Quantum Monte Carlo study of the two-dimensional ferromagnet

We present quantum Monte Carlo calculations that probe the paramagnet-ferromagnet phase transition in a two-dimensional Stoner Hamiltonian. With a screened Coulomb interaction we observe a first-order ferromagnetic transition for short screening lengths, and a second-order transition with a longer screening length, accompanied by a rising critical interaction strength. Finally, we discuss the consequences of our results for an ultracold atomic gas with finite ranged interactions.

Figure 1

The phase diagram for the system from the paramagnetic to ferromagnetic phase (shaded gray). The red solid line shows the first-order ferromagnetic transition and the dashed line the second-order transition with the tricritical point highlighted by the blue dot. The three lines denote boundaries with a screening length of $k_{\text{F}}^2{b}^2=0$, $k_{\text{F}}^2{b}^2=0.2$, and $k_{\text{F}}^2{b}^2=0.4$.

Figure 2

The error in energy $\Delta E$ for the $\mathbf{k}=\mathbf{0}$ (blue) and twist averaged (red) calculations with system size ($N$) for the screened Coulomb potential with $b=0$, $g/g_{\text{MF}}=0.5$. The red dotted line shows the $\sim 1/N^2$ scaling. The green dashed line shows the energy resolution required to determine the order of the ferromagnetic transition.

Figure 3

(a) and (b) Energy bands as a function of magnetization $M$ at the interaction strengths shown in (c) and (d) for the screened Coulomb potential (the lowest band is the weakest interaction strength). The red points highlight the minimum point in each energy band. The error bars are approximately the line width. (c) and (d) Magnetization as a function of interaction strength across the transition. The left-hand plots are at $k_{\text{F}}^2{b}^2=0$, and the right-hand plots at $k_{\text{F}}^2{b}^2=0.2$.

Figure 4

(a) Zero temperature phase behavior. Phase boundary for the emergence of ferromagnetism with varying interaction strength ($g/g_{\text{MF}}$, primary $x$ axis) and screening length ($k_{\text{F}}^2{b}^2$, primary $y$ axis) at zero temperature. The secondary $x$ axis and $y$ axis show the parameters relevant for cold atom gases. The solid line corresponds to a first-order transition, and the dashed line a second-order transition. The red line shows the boundary for the screened Coulomb interaction, and the blue line for the square potential. The magenta solid line shows the analytical result for the first-order transition with a screened Coulomb interaction. (b) Zero temperature first-order transition. The magnetization at the first-order transition as a function of Coulomb screening length calculated with QMC (red) and analytically (blue). The green line shows the minimum magnetization of a first-order transition that can be resolved within QMC. (c) Finite temperature phase behavior. The shift in the phase boundary interaction strength, $g-g_{T=0}$ from the screened Coulomb interaction zero temperature analytical phase boundary shown in [(a), magenta solid line] at finite temperatures $T\in \{0,0.1,0.2,0.3\}{T}_{\text{F}}$ calculated using the analytical formalism. The red solid lines denote a first-order transition and the dotted red line a second-order transition, and the green dots the tricritical points. The inset shows the tricritical point temperature with screening length.