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arXiv:1105.6021 · Physical Review B

Symmetry-protected topological orders of one-dimensional spin systems with D2+T symmetry

Zheng-Xin Liu, Xie Chen, Xiao-Gang Wen

Published Nov 29, 2011• 22 citations
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Abstract

In Liu et al. [Phys. Rev. B 84, 075135 (2011)] we studied eight gapped symmetric quantum phases in $S=1$ spin chains/ladders which respect a discrete spin rotation ${D}_{2}\ensuremath{\subset}SO(3)$ and time reversal $T$ symmetries. In this paper, using a generalized approach, we study all the 16 possible gapped symmetric quantum phases of one-dimensional (1D) integer spin systems with ${D}_{2}+T$ symmetry. Those phases are beyond Landau symmetry breaking theory and cannot be characterized by local order parameters since they do not break any symmetry. They correspond to 16 symmetry protected topological (SPT) orders. We show that all the 16 SPT orders can be fully characterized by the physical properties of the symmetry protected degenerate boundary states at the ends of a chain segment. So we can measure and distinguish all the 16 SPT orders experimentally. We also show that all these SPT orders can be realized in $S=1$ spin chain or spin ladder models. The gapped symmetric phases protected by subgroups of ${D}_{2}+T$ are also studied. Again, all these phases can be distinguished by physically measuring their end ``spins.''

