Magnetic phase transition in disordered interacting Dirac fermion systems via the Zeeman field

Using the determinant quantum Monte Carlo method, we investigate the antiferromagnetic phase transition that is induced by the Zeeman field in a disordered interacting two-dimensional Dirac fermion system. At a fixed interaction strength $U$, the antiferromagnetic correlation is enhanced as the magnetic field increases and, when the magnetic field is larger than a $B_c\left(U\right)$, the antiferromagnetic correlation shall be suppressed by the increased magnetic field. The impact of Zeeman field $B$, Coulomb repulsion $U$, and disorder $\mathrm{\Delta }$ is not isolated. The intensity of magnetic field effect on the antiferromagnetic correlation shall be strongly suppressed by disorder. Differently, it will be promoted by weak interaction, but, when $U$ becomes larger than $U_c=4.5$, the increased interaction will suppress the intensity of this effect and here $U_c=4.5$ coincides with the critical strength inducing the metal-Mott insulator transition in a clean system. Moreover, at a fixed magnetic field $B$, strong interaction shall suppress the antiferromagnetic phase rather than promote it.

Figure 1

Staggered transverse antiferromagnetic (AFM) structure factor $S_{\mathrm{AFM}}^{\mathrm{xx}}$. (a) As a function of the lattice size $L$ for various values of the magnetic field $B$. As the magnetic field increases, $S_{\mathrm{AFM}}^{\mathrm{xx}}$ is increased at each $L$. As the curve intercept gradually increases from zero to positive, the system reaches the AFM phase. The critical strength for the magnetic phase transition is approximately $B=0.45$. (b) As a function of $B$ for various values of $L$. The pink curve, which represents $L=\infty$, namely, $1/L=0$, is made up of intercepts that were obtained by $S_{\mathrm{AFM}}^{\mathrm{xx}}\left(L\right)$ curve fitting under each magnetic field. This curve shows the appearance/disappearance of the AFM phase. The induction of the AFM phase requires a large value of $B$ and will be eliminated by the magnetic field if its strength continues to increase. Calculations are performed on $2\times L\times L$ lattices for $U=3.0$ and $\mathrm{\Delta }=0.0$.

Figure 2

(a) Staggered transverse AFM structure factor $S_{\mathrm{AFM}}^{\mathrm{xx}}$ as a function of the spin polarization $P$ at various values of $L$. With $P$ as the abscissa, the curve has a more dramatic change. (b) $P$ as a function of $B$, where the function relationship depends on the interaction $U$: the $P\left(B\right)$ curve under a large value of $U$ enters the ascending stage faster and reaches 1 (the fully spin-polarized state) first. (c) $S_{\mathrm{AFM}}^{\mathrm{xx}}$ as a function of $L$ for various values of the inverse temperature $\beta$. The temperature effect is exhibited as the AFM phase is eliminated by the large $T$, which is accompanied by the disappearance of the intercept. (d) $S_{\mathrm{AFM}}^{\mathrm{xx}}$ as a function of $B$ for various values of $\beta$. The influence of $T$ is obvious as $S_{\mathrm{AFM}}^{\mathrm{xx}}$ increases with increasing $\beta$ under each $B$.

Figure 3

Staggered longitudinal antiferromagnetic structure factor $S_{\mathrm{AFM}}^{\mathrm{zz}}$ in the clean system. (a) As a function of $B$ at various values of $U$. $S_{\mathrm{AFM}}^{\mathrm{zz}}$ are effectively suppressed and reduced to 0 by the magnetic field. (b) As a function of $L$ for various values of $B$. The suppression of $B$ in the AFM phase is more clearly shown through the intercept of the $S_{\mathrm{AFM}}^{\mathrm{zz}}\left(L\right)$ curve, which quickly drops to 0 even under a weak magnetic field.

