Short-Range Vector Spin Glasses: Parisi Hierarchies and Real-Space Ultrametricity
Abstract
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≤4 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≤2 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 ξ(q) describes how far the differences between their frozen magnetization patterns remain correlated. Throughout a continuous sector of the Parisi hierarchy, ξ(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.
Related
- Simplified Parisi Ansatz III: Constructive Cavity Method and Ultrametricity
- Free energy of spherical vector spin glasses
- Parisi Formula for the ground state energy of quantum p-Spin Hamiltonians
- Kardar-Parisi-Zhang superdiffusion in chaotic quantum circuits
- Kardar--Parisi--Zhang density hydrodynamics in a coherent XXZ--ASEP Lindbladian
BibTeX
@misc{vijay2026shortrange,
title = {{Short-Range Vector Spin Glasses: Parisi Hierarchies and Real-Space Ultrametricity}},
author = {Akash Vijay and Jaewon Kim and Jong Yeon Lee},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12441},
eprint = {2610.12441},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12441},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Short-Range Vector Spin Glasses: Parisi Hierarchies and Real-Space Ultrametricity
AU - Akash Vijay
AU - Jaewon Kim
AU - Jong Yeon Lee
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.12441
UR - https://arxiv.org/abs/2610.12441
AB - 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≤4 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≤2 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 ξ(q) describes how far the differences between their frozen magnetization patterns remain correlated. Throughout a continuous sector of the Parisi hierarchy, ξ(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.
ER -