Galois Conjugates of Topological Phases
Galois Conjugates of Topological Phases
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拓扑相的伽罗瓦共轭
DOI：
10.1103/physrevb.85.045414
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发表时间：
2011
影响因子：
3.7
通讯作者：
Zhenghan Wang
中科院分区：
文献类型：
--
作者：
M. Freedman;J. Gukelberger;M. Hastings;S. Trebst;M. Troyer;Zhenghan Wang
伽罗瓦共轭将酉共形场论和拓扑量子场论（TQFT）与它们的非酉对应物联系起来。在这里，我们研究量子双模型的伽罗瓦共轭，如莱文-温模型。虽然这些伽罗瓦共轭哈密顿通常是非厄米的，我们发现，他们的基态波函数仍然服从一个广义版本的通常的代码属性（本地运营商不作用于基态流形），因此享有广义拓扑保护。本文讨论的关键问题是，这种非幺正拓扑相是否也可以作为厄米哈密顿算符的基态出现。具体的尝试，在建设厄米哈密顿与这些基态导致的代码属性和拓扑保护的退化基态的损失。除此之外，我们严格证明，没有本地的基的变化可以将伽罗瓦共轭双斐波那契理论的基态转换成拓扑模型的基态，其厄米哈密顿满足Lieb-Robinson界限。这些包括所有有缺口的本地或准本地哈密顿。类似的陈述适用于许多其他非酉TQFT。一个结果是，这些非酉TQFT不描述拓扑相位的物理实现。特别地，这意味着“加夫尼”波函数不能是带隙分数量子霍尔态的基态。
Galois conjugation relates unitary conformal field theories and topological quantum field theories (TQFTs) to their nonunitary counterparts. Here we investigate Galois conjugates of quantum double models, such as the Levin-Wen model. While these Galois-conjugated Hamiltonians are typically non-Hermitian, we find that their ground-state wave functions still obey a generalized version of the usual code property (local operators do not act on the ground-state manifold) and hence enjoy a generalized topological protection. The key question addressed in this paper is whether such nonunitary topological phases can also appear as the ground states of Hermitian Hamiltonians. Specific attempts at constructing Hermitian Hamiltonians with these ground states lead to a loss of the code property and topological protection of the degenerate ground states. Beyond this, we rigorously prove that no local change of basis can transform the ground states of the Galois-conjugated doubled Fibonacci theory into the ground states of a topological model whose Hermitian Hamiltonian satisfies Lieb-Robinson bounds. These include all gapped local or quasilocal Hamiltonians. A similar statement holds for many other nonunitary TQFTs. One consequence is that these nonunitary TQFTs do not describe physical realizations of topological phases. In particular, this implies that the ``Gaffnian'' wave function can not be the ground state of a gapped fractional quantum Hall state.