Evidence of attraction between charge carriers in a doped Mott insulator

Recent progress in optically trapped ultracold atomic gases is now making it possible to access microscopic observables in doped Mott insulators, which are the parent states of high-temperature superconductors. This makes it possible to address longstanding questions about the temperature scales at which attraction between charge carriers are present, and their mechanism. Controllable theoretical results for this problem are not available at low temperature due to the sign problem. In this work, we make important progress with this problem by employing worm-algorithm Monte Carlo, which allows us to obtain completely unbiased results for two charge carriers in a Mott insulator. Our method gives access to lower temperatures than what is currently possible in experiments, and provides evidence for attraction between dopants at a temperature scale that is now feasible in ultracold atomic systems. We also report on spin correlations in the presence of charge carriers, which are directly comparable to experiments.

Figure 1

Comparison between worm-algorithm Monte Carlo and exact diagonalization. Computing the kinetic energy for a single carrier in a $4\times 4$ system using the worm algorithm, we can benchmark it against exact diagonalization. As expected for an unbiased technique, we obtain perfect agreement. Here, $t/J=10/3$.

Figure 2

Illustration of spin-spin correlators. The spin-spin correlators $C_{\left|\mathbf{d}\right|,\mathbf{s}}^{\mathrm{s}}\left(\mathbf{r}\right)$ defined in Eq. (A2) illustrated in the cases of $\left|\mathbf{d}\right|=1$, $\left|\mathbf{d}\right|=\sqrt{2}$, and $\left|\mathbf{d}\right|=2$. Here, the bond distance $\mathbf{r}$ is defined as the distance from one of the carriers to the point between the spins for which the correlation is considered and $\mathbf{s}$ is the distance to the second carrier.

Figure 3

Hole-hole correlations. (a)–(c) show contour plots of the correlator $C^{\mathrm{h}}\left(\mathbf{s}\right)$ [Eq. (A1)] for the case $t/J=2$ and $\beta J=2.1,2.4,2.7$ respectively. The radial components of these are given in (g). For this parameter set, we observe a peak in the probability distribution at small separation which is visible already at temperatures corresponding to $J\beta =2.1$. In (a) the effect is small, with a peak value of $C^{\text{h}}\approx (2.3\pm 0.27)\times {10}^{-2}$ at $s\approx 4.12$. Decreasing the temperature bolsters the effect, and in (c) we observe a maximum value of $C^{\text{h}}\approx (6.0\pm 1.7)\times {10}^{-2}$, also at a separation of $s\approx 4.12$. This scenario corresponds to $U=8t$ in the Hubbard model, which is the parameter set that was realized in the experiment [37]. Slightly decreasing $t/J$ increases the height of the peak of $C^{\text{h}}$. In (d)–(f), contour plots of $C^{\mathrm{h}}\left(\mathbf{s}\right)$ are given for $t/J=5/3$ and $\beta J=2.1,2.4,2.7$ respectively. The corresponding radial components are again shown in (g). At the lowest temperature (f), the peak in $C^{\text{h}}$ is found when the holes are next-nearest neighbors.

Figure 4

Parameter regions with attraction. Discs denote parameters for which the correlator $C^{\text{h}}$ exhibits a peak at a carrier separation ${\mathbf{s}}_{\text{max}}$, which indicates attraction. The bar intersecting the disk gives the orientation of ${\mathbf{s}}_{\text{max}}$, while the color gives its magnitude according to the legend. The shaded region is a guide to the eye. For comparison, we also indicate the parameter sets of three recent experiments. In Koepsell et al. [36], the internal structure of a single polaron was mapped out using quantum gas microscopy. In Chiu et al. [37], correlations between dopants in an ultracold atomic gas were examined. In Mazurenko et al. [33], spin correlations in the Hubbard model were examined, though this parameter set, strictly speaking, does not correspond to a Mott insulator.

Figure 5

Spin correlations $C_{\left|\mathbf{d}\right|,\mathbf{s}}^{\mathrm{s}}\left(\mathbf{r}\right)$ in the proximity of carriers. Left column depicts a single hole, while right column shows two dopants which are next nearest neighbors. The top row (a), (b) gives $C_{\left|\mathbf{d}\right|=1,\mathbf{s}}^{\mathrm{s}}\left(\mathbf{r}\right)$, middle row (c), (d) gives $C_{\left|\mathbf{d}\right|=\sqrt{2},\mathbf{s}}^{\mathrm{s}}\left(\mathbf{r}\right)$, and bottom row (e), (f) gives $C_{\left|\mathbf{d}\right|=2,\mathbf{s}}^{\mathrm{s}}\left(\mathbf{r}\right)$. Model parameters are $J\beta =2.4$ and $t/J=2$. Additional examples of spin-correlations are provided in the Appendix. Animations can be found here [46].

Figure 6

Superexchange and kinetic-magnetic interaction. In the presence of a single hole (a), the spins $S_1$ and $S_2$ interact indirectly via super-exchange that is mediated by $S_3$. This gives rise to an effective ferromagnetic interaction. However, delocalization of the hole (a)–(c) brings $S_1$ and $S_2$ into direct contact, where interactions are antiferromagnetic. We refer to this as kinetic-magnetic interaction. The result is a high level of frustration and almost vanishing spin-correlations near the carrier. When two holes are present (d)–(f), only kinetic-magnetic interaction is present so that $S_1$ and $S_2$ become anticorrelated.

