Finite temperature dopant-induced spin reorganization explored via tensor networks in the two-dimensional $t$-$J$ model

Abstract

Doped Mott insulators host intertwined spin-charge phenomena that evolve with temperature and can culminate in stripe order or superconductivity at low temperatures. The two-dimensional $t$-$J$ model captures this interplay yet finite-temperature, infinite-size calculations remain difficult. Using purification represented by a tensor network - an infinite projected entangled-pair state (iPEPS) ansatz - we simulate the $t$-$J$ model at finite temperature directly in the thermodynamic limit, reaching temperatures down to one tenth of the hopping rate and hole concentrations up to one quarter of the lattice sites. Beyond specific heat, uniform susceptibility, and compressibility, we introduce dopant-conditioned multi-point correlators that map how holes reshape local exchange. Nearest-neighbor hole pairs produce a strong cooperative response that reinforces antiferromagnetism on the adjacent parallel bonds, and single holes weaken nearby antiferromagnetic bonds; d-wave pairing correlations remain short-ranged over the same window. These results provide experiment-compatible thermodynamic-limit benchmarks and establish dopant-conditioned correlators as incisive probes of short-range spin-texture reorganization at finite temperature.

Figures (13)

• Figure 1: Schematics of t-J model dynamics and dopant-induced spin correlations -- (A) Schematic of the two-dimensional $t$-$J$ model on a square lattice with the nearest neighbor hopping $t$ and exchange interaction $J$. In a representative classical picture, up (red) and down (blue) spins can hop around to sites with holes (gray circles) while double occupation of single lattice site is forbidden. (B) Hole-conditioned spin-spin correlation. Nearest neighbor (NN) bonds are antiferromagnetic (AFM; NN $\langle S^{z}_{i}S^{z}_{j}\rangle <0$) and conditioned correlation measure where we remove the background AFM correlation is trvially $0$ as we see in the middle dotted box. The left dotted box shows that conditioned on a single hole at the black site, the NN AFM bonds nearby are weakened and become more ferromagnetic compared to the background (represented by red solid lines). With two adjacent holes, AFM is reinforced on the plaquette edge parallel to the pair, i.e., the corresponding NN bonds become more antiferromagnetic compared to the background (represented by the blue solid line) as shown in the right dotted box.
• Figure 2: Thermodynamic observables -- (A) shows specific heat $C_v(T)$. Inset: $C_v$ linearly depends on the doping $1-n$ in the high-temperature regime as predicted in Ref. rigol2007. (B) shows uniform spin susceptibility $\chi(T)$, obtained from Eq. (\ref{['eq:chi_est']}). (C) shows charge compressibility $\kappa = \partial n/\partial \mu$ versus $\mu$ at several fixed temperatures. The inset shows $n(\mu)$ vs $\mu$.
• Figure 3: 3P correlator -- (A) shows 3P correlators of different temperatures and fillings in a $7\times7$ window. The black dot sitting at the center indicates the position of the hole projector $h$. $S^z$ operators are applied on the nearest neighboring sites (with a distance of $d=1$) or the next nearest neighboring sites (with a distance of $d=\sqrt{2}$). The intensities of the colors of different segments represent the magnitudes of different 3P correlators. The segment is red (blue) if the corresponding correlator is positive (negative). The correlators whose absolute values are smaller than $10^{-3}$ are shown on a linear scale, while others are on a logarithmic scale. The range of the 3P correlators increases when the temperature is lowered for a fixed filling. The signs of some long-range correlators alter when changing the filling or the temperature. (B) and (C) show how the correlators depend on $r$ at $T=1/10$, which represents the distance between the hole and the bond linking two $S^z$ operators of the 3P correlator, for $d=1$ and $d=\sqrt{2}$ respectively. (D) shows the $r$ dependence of $d=1$ at various temperatures for filling $n=0.85$.
• Figure 4: 3P correlator versus 4P correlator -- Green, blue, yellow insets show the configuration of $C^{(3)}_L$, $C^{(4)}_T$, and $C^{(4)}_\parallel$ respectively. Different markers represent different temperatures. Different correlators are plotted in accordance with the color of the corresponding insets.
• Figure 5: $d$-wave correlator -- $d$-wave correlators at different temperatures being studied decay exponentially.