Dynamics of order parameters of nonstoquastic Hamiltonians in the adaptive quantum Monte Carlo method

We derive macroscopically deterministic flow equations with regard to the order parameters of the ferromagnetic $p$-spin model with infinite-range interactions. The $p$-spin model has a first-order phase transition for $p>2$. In the case of $p\ge 5$, the $p$-spin model with antiferromagnetic $XX$ interaction has a second-order phase transition in a certain region. In this case, however, the model becomes a nonstoquastic Hamiltonian, resulting in a negative sign problem. To simulate the $p$-spin model with antiferromagnetic $XX$ interaction, we utilize the adaptive quantum Monte Carlo method. By using this method, we can regard the effect of the antiferromagnetic $XX$ interaction as fluctuations of the transverse magnetic field. A previous study [J. Inoue, J. Phys. Conf. Ser. 233, 012010 (2010)] derived deterministic flow equations of the order parameters in the quantum Monte Carlo method. In this study, we derive macroscopically deterministic flow equations for the magnetization and transverse magnetization from the master equation in the adaptive quantum Monte Carlo method. Under the Suzuki-Trotter decomposition, we consider the Glauber-type stochastic process. We solve these differential equations by using the Runge-Kutta method, and we verify that these results are consistent with the saddle-point solution of mean-field theory. Finally, we analyze the stability of the equilibrium solutions obtained by the differential equations.

Figure 1

Dynamics of order parameters without antiferromagnetic $XX$ interaction $\gamma =0$, given initial magnetization $m_0=0.2,0.4,0.6,0.8,1.0$ and initial transverse magnetization $m_x=\sqrt{1-m_0^2}$. The horizontal axis denotes the time $t$ of the deterministic flow equation, and the vertical axis denotes the longitudinal magnetization. The experimental settings are (a) $\mathrm{\Gamma }=0.5$, (b) $\mathrm{\Gamma }=1.3$, and (c) $\mathrm{\Gamma }=2.5$.

Figure 2

Dynamics of order parameters without antiferromagnetic $XX$ interaction, given initial magnetization $m_0=0.2,0.4,0.6,0.8,1.0$ and initial transverse magnetization $m_x=\sqrt{1-m_0^2}$. The horizontal axis denotes the time $t$ of the deterministic flow equation, and the vertical axis is the transverse magnetization. The experimental settings are the same as those in Fig. 1.

Figure 3

Landscape of the pseudo free energy (30) without antiferromagnetic $XX$ interaction. The horizontal axis is the longitudinal magnetization, and the vertical axis denotes the pseudo free energy. The experimental settings are the same as those for Figs. 1 and 2.

Figure 4

Order parameters without antiferromagnetic $XX$ interaction from the master equation and from the saddle-point equations. The figure on the left shows the longitudinal magnetization, and that on the right shows the transverse magnetization. The vertical axis denotes these order parameters. The horizontal axis denotes the strength of the transverse magnetic field. The circle denotes the solution obtained by the master equation, whereas the solid line denotes the solution obtained by the saddle-point equations. The blue dashed line denotes the critical point where the pseudo free energy takes the same value. The green dotted line denotes the spinodal point. (a) Longitudinal magnetization, (b) transverse magnetization.

Figure 5

Dynamics of order parameters with antiferromagnetic $XX$ interaction $\gamma =18$, given initial magnetization $m_0=0.2,0.4,0.6,0.8,1.0$ and initial transverse magnetization $m_x=\sqrt{1-m_0^2}$. The horizontal axis denotes the time $t$ of the deterministic flow equation, and the vertical axis denotes the longitudinal magnetization. The experimental settings are (a) $\mathrm{\Gamma }=5$, (b) $\mathrm{\Gamma }=15$, and (c) $\mathrm{\Gamma }=25$.

Figure 6

Dynamics of order parameters with antiferromagnetic $XX$ interaction, given initial magnetization $m_0=0.2,0.4,0.6,0.8,1.0$ and initial transverse magnetization $m_x=\sqrt{1-m_0^2}$. The horizontal axis denotes the time $t$ of the deterministic flow equation, and the vertical axis is the transverse magnetization. The experimental settings are the same as those in Fig. 5.

Figure 7

Landscape of the pseudo free energy (30) with antiferromagnetic $XX$ interaction. The horizontal axis is the longitudinal magnetization. The vertical axis is the pseudo free energy. The experimental settings are the same as those in Figs. 5 and 6.

Figure 8

Order parameters with antiferromagnetic $XX$ interaction from the master equation and from the saddle-point equations. The figure on the left shows the longitudinal magnetization, and that on the right shows the transverse magnetization. The vertical axis denotes these order parameters. The horizontal axis denotes the strength of the transverse magnetic field. The circle and the solid line denote what they do in Fig. 4. (a) Longitudinal magnetization, (b) transverse magnetization.

Figure 9

Trace of Jacobian matrix, determinant, and condition with real roots of eigenvalues utilizing the solutions $(m^{*},m_x^{*})$ of Eqs. (22) and (23). The horizontal axis denotes the strength of the transverse magnetic field. In (a) and (b) without antiferromagnetic $XX$ interaction $\gamma =0$, the solutions do and do not include the metastable solutions, respectively. In (c), we consider the case with antiferromagnetic $XX$ interaction $\gamma =18$. (a) With no metastable solutions in $\gamma =0$; (b) with metastable solutions in $\gamma =0$ and (c) $\gamma =18$.