Spectral Signatures of Order by Disorder in Finite-Size Frustrated Quantum Magnets: an Anderson Tower of States Analysis

Competing interactions or ‘frustration’ often leads to an accidental ground state degeneracy, in that, this degeneracy is not a consequence of any global symmetry of the Hamiltonian. Consequently, thermal or quantum fluctuations may lift it, often resulting in an ordered state, a phenomenon known as ‘order by disorder’ (ObD). While this phenomenon is well-understood in spin systems in the thermodynamic limit using spin-wave theory, its behavior in finite-size quantum spin-half systems remains less clear. This prompts the question: what are the intriguing finite-size manifestations of ObD in quantum magnets?

For ObD arising from quantum fluctuations at zero temperature, we show that there exists a low-lying tower of states in the exact diagonalization (ED) spectrum, akin to the ‘Anderson tower of states’ associated with spontaneous symmetry breaking, and the spectrum also exhibits characteristic splittings in certain higher energy levels. We provide a phenomenological description in terms of a quantum rotor model that successfully captures the subtle ObD mechanism at play in finite-size systems. This model addresses the rotor dynamics within the degenerate classical ground state manifold in presence of the ObD induced potential, which has minima at the anticipated ObD selected states. This potential competes with rotor tunneling between the minima, with the rotor delocalizing across the ground state manifold for small system sizes and localizing at the potential minima for larger sizes. In the limit of small systems, the potential acts as a perturbation to the rotor kinetic term, imprinting a signature of ObD in the splittings of the excited levels of the Anderson tower. We illustrate these ideas in various spin systems, including the Heisenberg-Kitaev chain, the Heisenberg-compass model on a square lattice, and the nearest-neighbor anisotropic exchange model on the pyrochlore lattice. Our study further enriches the understanding of ObD in quantum magnets.