论文标题
Virasoro代数在量子自旋链中的作用。 I.非理性案件
The action of the Virasoro algebra in quantum spin chains. I. The non-rational case
论文作者
论文摘要
我们研究了关键的XXZ量子量子链中，我们研究了由晶格templey-lieb代数（“ koo-saleur发电机”）（“ koo-saleur发电机”）构建的离散的Virasoro发电机的作用。我们探索了XXZ顶点模型的通用中心电荷下的连续限制virasoro模块的结构，对于循环模型，并行[ARXIV：2007.11539]。我们再次发现了不可塑性的模块，但这一次不是对数。 temperley-lieb模块的限制$ \ mathcal {w} _ {j，1} $ for $ j \ neq 0 $包含一对带有保量重量$ $（h_ {r，s}的偶联状态”对（h_ {r，s}，h_ {r，h_ {r，-s}）$（h_ {r，-s}）$（ Verma或Co-Verma模块。 $ \ Mathcal {w} _ {0，\ Mathfrak {q}^{\ pm2}} $的极限包含对角线字段$（h_ {r，1}，h_ {r，1}）$，并仅引起，并且仅引起n verma or co-verma模块，根据$ pmer { 2} $。为了获得大型系统尺寸$ n $的koo-saleur发电机的矩阵元素，我们使用bethe ansatz和量子逆散射方法，计算三个相邻旋转算子的相关组合的形式。形式因素之间的关系确保上述双重性已经存在于晶格级别。我们还研究了koo-saleur发电机融合了Virasoro发电机的哪种意义。我们考虑在虚弱的意义上融合，研究限制的换向器是否与换向器的极限相同？我们发现它仅重合到中心任期。作为侧面结果，我们计算了XXZ自旋链中两个相邻templey-Lieb发电机的基础期望值。
We investigate the action of discretized Virasoro generators, built out of generators of the lattice Temperley-Lieb algebra ("Koo-Saleur generators"[arXiv:hep-th/9312156]), in the critical XXZ quantum spin chain. We explore the structure of the continuum-limit Virasoro modules at generic central charge for the XXZ vertex model, paralleling [arXiv:2007.11539] for the loop model. We find again indecomposable modules, but this time not logarithmic ones. The limit of the Temperley-Lieb modules $\mathcal{W}_{j,1}$ for $j\neq 0$ contains pairs of "conjugate states" with conformal weights $(h_{r,s},h_{r,-s})$ and $(h_{r,-s},h_{r,s})$ that give rise to dual structures: Verma or co-Verma modules. The limit of $\mathcal{W}_{0,\mathfrak{q}^{\pm2}}$ contains diagonal fields $(h_{r,1},h_{r,1})$ and gives rise to either only Verma or only co-Verma modules, depending on the sign of the exponent in $\mathfrak{q}^{\pm 2}$. In order to obtain matrix elements of Koo-Saleur generators at large system size $N$ we use Bethe ansatz and Quantum Inverse Scattering methods, computing the form factors for relevant combinations of three neighbouring spin operators. Relations between form factors ensure that the above duality exists already at the lattice level. We also study in which sense Koo-Saleur generators converge to Virasoro generators. We consider convergence in the weak sense, investigating whether the commutator of limits is the same as the limit of the commutator? We find that it coincides only up to the central term. As a side result we compute the ground-state expectation value of two neighbouring Temperley-Lieb generators in the XXZ spin chain.
