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
Tc and nuclear magnetic resonance (NMR) signatures of charge clustering in the underdoped regime.
We present accurate numerical results for low-lying one- and two-hole states in the t-J model on a 4\ifmmode\times\else\texttimes\fi{}4 lattice. We find six level crossings in the one-hole ground state for 0t/J\ensuremath{\infty}; accurate t/J values of these crossings and the associated ground-state quantum numbers are given. A degeneracy of k=(0,0) S=1/2, and S=3/2 one-hole levels at t/J=1/2 is noted, which is consistent with a recent analytical result. For small t/J, the S=1/2 one-hole and S=0 two-hole bandwidths on the 4\ifmmode\times\else\texttimes\fi{}4 lattice are
Wh=[1.190 445 7(1)]t and
Whh=[2.575(4)]
t2/J, respectively. The origin of these qualitatively different behaviors is discussed, and a simple relation is found between the small-(t/J) one-hole bandwidth and a static-hole ground-state matrix element. The linear-t term in
Wh is apparently a finite-lattice artifact. As a measure of finite-size effects we determined the rms hole-hole separation in the two-hole ground states; we find evidence of important finite-size effects for t/J\ensuremath{\gtrsim}1, for which the rms hole-hole separation is clearly constrained by the 4\ifmmode\times\else\texttimes\fi{}4 lattice. Intermediate-(t/J) hole separations and binding energies for 0.3\ensuremath{\lesssim}t/J\ensuremath{\lesssim}1, however, scale approximately as powers of t/J, and can be used to give bulk-limit estimates for t/J=3. In particular, we estimate that the bulk-limit ground-state rms hole-hole separation at t/J=3 is \ensuremath{\approxeq}1.8
a0, corresponding to 7 \AA{} in the high-temperature superconductors. The similarity to the observed in-plane coherence length of ${\ensuremath{\xi}}_{\mathit{a}\mathit{b}}$\ensuremath{\approxeq}14 \AA{} supports the identification of t-J model hole pairs with the Cooper pairs of high-temperature superconductivity.