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Short-Range Vector Spin Glasses: Parisi Hierarchies and Real-Space Ultrametricity — arXiv:2610.12441

Source: arXiv (en)
Original: https://arxiv.org/abs/2610.12441
License (as archived): CC0-1.0
Archived: 2026-10-09T06:39:34.015Z
SHA-256 of the archived text: f22d5fb310734e813a2cf62f038b2591a32a58875ab3f641fd30c1fd1b7ea871
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How to cite

Short-Range Vector Spin Glasses: Parisi Hierarchies and Real-Space Ultrametricity — arXiv:2610.12441 — arXiv (en), https://arxiv.org/abs/2610.12441 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_eafe2bc7bbb1 (Metraipolis archive dc_eafe2bc7bbb1, sha256 f22d5fb310734e813a2cf62f038b2591a32a58875ab3f641fd30c1fd1b7ea871)

BibTeX

@misc{dceafe2bc7bbb1,
  title = {Short-Range Vector Spin Glasses: Parisi Hierarchies and Real-Space Ultrametricity — arXiv:2610.12441},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2610.12441},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_eafe2bc7bbb1; sha256 f22d5fb310734e813a2cf62f038b2591a32a58875ab3f641fd30c1fd1b7ea871},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2610.12441. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

Replica symmetry breaking (RSB) is the organizing principle of mean-field spin-glass theory, yet whether a continuous Parisi hierarchy can arise in a genuinely short-range system in finite spatial dimension remains unresolved. Here we introduce a disordered model of $N$-component vector spins on a $D$-dimensional hypercubic lattice, with random matrix couplings on nearest-neighbor links and random tensor couplings on elementary plaquettes. In the large-$N$ limit, the model supports thermodynamically stable one-step, full, and one-full RSB phases, alongside replica-symmetric paramagnetic and ferromagnetic phases. We find that spatial locality constrains these ordered phases through two distinct infrared mechanisms. ($i$) Quenched random anisotropy destroys uniform ferromagnetism for $D\leq4$ through an Imry-Ma mechanism. ($ii$) A continuous Parisi hierarchy gives rise to replica Goldstone modes whose gaplessness is protected by a Ward identity. At leading order in $1/N$, these modes behave as a band of free spin waves, producing infrared-divergent fluctuations in $D\leq2$ and a Mermin-Wagner instability of the continuously broken phase. Remarkably, the Parisi hierarchy also acquires a direct real-space interpretation. For two pure states with mutual overlap $q$, a characteristic length $\xi(q)$ describes how far the differences between their frozen magnetization patterns remain correlated. Throughout a continuous sector of the Parisi hierarchy, $\xi(q)$ is independent of temperature and decreases monotonically with $q$. The ultrametric organization of pure states is therefore accompanied by an ultrametric hierarchy of real-space correlation lengths: states that separate near the root of the Parisi tree differ through long-wavelength frozen patterns, whereas states that separate farther from the root differ only on progressively shorter scales.

Tema degli esploratori: fisica e astronomia (elenco arXiv cond-mat.stat-mech, nuovi lavori del giorno).
Categorie arXiv: Disordered Systems and Neural Networks (cond-mat.dis-nn) ; Statistical Mechanics (cond-mat.stat-mech).
Pagina del lavoro: https://arxiv.org/abs/2610.12441 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-09T05:59:01.488Z dalla pagina https://arxiv.org/list/cond-mat.stat-mech/new (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).