Theory of phonon-assisted forbidden optical transitions in spin-gap systems

We consider the absorption of light with emission of one $S_{tot}=1$ magnetic excitation in systems with a spin gap induced by quantum fluctuations. We argue that an electric-dipole transition is allowed on the condition that a virtual phonon instantaneously breaks the inversion symmetry. We derive an effective operator for the transition and argue that the proposed theory explains the polarized experiments in ${\text{CuGeO}}_3$ and ${\text{SrCu}}_2{\left({\text{BO}}_3\right)}_2$.

Figure 1

Three examples of distortions of the ${\text{Cu}}_2{\mathrm{O}}_2$ cluster with associated Dzyaloshinski-Moriya vectors. On the left, the motion along the $y$ axis creates a Dzyaloshinski-Moriya interaction whose vector is along $z$ [the mirror plane ${\text{Cu}}_2{\mathrm{O}}_2$ and the one perpendicular which has the O(2) atoms]; on the middle, the atoms move out of the plane and the Dzyaloshinski-Moriya vector is along $y$ (mirror plane $xz$ passing through the bond Cu-Cu); on the right, these distortions break the inversion center at the middle of the Cu-Cu bond. However, the two perpendicular mirror planes $xy$ and $xz$ imply that the Dzyaloshinski-Moriya interaction actually vanishes. For ${\text{CuGeO}}_3$, the $x$ axis is the $c$ axis and the $xy$ plane is the plane of the ${\text{CuO}}_2$ chains.

Figure 2

Excitation spectrum in ${\text{SrCu}}_2{\left({\text{BO}}_3\right)}_2$ for two directions of the external magnetic field (from Ref. 33). Definitions of the intensities of the optical transitions are also given.

Figure 3

Configuration $\mathbf{E}\left(t\right)\Vert \left(ab\right)·\mathbf{H}\Vert \left(ab\right)$. Intensity of the optical transitions from the ground state to the first split triplet state in ${\text{SrCu}}_2{\left({\text{BO}}_3\right)}_2$ in the electric-dipole approximation $\left(\mathit{\delta }\Vert c\right)$. $(+)+(-)$ is the sum of the intensity of the upper and lower mode and (0) is the middle mode. The theoretical curves are given by Eqs. (23, 24). There are no fitting parameters except an overall amplitude. The experimental data are from Ref. 18. Note the intensity of the $(+)+(-)$ mode is here twice as large as shown in the original experimental paper.[34].

Figure 4

Configuration $\mathbf{E}\left(t\right)\Vert c·\mathbf{H}\Vert \left(ab\right)$. Intensity of the optical transitions from the ground state to the first split triplet state in ${\text{SrCu}}_2{\left({\text{BO}}_3\right)}_2$ in the electric-dipole approximation $\left(\mathbf{\delta }\perp c\right)$. $(+)$, $(-)$, and (0) are, respectively, the upper, lower, and middle mode. The theoretical curves are given by Eqs. (32, 32). There are no fitting parameters except an overall amplitude. The experimental data are from Ref. 18.