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arXiv:1412.3370 · Physical Review B

All spin-1 topological phases in a single spin-2 chain

Augustine Kshetrimayum, Hong-Hao Tu, Román Orús

Published May 19, 2015• 14 citations
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Abstract

Here we study the emergence of different symmetry-protected topological (SPT) phases in a spin-2 quantum chain. We consider a Heisenberg-like model with bilinear, biquadratic, bicubic, and biquartic nearest-neighbor interactions, as well as uniaxial anisotropy. We show that this model contains four different effective spin-1 SPT phases, corresponding to different representations of the $({\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2})+T$ symmetry group, where ${\mathbb{Z}}_{2}$ is some $\ensuremath{\pi}$ rotation in the spin internal space and $T$ is time reversal. One of these phases is equivalent to the usual spin-1 Haldane phase, while the other three are different but also typical of spin-1 systems. The model also exhibits an $SO(5)$-Haldane phase. Moreover, we also find that the transitions between the different effective spin-1 SPT phases are continuous and can be described by a $c=2$ conformal field theory. At such transitions, indirect evidence suggests a possible effective field theory of four massless Majorana fermions. The results are obtained by approximating the ground state of the system in the thermodynamic limit using matrix product states via the infinite time-evolving block decimation method, as well as by effective field theory considerations. Our findings show that different large effective spin-1 SPT phases separated by continuous quantum phase transitions can be stabilized in a simple quantum spin chain.

