Sign-Problem-Free Quantum Monte Carlo Study on Thermodynamic Properties and Magnetic Phase Transitions in Orbital-Active Itinerant Ferromagnets

The microscopic mechanism of itinerant ferromagnetism is a long-standing problem due to the lack of nonperturbative methods to handle strong magnetic fluctuations of itinerant electrons. We nonpertubatively study thermodynamic properties and magnetic phase transitions of a two-dimensional multiorbital Hubbard model exhibiting ferromagnetic ground states. Quantum Monte Carlo simulations are employed, which are proved in a wide density region free of the sign problem usually suffered by simulations for fermions. Both Hund’s coupling and electron itinerancy are essential for establishing the ferromagnetic coherence. No local magnetic moments exist in the system as a priori; nevertheless, the spin channel remains incoherent showing the Curie-Weiss-type spin magnetic susceptibility down to very low temperatures at which the charge channel is already coherent, exhibiting a weakly temperature-dependent compressibility. For the SU(2) invariant systems, the spin susceptibility further grows exponentially as approaching zero temperature in two dimensions. In the paramagnetic phase close to the Curie temperature, the momentum space Fermi distributions exhibit strong resemblance to those in the fully polarized state. The long-range ferromagnetic ordering appears when the symmetry is reduced to the Ising class, and the Curie temperature is accurately determined. These simulations provide helpful guidance to searching for novel ferromagnetic materials in both strongly correlated $d$-orbital transition-metal oxide layers and the $p$-orbital ultracold atom optical lattice systems.

Popular Summary

Itinerant ferromagnetism is one of the major challenges of contemporary condensed-matter physics. Unlike superconductivity, which has a well-controlled weak-coupling Bardeen-Cooper-Schrieffer theory as a starting point, itinerant ferromagnetism is an intrinsically strong correlation phenomenon. Ferromagnetism is favored by repulsive interactions, however, and even under very strong repulsions electrons typically still remain unpolarized because of the large kinetic energy cost required for spin polarizations. In particular, the finite temperature thermodynamic properties and magnetic phase transitions are long-standing problems of itinerant ferromagnetism characterized by strong dynamic fluctuations of ferromagnetic domains. These difficulties are typically beyond the scope of perturbative methods. We conduct a nonperturbative simulation study of itinerant ferromagnetism using sign-problem-free quantum Monte Carlo simulations for a wide range of electron densities.

The method of quantum Monte Carlo simulations is ideal for studying strongly correlated systems, and it produces asymptotically exact results. However, it typically suffers from the sign problem in fermion systems, which leads to uncontrollable numerical errors. Performing sign-problem-free quantum Monte Carlo simulations for itinerant fermion systems and ferromagnetic phase transitions is a significant challenge in strong correlation physics. We demonstrate a remarkable property of a multiorbital Hubbard model possessing ferromagnetic ground states: The sign problem is absent for all of the filling densities at any temperatures. This fact opens up new opportunities for well-controlled studies of the thermodynamics of itinerant ferromagnetism with asymptotic exactness. In particular, we find unambiguously that ferromagnetic metals exhibit spin incoherence, while simultaneously remaining coherent in the charge channel, without local moments. We perform critical scaling of magnetic ordering and accurately determine the Curie temperature in an itinerant model of ferromagnetism.

Our study provides a well-controlled reference point for the study of strongly correlated itinerant ferromagnetic systems. Our results provide guidance and benchmarks for the study of the mechanism of itinerant ferromagnetism and for current experimental efforts to search for novel itinerant ferromagnetic states in both condensed matter and ultracold atom systems.

Figure 1

(a) ${\chi }^{-1}(T)$ exhibits the CW law at different values of $n$. The inset shows interceptions corresponding to the mean-field value of Curie temperature $T_0$. (b) The compressibility $\kappa (T)$ at different values of $n$. The dashed lines represent $\kappa (T)$ of 1D spinless fermions at the same densities for comparison. Values of $n$ in (a) and (b) are represented by the same legend. $V=0$ and $J=2$ for both figures. The error bars of the QMC data are smaller than the symbols.

Figure 2

(a) ${\chi }^{-1}(T)$ and (b) $\kappa (T)$ at a large value of $V=8$ in the temperature regime of $1/6<T<1/2$. Different values of $n$ are shown in the legend in (b). At $n=1$, the system is in the Mott-insulating state, and thus $\kappa (T)$ drops to nearly zero at low temperatures. The error bars of the QMC data are smaller than the symbols.

Figure 3

(a) The density dependence of the Curie temperature $T_0(n)$ at $V=0$ and 8 with $J=2$, respectively. (b) The density dependence of the reduced Curie constant: $C/n$ versus $n$. The lower and upper bold lines represent the limits of the spin-$\frac{1}{2}$ and spin-1 moments, respectively. Plots are based on the results of $\chi (T)$ in Figs. 1 and 2.