Figure 4

$S_{\mathrm{AFM}}^{\mathrm{xx}}$ as a function of $B$ at $L=12, \mathrm{\Delta }=0.0$, and $U$ in the following ranges: (a) $U=1.0$ to 4.5. Each $S_{\mathrm{AFM}}^{\mathrm{xx}}\left(B\right)$ curve has a special magnetic field strength $B_c$ for the peak value $S_{max}$. As $U$ increases from 0 to 4.5, $S_{max}$ always increases, which is accompanied by a decrease in $B_c$. (b) $U=$ 4.5 to 6.0. When the increased interaction exceeds 4.5, although $B_c$ continues to decrease, $S_{max}$ has a decreasing trend, in contrast to the previous stage. The point that separates the two stages is at $U=4.5$, which is the critical value for the interaction-induced AFM phase transition. (c) $S_{\mathrm{AFM}}^{\mathrm{xx}}$ as a function of $L$ at $\mathrm{\Delta }=0.0$. At $B=0.6$, starting from the AFM phase at $U=3.0$, the effect of the interaction on the AFM phase presents a transition from prompting to inhibiting (the intercept of the $S_{\mathrm{AFM}}^{\mathrm{xx}}$ curve initially increases and subsequently decreases). (d) $S_{\mathrm{AFM}}^{\mathrm{xx}}$ as a function of $U$ at various values of $B$. As $B$ increases, the effect of the interaction promoting the AFM phase is gradually reversed. When $B\ne 0$, the $S_{\mathrm{AFM}}^{\mathrm{xx}}\left(U\right)$ curve shows a trend of rising initially and subsequently falling instead of a monotonic increase.

Figure 5

(a) $S_{\mathrm{AFM}}^{\mathrm{xx}}$ as a function of $B$ for various values of $\mathrm{\Delta }$ at $U=3.0$ and $L=12$. As $\mathrm{\Delta }$ increases from 0, $S_{\mathrm{AFM}}^{\mathrm{xx}}$ is suppressed, and the peak of the curve gradually disappears. When $\mathrm{\Delta }=1.2$, the magnetic field has almost no effect on $S_{\mathrm{AFM}}^{\mathrm{xx}}$. (b) $S_{\mathrm{AFM}}^{\mathrm{xx}}$ as a function of $\mathrm{\Delta }$ for various values of $L$. Starting from the AFM phase at $U=3.0, \mathrm{\Delta }=0.0$, and $B=0.6$ in Fig. 1, an increase in $\mathrm{\Delta }$ leads to a decrease in $S_{\mathrm{AFM}}^{\mathrm{xx}}$ under each $L$; namely, disorder fully suppresses the AFM phase.

Figure 6

$S_{\mathrm{AFM}}^{\mathrm{xx}}$ as a function of $B$ for various values of $L$. The pink curve, which represents $L=\infty$, is the combination of the values to which $S_{\mathrm{AFM}}^{\mathrm{xx}}-1/L$ curves (under each value of $B$) tend when $1/L=0$. Therefore, its decline due to increasing $\mathrm{\Delta }$ intuitively represents the suppression of the AFM phase by disorder. Calculations are performed at $U=3.0$ and $\mathrm{\Delta }=$ 0.3 (a), 0.6 (b), 0.9 (c), and 1.2 (d).

Figure 7

$S_{\mathrm{AFM}}^{\mathrm{xx}}$ as a function of $L$ for various values of $\mathrm{\Delta }$ at $U=3.0$. The magnetic fields in panels (a), (b), (c), and (d) are 0.0, 0.3, 0.6, and 0.9, respectively, and the systems without $\mathrm{\Delta }$ are the metallic phase (a), band insulating phase (b), and AFM phase (c),(d), respectively. The disorder effect only appears under a strong enough $B$, which is manifested as a decrease in the intercepts of curves.

Figure 8

Conductivity ${\sigma }_{\mathrm{dc}}$ is shown as a function of (a) temperature $T$ and (b) magnetic field $B$ at half filling. Conductivity has different behaviors with the magnetic field greater/less than the critical value $B_c$ for the phase transition and $B_c$ at $U=3.0,\mathrm{\Delta }=0$ is about 0.14.

Figure 9

$S_{\mathrm{AFM}}^{\mathrm{xx}}$ computed in the system with $\beta =12, U=5, \mathrm{\Delta }=0.5$, and $B=0.5$. For a specified value of $L$, the data that are obtained from an ensemble with an increasing number of disorder realizations are consistent within the statistical errors.