横截面Ising环的A-圈问题
The A-Cycle Problem for Transverse Ising Ring
摘要：
传统的方法是将横截面伊辛模型映射到费米c-循环问题上，由于热力学极限的限制，忽略了边界效应。如果坚持一个完美的周期边界条件，我们可以得到一个所谓的a-循环问题，迄今为止还没有得到认真对待（Lieb et al.，1961 \textit{Ann. of Phys.} \textbf{16} 407）。在这项工作中，我们在这方面展示了一个有点令人惊讶但确切的结果。我们发现，奇数的晶格位数，$N$，在a-循环问题中起着意想不到的作用，即使在热力学极限，$N\rightarrow\infty$，由于边界约束。我们特别关注的系统与$N(\in Odd)\rightarrow\infty$，这是与一个与$N(\in Even)\rightarrow\infty$，因为前者遭受环挫折。作为一种新的效应，我们发现环阻挫在基态以上产生了一个低能的无带隙谱。通过证明一个新的Toeplitz行列式定理，我们证明了无带隙区的基态具有特殊的纵向自旋-自旋关联。通过计算基态的纠缠熵，揭示了基态的纠缠特性。在低温下，比热的新的行为进行了预测。我们还提出了一个实验方案，观察新的现象，由于环挫折。
英文摘要：
Traditionally, the transverse Ising model is mapped to the fermionic c-cycle problem, which neglects the boundary effect due to thermodynamic limit. If persisting on a perfect periodic boundary condition, we can get a so-called a-cycle problem that has not been treated seriously so far (Lieb et al., 1961 \textit{Ann. of Phys.} \textbf{16} 407). In this work, we show a little surprising but exact result in this respect. We find the odevity of the number of lattice sites, $N$, in the a-cycle problem plays an unexpected role even in the thermodynamic limit, $N\rightarrow\infty$, due to the boundary constraint. We pay a special attention to the system with $N(\in Odd)\rightarrow\infty$, which is in contrast to the one with $N(\in Even)\rightarrow\infty$, because the former suffers a ring frustration. As a new effect, we find the ring frustration induces a low-energy gapless spectrum above the ground state. By proving a theorem for a new type of Toeplitz determinant, we demonstrate that the ground state in the gapless region exhibits a peculiar longitudinal spin-spin correlation. The entangled nature of the ground state is also disclosed by the evaluation of its entanglement entropy. At low temperatures, new behavior of specific heat is predicted. We also propose an experimental protocol for observing the new phenomenon due to the ring frustration.
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