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

Abstract

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.

Figures (13)

• Figure 1: Schematic evolution from cluster-localized pairing to stripe-coherent superconductivity in the $t$--$t'$--$J$ model.a. At intermediate temperatures, after static stripe order has melted, doped holes form hole-rich clusters embedded in an AFM background. The dominant pair wavefunctions are localized on these hole-rich clusters. b. Upon further cooling, the charge clusters reorganize into stripes and the leading pair wavefunction becomes increasingly delocalized across the system, yielding a coherent $d$-wave like structure. The blue shading indicates the hole-dense regions, the red and blue arrows indicate local magnetic moments, and the green envelopes schematically indicate the spatial structure of the dominant pair wavefunctions.
• Figure 2: Pair wavefunctions in a single METTS snapshot. (a) Eigenvalues $\epsilon_n^{(s)}$ and (b--d) the three leading pair wavefunctions $\chi_n^{(s)}$ obtained by diagonalizing the snapshot singlet-pair correlation matrix $\rho_2^{(s)}$ [cf. Eqs. \ref{['eq:pair_matrix_snapshot']} and \ref{['eq:pair_diag']}] for one representative METTS configuration for doping $p=1/16$, $J/t=0.4$ and $t'/t=0.2$ on an $L=16$, $W=6$ cylinder at $T/t=0.04$. In (a), the three markers identify the eigenvalues corresponding to the eigenvectors displayed in (b--d): the largest positive eigenvalue is marked by a blue star, the second by an orange triangle, and the third by a violet square; the subleading eigenvalues are shown as gray crosses. In (b--d), circles on lattice sites encode the snapshot hole density $n_h^{(s)}(\mathbf r)$ [Eq. \ref{['hole_density_snapshot']}], with both circle area and grayscale intensity increasing with $n_h^{(s)}(\mathbf r)$ (right colorbar beside (a)). Colored nearest-neighbour bonds encode the signed amplitude of the pair wavefunction $\chi_n^{(s)}$ (left colorbar beside (a)). The leading pair wavefunctions are concentrated on the hole-rich regions.
• Figure 3: Thermal pairing spectrum of the $t$--$t'$--$J$ model. Temperature dependence of the leading eigenvalues $\epsilon_n$ of the thermally averaged singlet pairing density matrix $\rho_{2}$ (Eq. \ref{['eq:pair_matrix']}) on cylinders $16\times 6$ (panel (a)) and $24\times 4$ (panel (b)) for doping $p=1/16$, $J/t=0.4$ and $t'/t=0.2$. Curves are color-coded by eigenvalue index; the three leading eigenvalues ($n=1,2,3$) are highlighted with distinct markers, while sub-leading eigenvalues are shown in gray. The largest positive eigenvalue is marked by a blue circle, the second by an orange square, and the third by a violet triangle. Shaded bands denote statistical uncertainties from bootstrap resampling over METTS samples. Insets show the eigenvalue splitting $w=\epsilon_1-\epsilon_3$ which characterizes the Josephson coupling between the condensates, and they start increasing sharply below $T/t \lesssim0.2$. The $T/t=0$ DMRG reference is indicated by a dashed line. Right subpanels (c--e) display the corresponding pair wavefunctions at temperature $T/t=0.005$, together with the averaged hole-densities.
• Figure 4: Forestalled phase separation from hole-cluster statistics. Density-weighted cluster-size distributions $p_m$ extracted from individual METTS snapshots using the on-site hole density $n_h^{(s)}(\mathbf r)$ [Eq. \ref{['hole_density_snapshot']}], and the cluster-identification protocol of Ref. Sinha2025 for doping $p=1/16$, $J/t=0.4$ and $t'/t=0.2$. Panels (a--e) show $24\times4$ cylinders at $T/t=2.00,\,0.20,\,0.15,\,0.05,\,0.01$, and panels (f--i) show $16\times6$ cylinders at $T/t=2.00,\,0.20,\,0.15,\,0.05$ (left to right). Bars at fixed $m$ are stacked by cluster hole-mass bins $I_{k}=[k-1,k)$; the total stacked height at fixed $m$ is $p_m=\sum_{k\ge 1}p_m^{(k)}$ [Eq. \ref{['eq:clust_pm_binned']}]. The mean cluster size $\bar{m}$ is annotated in each panel. Cooling transfers weight from $m=1$ to larger sizes and increases $\bar{m}$, consistent with forestalled phase separation. We find a stripe-like reorganization at the lowest shown temperature, $T/t=0.01$, on $24\times4$ with most weight concentrated on $m=12$.
• Figure 5: Properties of the leading superconducting pair wavefunction.(a,b) Inverse participation ratio $\mathrm{IPR}_1$ [Eq. \ref{['eq:ipr_def']}] of the leading pair wavefunction $\chi_{1}(\alpha)$ for cylinders of size $24\times4$ and $16\times6$, respectively, for doping $p=1/16$, $J/t=0.4$ and $t'/t=0.2$. (c,d) Pair-charge locking coefficient $\Lambda$ [Eq. \ref{['eq:r_def']}] between the on-site hole density $n_h(\mathbf r)$ and the site-centered $d$-wave amplitude $|\Delta_{d,1}(\mathbf r)|$ [Eqs. \ref{['eq:dfield_def']}--\ref{['eq:r_def']}] constructed from the leading pair wavefunction. Error bars show statistical uncertainty from METTS sampling; colors compare bond dimensions.