Many-body Landau-Zener effect at fast sweep

The asymptotic staying probability $P$ in the Landau-Zener effect with interaction is analytically investigated at fast sweep $\epsilon =\pi {\Delta }^2∕\left(2\hslash v\right)⪡1$. We have rigorously calculated the value of $I_0$ in the expansion $P\cong 1-\epsilon +{\epsilon }^2∕2+{\epsilon }^2{I}_0$ for arbitrary couplings and relative resonance shifts of individual tunneling particles. The results essentially differ from those of the mean-field approximation. It is shown that strong long-range interactions such as dipole-dipole interaction generate huge values of $I_0$ because a flip of one particle strongly influences many others. However, in the presence of strong static disorder, making the resonances for individual particles shifted with respect to each other, the influence of interactions is strongly reduced. In molecular magnets the main source of static disorder is the coupling to nuclear spins. Our calculations using the actual shape of the ${\mathrm{Fe}}_8$ crystal studied in the Landau-Zener experiments [Wernsdorfer et al., Europhys. Lett. 50, 552 (2000)] yield a value of $I_0$ that is in good agreement with the value extracted from the experimental data.

Figure 1

A pair of tunnel-split levels vs energy bias $W\left(t\right)$. Here $E_{\downarrow }$ and $E_{\uparrow }$ are the bare energy levels $(\Delta =0)$, whereas $E_{\pm }$ are the exact adiabatic energy levels of Eq. (4). $P$ denotes the probability to remain in the (bare) state ∣↓⟩ after crossing the resonance.

Figure 2

Bare energy levels vs the sweep field $W$ for a ring of ten two-level systems with $V_i=0$, coupled by an antiferromagnetic nearest-neighbor interaction. All the lines except the two with the highest slopes, which correspond to the states with all spins up and all spins down, are strongly degenerate. The energy levels for the ferromagnetic coupling can be obtained by the upside-down transformation of this graph.

Figure 3

(Color online) $I_{ij}^{\left(\mathrm{sym}\right)}$ vs its independent arguments ${\beta }_{ij}$ and ${\tilde{\delta }}_{ij}$.

Figure 4

$I_{ij}^{\left(\mathrm{sym}\right)}$ as a function of the resonance shift ${\tilde{\delta }}_{ij}$ for ${\beta }_{ij}=\pm 5$.

Figure 5

${\overline{I}}_{ij}^{\left(\mathrm{sym}\right)}$ as a function of $\beta$ for different strengths of static disorder $\sigma$.

Figure 6

$G$ of Eq. (69) for the sphere vs width of distribution of individual resonances $\sigma$. Dashed lines on the left and right are asymptotes of Eqs. (73, 74), respectively.

Figure 7

(Color online) Dimensionless macroscopic internal field $D_{\mathbf{m}\mathbf{m}}$ of Eq. (88) in a sample of rectangular shape with $L_x=L_y=1$ and $L_z=3$, uniformly magnetized along the $z$ axis, plotted vs $x$ and $z$ for $y=0$.

Figure 8

Replotting experimental results of Ref. 11 vs $\epsilon$. (a) Staying probability $P$. (b) ${\Delta }^{\mathrm{eff}}$ defined by Eq. (9). Fitting with Eq. (10) at $\epsilon ⪡1$ yields $I_0\simeq 11$.