Tunable nanomagnetism in moderately cold fermions on optical lattices

Localized defects, unavoidable in real solids, may be simulated in (generically defect-free) cold-atom systems, e.g., via modifications of the optical lattice. We study the Hubbard model on a square lattice with single impurities, pairs of nearby impurities, or lines of impurities using numerically exact determinantal quantum Monte Carlo simulations. In all cases, correlations on the “impurity” sites are enhanced either by larger on-site interactions or by a reduced coupling to the environment. We find highly nontrivial magnetic correlations, which persist at elevated temperatures and should be accessible in cold-atom systems, with current experimental techniques. With improved cooling techniques, these features could be followed towards generic quantum antiferromagnetism in the homogeneous limit. More generally, tunable crossing points between different correlation functions could be used, in a quantum steelyard balance setup, as robust thermometers.

Figure 1

The effect of single or paired impurity sites with on-site interaction $U^{\prime }=8t$ introduced into the $U=4t$ medium at $T=0.25t$. Nearest-neighbor spin correlations $\langle {\mathbf{S}}_i·{\mathbf{S}}_j\rangle$ (first row) and double occupancy $D$ (second row) from DQMC calculations as compared to $\langle {\mathbf{S}}_i·{\mathbf{S}}_j\rangle$ (third row) and $D$ (fourth row) from RDMFT calculations. Impurity sites are marked with bright white dots.

Figure 2

First row: The sum of the effects of two single impurities $\sum \left(\Delta {\langle {\mathbf{S}}_i·{\mathbf{S}}_j\rangle }_0\right)$. For ease of comparison we consider the change in NN spin correlations with respect to the homogeneous impurity-free value ${\langle {\mathbf{S}}_i·{\mathbf{S}}_j\rangle }_{\mathrm{hom}}$: $\Delta \langle {\mathbf{S}}_i·{\mathbf{S}}_j\rangle =\langle {\mathbf{S}}_i·{\mathbf{S}}_j\rangle -{\langle {\mathbf{S}}_i·{\mathbf{S}}_j\rangle }_{\mathrm{hom}}$. Red (green) colors indicate enhanced (suppressed) spin correlations (relative to the impurity-free homogeneous value). Second row: Difference of $\Delta \langle {\mathbf{S}}_i·{\mathbf{S}}_j\rangle$ and $\sum \left(\Delta {\langle {\mathbf{S}}_i·{\mathbf{S}}_j\rangle }_0\right)$. All parameters are as in Fig. 1. Impurity sites are marked with bright white dots.

Figure 3

The effect of impurity sites introduced into a $U=4t$ medium at $T/t=0.25$ on nearest-neighbor spin correlations $\langle {\mathbf{S}}_i·{\mathbf{S}}_j\rangle$. First row: $U^{\prime }=8t=2U$, $t^{\prime }=t$; second row: $U^{\prime }=4t=U$, $t^{\prime }=0.5t$; third row: $U^{\prime }=4t=U$, $t^{\prime }=0.7t$. Red (green) colors indicate enhanced (suppressed) spin correlations (relative to the impurity-free homogeneous value). A red frame highlights the system with two impurities on neighboring lattice sites that is discussed in detail in the main text. Impurity sites are marked with bright white dots.

Figure 4

The effect of impurity sites with $U^{\prime }=4t=U$, $t_{\mathrm{imp},j}=t^{\prime }$ introduced into a medium with $U=4t$ at $T/t=0.25$ on NN spin correlations: $\langle {\mathbf{S}}_i·{\mathbf{S}}_j\rangle$ as a function of $t^{\prime }$ for a system with two impurities on the neighboring lattice sites as in the second column of Fig. 3 (dashed lines and open symbols) or single line of impurities (solid lines and solid symbols). The inset illustrates the choice of the bonds $i\text{−}j$ for the case of an impurity pair. Black solid line: NN spin correlations ${\langle {\mathbf{S}}_i·{\mathbf{S}}_j\rangle }_{\mathrm{hom}}$ for the homogeneous system with $U/t=4$, $T/t=0.25$. Dotted line: NN spin correlations for a homogeneous system with the parameters of the impurity (LDA), ${\langle {\mathbf{S}}_i·{\mathbf{S}}_j\rangle }_{\mathrm{LDA}}$.

