Symmetry-Breaking and the Classification of States

Xie Chen, Zheng‐Cheng Gu, Xiao-Gang Wen

Quantum phases with different orders exist with or without breaking the symmetry of the system. Recently, a classification of gapped quantum phases which do not break time reversal, parity, or on-site unitary symmetry has been given for 1D spin systems by X. Chen, Z.-C. Gu, and X.-G. Wen [Phys. Rev. B 83, 035107 (2011)]. It was found that such symmetry-protected topological (SPT) phases are labeled by the projective representations of the symmetry group which can be viewed as a symmetry fractionalization. In this paper, we extend the classification of 1D gapped phases by considering SPT phases with combined time reversal, parity, and/or on-site unitary symmetries and also the possibility of symmetry breaking. We clarify how symmetry fractionalizes with combined symmetries and also how symmetry fractionalization coexists with symmetry breaking. In this way, we obtain a complete classification of gapped quantum phases in 1D spin systems. We find that in general, symmetry fractionalization, symmetry breaking, and long-range entanglement (present in 2 or higher dimensions) represent three main mechanisms to generate a very rich set of gapped quantum phases. As an application of our classification, we study the possible SPT phases in 1D fermionic systems, which can be mapped to spin systems by Jordan-Wigner transformation.

Domenico Giulini

Cheng-Ju Lin, Liujun Zou

A spontaneous symmetry-breaking order is conventionally described by a tensor-product wavefunction of some few-body clusters; some standard examples include the simplest ferromagnets and valence bond solids. We discuss a type of symmetry-breaking orders, dubbed entanglement-enabled symmetry-breaking orders, which cannot be realized by any such tensor-product state. Given a symmetry breaking pattern, we propose a criterion to diagnose if the symmetry-breaking order is entanglement-enabled, by examining the compatibility between the symmetries and the tensor-product description. For concreteness, we present an infinite family of exactly solvable gapped models on one-dimensional lattices with nearest-neighbor interactions, whose ground states exhibit entanglement-enabled symmetry-breaking orders from a discrete symmetry breaking. In addition, these ground states have gapless edge modes protected by the unbroken symmetries. We also propose a construction to realize entanglement-enabled symmetry-breaking orders with spontaneously broken continuous symmetries. Under the unbroken symmetries, some of our examples can be viewed as symmetry-protected topological states that are beyond the conventional classifications.

Naoto Nagaosa

T. I. Schelkacheva, Elena E. Tareyeva, N. M. Chtchelkatchev

A short survey is presented on spin-glass-like states characteristics in complex nonmagnetic systems. We discuss the interplay of the interaction structure and symmetry with the classification of scenarios of the replica symmetry breaking. It is shown that the kind of the transition to the nonergodic state depends not only on the presence or absence of the reflection symmetry but on the number of interacting operators and their individual characteristics.

V. N. Ivanov