Magnetism, transport, and thermodynamics in two-dimensional half-filled Hubbard superlattices

We study magnetic, transport, and thermodynamic properties of the half-filled two-dimensional $\left(2D\right)$ Hubbard model with layered distributed repulsive interactions using unbiased finite temperature quantum Monte Carlo simulations. Antiferromagnetic long-ranged correlations at $T=0$ are confirmed by means of the magnetic structure factor and the onset of short-ranged ones is at a minimum temperature, which can be obtained by peaks in susceptibility and specific heat following a random-phase-approximation (RPA) prediction. We also show that transport is affected in the large interaction limit and is enhanced in the nonrepulsive layers suggesting a change of dimensionality induced by increased interactions. Lastly, we show that by adiabatically switching the interactions in layered distributed patterns reduces the overall temperature of the system with a potential application in cooling protocols in cold atoms systems.

Figure 1

Cartoon of the regularly distributed on-site repulsive interactions for the $L_{\mathrm{U}}=2$ $L_0=1$ case in a $12\times 12$ lattice. Throughout this work, $x$ represents the direction along the layers and $y$ perpendicular to it.

Figure 2

Local moment profile throughout layers for different SL's with $L=12$ and $U/t=4$ at temperature $T/t=0.1$. Filled and empty symbols denote repulsive and free $(U=0)$ sites, respectively.

Figure 3

Local moment dependence with $U/t$ for different SL's with $L=12$ at temperature $T/t=0.1$. Filled and empty symbols denote repulsive and free $(U=0)$ sites, respectively.

Figure 4

(a) Local moment in free sites as a function of $U/t$ for different SL's with $L=12$ at temperature $T/t=0.1$. Empty symbols represent sites at the edge of the free layers whereas crossed symbols represent sites at the central line of the free layer: see cartoon in (b) with the example for the SL configuration $L_{\mathrm{U}}=1L_0=3$.

Figure 5

Nearest-neighbor spin-spin correlation function as a function of $U/t$ at $T/t=0.10$ for $L=12$ lattices. Correlations are calculated along (a) repulsive (filled symbols) and (b) free (empty symbols) layers, and (c) perpendicular to the layers, between a free and a repulsive site (half-filled symbols). Stars represent the corresponding homogeneous system. The dashed line represent the Heisenberg model nearest-neighbor spin-spin correlations QMC result [45].

Figure 6

(a) and (b) [(c) and (d)] display the local moment and NN spin-spin correlation function inside the repulsive [free] layer as a function of the temperature for three different SL structures. (e) also provides the temperature dependence but for the NN spin-spin correlation function where one site is at the repulsive layer and the other at the free one, computing, essentially, the coupling between the two types of layers.

Figure 7

Structure factor along a path in momentum space for different SL's with $L=12$, $U/t=4$, and $T/t=0.0625$. All SL's shown display a dominant $\mathbf{q}=(\pi ,\pi )$ related with an overall AF ordering in both directions neglecting the underlying SL structure.

Figure 8

AF structure factor vs inverse temperature $\beta$ for different SL's with $L=12$ and $U_{\mathrm{eff}}/t=4$. It is clearly seen that an effective $U$ model does not explain all the features of magnetism in SL's. Finite-size dependence of AF structure factor for the SL's $L_{\mathrm{U}}=1$, $L_0=1$, and various $U_{\mathrm{eff}}/t$. The linear extrapolation to limit $L\to \infty$ shows that the AF order is long ranged.

Figure 9

Staggered magnetization $m_{\mathrm{AF}}$ dependence with $U_{\mathrm{eff}}/t$ and four different SL's compared with homogeneous result obtained from Ref. [47]. The continuous line is obtained from RPA approximation [48] and the dashed one is the QMC Heisenberg [45] result.

Figure 10

Temperature dependence of the ratio of kinetic energies of superlattices to the noninteracting result. The lattice size is $12\times 12$ and the interactions in the repulsive sites is $U/t=8$.