DOI
10.1103/physrevb.84.195145

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In Liu et al. [Phys. Rev. B 84, 075135 (2011)] we studied eight gapped symmetric quantum phases in S=1S=1S=1 spin chains/ladders which respect a discrete spin rotation D2\ensuremath⊂SO(3){D}_{2}\ensuremath{\subset}SO(3)D2​\ensuremath⊂SO(3) and time reversal TTT symmetries. In this paper, using a generalized approach, we study all the 16 possible gapped symmetric quantum phases of one-dimensional (1D) integer spin systems with D2+T{D}_{2}+TD2​+T symmetry. Those phases are beyond Landau symmetry breaking theory and cannot be characterized by local order parameters since they do not break any symmetry. They correspond to 16 symmetry protected topological (SPT) orders. We show that all the 16 SPT orders can be fully characterized by the physical properties of the symmetry protected degenerate boundary states at the ends of a chain segment. So we can measure and distinguish all the 16 SPT orders experimentally. We also show that all these SPT orders can be realized in S=1S=1S=1 spin chain or spin ladder models. The gapped symmetric phases protected by subgroups of D2+T{D}_{2}+TD2​+T are also studied. Again, all these phases can be distinguished by physically measuring their end ``spins.''
We study different quantum phases in integer spin systems with on-site ${D}_{2h}={D}_{2}\ensuremath{\bigotimes}{Z}_{2}$ and translation symmetry. We find four distinct nontrivial phases in S=1S=1S=1 spin chains despite the fact that they all have the same symmetry. All four phases have gapped bulk excitations, doubly degenerate end states, and the doubly degenerate entanglement spectrum. These nontrivial phases are examples of symmetry-protected topological (SPT) phases introduced by Gu and Wen [Phys. Rev. B 80, 155131 (2009)]. One of the SPT phases corresponds to the Haldane phase and the other three are new. These four SPT phases can be distinguished experimentally by their different responses of the end states to weak external magnetic fields. According to the Chen--Gu--Wen classification, the D2h{D}_{2h}D2h​ symmetric spin chain can have a total of 64 SPT phases that do not break the symmetry. Here we constructed seven nontrivial phases from the seven classes of nontrivial projective representations of the D2h{D}_{2h}D2h​ group. Four of these are found in S=1S=1S=1 spin chains and studied in this paper.
Symmetry protected topological (SPT) phases are gapped short-range-entangled quantum phases with a symmetry GGG. They can all be smoothly connected to the same trivial product state if we break the symmetry. The Haldane phase of spin-1 chain is the first example of SPT phases which is protected by SO(3)\mathit{SO}(3)SO(3) spin rotation symmetry. The topological insulator is another example of SPT phases which are protected by U(1)U(1)U(1) and time-reversal symmetries. In this paper, we show that interacting bosonic SPT phases can be systematically described by group cohomology theory: Distinct ddd-dimensional bosonic SPT phases with on-site symmetry GGG (which may contain antiunitary time-reversal symmetry) can be labeled by the elements in H1+d[G,UT(1)]{\mathcal{H}}^{1+d}[G,{U}_{T}(1)]H1+d[G,UT​(1)], the Borel (1+d)(1+d)(1+d)-group-cohomology classes of GGG over the GGG module UT(1){U}_{T}(1)UT​(1). Our theory, which leads to explicit ground-state wave functions and commuting projector Hamiltonians, is based on a new type of topological term that generalizes the topological $\ensuremath{\theta}$ term in continuous nonlinear $\ensuremath{\sigma}$ models to lattice nonlinear $\ensuremath{\sigma}$ models. The boundary excitations of the nontrivial SPT phases are described by lattice nonlinear $\ensuremath{\sigma}$ models with a nonlocal Lagrangian term that generalizes the Wess-Zumino-Witten term for continuous nonlinear $\ensuremath{\sigma}$ models. As a result, the symmetry GGG must be realized as a non-on-site symmetry for the low-energy boundary excitations, and those boundary states must be gapless or degenerate. As an application of our result, we can use ${\mathcal{H}}^{1+d}[U(1)\ensuremath{\rtimes}{Z}_{2}^{T},{U}_{T}(1)]$ to obtain interacting bosonic topological insulators (protected by time reversal Z2T{Z}_{2}^{T}Z2T​ and boson number conservation), which contain one nontrivial phase in one-dimensional (1D) or 2D and three in 3D. We also obtain interacting bosonic topological superconductors (protected by time-reversal symmetry only), in term of H1+d[Z2T,UT(1)]{\mathcal{H}}^{1+d}[{Z}_{2}^{T},{U}_{T}(1)]H1+d[Z2T​,UT​(1)], which contain one nontrivial phase in odd spatial dimensions and none for even dimensions. Our result is much more general than the above two examples, since it is for any symmetry group. For example, we can use ${\mathcal{H}}^{1+d}[U(1)\ifmmode\times\else\texttimes\fi{}{Z}_{2}^{T},{U}_{T}(1)]$ to construct the SPT phases of integer spin systems with time-reversal and U(1)U(1)U(1) spin rotation symmetry, which contain three nontrivial SPT phases in 1D, none in 2D, and seven in 3D. Even more generally, we find that the different bosonic symmetry breaking short-range-entangled phases are labeled by the following three mathematical objects: $({G}_{H},{G}_{\ensuremath{\Psi}},{\mathcal{H}}^{1+d}[{G}_{\ensuremath{\Psi}},{U}_{T}(1)])$, where GH{G}_{H}GH​ is the symmetry group of the Hamiltonian and ${G}_{\ensuremath{\Psi}}$ the symmetry group of the ground states.
Symmetry protected topological (SPT) phases are gapped short-range-entangled quantum phases with a symmetry G. They can all be smoothly connected to the same trivial product state if we break the symmetry. The Haldane phase of spin-1 chain is the first example of SPT phase which is protected by SO(3) spin rotation symmetry. The topological insulator is another exam- ple of SPT phase which is protected by U(1) and time reversal symmetries. It has been shown that free fermion SPT phases can be systematically described by the K-theory. In this paper, we show that interacting bosonic SPT phases can be systematically described by group cohomology theory: distinct d-dimensional bosonic SPT phases with on-site symmetry G (which may contain anti-unitary time reversal symmetry) can be labeled by the elements in H^{1+d}[G, U_T(1)] - the Borel (1 + d)-group-cohomology classes of G over the G-module U_T(1). The boundary excitations of the non-trivial SPT phases are gapless or degenerate. Even more generally, we find that the different bosonic symmetry breaking short-range-entangled phases are labeled by the following three mathematical objects: (G_H, G_{\Psi}, H^{1+d}[G_{\Psi}, U_T(1)], where G_H is the symmetry group of the Hamiltonian and G_{\Psi} the symmetry group of the ground states.
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 onsite unitary symmetry has been given for 1D spin systems in [X. Chen, Z.-C. Gu, and X.-G. Wen, Phys. Rev. B 83, 035107 (2011); arXiv:1008.3745]. 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 considering the possibility of 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.
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.
We study symmetric gapped ground states of one-dimensional spin SSS chains with a continuous exponential symmetry U(1)⋊Z2U(1)\rtimes\mathbb Z_2U(1)⋊Z2​ with integer base qqq, where the U(1) charge has a modulated unit of charge qL−jq^{L-j}qL−j decaying exponentially with site jjj. The simplest model is a quantum breakdown model that can be viewed as a qqq-deformation of the XXZ Haldane spin chain. We show that the topology of their symmetric ground states crucially depends on SSS and qqq. For integer SSS, odd qqq allows a nontrivial Z2\mathbb Z_2Z2​ Haldane-type symmetry-protected topological (SPT) phase with protected edge modes, while even qqq allows no nontrivial SPT phase. For half-odd integer SSS, odd qqq prohibits any unique symmetric gapped ground state due to a Lieb-Schultz-Mattis obstruction, whereas even qqq enforces a novel symmetric gapped phase which we name as the ``half-SPT" phase. Unconventionally, the half-SPT phase only has a symmetry protected edge mode on one edge, and has half of the string orders of an SPT phase. Our results uncover a new type of symmetry-protected gapped phases beyond existing classification schemes.