ii) pairing nucleates on these clusters; (iii) upon further cooling, the clusters reorganise into stripes, allowing Cooper pairs to tunnel between them and establish long‑range phase coherence, yielding the observed stripe‑ordered superconducting ground state.

Overall, the paper provides a detailed, temperature‑resolved view of how charge inhomogeneity evolves into stripe order and how this evolution controls the emergence of superconductivity in the t‑t′‑J model. By linking the eigenstructure of the 2RDM to real‑space charge patterns, the authors offer a powerful diagnostic that could be applied to other strongly correlated models and help bridge the gap between theoretical simulations and experimental observations in cuprate superconductors.