Figure 5

Temperature dependence of impurity effects on NN spin correlations $\langle {\mathbf{S}}_i·{\mathbf{S}}_j\rangle$. First column: $U^{\prime }=8t=2U$, $t^{\prime }=t$; second column: $U^{\prime }=4t=U$, $t^{\prime }=0.5t$; third column: $U^{\prime }=4t=U$, $t^{\prime }=0.7t$. Amplified spin correlations (relative to the impurity-free homogeneous value for specific $T/t)$ are shown in red, and the suppressed ones in green. The data in the red frame are repeated from Fig. 3. Impurity sites are marked with bright white dots.

Figure 6

Temperature dependence of NN spin correlation functions $\langle {\mathbf{S}}_i·{\mathbf{S}}_j\rangle$ for a system with two impurities on neighboring lattice sites (solid symbols and dashed lines) or with a single line of impurities (open symbols and solid lines). $U^{\prime }=4t=U$, $t_{\mathrm{imp},j}=t^{\prime }$ with (a) $t^{\prime }=0.5t$ and (b) $t^{\prime }=0.7t$. Bonds are denoted as in Fig. 4. Black solid line: ${\langle {\mathbf{S}}_i·{\mathbf{S}}_j\rangle }_{\mathrm{hom}}\left(T\right)$ for the homogeneous impurity-free system with $U/t=4$. Gray dotted line: NN spin correlations ${\langle {\mathbf{S}}_i·{\mathbf{S}}_j\rangle }_{\mathrm{LDA}}$ for a homogeneous system with the parameters of the impurity (LDA). The arrows are pointing to the crossing temperatures $T_1$ and $T_2$ discussed in the main text. Inset: $T_1$ as a function of $t^{\prime }/t$.

Figure 7

(a) Excess entropy $\Delta S_{\mathrm{imp}}=S-S_{\mathrm{hom}}$ of a system with two NN impurities [$U^{\prime }=4t=U$, $t^{\prime }=0.7t$ (circles), $t^{\prime }=0.5t$ (squares), or $t^{\prime }/t\to 0$ (triangles)] vs temperature $T$. (b) Entropy $S_{\mathrm{at}}$ of an isolated “atomic” site $(U/t=4,t^{\prime }/t=0$; dashed-dotted line) and entropy per particle ${(S/N)}_{\mathrm{hom}}$ of the reference homogeneous impurity-free system with $U=4t$ (solid line); their difference defines the excess entropy $\Delta S_{\mathrm{at}}$ per decoupled site (dotted line). (c) Scaling of the excess entropy [data of (a)] with the relative bandwidth change $1-{(t^{\prime }/t)}^2$ of each of the two NN impurities; in the limit $t^{\prime }\to 0$ (black triangles), $\Delta S_{\mathrm{imp}}$ includes $2\Delta S_{\mathrm{at}}$ (two decoupled atomic sites) plus a surface contribution $\Delta S_{\mathrm{surf}}$ (shaded; with $N_{\mathrm{surf}}=6$ surface sites; cf. inset). (d) Excess entropy of a single-impurity system at $t^{\prime }=0$ (crosses); here, $\Delta S_{\mathrm{imp}}$ includes $\Delta S_{\mathrm{at}}$ (one decoupled atomic site) plus surface effects $\Delta S_{\mathrm{surf}}$ (with $N_{\mathrm{surf}}=4$ surface sites; cf. inset). (e) Surface contribution $\Delta S_{\mathrm{surf}}$ (per surface site $N_{\mathrm{surf}}$) to the excess entropy for systems with two NN impurities [triangles; cf. (a) and (c)] or a single impurity [crosses; cf. (d)].