Testing the Monte Carlo–mean field approximation in the one-band Hubbard model

The canonical one-band Hubbard model is studied using a computational method that mixes the Monte Carlo procedure with the mean field approximation. This technique allows us to incorporate thermal fluctuations and the development of short-range magnetic order above ordering temperatures, contrary to the crude finite-temperature Hartree-Fock approximation, which incorrectly predicts a Néel temperature $T_N$ that grows linearly with the Hubbard $U/t$. The effective model studied here contains quantum and classical degrees of freedom. It thus belongs to the “spin fermion” model family widely employed in other contexts. Using exact diagonalization, supplemented by the traveling cluster approximation, for the fermionic sector, and classical Monte Carlo for the classical fields, the Hubbard $U/t$ vs temperature $T/t$ phase diagram is studied employing large three- and two-dimensional clusters. We demonstrate that the method is capable of capturing the formation of local moments in the normal state without long-range order, the nonmonotonicity of $T_N$ with increasing $U/t$, the development of gaps and pseudogaps in the density of states, and the two-peak structure in the specific heat. Extensive comparisons with determinant quantum Monte Carlo results suggest that the present approach is qualitatively, and often quantitatively, accurate, particularly at intermediate and high temperatures. Finally, we study the Hubbard model including plaquette diagonal hopping (i.e., the $t\text{−}{t}^{\prime }$ Hubbard model) in two dimensions and show that our approach allows us to study low-temperature properties where determinant quantum Monte Carlo fails due to the fermion sign problem. Future applications of this method include multiorbital Hubbard models such as those needed for iron-based superconductors.

Figure 1

The $T/t-U/t$ phase diagram for the one-band Hubbard model. The solid red squares show the dependence of $T_N$ on $U/t$ obtained using the MCMF technique on $4^3$ clusters. The crosses are estimations of $T_N$ obtained from finite-size scaling. The AF-I region denotes the Néel type AFM phase with long-range order and insulating characteristics. The open squares are the $T_N$ obtained from the DQMC method, from Ref. [48]. The light blue region depicts the regime of preformed local moments above the AF-I phase. The dashed line shows the $T_N$ obtained from the simplistic Hartree-Fock calculation at finite temperature where the critical temperature incorrectly grows linearly with $U/t$ at large $U/t$. The determination of the crossover between the gray and blue regions, and the fact that the MCMF local moment region coincides with HF $T_N$ at temperatures much larger than typical $T_N$ scales, are discussed in the text.

Figure 2

(a) The magnetic structure factor $S\left(\mathbf{q}\right)$ for $\mathbf{q}=(\pi ,\pi ,\pi )$, obtained at various $U/t$'s as indicated. The data are from $4{}^3$ clusters with 4000 MC sweeps at every temperature, while cooling the system down from high to low temperatures, as described in Sec. 2. (b) The corresponding local moments $M$ vs temperature. We capture the feature that at large $U/t$ the peak in the moment size shifts to nonzero temperature. This effect, due to the setting of long-range order, was reported before in the DQMC studies of Ref. [51]. At small $U$ the overall shape is also in good agreement with the DQMC data, indicating that the MCMF method indeed captures the essence of the problem. Panel (c) shows the expectation value of double occupation for the various $U/t$'s indicated. The thin dashed line indicates a cutoff discussed in the text.

Figure 3

(a) Specific heat vs temperatures for different $U$ values. Two peak structures are observed: at large $U/t$ the high-temperature peak corresponds to the moment formation while the lower one corresponds to moment ordering. The inset shows the universal crossing of the different $C_v$ curves at $T/t\sim 2.0$. Panel (b) shows the position of the high-temperature peak varying $U$. The Hartree-Fock results are shown with open squares. At large $U$, beyond $6t$, these peak positions are close to the mean field results. At low $U$, the high-temperature peak position saturates to $1t$, while the low-$T$ peak approaches zero. The nonmerging of the two peaks in three dimensions is in agreement with DQMC data in two dimensions. The low-temperature peak corresponds to $T_N$ in Fig. 1. The open circles in (b) show the crossover temperature from the no local moments regime to a region of preformed moments as obtained from the data on double occupation shown in Fig. 2. In (a) the data shown are a smoothed version of the actual data to reduce statistical fluctuations.

