Phase transitions and ordering structures of a model of a chiral helimagnet in three dimensions

Phase transitions in a classical Heisenberg spin model of a chiral helimagnet with the Dzyaloshinskii-Moriya interaction in three dimensions are numerically studied. By using the event-chain Monte Carlo algorithm recently developed for particle and continuous spin systems, we perform equilibrium Monte Carlo simulations for large systems up to about ${10}^6$ spins. Without magnetic fields, the system undergoes a continuous phase transition with critical exponents of the three-dimensional XY model, and a uniaxial periodic helical structure emerges in the low-temperature region. In the presence of a magnetic field perpendicular to the axis of the helical structure, it is found that there exists a critical point on the temperature and magnetic-field phase diagram and that above the critical point the system exhibits a phase transition with strong divergence of the specific heat and the uniform magnetic susceptibility.

Figure 1

A schematic picture of the chiral-soliton-lattice structure. Each arrow represents a spin. The magnetic field is in the upward direction of the figure.

Figure 2

Autocorrelation function $C_{{\left|\mathit{m}\left({\mathit{q}}_{\mathrm{chiral}}\right)\right|}^2}\left(t\right)$ of the square of the wave-vector-dependent magnetization ${\left|\mathit{m}\left({\mathit{q}}_{\mathrm{chiral}}\right)\right|}^2$ of the chiral helimagnetic model in three dimensions without magnetic fields. Open and filled symbols represent $C_{{\left|\mathit{m}\left({\mathit{q}}_{\mathrm{chiral}}\right)\right|}^2}\left(t\right)$ with and without the ECMC algorithm, respectively. The temperature is set to the critical temperature $T_{\mathrm{c}}$ estimated in Sec. 4a.

Figure 3

Temperature dependence of the specific heat $c$ of the chiral helimagnetic model in three dimensions without magnetic fields.

Figure 4

Wave-number dependence of ${\chi }^{\parallel }\left(q\right)$ of the three-dimensional chiral helimagnetic model without magnetic fields (a) for various system sizes at $T/J=1.680645$, which is close to the critical temperature, and (b) with $L=32$ at various temperatures above and below the critical temperature.

Figure 5

Temperature dependence of the finite-size correlation length ${\xi }_L\left(q_{\mathrm{chiral}}\right)$ divided by $\alpha L$ of the three-dimensional chiral helimagnetic model without magnetic fields. The inset presents an enlarged view around the critical temperature.

Figure 6

A finite-size scaling plot of the finite-size correlation length ${\xi }_L^{\parallel }\left(q_{\mathrm{chiral}}\right)$ divided by $\alpha L$ of the three-dimensional chiral helimagnetic model without magnetic fields. The smallest system size of this FSS plot is $L_{\min }=16$. The critical temperature $T_{\mathrm{c}}$ and the critical exponent $\nu$ are estimated as $T_{\mathrm{c}}/J=1.68672\left(4\right)$ and $\nu =0.676\left(3\right)$, respectively.

Figure 7

(a) Temperature dependence of the wave-vector-dependent magnetic susceptibility ${\chi }^{\parallel }\left(q_{\mathrm{chiral}}\right)$ of the chiral helimagnetic model in three dimensions without magnetic fields. (b) A finite-size scaling plot of ${\chi }^{\parallel }\left(q_{\mathrm{chiral}}\right)$ of the chiral helimagnetic model in three dimensions without magnetic fields. The value of the critical temperature $T_{\mathrm{c}}$ estimated by the finite-size scaling analysis of the finite-size correlation length ratio ${\xi }_L^{\parallel }\left(q_{\mathrm{chiral}}\right)/\alpha L$ is used.

Figure 8

Wave-number dependence of ${\chi }^{\parallel }\left(\mathit{q}\right)$ of the chiral helimagnetic model in three dimensions for various temperatures with $L=32$. The values of the magnetic fields perpendicular to the DM vector are (a) $h/J=0.1$, (b) $h/J=0.2$, and (c) $h/J=0.3$. The vertical line represents $q_{\mathrm{chiral}}/2\pi$.

Figure 9

Wave-number dependence of ${\chi }^{\parallel }\left(\mathit{q}\right)$ of the chiral helimagnetic model in three dimensions for various system sizes near the estimated transition temperature depending on the magnetic field. The values of the magnetic fields perpendicular to the DM vector are (a) $h/J=0.1$, (b) $h/J=0.2$, and (c) $h/J=0.3$.

Figure 10

System-size dependence of the peak value of the susceptibility ${\chi }_{*}^{\parallel }\left(0\right)$ (a) and the specific heat $c_{*}$ (b) of the chiral helimagnetic model in three dimensions with a magnetic field perpendicular to the DM vector $h/J=0$, 0.1, 0.2, and 0.3. The black dotted lines are proportional to $L^3$. The insets show enlarged views.

Figure 11

Temperature dependence of specific heat $c$ of the chiral helimagnetic model in three dimensions with a magnetic field perpendicular to the DM vector $h/J=0.3$.

Figure 12

The energy-density distribution function $P\left(e\right)$ of the chiral helimagnetic model in three dimensions with a magnetic field perpendicular to the DM vector $h/J=0.3$. The system size $L=64$ is the largest size in our simulations and the temperatures are close to the transition temperature.

Figure 13

A possible magnetic phase diagram of the chiral helimagnetic model in three dimensions. In the phase diagram, “CSL” and “P” denote the chiral soliton lattice phase and paramagnetic phase, respectively. The filled squares are the estimated transition temperature in this work and the circle represents an expected critical point whose precise location is not determined.