Semiclassical analysis of spin systems near critical energies

The spectral properties of su(2) Hamiltonians are studied for energies near the critical classical energy ${\epsilon }_c$ for which the corresponding classical dynamics presents hyperbolic points. A general method leading to an algebraic relation for eigenvalues in the vicinity of ${\epsilon }_c$ is obtained in the thermodynamic limit, when the semiclassical parameter $n^{-1}={\left(2s\right)}^{-1}$ goes to zero (where $s$ is the total spin of the system). Two applications of this method are given and compared with numerics. Matrix elements of observables, computed between states with energy near ${\epsilon }_c$, are also computed and shown to be in agreement with the numerical results.

Figure 1

Phase space portrait of a classical trajectory ${\mathcal{C}}_0$ (full lines) describing a critical orbit passing through two HP ${\overline{\alpha }}_i$ and ${\overline{\alpha }}_j$. For $O\left(n^{-1/2}\right)<\left|\overline{\alpha }-{\overline{\alpha }}_i\right|<O\left(n^0\right)$ both the linearized solutions around HP point and the WKB solutions coexist (dark gray region), permitting to identify both asymptotic behaviors. The “in” and “out” solutions are connected via the $T^{\left(i\right)}$ matrices. Branch cuts of the WKB solutions are displayed as broken lines.

Figure 2

(Color online) Homoclinic case (up): $h=1;{\gamma }_x=4;{\gamma }_y=1/4;\mu =5$. Heteroclinic case (down): $h=1;{\gamma }_x=5;{\gamma }_y=2;\mu =6$. Left: comparison between the zeros of $D$ (blue line) and the eigenvalues of Eq. (20) computed numerically (black dots) for $n=500$. We define the renormalized energy $\eta =-\frac{h+{\epsilon }_1}{2\sqrt{\left({\gamma }_x-h\right)\left(h-{\gamma }_y\right)}}$; $\eta =\frac{{\eta }_1+{\eta }_2}{2}=\frac{-\left(\lambda +{\gamma }_y\right)\sqrt{{\gamma }_y}}{2\sqrt{\left({\gamma }_x-{\gamma }_y\right)\left({\gamma }_y^2-h^2\right)}}$, respectively, for the homoclinic and heteroclinic cases. Middle: stereographic projection of the critical classical orbit. Right: critical orbit on the Riemann Sphere, the zeroes of $\Psi \left(\overline{\alpha }\right)$ (black dots) are plotted for $n=120$, in the semiclassical limit they condense in branch cuts of $G_0$ [8, 22].

Figure 3

(Color online) Matrix elements of the operator ${\widehat{s}}_z=\frac{{\widehat{S}}_z}{s}$ between two states near the critical energy. ${\epsilon }^{\left(c\right)}$ is chosen to be the energy closest to the critical classical energy ${\epsilon }_c$. The agreement of the numerical data (dots) with the predictions of Eq. (25) (circles) gets better for $n$ big. The logarithmic downward shift as $n$ increases is due to the fact that $\mu \left(i\right)\sim {\text{ln}}^{-1}n$.