Figure 4

Density of states $N\left(\omega \right)$ for (a) $U/t=12$ and (b) $U/t=4$ at the temperatures indicated in panel (a). At large $U/t=12$, the Hubbard gap is gradually filled up due to thermal fluctuations. The weight at $\omega -\mu =0$ monotonically increases with increasing temperature. A DOS pseudogap is seen above $T_N\sim 0.2t$. At a smaller coupling $U/t(=4)$, while the gap is filled similarly to the large $U$ case with the increase of temperature, above $T_N$ we observe a nonmonotonicity in the dependence of the zero-energy weight with temperature. (c) Magnitude of the auxiliary classical fields averaged over the lattice $\left(\langle \left|\mathbf{m}\right|\rangle \right)$ vs temperature for the $U$ values in (d). At large temperature the thermal fluctuations cause $\left(\langle \left|\mathbf{m}\right|\rangle \right)$ to grow linearly with temperature for all the $U$'s shown (inset). The reason for this temperature dependence of $\langle \left|\mathbf{m}\right|\rangle$ and its correlation with $N(\omega =0)$ is discussed in the text. (d) Shows the $N(\omega =0)$ feature remarked in panel (b) for different $U$ values. This nonmonotonicity was reported before in a DQMC study; see Ref. [51].

Figure 5

(a) Real-space spin-spin correlations $C\left(\right|\mathbf{r}\left|\right)$, for $\left|\mathbf{r}\right|=0,1,\sqrt{2},\sqrt{3},2$, at $T/t=0.4$, i.e., a temperature above $T_N$. See the text for the definition of $C\left(\right|\mathbf{r}\left|\right)$. The $\left|\mathbf{r}\right|=0$ curve corresponds to the square of the local moment and shows that the size of the preformed magnetic moment increases with $U/t$ and saturates beyond $U/t\sim 8$, i.e., where $T_N$ is maximized. The rest of the curves show the real-space AFM correlations, $\mathbf{q}=(\pi ,\pi ,\pi )$, among the moments. Again the correlations are the largest for $U/t\sim 8$. On the large $U$ side the decrease in the correlation results from thermal fluctuations competing with the AFM spin order stiffness which scales as $t^2/U$. (b) Shows the dependence of $C\left(\right|\mathbf{r}\left|\right)$ on temperature for large $U/t(=14)$. The magnitude of the moment is almost independent of the temperature, while there is a clear short-range AFM correlation between the moments at all temperatures shown. Longer range correlations for $\left|\mathbf{r}\right|>1$ are suppressed rapidly above $T_N\sim 0.12t$. (c) The real-space correlations $C\left(\right|\mathbf{r}|=1)$ using only the $x,y$, or $z$ components of the spin. The data confirm explicitly the rotational invariance expected to exist in $H_{\mathrm{eff}}$. The AFM structure factor is also displayed for comparison. Results are similar for $\left|\mathbf{r}\right|>1$ as well.

Figure 6

Comparison of results for an $8{}^2$ system at $U/t=6$, obtained using ED+MC (solid line) and TCA with two different sizes of traveling cluster sizes, $4^2$ (squares) and $8^2$ (triangles). Panel (a) shows $S(\pi ,\pi )$, (b) and (c) show the DOS, while (d) and (e) display the $P_q\left(\right|m\left|\right)$, for the three cases. The DOS and $P_q\left(\right|m\left|\right)$ are shown at low $T(=0.01t)$ and high $T(=0.5t)$ temperatures, as indicated.