DOI
10.1103/physrevb.91.205118

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Here we study the emergence of different symmetry-protected topological (SPT) phases in a spin-2 quantum chain. We consider a Heisenberg-like model with bilinear, biquadratic, bicubic, and biquartic nearest-neighbor interactions, as well as uniaxial anisotropy. We show that this model contains four different effective spin-1 SPT phases, corresponding to different representations of the (Z2\ifmmode×\else\texttimes\fiZ2)+T({\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2})+T(Z2​\ifmmode×\else\texttimes\fiZ2​)+T symmetry group, where Z2{\mathbb{Z}}_{2}Z2​ is some \ensuremathπ\ensuremath{\pi}\ensuremathπ rotation in the spin internal space and TTT is time reversal. One of these phases is equivalent to the usual spin-1 Haldane phase, while the other three are different but also typical of spin-1 systems. The model also exhibits an SO(5)SO(5)SO(5)-Haldane phase. Moreover, we also find that the transitions between the different effective spin-1 SPT phases are continuous and can be described by a c=2c=2c=2 conformal field theory. At such transitions, indirect evidence suggests a possible effective field theory of four massless Majorana fermions. The results are obtained by approximating the ground state of the system in the thermodynamic limit using matrix product states via the infinite time-evolving block decimation method, as well as by effective field theory considerations. Our findings show that different large effective spin-1 SPT phases separated by continuous quantum phase transitions can be stabilized in a simple quantum spin chain.
Symmetry-protected trivial (SPt) phases of matter are the product-state analog of symmetry-protected topological (SPT) phases. This means, SPt phases can be adiabatically connected to a product state by some path that preserves the protecting symmetry. Moreover, SPt and SPT phases can be adiabatically connected to each other when interaction terms that break the symmetries protecting the SPT order are added in the Hamiltonian. It is also known that spin-1 SPT phases in quantum spin chains can emerge as effective intermediate phases of spin-2 Hamiltonians. In this paper we show that a similar scenario is also valid for SPt phases. More precisely, we show that for a given spin-2 quantum chain, effective intermediate spin-1 SPt phases emerge in some regions of the phase diagram, these also being adiabatically connected to nontrivial intermediate SPT phases. We characterize the phase diagram of our model by studying quantities such as the entanglement entropy, symmetry-related order parameters, and 1-site fidelities. Our numerical analysis uses matrix product states and the infinite time evolving block decimation method to approximate ground states of the system in the thermodynamic limit. Moreover, we provide a field theory description of the possible quantum phase transitions between the SPt phases. Together with the numerical results, such a description shows that the transitions may be described by conformal field theories with central charge c=1c=1c=1. Our results are in agreement with, and further generalize, those of Y. Fuji, F. Pollmann, and M. Oshikawa [Phys. Rev. Lett. 114, 177204 (2015)].
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.
We study different phases of the one-dimensional bond-alternating spin-1/21/21/2 Heisenberg model by using the symmetry fractionalization mechanism. We employ the infinite matrix-product state representation of the ground state (through the infinite-size density matrix renormalization group algorithm) to obtain inequivalent projective representations of the (unbroken) symmetry groups of the model, which are used to identify the different phases. We find that the model exhibits trivial as well as symmetry-protected topological phases. The symmetry-protected topological phases are Haldane phases on even/odd bonds, which are protected by the time-reversal (acting on the spin as σ→−σ\sigma\rightarrow-\sigmaσ→−σ), parity (permutation of the chain about a specific bond), and dihedral (π\piπ-rotations about a pair of orthogonal axes) symmetries. Additionally, we investigate the phases of the most general two-body bond-alternating spin-1/21/21/2 model, which respects the time-reversal, parity, and dihedral symmetries, and obtain its corresponding twelve different types of the symmetry-protected topological phases.
We study different phases of the one-dimensional bond-alternating spin-1/2 Heisenberg model by using the symmetry fractionalization mechanism. We employ the infinite matrix-product state representation of the ground state (through the infinite-size density matrix renormalization group algorithm) to obtain inequivalent projective representations and commutation relations of the (unbroken) symmetry groups of the model, which are used to identify the different phases. We find that the model exhibits trivial as well as symmetry-protected topological phases. The symmetry-protected topological phases are Haldane phases on even/odd bonds, which are protected by the time-reversal (acting on the spin as σ → -σ), parity (permutation of the chain about a specific bond), and dihedral (π-rotations about a pair of orthogonal axes) symmetries. Additionally, we investigate the phases of the most general two-body bond-alternating spin-1/2 model, which respects the time-reversal, parity, and dihedral symmetries, and obtain its corresponding twelve different types of the symmetry-protected topological phases.
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.
We study Z3{\mathbb{Z}}_{3}Z3​ symmetry-protected topological (SPT) phases in one-dimensional spin systems with ${Z}_{3}\ifmmode\times\else\texttimes\fi{}{Z}_{3}$ symmetry. We construct ground-state wave functions of the matrix product form for nontrivial Z3{\mathbb{Z}}_{3}Z3​ phases and their parent Hamiltonian from a cocycle of the group cohomology ${H}^{2}({Z}_{3}\ifmmode\times\else\texttimes\fi{}{Z}_{3},U(1))$. The Hamiltonian is an SU(3) version of the Affleck-Kennedy-Lieb-Tasaki (AKLT) model, consisting of bilinear and biquadratic terms of su(3) generators in the adjoint representation. A generalization to the SU(N)\mathrm{SU}(N)SU(N) case, the SU(N)\mathrm{SU}(N)SU(N) AKLT Hamiltonian, is also presented, which realizes nontrivial ZN{\mathbb{Z}}_{N}ZN​ SPT phases. We use the infinite-size variant of the density matrix renormalization group (iDMRG) method to determine the ground-state phase diagram of the SU(3) bilinear-biquadratic model as a function of the parameter $\ensuremath{\theta}$ controlling the ratio of the bilinear and biquadratic coupling constants. The nontrivial Z3{\mathbb{Z}}_{3}Z3​ SPT phase is found for a range of the parameter $\ensuremath{\theta}$ including the point of vanishing biquadratic term $(\ensuremath{\theta}=0)$ as well as the SU(3) AKLT point [$\ensuremath{\theta}=arctan(2/9)$]. A continuous phase transition to the SU(3) dimer phase takes place at $\ensuremath{\theta}\ensuremath{\approx}\ensuremath{-}0.027\ensuremath{\pi}$, with a central charge $c\ensuremath{\approx}3.2$. For SU(3) symmetric cases, we define string order parameters for the Z3{\mathbb{Z}}_{3}Z3​ SPT phases in a similar way to the conventional Haldane phase. We propose simple spin models that effectively realize the SU(3) and SU(4) AKLT models.