Metraipolis › archive › dc_45147cdfffdd

The spin-1/2 Heisenberg XXZ chain and the Lorentz mirror model with loop weight 2 — arXiv:2608.21306

Source

arXiv (en)

Original

https://arxiv.org/abs/2608.21306

License (as archived)

CC0-1.0

Archived

2026-10-18T10:39:42.770Z

SHA-256 of the archived text

51d703fbc949e177481625b17409a23e816c9ec88b5c113a250f0edc9e7be439

Stable URL of this page

https://metraipolis.com/documento/dc_45147cdfffdd

How to cite

The spin-1/2 Heisenberg XXZ chain and the Lorentz mirror model with loop weight 2 — arXiv:2608.21306 — arXiv (en), https://arxiv.org/abs/2608.21306 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_45147cdfffdd (Metraipolis archive dc_45147cdfffdd, sha256 51d703fbc949e177481625b17409a23e816c9ec88b5c113a250f0edc9e7be439)

@misc{dc45147cdfffdd,
  title = {The spin-1/2 Heisenberg XXZ chain and the Lorentz mirror model with loop weight 2 — arXiv:2608.21306},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2608.21306},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_45147cdfffdd; sha256 51d703fbc949e177481625b17409a23e816c9ec88b5c113a250f0edc9e7be439},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2608.21306. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

We prove that for the spin-1/2 Heisenberg XXZ chain in the range $\Delta\in[-1,1/2]$, the ground state on the torus of length $L$ converges to an infinite volume ground state $\langle\cdot\rangle$ as $L\to\infty$, and that the spin-spin correlation $\langle S_0^{(1)}S_x^{(1)}\rangle$ decays polynomially fast in $x$. In the range $\Delta\in[-1,0]$ we have the stronger results: that the convergence holds for several finite-volume, finite temperature states, that $L$ and $\beta=1/T$ can be taken to infinity in any order, that the convergence holds on (Euclidean) dynamic correlators, and that $\langle S_0^{(1)}S_x^{(1)}(t)\rangle$ decays to zero in $|(x,t)|$, and cannot decay exponentially fast. The convergence of the ground state in infinite volume is known rigorously by Bethe Ansatz methods, while our correlation decay results (apart from the points $\Delta=0,-1$), the convergence on dynamic correlators and the interchangeability of limits are new at the rigorous level. Moreover our methods are new and do not use any Bethe Ansatz techniques, using only the following related probabilistic models. In the Lorentz mirror model with loop weight 2, a model of random loops on the square lattice, in a large range of parameters we prove that connection probabilities tend to 0, and do so polynomially fast in the symmetric case of the model. Our proof uses two couplings of this mirror model with the six-vertex model, one of which is new. The main input is the delocalisation of the height function of the six-vertex model proved by several authors. We further prove that the height function of a certain space-time version of the six-vertex model delocalises and then prove through analogous couplings to those mentioned above that connection probabilities converge to 0 in the loop representation of the XXZ model introduced by Ueltschi.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-08-24 in math.PR).
Categorie arXiv: Mathematical Physics (math-ph) ; Probability (math.PR).
Pagina del lavoro: https://arxiv.org/abs/2608.21306 — 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 2026-10-18T10:38:26.410Z dalla pagina https://arxiv.org/catchup/math.PR/2026-08-24?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).