Competition of spin and charge excitations in the one-dimensional Hubbard model

Motivated by recent experiments with ultracold fermionic atoms in optical lattices, we study finite temperature magnetic correlations, as singlet and triplet correlations, and the double occupancy in the one-dimensional Hubbard model. We point out that for intermediate interaction strengths the double occupancy has an intriguing doubly nonmonotonic temperature dependence due to the competition between spin and charge modes, related to the Pomeranchuk effect. Furthermore, we determine properties of magnetic correlations in the temperature regimes relevant for current cold atom experiments and discuss effects of the trap on spatially integrated observables. We estimate the entropy and the temperature reached in the experiment by Greif, Uehlinger, Jotzu, Tarruell, and Esslinger [Science 340, 1307 (2013)].

Figure 1

Nearest-neighbor spin correlations $\langle {\widehat{S}}_i^z{\widehat{S}}_{i+1}^z\rangle$ as a function of $U/t$ and $k_{B}T/t$, from DMRG. One can notice that the spin correlations are stronger within the intermediate $U/t$ region at finite temperatures. For large $U/t$, the spin correlations decay on the scale of the antiferromagnetic exchange $J_{\mathrm{ex}}=4t^2/U$.

Figure 2

Nearest-neighbor spin correlations $\langle {\widehat{S}}_i^z{\widehat{S}}_{i+1}^z\rangle$ vs $k_{B}T/t$. Full lines: DMRG result for $U/t=1,2,3,5,7,10,20$ from top to bottom increasing $U/t$ (for low $k_{B}T/t$). Dashed lines: exact correlations for the Heisenberg chain of coupling $J_{\mathrm{ex}}=4t^2/U$ for $U/t=10,20$. For large interactions $U/t\gtrsim 10$, the spin correlations decay on the scale of the antiferromagnetic exchange $J_{\mathrm{ex}}=4t^2/U$. The agreement with correlations in the Heisenberg chain is excellent for $U/t\gtrsim 20$.

Figure 3

Temperature derivative of the double occupancy ${\displaystyle \partial n_d/\partial \left(k_{B}T\right)}$ vs $(U/t,k_{B}T/t)$, from DMRG. This quantity allows one to characterize the regimes of growths or decays of the double occupancy with temperature. The successive decay and growth of double occupancy, for large $U/t$, is the signature of the Pomeranchuk effect. For intermediate interactions $1\lesssim U/t\lesssim 4$, the double occupancy successively grows, decays, and grows again with temperature. This signals the presence of a double Pomeranchuk effect.

Figure 4

Double occupancy $n_d$ vs $k_{B}T/t$, from DMRG, for $U/t=1,2,3,5,7,10$ from top to bottom increasing $U/t$. For large $U/t$, the conventional Pomeranchuk effect is the successive decay and growth of double occupancy with growing temperature. For intermediate interactions $1\lesssim U/t\lesssim 4$, the double occupancy successively decays (not visible here), grows, decays, and grows again with temperature, which we refer to as a double Pomeranchuk effect.

Figure 5

The double occupancy as a function of temperature: (a) for $U/t=1$, (b) for $U/t=3$, and (c) for $U/t=10$. Thick red line: double occupancy from the phenomenological argument. The dashed and dotted lines denote the contribution of the spin and charge sectors, respectively. Thick black line: DMRG, same data as Fig. 4 but centered on the low-temperature regime. The density of states (up to an arbitrary factor) is also shown as a shaded area chart, which is estimated by the spinon, particle, and hole dispersion given by Bethe ansatz [25]. For $U/t=10$, the contribution from the charge sector is negligible, and the temperature dependence of $n_d$ is very well described by that of the spin sector in this temperature regime.

Figure 6

Spin correlations in a homogeneous chain at half filling for $U/t=1,2,3,5,7,10,20$ from bottom to top increasing $U/t$ (high $k_{B}T/t$). (a) Singlet correlations. The number of singlets is maximal in the ground state and decays monotonically with growing $T$. (b) Triplet $S^z=0$ correlations. At low temperature triplet excitations are created, and they decay again at higher temperatures.

Figure 7

(a) Spin correlations $\langle {\widehat{S}}_i^z{\widehat{S}}_{i+1}^z\rangle$ as a function of distance $|i-j|$, log-lin scale, $U/t=1.5$, $k_{B}T/t=0.20,0.48,1.11$ from top to bottom increasing $T$. The correlations decay exponentially for large distances and increase while lowering $T$. The DMRG relative error for distances $|i-j|\lesssim 12$ is below 1% comparing the number of retained states $M=192$ and 512. (b) Correlation length $\xi$ as a function of temperature $k_{B}T/t$ for $U/t=0,3,10$. The value of $\xi$ represented here is obtained with a number of retained states $M=512$. The deviations with respect to the number of states $M=192$ and 512 is below 1%; see Fig. 8 for the convergence with respect to the fit range.

Figure 8

Rescaled spin-correlation length $\xi k_{B}T/at$ vs temperature. In the Tomonaga-Luttinger liquid regime, the conformal field theory predicts that $\xi k_{B}T/at$ goes to a constant for low $T$. We compare it with the prediction of the Tomonaga-Luttinger liquid for $U/t=5,0,10$ in this order from top to bottom (arrows). The correlation length is estimated on a range $|i-j|$ in $[10,20]$ and the error bar is the difference with the estimate for a range $[6,12]$, the small ranges being the main source of error.

Figure 9

(a) Singlet and triplet correlations as a function of $k_{B}T/t$ averaged over the chain in the presence of a trapping potential, for a rescaled density of $\rho =1.61$ and $U/t=1.44$. (b) Density distribution, (c) local singlet, and (d) triplet correlations along the chain, $k_{B}T/t=0.5,1.5,3$ for the solid, dotted, and dash-dotted lines, respectively.

Figure 10

(a) Singlet and triplet correlations as a function of $k_{B}T/t$ averaged over a single trapped chain, for a rescaled density of $\rho =4.82$ and $U/t=1.44$. (b) Density distribution, (c) local singlet, and (d) triplet correlations along the chain, $k_{B}T/t=0.5,1.5,3$ for the solid, dotted, and dash-dotted lines respectively.

Figure 11

Singlet and triplet correlations as a function of the local doping $\delta =1-\langle \widehat{n}\rangle$, collapsed data for several trapped chains with different rescaled densities $0<\rho <5$ (various markers), $k_{B}T/t=1$. (a) $U/t=1.5$. (b) $U/t=10$. The Gutzwiller approximation for $U/t=10$ is also depicted as lines; see text.

Figure 12

The final entropy $S_f$ (a) and temperature $k_B{T}_f/t$ (b) for the experiment [20]. Both are deduced from the difference of the singlet and triplet correlations as a function of the experimental initial entropy $S_i$. The error bars on $S_i$ have been estimated in [20]. The uncertainty on $S_f$ and $k_B{T}_f/t$ is the propagated uncertainty of $\langle {\widehat{P}}_s-{\widehat{P}}_{t0}\rangle /2$ from [20], the DMRG errors being much smaller. We also take into account the uncertainty due to the fluctuations of the number of atoms $N=66000\pm 6000$ and they are found to add very little to the estimated uncertainty. Experimental data courtesy of Greif et al. [20].

Figure 13

The singlet-triplet population imbalance as a function of the final entropy $S_f$ for $t/t_{\perp }=7.26,5.03$. The dots denote experimental singlet-triplet population imbalance from [20] and the deduced final entropy. The error bars on the population are from the experimental ones [20] and the propagated error on the entropy is represented. All other uncertainties are neglected (see Fig. 12 for details).