一维局域模型中任意温度下的自发对称性破缺
Spontaneous symmetry breaking at any temperature in a local one-dimensional model
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中文总结 AI 辅助
本文提出一个一维经典统计模型，其螺旋对称群在任意温度下自发破缺，构造了无穷多个 Gibbs 态，并通过有限系统正则化支持无限线上的对称性破缺解释。
中文摘要 AI 辅助
我们提出一个在一维无限格点上具有最近邻相互作用和局域态空间 $\mathbb{Z}^2$ 的经典统计物理模型，其具有一个“螺旋”对称群 $\mathbb{Z}^2\rtimes \mathbb{Z}$，该对称群混合了味对称与空间平移。对于任意 $0<\beta<\infty$，我们构造了一个由 $\mathbb{R}^2$ 参数化的不可数无穷多个 DLR（Gibbs）态，并具有自由的 $\mathbb{Z}^2$ 味对称群作用；因此，我们将该模型解释为在任意温度下表现出自发对称性破缺。我们构造的最一般的 DLR 态被解释为局域畴壁的无限序列，分隔一对渐近的“对称扇区”。尽管任何可归一化的 Gibbs 态必然破缺一个可数无穷的自由作用对称群，但对该问题在（任意）大但具有周期性边界条件的有限系统上的自然正则化，其测度集中在不相交且远分离的对称相关簇中（在现在唯一的 Gibbs 态中），这证明了我们在无限线上对自发对称性破缺的解释。对局域态空间为 $\mathbb{R}^2$ 的模型进行修改，必须在任何温度下破缺一个连续的螺旋对称性。这与 Mermin-Wagner 定理并不矛盾，因为该连续螺旋对称性是非紧致的，并且不与平移交换。
英文摘要
We present a one-dimensional classical statistical physics model on an infinite lattice with nearest-neighbor interactions and local state space $\mathbb{Z}^2$, with a ``helical" symmetry group $\mathbb{Z}^2\rtimes \mathbb{Z}$ mixing flavor symmetry with spatial translation. For any $0<β<\infty$, we construct an uncountable infinity of DLR (Gibbs) states parameterized by $\mathbb{R}^2$, with a free $\mathbb{Z}^2$ flavor symmetry group action; as such, we interpret the model as exhibiting spontaneous symmetry breaking at any temperature. The most general DLR state we construct is interpreted as an infinite sequence of localized domain walls separating a pair of asymptotic ``symmetry sectors". Although any normalizable Gibbs state necessarily breaks a countably infinite freely-acting symmetry group, a natural regularization of the problem to (arbitrarily) large but finite systems with periodic boundary conditions has measure concentration in disjoint and far-separated symmetry-related clusters in the (now unique) Gibbs state, which justifies our interpretation of spontaneous symmetry breaking on the infinite line. A modification of the model with local state space $\mathbb{R}^2$ must break a continuous helical symmetry at any temperature. This is not a contradiction with the Mermin-Wagner Theorem, because this continuous helical symmetry is non-compact and does not commute with translation.
发表机构
- University of Colorado, Boulder（科罗拉多大学博尔德分校）
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