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arXiv:1912.11383 · Physical review. B./Physical review. B

Symmetry-protected trivial phases and quantum phase transitions in an anisotropic antiferromagnetic spin-1 biquadratic model

Xihao Chen, Ian P. McCulloch, Murray T. Batchelor, Huan-Qiang Zhou

Published Aug 24, 2020• 4 citations
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Abstract

The ground-state phase diagram is obtained for an antiferromagnetic spin-1 anisotropic biquadratic model. With the help of symmetry and duality transformations, three symmetry-protected trivial phases and one dimerized symmetry-breaking phase are found. Local and nonlocal order parameters are identified to characterize these phases. Quantum phase transitions between the symmetry-protected phases belong to the Gaussian universality class with central charge $c=1$, and quantum phase transitions from the symmetry-protected trivial phases to the dimerized phase belong to the Ising universality class with central charge $c=1/2$. In addition, the model admits three characteristic lines of factorized ground states, which are located in the symmetry-protected trivial phases instead of a symmetry-breaking phase, in sharp contrast to other known cases.

DOI
10.1103/physrevb.102.085146

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The ground-state phase diagram is obtained for an antiferromagnetic spin-1 anisotropic biquadratic model. With the help of symmetry and duality transformations, three symmetry-protected trivial phases and one dimerized symmetry-breaking phase are found. Local and nonlocal order parameters are identified to characterize these phases. Quantum phase transitions between the symmetry-protected phases belong to the Gaussian universality class with central charge c=1c=1c=1, and quantum phase transitions from the symmetry-protected trivial phases to the dimerized phase belong to the Ising universality class with central charge c=1/2c=1/2c=1/2. In addition, the model admits three characteristic lines of factorized ground states, which are located in the symmetry-protected trivial phases instead of a symmetry-breaking phase, in sharp contrast to other known cases.
The ground-state phase diagram is mapped out for an alternative anisotropic extension of quantum spin-1 ferromagnetic biquadratic model, which accommodates twelve distinct phases: three degenerate fractal phases, six Luttinger liquid phases and three symmetry-protected trivial phases. It is found that distinct types of quantum phase transitions are involved between them. In particular, one type arises from an instability of a Luttinger liquid towards a degenerate fractal phase, and the other type describes spontaneous symmetry breaking with type-B Goldstone modes from one degenerate fractal phase to another degenerate fractal phase, with the fractal dimension dfd_fdf​ being identical to the number of the type-B Goldstone modes, both of which turn out to be one. In addition, quantum phase transitions from the Luttinger liquid phases to the symmetry-protected trivial phases are identified to be in the Kosterlitz-Thouless universality class, with central charge being one.
By the infinite time-evolving block decimation method, the ground state properties of the S=1 Heisenberg–Ising alternating chain have been investigated in the thermodynamic limit. A ground-state phase diagram, which includes three different phases, i.e., an antiferromagnetic strip phase, a dimer phase, and an antiferromagnetic phase, is obtained. A modified definition of the string order is adopted to describe the dimer phase, and it is found to have nonzero antiferromagnetic topological string order. The quantum phase transitions (QPTs) in such a model can be well described by the singularities of bipartite entanglement and fidelity bifurcations, and two second-order QPTs points are determined to be at JI≃±0.66983. The central charges at both QPTs points are determined to be ∼1/2. Therefore, both QPTs belong to the Ising universality class with central charges c=c‾=1/2.
An extensive investigation, both numerical and analytical, is performed for an anisotropic extension of the ferromagnetic spin-1 biquadratic model. The ground state phase diagram accommodates three symmetry-protected trivial phases, three coexisting fractal phases and six Luttinger liquid phases. A novel universality class arises from an instability of a Luttinger liquid towards an exotic quantum state of matter with infinitely degenerate ground states. The latter in turn is a scale-invariant quantum state of matter, which may be attributed to the coexistence of SU(2){\rm SU}(2)SU(2) spontaneous symmetry breaking with one type-B Goldstone mode on the characteristic line: Jy=JzJ_y=J_zJy​=Jz​, and U(1){\rm U}(1)U(1) spontaneous symmetry breaking without any gapless Goldstone mode on the characteristic line Jx/Jz=0J_x/J_z=0Jx​/Jz​=0, together with their cyclic permutations with respect to xxx, yyy and zzz.
The S=1/2S=1/2S=1/2 and S=1S=1S=1 two-dimensional quantum Heisenberg antiferromagnets on the anisotropic dimerized square lattice are investigated by the quantum Monte Carlo method. By finite-size-scaling analyses on the correlation lengths, the ground-state phase diagram parametrized by strengths of the dimerization and of the spatial anisotropy is determined much more accurately than the previous works. It is confirmed that the quantum critical phenomena on the phase boundaries belong to the same universality class as that of the classical three-dimensional Heisenberg model. Furthermore, for S=1,S=1,S=1, we show that all the spin-gapped phases, such as the Haldane and dimer phases, are adiabatically connected in the extended-parameter space, though they are classified into different classes in terms of the string order parameter in the one-dimensional, i.e., the zero-interchain-coupling, case.
Duality and ground-state phase diagram for the quantum XYZ model with arbitrary spin sss in one spatial dimension
Five duality transformations are unveiled for the quantum XYZ model with arbitrary spin sss in one spatial dimension. The presence of these duality transformations drastically reduces the entire ground-state phase diagram to two {\it finite} regimes - the principal regimes, with all the other ten regimes dual to them. Combining with the determination of critical points from the conventional order parameter approach and/or the fidelity approach to quantum phase transitions, we are able to map out the ground-state phase diagram for the quantum XYZ model with arbitrary spin sss. This is explicitly demonstrated for s=1/2,1,3/2s=1/2,1,3/2s=1/2,1,3/2 and 2. As it turns out, all the critical points, with central charge c=1c=1c=1, are self-dual under a respective duality transformation for half-integer as well as integer spin sss. However, in the latter case, the presence of the so-called symmetry protected topological phase, i.e., the Haldane phase, results in extra lines of critical points with central charge c=1/2c=1/2c=1/2, which is not self-dual under any duality transformation.
An anyon-chain-like lattice model with symmetry described by the Ising fusion category is studied. Combining numerical and analytical studies, we uncover a rich phase diagram that contains three phases: a symmetric critical phase and two categorical symmetry breaking phases. The symmetric phase lies in the same universality class as the usual critical Ising model. The first symmetry-breaking phase, dubbed the \emph{categorical ferromagnetic} phase, has the Ising fusion category fully broken and exhibits a threefold ground-state degeneracy, as expected from the generalized Landau paradigm. The other symmetry-breaking phase is analogous to a conventional antiferromagnet: it breaks lattice translation and part of the Ising fusion category, and therefore is termed the \emph{categorical antiferromagnetic} phase. Unlike ordinary antiferromagnetic states associated with finite invertible symmetry breaking, this phase itself is critical, being described by a fourfold degenerate Ising conformal field theory. We argue more generally that antiferromagnetic states associated with broken non-invertible symmetries have a large low-energy manifold that grows exponentially in system size, due to the greater-than-one quantum dimension of domain walls. We also numerically study the transitions between the three phases. The transition between the symmetric and categorical ferromagnetic phase is described by the c=7/10c=7/10c=7/10 tricritical Ising CFT, while the transition between the symmetric and categorical antiferromagnetic phases is less understood. Our numerical data suggest that the latter transition is continuous and described by a conformal field theory with central charge c=3/2c=3/2c=3/2.