自旋-1/2海森堡XXZ链与圈权重为2的洛伦兹镜像模型
The spin-1/2 Heisenberg XXZ chain and the Lorentz mirror model with loop weight 2
AI总结：
该研究严格证明了自旋-1/2海森堡XXZ链的关联衰减等性质，利用全新的镜像模型与六顶点模型耦合方法，得到了多项此前未证明的结果。
AI中文摘要：
我们证明，对于处于Δ∈[-1,1/2]范围的自旋-1/2海森堡XXZ链，长度为L的环面上的基态会在L→∞时收敛到无限体积基态⟨·⟩，且自旋-自旋关联⟨S₀⁽¹⁾Sₓ⁽¹⁾随x呈多项式衰减。在Δ∈[-1,0]范围内，我们得到更强的结果：该收敛对若干有限体积、有限温度态成立；L和β=1/T可按任意顺序趋于无穷；该收敛在（欧几里得）动态关联子上也成立；且⟨S₀⁽¹⁾Sₓ⁽¹⁾(t)随|(x,t)|衰减至0，且无法呈指数衰减。无限体积基态的收敛已通过Bethe Ansatz方法得到严格证明，而我们的关联衰减结果（除Δ=0、-1点外）、动态关联子的收敛以及极限的可交换性均为严格层面的新结果。此外，我们的方法是全新的，未使用任何Bethe Ansatz技术，仅利用以下相关概率模型：在圈权重为2的洛伦兹镜像模型（一种方格上的随机圈模型）中，我们证明在很大的参数范围内，连接概率趋于0，且在该模型的对称情形下呈多项式衰减。我们的证明使用了该镜像模型与六顶点模型的两种耦合，其中一种是新的，主要输入是多位作者证明的六顶点模型高度函数的非局域化。我们进一步证明，六顶点模型的某一时空版本的高度函数发生非局域化，然后通过与上述类似的耦合证明，XXZ模型的圈表示（由Ueltschi引入）中的连接概率趋于0。
英文摘要：
We prove that for the spin-1/2 Heisenberg XXZ chain in the range $Δ\in[-1,1/2]$, 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 $Δ\in[-1,0]$ we have the stronger results: that the convergence holds for several finite-volume, finite temperature states, that $L$ and $β=1/T$ 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 $Δ=0,-1$), 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.
