Macroscopic quantum spin tunnelling with two interacting spins

TL;DR

This paper investigates quantum spin tunnelling in a system of two coupled large spins, revealing how the ground state and energy splitting depend on spin parity and exchange interaction, using spin coherent state path integrals.

Contribution

It provides an exact analysis of the ground state structure and energy splitting in a two-spin system with antiferromagnetic coupling, highlighting differences between integer and half-integer spins.

Findings

Ground state is non-degenerate for antiferromagnetic coupling.

Energy splitting scales as ^{2s} with exchange constant .

Ground state is + for integer spins and - for half-integer spins.

Abstract

We study the simple Hamiltonian, $H=-K(S_{1z}^2 +S_{2z}^2)+ \lambda\vec S_1\cdot\vec S_2$, of two, large, coupled spins which are taken equal, each of total spin $s$ with $\lambda$ the exchange coupling constant. The exact ground state of this simple Hamiltonian is not known for an antiferromagnetic coupling corresponding to the $\lambda>0$. In the absence of the exchange interaction, the ground state is four fold degenerate, corresponding to the states where the individual spins are in their highest weight or lowest weight states, $|\hskip-1 mm\uparrow, \uparrow\rangle, |\hskip-1 mm\downarrow, \downarrow\rangle, |\hskip-1 mm\uparrow, \downarrow\rangle, |\hskip-1 mm\downarrow, \uparrow\rangle$, in obvious notation. The first two remain exact eigenstates of the full Hamiltonian. However, we show the that the two states $ |\hskip-1 mm\uparrow, \downarrow\rangle, |\hskip-1 mm\downarrow, \uparrow\rangle$ organize themselves into the combinations $|\pm\rangle=\frac{1}{\sqrt 2} (|\hskip-1 mm\uparrow, \downarrow\rangle \pm |\hskip-1 mm\downarrow \uparrow\rangle)$, up to perturbative corrections. For the anti-ferromagnetic case, we show that the ground state is non-degenerate, and we find the interesting result that for integer spins the ground state is $|+\rangle$, and the first excited state is the anti-symmetric combination $|-\rangle$ while for half odd integer spin, these roles are exactly reversed. The energy splitting however, is proportional to $\lambda^{2s}$, as expected by perturbation theory to the $2s^{\rm th}$ order. We obtain these results through the spin coherent state path integral.