AI 中文总结
研究二阶半无限有根凯莱树上的混合量子伊辛 - XY 模型，利用树索引量子马尔可夫链相容性准则等推导边界方程、计算局部转移算子，证明有唯一正平移不变解及相关结论，还得出约化边界律动力学性质并计算局部两体纠缠。
AI 中文摘要
我们研究二阶半无限有根凯莱树上的混合量子伊辛 - XY 模型。对于每个顶点\(u\)，边\(\langle u,(u,1)\rangle\)带有\(XY\)相互作用，边\(\langle u,(u,2)\rangle\)带有伊辛相互作用。利用树索引量子马尔可夫链的相容性准则并结合归一化迹，我们推导平移不变边界方程并明确计算相关局部转移算子。证明了对于所有\(J_I,J_{XY}\in\mathbb R\)和\(\beta>0\)，边界方程有唯一正平移不变解，模型有唯一由正平移不变边界条件生成的平移不变量子马尔可夫链。还表明约化边界律动力学无大于 1 周期的可允许周期点，并计算了树上自然三顶点簇上的局部两体纠缠。
英文摘要
We study a mixed quantum Ising-$XY$ model on the semi-infinite rooted Cayley tree of order two. For every vertex $u$, the edge $\langle u,(u,1)\rangle$ carries an $XY$ interaction and the edge $\langle u,(u,2)\rangle$ carries an Ising interaction. Using the compatibility criterion for tree-indexed quantum Markov chains and consistently working with the normalized trace, we derive the translation-invariant boundary equation and compute explicitly the associated local transfer operator, namely the one-step partial-trace map which propagates successor boundary data to the parent vertex. We prove that the boundary equation has a unique positive translation-invariant solution for all $J_I,J_{XY}\in\mathbb R$ and $β>0$. Hence the model admits a unique translation-invariant quantum Markov chain generated by a positive translation-invariant boundary condition. We also show that the reduced boundary-law dynamics, i.e. the induced finite-dimensional recursion for the boundary-law parameters, has no admissible periodic points of period greater than one and compute the local two-site entanglement on the natural three-site cluster of the tree.
Comments33 pages
