Scaling analysis and instantons for thermally assisted tunneling and quantum Monte Carlo simulations

We develop an instantonic calculus to derive an analytical expression for the thermally assisted tunneling decay rate of a metastable state in a fully connected quantum spin model. The tunneling decay problem can be mapped onto the Kramers escape problem of a classical random dynamical field. This dynamical field is simulated efficiently by path-integral quantum Monte Carlo (QMC). We show analytically that the exponential scaling with the number of spins of the thermally assisted quantum tunneling rate and the escape rate of the QMC process are identical. We relate this effect to the existence of a dominant instantonic tunneling path. The instanton trajectory is described by nonlinear dynamical mean-field theory equations for a single-site magnetization vector, which we solve exactly. Finally, we derive scaling relations for the “spiky” barrier shape when the spin tunneling and QMC rates scale polynomially with the number of spins $N$ while a purely classical over-the-barrier activation rate scales exponentially with $N$.

Figure 1

(Left) Plot of the effective potential $U_{\ell }\left(m\right)$ as a function of the magnetization $m$ for the Curie-Weiss model (12) for $\ell =1, \mathrm{\Gamma }=0.4$, and $h=0.015$. The red line depicts the tunneling path (instanton) at zero temperature. Barrier energies above the instanton energy are shown with blue filling. (Right) Plots of the effective potential for the Curie-Weiss model at different values of $\ell$ shown with green lines with $h=0.1$ and $\mathrm{\Gamma }=0.21$. The colored dashed lines correspond to the local minimum (blue), the maximum (red), and the global minimum (black) of $U_{\ell }\left(m\right)$ for different values of $\ell$. The total spin parameter $\ell$ changes in the range from ${\ell }_c$ to 1 where ${\ell }_c={(h^{2/3}+{\mathrm{\Gamma }}^{2/3})}^{3/2}$ is the smallest value of $\ell$ beyond which the effective potential is monostable. The domain of the effective potential for each $\ell$ is $m\in (-\ell ,\ell )$. The inset in the right figure shows the plot of interaction energy density $-g\left(m\right)$. We see by comparison of figures in left and right panels that for a given $\ell$ the turning point at the barrier exit satisfies the condition $a_1\le \ell$ where the equality is reached only for $\ell =1$.

Figure 2

Schematic illustration of the stochastic Kramers escape paths $m(\tau ,t)$ depicted with blue lines. The most probable escape path is shown with a solid blue line. It starts near $m_0$ and arrives at $m_1$ crossing the vicinity of saddle point $m_z\left(\tau \right)$ shown with a red dot. The dashed line depicts the boundary of the region where $\mathcal{F}\left[m\left(\tau \right)\right]\le \mathcal{F}\left[m_z\left(\tau \right)\right]$. Inside this region $\mathcal{F}\left[m\left(\tau \right)\right]-\mathcal{F}\left[m_z\left(\tau \right)\right]\gg T=\hslash /\beta$.

Figure 3

Dependence on the inverse temperature $\beta$ of the the scaling exponent $\alpha$ ($y$ axis) in the analytical expression for the tunneling decay rate (20) and the scaling exponent for the QMC escape rate obtained via continuous path-integral quantum Monte Carlo simulations. Both results are obtained for the the Curie-Weiss model (12) for different values of transverse field $\mathrm{\Gamma }$ and bias $h$. The $x$ axis corresponds to different values for the inverse temperature $\beta$. (a) Different colors correspond to different values of the transverse field $\mathrm{\Gamma }\in \{0.4,0.5,0.6\}$ for zero bias $h=0$. (b) Different colors correspond to different values of $\mathrm{\Gamma }\in \{0.3,0.4,0.5\}$ with $h=0.1$. In both figures, the $·$ symbols with dashed lines correspond to the analytical values while the $\times$ symbols with dotted lines correspond to numerical values. Error bars correspond to the numerical fitting of the exponent $\alpha$ in the numerical data.

Figure 4

Cartoon of the energy function $-g\left(m\right)$ for a problem with small and narrow barrier. The barrier dimensions obey $\mathrm{\Delta }m\ll \mathrm{\Delta }g\ll 1$ [cf. (84)].