Title: Emptiness formation probability and Painlevé V equation in the XY spin chain Show Chinese title

Title: 空缺形成概率与XY自旋链中的Painlevé V方程

Abstract: We reconsider the problem of finding $L$ consecutive down spins in the ground state of the XY chain, a quantity known as the Emptiness Formation Probability. Motivated by new developments in the asymptotics of Toeplitz determinants, we show how the crossover between the critical and off-critical behaviour of the Emptiness Formation Probability is exactly described by a $\tau$ function of a Painlev\'e V equation. Following a recent proposal, we also provide a power series expansion for the $\tau$ function in terms of irregular conformal blocks of a Conformal Field Theory with central charge $c=1$. Our results are tested against lattice numerical calculations, showing excellent agreement. We finally rediscuss the free fermion case where the Emptiness Formation Probability is characterized by a Gaussian decay for large $L$. Show Chinese abstract

Abstract: 我们重新考虑在XY链基态中寻找$L$个连续向下自旋的问题，这一量被称为空缺形成概率。 受Toeplitz行列式渐近新发展的启发，我们展示了空缺形成概率在临界和非临界行为之间的转换如何由Painlevé V方程的一个$\tau$函数精确描述。 根据最近的一项提议，我们还提供了$\tau$函数的幂级数展开，该展开以具有中心电荷$c=1$的共形场论的不规则共形块表示。 我们的结果与格点数值计算进行了对比，显示出极好的一致性。 最后，我们重新讨论了自由费米子的情况，在这种情况下，空缺形成概率对于大的$L$表现出高斯衰减。