Figure 7

Average sign. This plot shows the exponential decay of ${\langle s\rangle }^{\prime }$ with increased values of $t\beta$ and $J\beta$. Only values ${\langle s\rangle }^{\prime }>{10}^{-3}$ are shown due to an otherwise poor SNR.

Figure 8

Determining hole-hole attraction. Illustrations of how hole-hole attraction is inferred using radially projected hole-hole correlations obtained using WAMC. If the difference $\mathrm{\Delta }{C}^{*}=\text{max}\left[C(s<s_{\text{bg}})\right]-\text{max}\left[C(s\ge s_{\text{bg}})\right]-2\times \text{noise}>0$ [(a) and (c)], the data are said to indicate attraction, otherwise (b) attraction is deemed to be absent. The partition divider $s_{\text{bg}}=6$ is chosen such that it exceeds separation distances where a peak is observed.

Figure 9

Attractive parameter region. The parameter region is scanned using WAMC. Red dots indicate observed hole-hole attraction, blue dots signify no observed attraction, and gray crosses mark points deemed to have inadequate SNR. The reason for large parameter values suffering from a poor SNR can be explained by the sign problem (cf. Fig. 7). On the other hand, the low SNR for small parameter values is caused by a weak signal.

Figure 10

Extent of simulation. This figure illustrates the number of sampled world-line configurations in the partition function sector for different values of $t\beta$ and $J\beta$. The combined total simulation time amounts to several million core-hours.

Figure 11

Hole-hole correlations. (a)–(c) show contour plots of the correlator $C^{\mathrm{h}}\left(\mathbf{s}\right)$ for the case $t/J=10/3$ and $\beta J=1.35,1.65,1.95$, respectively. The radial components of these are given in (d). No attraction is found for these parameter values.

Figure 12

Hole-hole correlations. (a)–(f) show contour plots of the correlator $C^{\mathrm{h}}\left(\mathbf{s}\right)$ for the case $t/J=5/6$ and $\beta J=0.90,1.50,2.10,2.70,3.30,3.90$, respectively. The radial components of these are given in (g). Attraction is present for all of these parameter values.

Figure 13

Spin correlations $C_{\left|\mathbf{d}\right|,\mathbf{s}}^{\mathrm{s}}\left(\mathbf{r}\right)$. The left column depicts a single hole, while the remaining columns depict two dopants located in the close vicinity of one another. The top row (a)–(e) gives $C_{\left|\mathbf{d}\right|=1,\mathbf{s}}^{\mathrm{s}}\left(\mathbf{r}\right)$, middle row (f)–(j) gives $C_{\left|\mathbf{d}\right|=\sqrt{2},\mathbf{s}}^{\mathrm{s}}\left(\mathbf{r}\right)$, and bottom row (k)–(o) gives $C_{\left|\mathbf{d}\right|=2,\mathbf{s}}^{\mathrm{s}}\left(\mathbf{r}\right)$. Model parameters are $t\beta =2.75$ and $J\beta =3.3$.

Figure 14

Spin correlations $C_{\left|\mathbf{d}\right|,\mathbf{s}}^{\mathrm{s}}\left(\mathbf{r}\right)$. The left column depicts a single hole, while the remaining columns depict two dopants located in the close vicinity of one another. The top row (a)–(e) gives $C_{\left|\mathbf{d}\right|=1,\mathbf{s}}^{\mathrm{s}}\left(\mathbf{r}\right)$, middle row (f)–(j) gives $C_{\left|\mathbf{d}\right|=\sqrt{2},\mathbf{s}}^{\mathrm{s}}\left(\mathbf{r}\right)$, and bottom row (k)–(o) gives $C_{\left|\mathbf{d}\right|=2,\mathbf{s}}^{\mathrm{s}}\left(\mathbf{r}\right)$. Model parameters are $t\beta =4.5$ and $J\beta =2.7$.

Figure 15

Spin correlations $C_{\left|\mathbf{d}\right|,\mathbf{s}}^{\mathrm{s}}\left(\mathbf{r}\right)$. The left column depicts a single hole, while the remaining columns depict two dopants located in the close vicinity of one another. The top row (a)–(e) gives $C_{\left|\mathbf{d}\right|=1,\mathbf{s}}^{\mathrm{s}}\left(\mathbf{r}\right)$, middle row (f)–(j) gives $C_{\left|\mathbf{d}\right|=\sqrt{2},\mathbf{s}}^{\mathrm{s}}\left(\mathbf{r}\right)$, and bottom row (k)–(o) gives $C_{\left|\mathbf{d}\right|=2,\mathbf{s}}^{\mathrm{s}}\left(\mathbf{r}\right)$. Model parameters are $t\beta =5.4$ and $J\beta =2.7$.

Figure 16

Kinetic energy. A single carrier's average kinetic energy, as a function of temperature, in a system of $4\times 4$ lattice sites with periodic boundary conditions. The ratio of hopping strength to super-exchange strength is $t/J=10/3$. While error bars are given for the WAMC data, they are smaller than the marker size. The data indicate a perfect agreement between WAMC and ED.