Topological and Many-body Localization Phases in Spinor Bose-Hubbard Models

在此之前，我们重新讨论了Kennedy-Tasaki对偶，揭示了KT对偶为拓扑临界性提供了一种“隐藏的对称破缺”解释。我们还注意到KT的自对偶性与Lieb-Schultz-Mattis(LSM)反常现象密切相关，但是LSM反常现象的性质还不是很清楚。因此，我们把重点放在了对突发异常的调查上。利用微扰理论，我们发现我们的自旋1模型在临界自对偶点附近的低能理论等价于自旋1/2的XYZ链。这意味着，在自对偶点附近，完整理论的对称性Z2y×Z2z×Ztrn在低能理论中退化为Zy‘×Zz’×Ztrn。换言之，在低能理论中，Z4y×Z2z×Ztrn会导致LSM反常，从而导致不存在唯一的有隙基态。然而，在自旋为1的希尔伯特空间中，Z4y倍Z4z×Ztrn没有反常。换句话说，自对偶点附近的LSM反常实际上是浮现的。由于自旋为1的希尔伯特空间中的完整理论是无反常的，浮现反常必须通过某种机制来抵消。注意，对于带隙对称Z2y，非平凡群元(π旋转)等于低能理论中的-1。这表明基态“堆积”在受Z2y×Ztrn保护的带隙(弱)对称保护的拓扑(SPT)相上。正是这个SPT阶段抵消了紧急异常。我们用场论来证明我们的论点。*\r时间意味着半直积。

Previously, we revisited the Kennedy-Tasaki duality and revealed the KT duality provides a “hidden symmetry breaking” interpretation for the topological criticality. We also noticed that the KT self-duality is closely related to an emergent Lieb-Schultz-Mattis (LSM) anomaly.However, the nature of the emergent LSM anomaly was not very clear to us before. Therefore, we focused on investigating the emergent anomaly. Using perturbation theory, we find that the low-energy theory of our spin-1 model near the critical self-dual point is equivalent to a spin-1/2 XYZ chain. This means that, near the self-dual point, the symmetry Z2y \rtimes Z2z × Ztrn of the complete theory reduces to Zy′ × Zz′ × Ztrn in the low-energy theory. In other words, in the low-energy theory, Z4y \rtimes Z2z × Ztrn leads to an LSM anomaly, which results in the absence of a unique gapped ground state. However, Z4y \rtimes Z4z × Ztrn in the spin-1 Hilbert space has no anomaly. In other words, the LSM anomaly around the self-dual point is actually emergent. Since the complete theory in the spin-1 Hilbert space is anomaly-free, the emergent anomaly has to be cancelled by some mechanism. Note that for the gapped symmetry Z2y, the nontrivial group element (π rotation) is identical to －1 in the low-energy theory. This indicates that the ground state is "stacked" on a gapped (weak) symmetry-protected topological (SPT) phase protected by Z2y × Ztrn. It is this SPT phase that cancels the emergent anomaly. We use field theory to demonstrate our argument.* \rtimes means semi-direct product.