Figure 7

Finite-size scaling analysis for the 3D case. Panel (a) shows $S(\pi ,\pi ,\pi )$ for $U/t=8$ for three system sizes obtained using TCA. To find $T_N$ in the thermodynamic limit we fit $T_N\left(L\right)$, the Néel temperature for a cluster of size $L^3$, against 1/$L$. Fitting to a scaling form (see text) provides $T_N^{\mathrm{Thermo}}$, the Néel temperature in the thermodynamic limit. As a typical example, in (b) we show the MCMF $T_N\left(L\right)$ data and the fit using the scaling form for $U=8t$. The dashed line is a guide to the eye. See text for discussion.

Figure 8

(a) $T_N$ vs $U/t$ and (b) representative $S(\pi ,\pi )$'s, both for a $32{}^2$ system. The results are obtained using a $4{}^2$ traveling cluster.

Figure 9

The specific-heat vs temperature data in two dimensions for various $U$ values. The twin peak structure and universal crossing are clearly seen. The corresponding loci of the high- and low-temperature peaks are shown in (b). Here we also show DQMC data for the high-temperature peaks. For comparison with DQMC, we show these data for a $6{}^2$ system size. The dashed line is a guide to the eye.

Figure 10

Spatial snapshots of $\left\{\right|{\mathbf{m}}_i\left|\right\}$ for $U/t=4$ (top) and $U/t=14$ (bottom). Panels (a)–(d) and (e)–(h) show the snapshots for $T/t=0.37$, 0.12, 0.08, and 0.01 for the two cases, respectively. At $U/t=14$ (bottom), values as high as $\left|\mathbf{m}\right|\sim 1.0$ exist at all temperatures shown, much above $T_N(\sim 0.06t)$. $\left|\mathbf{m}\right|$, however, grows with reducing temperature and shows thermal fluctuations at the higher temperatures for $U/t=4$ (top). In the figure, yellow implies $\left|\mathbf{m}\right|=1$ and black $\left|\mathbf{m}\right|=0$. The snapshots are for a ${32}^2$ system size.

Figure 11

$P_q\left(\right|m\left|\right)$ for $U/t=4$ and 14 at various temperatures indicated on the right. (a) At $U/t=4$ and high temperature, lattice sites acquire values of $\left|\mathbf{m}\right|$ between 0 and 1 in a uniform manner. This distribution starts peaking at $T/t\sim 0.08$ which is close to $T_N$. At lower $T$, the auxiliary field distribution peaks at about $\left|\mathbf{m}\right|=0.7$. (b) At large $U/t$, the $\left|\mathbf{m}\right|$ values are well defined moments (about 1) at all sites at the temperatures shown, much above $T_N$. There is only a small thermal broadening even at $T\sim 10T_N$.

Figure 12

DQMC results showing $S(\pi ,\pi )$ for several $U$'s at (a) $t^{\prime }=0$ and (b) $t^{\prime }=-0.3t$. The inset in (b) contains the average sign vs temperature. For $t^{\prime }=-0.3t$, sign error increases rapidly preventing the access to low temperatures. Panels (c) and (d) show the $S(\pi ,\pi )$ obtained from MCMF. The data shown here are for a $6{}^2$ system.

Figure 13

MCMF results for $t^{\prime }\ne 0$. Panel (a) contains the evolution of $T_N$ with increasing $t^{\prime }/t$ for different values of $U/t$ and for $\mathbf{q}=(\pi ,\pi )$ and $\mathbf{q}=(0,\pi )$. The case $\mathbf{q}=(\pi ,0)$ is identical to $\mathbf{q}=(0,\pi )$ and it is not shown. Panel (b) displays the typical $S(\pi ,\pi )$ for $U/t=16$ at various $t^{\prime }/t$, as indicated. Panels (c) and (d) show $N\left(\omega \right)$ for $U/t=4$ and $U/t=16$ respectively, at different values of $t^{\prime }/t$.