临界量子自旋链中Kac-Moody对称性的出现
Emergence of Kac-Moody Symmetry in Critical Quantum Spin Chains
摘要：
给定具有微观李群对称性的临界量子自旋链，例如对应于$U(1)$或$SU(2)$自旋各向同性，我们数值研究了低能长距离下Kac-Moody对称性的出现。在这种情况下，一个这样的临界量子自旋链是由共形场论描述的，其中通常与共形不变性相关的维拉索罗代数用与守恒电流相关的卡茨-穆迪代数来扩充。具体来说，我们首先提出了一种方法来构造相应的Kac-Moody生成元格算子。然后，我们数值表明，当投影到低能量态的量子自旋链，这些运营商确实近似满足卡茨-穆迪代数。晶格版本的Kac-Moody生成器允许我们计算所谓的能级常数，并将晶格哈密顿量的低能本征态组织成Kac-Moody塔。我们用XXZ模型和海森堡模型说明了这个建议。
英文摘要：
Given a critical quantum spin chain with a microscopic Lie-group symmetry, corresponding e.g. to $U(1)$ or $SU(2)$ spin isotropy, we numerically investigate the emergence of Kac-Moody symmetry at low energies and long distances. In that regime, one such critical quantum spin chain is described by a conformal field theory where the usual Virasoro algebra associated to conformal invariance is augmented with a Kac-Moody algebra associated to conserved currents. Specifically, we first propose a method to construct lattice operators corresponding to the Kac-Moody generators. We then numerically show that, when projected onto low energy states of the quantum spin chain, these operators indeed approximately fulfill the Kac-Moody algebra. The lattice version of the Kac-Moody generators allow us to compute the so-called level constant and to organize the low-energy eigenstates of the lattice Hamiltonian into Kac-Moody towers. We illustrate the proposal with the XXZ model and the Heisenberg model with a next-to-nearest-neighbor coupling.
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