Figure 11

(a) Effective hopping as a function of interaction strength $U/t$, for the homogeneous system and different superlattices with $L=12$ and $T/t=0.10$. (b) and (c) with the effective hopping contribution split along direction along and across layers, respectively.

Figure 12

Ratio between the $x$ component of the kinetic energy at a finite $U/t$ to the $U=0$ case for $L=12$ lattice with $T/t=0.10$ for different SLs. In (a), we plot the hopping contribution from the free layers and in (b) the same for the repulsive ones. The filled (empty) symbols denote electronic transport between repulsive (free) sites. To avoid misleading interpretation due to the different number of repulsive $\left(N_{\mathrm{U}}\right)$ and free $\left(N_{\mathrm{U}=0}\right)$ sites, in this case, we have normalized the results by the number of sites of each type. In (b), we also include the analytical results in the extreme limits of $U/t\ll 1$ (perturbation theory) and $U/t\gg 1$ (strong coupling) [49] and the contribution to the kinetic energy between repulsive sites smoothly interpolates between both limits albeit the layered distributed interactions.

Figure 13

Uniform spin susceptibility $\chi$ as a function of temperature for various SL's with $U/t=4$ and for homogeneous systems with $U/t=0$ (line) and 4 (stars) in $L=12$ lattices. The peak in this quantity defines $T_{\mathrm{spin}}$, the onset of antiferrogmagnetic fluctuations.

Figure 14

Temperature dependence of specific heat for four different SL configurations with $L=12$ and the corresponding homogeneous cases with $U/t=0$ and 4. While the high-temperature peak position is roughly constant at $T_{\mathrm{high}}\approx t$, the low-temperature one significantly varies for the SL's presented and displays increasing $T_{\mathrm{low}}$ as the $U_{\mathrm{eff}}$ becomes higher and closer to the homogeneous $U/t=4$ case. Lines are guides to the eye.

Figure 15

Position of the $\text{low-}T$ peak of the specific heat (full symbols) and peak of the spin-susceptibility (open symbols) as a function of $U_{\mathrm{eff}}/t$ for different SL's with $U/t=4$ together with the homogeneous lattice results. Dashed line corresponds to an RPA-like form for the temperature scale in which antiferromagnetic fluctuations occur: $T\propto texp[-2\pi \sqrt{t/U_{\mathrm{eff}}}]$.

Figure 16

Position of the $\text{high-}T$ peak of specific heat for different SL's as a function of $U_{\mathrm{eff}}$, the dashed line denotes the linear extrapolation to the data presented in the strong coupling limit $(T\sim U/4.1)$. Error bars denote the confidence interval of estimating the peak from the data of the specific heat.

Figure 17

Entropy as a function of temperature for fixed $L_{\mathrm{U}}=1$ and $L_0=3$ and $U/t=2$, 4, 8, and 16 (a) and for fixed $U/t=8$, and different SL configurations in a lattice with $L=8$. Dotted (dashed) lines represent adiabats with $S/\left(N{k}_{\mathrm{B}}\right)=0.8$ (0.5).

Figure 18

(a) and (c) [(b) and (d)] show the temperature dependence of the kinetic [potential] energy contributions to the specific heat for a $L=8$ lattice. (a) and (b) focus in a given SL with configuration $L_{\mathrm{U}}=1$ and $L_0=3$ and different interactions, while (c) and (d) compare the homogeneous result with a SL ($L_{\mathrm{U}}=1$ and $L_0=7$) for a given $U/t=8$. Dashed lines depict the noninteracting result.

Figure 19

Temperature as a function of $U_{\mathrm{eff}}/t$ for fixed $S/\left(N{k}_{\mathrm{B}}\right)=0.5$ (dashed lines) and 0.8 (dotted lines), for fixed $L_{\mathrm{U}}=1$ and $L_0=3$ (squares and up triangles) and for fixed $U/t=8$ (circles and down triangles).