Dissipative dynamics of topological defects in frustrated Heisenberg spin systems

We study the dynamics of topological defects of a frustrated spin system displaying spiral order. As a starting point we consider the $SO\left(3\right)$ nonlinear sigma model to describe long-wavelength fluctuations around the noncollinear spiral state. Besides the usual spin-wave magnetic excitations, the model allows for topologically nontrivial static solutions of the equations of motion, associated with the change of chirality (clockwise or counterclockwise) of the spiral. We consider two types of these topological defects, single vortices and vortex-antivortex pairs, and quantize the corresponding solutions by generalizing the semiclassical approach to a non-Abelian field theory. The use of the collective coordinates allows us to represent the defect as a particle coupled to a bath of harmonic oscillators, which can be integrated out employing the Feynman-Vernon path-integral formalism. The resulting effective action for the defect indicates that its motion is damped due to the scattering by the magnons. We derive a general expression for the damping coefficient of the defect, and evaluate its temperature dependence in both cases, for a single vortex and for a vortex-antivortex pair. Finally, we consider an application of the model for cuprates, where a spiral state has been argued to be realized in the spin-glass regime. By assuming that the defect motion contributes to the dissipative dynamics of the charges, we can compare our results with the measured inverse mobility in a wide range of temperature. The relatively good agreement between our calculations and the experiments confirms the possible relevance of an incommensurate spiral order for lightly doped cuprates.

Figure 1

Incommensurate magnetic response for the spiral spin modulation (1). The magnetic susceptibility corresponding to the spiral order with the wave vector ${\mathbf{k}}_s$ exhibits two peaks at the points $(\pi ∕a,\pi ∕a)\pm \mathbf{Q}$ marked by a circle in the figure. In this case $\mathbf{Q}$ has finite components in both the $x$ and $y$ directions, and the distance between the peaks is twice the modulus of $\mathbf{Q}$.

Figure 2

Spin background corresponding to Eq. (1) and to the case depicted in Fig. 1. Here, a value of $\mathbf{Q}=(2\pi ∕20a,-2\pi ∕20a)$ has been chosen.

Figure 3

Spin background corresponding to the one-vortex solution of Eq. (10). The center of the vortex is marked by the circle. The spiral incommensurability $\mathbf{Q}$ is the same used in Fig. 2.

Figure 4

Spin background corresponding to the vortex-antivortex solution of Eq. (11). The centers of the vortices are marked by a circle. The spiral incommensurability $\mathbf{Q}$ is the same used in Fig. 2.

Figure 5

(Color online) Inverse mobility in units of ${\mu }_0^{-1}$ as a function of the rescaled temperature $T∕E_c$. Left panel: inverse mobility of a vortex-antivortex pair, according to Eq. (81), at several values of $\alpha$. Right panel: inverse mobility of a single-vortex defect, according to Eq. (83).

Figure 6

(Color online) Contribution (81) of the motion of a vortex-antivortex pair to the inverse mobility of the holes in cuprates.

Figure 7

(Color online) Contribution (81) of the motion of a single-vortex defect to the inverse mobility of the holes in cuprates.