圆柱上的可解临界稠密聚合物
Solvable Critical Dense Polymers on the Cylinder
摘要：
在有限周长圆柱上精确求解了临界致密聚合物的格点模型。该模型是对数极小模型的Yang-Baxter可积级数的第一个成员LM（1，2）。圆柱体拓扑结构允许具有逸度α的不可收缩环缠绕在圆柱体周围，或者允许任意数量的缺陷沿着圆柱体的全长传播。使用一个扩大的周期Temperley-Lieb代数，我们建立了交换传递矩阵作用于状态，其链接被认为是不同的连接周围的前面或后面的圆柱。这些传递矩阵满足一个函数方程的形式的反演单位。对于偶数N，这涉及到一个不可对角化的辫子算子J和一个对合R=-（J^3- 12 J）/16=（-1）^{F}，特征值R=（-1）^{ell/2}。缺陷的数量可以将理论分成几个部分。对于圈逸度α =2的情形，反演恒等式在有限几何中精确求解。特征值的分类的物理组合学的模式的零在复杂的频谱参数平面产生的选择规则。有限尺寸修正由Euler-Maclaurin公式得到。在标度极限下，得到了共形的配分函数，并确定了中心荷c=-2和Delta_t=（t^2-1）/8.这里，在具有Kac标签r= 1，2，3，.的ell偶数扇区中，t=ell/2和t= 2 r-s; s= 1，2，而t在e11奇数扇区中是半整数。引人注目的是，ell/2奇数扇区表现出W扩展对称性，但ell/2偶数扇区没有。此外，在所有偶数扇区上的朴素迹求和不会产生模不变量。
英文摘要：
A lattice model of critical dense polymers is solved exactly on a cylinder with finite circumference. The model is the first member LM(1,2) of the Yang-Baxter integrable series of logarithmic minimal models. The cylinder topology allows for non-contractible loops with fugacity alpha that wind around the cylinder or for an arbitrary number ell of defects that propagate along the full length of the cylinder. Using an enlarged periodic Temperley-Lieb algebra, we set up commuting transfer matrices acting on states whose links are considered distinct with respect to connectivity around the front or back of the cylinder. These transfer matrices satisfy a functional equation in the form of an inversion identity. For even N, this involves a non-diagonalizable braid operator J and an involution R=-(J^3-12J)/16=(-1)^{F} with eigenvalues R=(-1)^{ell/2}. The number of defects ell separates the theory into sectors. For the case of loop fugacity alpha=2, the inversion identity is solved exactly for the eigenvalues in finite geometry. The eigenvalues are classified by the physical combinatorics of the patterns of zeros in the complex spectral-parameter plane yielding selection rules. The finite-size corrections are obtained from Euler-Maclaurin formulas. In the scaling limit, we obtain the conformal partition functions and confirm the central charge c=-2 and conformal weights Delta_t=(t^2-1)/8. Here t=ell/2 and t=2r-s in the ell even sectors with Kac labels r=1,2,3,...; s=1,2 while t is half-integer in the ell odd sectors. Strikingly, the ell/2 odd sectors exhibit a W-extended symmetry but the ell/2 even sectors do not. Moreover, the naive trace summing over all ell even sectors does not yield a modular invariant.
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