Figure 4

The on-site particle number fluctuation $\delta$ defined in Eq. (10). The parameters are $\beta =6$, $J=2$, and $L=30$.

Figure 5

The momentum space distribution $n_F(k)$ ($0\le k\le \pi$) at $\beta =10$ and $n=1$, and $n_F(k)$ of the noninteracting spinless fermion (red dashed line) is plotted for comparison. The system size $L=50$. The parameter values are $V=0$ and $J=2$. The error bars of the QMC data are smaller than the symbol.

Figure 6

The SU(2) invariant model in the critical region: ${\chi }^{-1}(T)$ at $n=1$, $J=2$, and $V=0$. It crosses over from the CW law to the exponential form of $\chi (T)=A{e}^{b(T_0/T)}$. The inset shows the linear scaling of $\ln \chi (T)$ versus $\beta$.

Figure 7

The FM long-range ordering of the Ising symmetric model with parameters $J_{\perp }=2$, $J_{\parallel }=4$, $n=1$, and $V=0$. (a) The finite-size scaling of $S_{\perp }/L^2$. The FM critical temperature is extracted as $T_c=1/{\beta }_c=0.134\pm 0.002$, with ${\beta }_c\approx 7.4–7.5$. (b) The critical scaling: $S_{\perp }{L}^{-2+\eta }$ versus $T$, with $\eta =\frac{1}{4}$. The crossing of curves at different values of $L$ yields $T_c\approx 0.134$. (c) The data collapse of the scaling form $S_{\perp }(0)L^{-2+\eta }=f[(T-T_c)L^{1/\nu }]$ fitted by the parameters of $\nu =1$, $\eta =\frac{1}{4}$, and $T_c=0.1337$. The error bars are typically smaller than symbols.

Figure 8

The average of the fermion sign versus $t_{\perp }$ at different values of $L$. The parameters are $V=0$, $J=2$, $n=1$, and $\beta =6$.

Figure 9

${\chi }^{-1}(T)$ at $t_{\perp }=0$ and 0.02, respectively, with the system size $L=20$. Periodical boundary conditions are used. The parameters are $V=0$, $J=2$, $n=1$.

Figure 10

The chemical potential $\mu (T)$ is tuned to maintain the filling $n$ unchanged at all temperatures. $V=0$ in (a), and $V=8$ in (b). The system size is $L=8$.

Figure 11

The finite-size scalings for $\chi (T,L)$ versus $1/L$ with the fitting $\chi (T,L)=aL{e}^{-L/b}+\chi (T)$. Values of $\beta =1/T$ are presented in the legend. The parameter value at $V=0$ and $J=2$.

Figure 12

The finite -size scaling of $\chi (T,L)$ versus $1/L$ for $V=8$. Scaling ansatz $\chi (T,L)=aL{e}^{-L/b}+\chi (T)$ is used. The legend is the same as in Fig. 11.

Figure 13

The finite-size scaling of $\chi (T,L)$ at $n=1$, $V=0$, and $J=2$ in the low-temperature critical region with the scaling ansatz $\chi (T,L)=aL{e}^{-L/b}+\chi (T)$ used. Values of $\beta =1/T$ are presented in the legend.

Figure 14

The QMC results for the normalized on-site spin moment $⟨S_z^2⟩/n$ at $\beta =6$, $J=2$, and $L=30$. Values of $V$ are marked in the inset.

Figure 15

The momentum space occupation $n_F(k)$ at $\beta =10$, $V=0$, $J=2$, and $n=1$. Results of $n_F(k)$ for both the (a) antiperiodical boundary condition and (b) periodical boundary condition are presented at two different sizes of $L=30$ and 50.

Figure 16

The finite-size scaling of $S_{\text{orb}}(\pi ,\pi )/L^2$ with the parameter values $J_{\perp }=J_{\parallel }=2$, $n=1$, and $V=8$. The system sizes range from $L=10$ to 60, and the values of inverse temperature $\beta$ are presented in the legend. The orbital ordering occurs at $\beta$ between 7.2 and 7.6, which corresponds to $T_{\text{orb}}/t_{\parallel }\approx 0.132–0.139$.

Figure 17

(a) The finite-size scaling of $\chi (T,L)$ versus $1/L$ in the off-critical region. Hund’s rule coupling is modified as $J_{\perp }=2J_{\parallel }=4$. (b) The CW behavior of the extrapolated ${\chi }^{-1}(T)$. The interception on the temperature axis yields $T_0/t_{\parallel }=0.20\pm 0.01$.