伊辛模型中质量激发的E8高斯圆半径
Radii of the E8 Gosset Circles as the Mass Excitations in the Ising Model
摘要：
Zamolodchikov猜想暗示了特殊李群E8的存在，这一猜想似乎被Coldea等人对一维伊辛模型的量子相变实验所证实。由仿射Toda场论得出的E8模型预测了8个束缚态，其质量关系为递增阶m1，m2= tau m1，m3，m4，m5，m6=tau m3，m7= tau m4，m8= tau m5，其中tau=（1+ \sqrt（5））/2表示黄金比例。Coxeter群W（H4）是Coxeter-Weyl群W（E8）的极大子群。这些质量原来是成比例的半径戈塞特圆的考克斯特平面上获得的正交投影的根系E8。我们还注意到，质量m1，m3，m4和m5对应于通过投影600胞腔的顶点而获得的圆的半径，600胞腔是非晶体学Coxeter群W（H4）的4D多面体。简单根在考克斯特平面上的特殊非正交投影导致束缚态质量的数值精确为0.4745，0.7678，0.9438，1.141，1.403，1.527，1.846和2.270。我们注意到的前两个数值的前两个质量的束缚态所确定的Coldea等惊人的平等。al.
英文摘要：
The Zamolodchikov's conjecture implying the exceptional Lie group E8 seems to be validated by an experiment on the quantum phase transitions of the 1D Ising model carried out by the Coldea et. al. The E8 model which follows from the affine Toda field theory predicts 8 bound states with the mass relations in the increasing order m1, m2= tau m1, m3, m4, m5, m6=tau m3, m7= tau m4, m8= tau m5, where tau= (1+\sqrt(5))/2 represents the golden ratio. Above relations follow from the fact that the Coxeter group W(H4) is a maximal subgroup of the Coxeter-Weyl group W(E8). These masses turn out to be proportional to the radii of the Gosset's circles on the Coxeter plane obtained by an orthogonal projection of the root system of E8 . We also note that the masses m1, m3, m4 and m5 correspond to the radii of the circles obtained by projecting the vertices of the 600-cell, a 4D polytope of the non-crystallographic Coxeter group W(H4). A special non-orthogonal projection of the simple roots on the Coxeter plane leads to exactly the numerical values of the masses of the bound states as 0.4745, 0.7678, 0.9438, 1.141, 1.403, 1.527, 1.846, and 2.270. We note the striking equality of the first two numerical values to the first two masses of the bound states determined by the Coldea et. al.
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