Quantum interactions of topological solitons from electrodynamics

The Casimir energy for the classically stable configurations of the topological solitons in 2D quantum antiferromagnets is studied by performing the path integral over quantum fluctuations. The magnon fluctuation around the solitons saturating the Bogomol'nyi inequality may be viewed as a charged scalar field coupled with an effective magnetic field induced by the solitons. The magnon-soliton coupling is closely related to the Pauli Hamiltonian, with which the effective action is calculated by adapting the worldline formulation of the derivative expansion for the $2+1$-dimensional quantum electrodynamics in an external field. The resulting framework is more flexible than the conventional scattering analysis based on the Dashen-Hasslacher-Neveu formula. We obtain a short-range attractive well and a universal long-range $1/r$-type repulsive potential between two solitons.

Figure 1

The magnetic field strengths $|B_3\left(x\right)|$ in (25) for two solitons of the same size $\lambda =1$ with various distances $d=0,1.5,3,4.5,7$. The region for each $d$ with $|B_3\left(x\right)|<0.03$ is excluded. They are identical to the topological charge densities of the optimal solitons that saturate the Bogomol'nyi bound for $q=2$. The density for $d\gg \lambda$ is asymptotically a superposition of two independent solitons in (24) each shifted by $\pm d$, while it is deformed as $d$ decreases and becomes eventually the ring configuration in (26) with $n=2$ at $d=0$.

Figure 2

The magnetic field strength $|B_3\left(x\right)|$ for two solitons of the same size $\lambda =1$ each located at $d=3$ apart from the origin with various phase angles (a) $\theta =0$, (b) $\theta =3\pi /5$, (c) $\theta =\pi$, given by (22) and (27), which also equals to the topological charge density that saturates the Bogomol'nyi bound. Classically, there are no forces between the solitons as they are optimized.

Figure 3

The quantum interaction potential $V(\lambda ,d)$ for two static solitons of size $\lambda$ subtracted by the self-energy $2V_1\left(\lambda \right)$ with a separation $2d$ (solid with markers). An attractive well of depth $\approx -16.7$ lies at $d=0$ for the relative orientation $\theta =\pi$. The short-range interaction depends on $\theta$ (see Fig. 2), while the long-range one (89) is universal. The leading repulsive $1/d$-law (dashed) and the next order (dash-dotted).