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arxiv: 2509.10610 · v2 · submitted 2025-09-12 · ❄️ cond-mat.str-el

Finite-Size Spectral Signatures of Order by Quantum Disorder: A Perspective from Anderson's Tower of States

Subhankar Khatua , Griffin C. Howson , Michel J. P. Gingras , Jeffrey G. Rau This is my paper

Pith reviewed 2026-05-18 17:14 UTC · model grok-4.3

classification ❄️ cond-mat.str-el
keywords order by quantum disorderexact diagonalizationAnderson tower of statesfrustrated magnetsquantum rotor modelfinite size effectsspin 1/2 models
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0 comments X

The pith

Effective quantum rotor model links finite-size energy splittings to order-by-quantum-disorder scale in frustrated magnets.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

In systems with many classical ground states, quantum fluctuations can select one particular ordered state through order by quantum disorder. The paper demonstrates that this selection leaves clear signatures in the exact diagonalization spectrum of finite clusters. An effective quantum rotor model is introduced to capture how the rotor localizes due to the selection energy while still tunneling between equivalent states at small sizes. The model connects the size-dependent splittings seen in the spectrum to the strength of the quantum ordering, allowing estimates that match spin-wave results. This framework is shown to apply to frustrated spin-1/2 models in one, two, and three dimensions.

Core claim

Order by quantum disorder occurs when quantum zero-point fluctuations lift the degeneracy of a subextensive manifold of classical ground states and select a unique long-range ordered phase. In exact diagonalization studies of finite systems this selection manifests as a low-energy tower whose splittings are described by an effective quantum rotor. The rotor Hamiltonian encodes the competition between tunneling among symmetry-related configurations and an ObQD-induced potential that favors localization into the selected state. Fitting the observed level structure to this rotor model extracts the ObQD energy scale, which can be directly compared with independent spin-wave calculations. The sa

What carries the argument

Effective quantum rotor model for the competition between ObQD localization and tunneling between symmetry-related ground states

Load-bearing premise

The low-energy spectrum of the microscopic model is faithfully represented by the effective quantum rotor for the sizes and models considered.

What would settle it

Observing that the energy splittings in larger ED clusters do not follow the predicted dependence on the ObQD scale obtained from spin-wave theory would falsify the rotor mapping.

Figures

Figures reproduced from arXiv: 2509.10610 by Griffin C. Howson, Jeffrey G. Rau, Michel J. P. Gingras, Subhankar Khatua.

Figure 1
Figure 1. Figure 1: FIG. 1. Illustration of order by quantum disorder (ObQD) selection, [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Ferromagnetic Heisenberg-compass model on the square lat [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Finite size scaling of the BLM spectrum of the Heisenberg [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Antiferromagnetic Heisenberg-Kitaev model on the honey [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
read the original abstract

In frustrated magnetic systems with a subextensive number of classical ground states, quantum zero-point fluctuations can select a unique long-range ordered state, a celebrated phenomenon referred to as \emph{order by quantum disorder} (ObQD). For frustrated spin-$\frac{1}{2}$ models, unbiased numerical methods able to expose ObQD are necessary. We show that ObQD can be identified from exact diagonalization (ED) calculations through an analysis akin to the Anderson tower of states associated with spontaneous symmetry breaking. By defining an effective quantum rotor model, we describe the competition between ObQD-induced localization of the rotor and its tunneling between symmetry-related ground states, identifying the crossover lengthscale from the finite-size regime where the rotor is delocalized, to the infinite system-size limit where it becomes localized. This rotor model relates the characteristic splittings in the ED energy spectrum to the ObQD selection energy scale, providing an estimate that can be compared to spin wave calculations. We demonstrate the general applicability of this approach in one-, two- and three-dimensional frustrated spin models that exhibit ObQD.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript claims that order by quantum disorder (ObQD) in frustrated spin-1/2 models can be identified from exact diagonalization (ED) spectra by constructing an effective quantum rotor model. This model captures the competition between ObQD-driven localization and tunneling between degenerate ground states, relating finite-size spectral splittings to the ObQD energy scale from spin-wave theory. The approach is demonstrated to apply across one-, two-, and three-dimensional models, providing a finite-size crossover lengthscale to the localized regime in the thermodynamic limit.

Significance. If the effective rotor model is shown to faithfully reproduce the ED spectra without additional fitting parameters, as suggested by the mapping from spin-wave dispersion and classical manifold geometry, this provides a significant tool for detecting ObQD in numerical studies of frustrated magnets. It extends the Anderson tower of states concept to ObQD selection and allows direct comparison between ED splittings and spin-wave predictions, which is particularly useful in systems where unbiased methods are required to resolve the selected ordered state.

minor comments (2)
  1. The mapping from the spin-wave dispersion and manifold geometry to the rotor inertia and ObQD gap (described in the effective rotor model construction) should include an explicit equation or appendix derivation to allow readers to verify the parameter-free relation to ED splittings.
  2. In the discussion of the crossover lengthscale, clarify how the ratio of rotor inertia to ObQD gap scales with system size for the 3D example, as this is central to the localization claim in the thermodynamic limit.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of our manuscript, accurate summary of the effective quantum rotor model approach, and recommendation for minor revision. The significance statement correctly identifies the utility of relating ED splittings to spin-wave ObQD scales across dimensions.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via independent spin-wave input

full rationale

The central construction defines an effective quantum rotor whose inertia and potential are fixed by the ObQD selection scale computed from spin-wave theory on the classical manifold geometry. This scale is obtained independently of the ED spectra; the rotor Hamiltonian is then solved to predict the finite-size tower splittings and the localization crossover. No parameter is fitted to the ED data, no self-citation supplies the load-bearing uniqueness or ansatz, and the mapping is derived from the microscopic spin-wave dispersion rather than assumed. The paper therefore compares two independent calculations (spin-wave scale vs. ED splittings) rather than recovering one from the other by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The approach rests on standard assumptions of quantum spin models and the validity of an effective low-energy rotor description; no explicit free parameters are named in the abstract, but the rotor parameters themselves are implicitly determined by the microscopic model.

axioms (2)
  • domain assumption The low-energy spectrum of the frustrated spin-1/2 Hamiltonian can be faithfully mapped onto a quantum rotor whose tunneling and localization are controlled by the ObQD selection energy.
    This mapping is the central modeling step invoked to relate ED splittings to the ObQD scale.
  • domain assumption Spin-wave calculations provide an independent estimate of the ObQD selection energy that can be directly compared to the rotor-derived value.
    The abstract states that the rotor estimate can be compared to spin-wave results, presupposing the accuracy of both.
invented entities (1)
  • effective quantum rotor no independent evidence
    purpose: To model the competition between ObQD-induced localization and tunneling between symmetry-related states in finite-size systems.
    The rotor is introduced as a simplified description of the low-energy manifold selected by quantum fluctuations.

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