Higher-form symmetry breaking at Ising transitions
Higher-form symmetry breaking at Ising transitions
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DOI：
10.1103/physrevresearch.3.033024
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发表时间：
2020-11
影响因子：
4.2
通讯作者：
Jiarui Zhao;Zheng Yan;M. Cheng;Z. Meng
中科院分区：
文献类型：
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作者：
Jiarui Zhao;Zheng Yan;M. Cheng;Z. Meng
近年来，超越朗道对称性破缺范式的新相态不断涌现，为了跟上这一快速发展，新的整体对称性概念被引入。其中，对称性荷在空间上扩展的高阶对称性可以用来描述拓扑有序相，即对称性的自发破缺，从而将非常规相和常规相统一在同一概念框架下。然而，除了某些可解模型外，这些概念工具还没有被投入定量测试，因此限制了它在更一般的量子多体系统中的使用。本文研究了量子Ising模型中的Z2高阶对称性，它与整体Ising对称性（0-形式）是对偶的。我们计算的伊辛无序算子，这是一个非局部的高阶对称性序参数的期望值，分析在自由标量理论和通过无偏量子Monte Carlo模拟的相互作用的不动点在（2+1）d。从这个扩展对象的标度形式，我们确认，高形式的对称性确实是自发打破顺磁，或量子无序相（在朗道意义上）内，但在铁磁/有序相保持对称。在伊辛临界点，我们发现，高形式的对称性也自发破缺，即使0-形式的对称性被保存。我们讨论的例子中，全球0-形式的对称性和对偶的高形式的对称性被保存，在系统的余维-1流形的动量空间中的无隙点。这些结果提供了高阶对称算子的非平凡工作实例，包括相互作用共形场论中单形式序参量的首次计算，并为它们在量子多体系统中的通用实现开辟了道路。
In recent years, new phases of matter that are beyond the Landau paradigm of symmetry breaking are mountaining, and to catch up with this fast development, new notions of global symmetry are introduced. Among them, the higher-form symmetry, whose symmetry charges are spatially extended, can be used to describe topologically ordered phases as the spontaneous breaking of the symmetry, and consequently unify the unconventional and conventional phases under the same conceptual framework. However, such conceptual tools have not been put into quantitative test except for certain solvable models, therefore limiting its usage in the more generic quantum manybody systems. In this work, we study Z2 higher-form symmetry in a quantum Ising model, which is dual to the global (0-form) Ising symmetry. We compute the expectation value of the Ising disorder operator, which is a non-local order parameter for the higher-form symmetry, analytically in free scalar theories and through unbiased quantum Monte Carlo simulations for the interacting fixed point in (2+1)d. From the scaling form of this extended object, we confirm that the higher-form symmetry is indeed spontaneously broken inside the paramagnetic, or quantum disordered phase (in the Landau sense), but remains symmetric in the ferromagnetic/ordered phase. At the Ising critical point, we find that the higher-form symmetry is also spontaneously broken, even though the 0-form symmetry is preserved. We discuss examples where both the global 0-form symmetry and the dual higher-form symmetry are preserved, in systems with a codimension-1 manifold of gapless points in momentum space. These results provide non-trivial working examples of higher-form symmetry operators, including the first computation of one-form order parameter in an interacting conformal field theory, and open the avenue for their generic implementation in quantum many-body systems.