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arXiv:1501.00422 · Modern Physics Letters B

Characterizing the Haldane phase in quasi-one-dimensional spin-1 Heisenberg antiferromagnets

Keola Wierschem, Pinaki Sengupta

Published Dec 23, 2014• 55 citations
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Abstract

We review the basic properties of the Haldane phase in spin-1 Heisenberg antiferromagnetic chains, including its persistence in quasi-one-dimensional (Q1D) geometries. Using large-scale numerical simulations, we map out the phase diagram for a realistic model applicable to experimental Haldane compounds. We also investigate the effect of different chain coupling geometries and confirm a general mean-field universality of the critical coupling times the coordination number of the lattice. Inspired by recent developments in the characterization of symmetry protected topological (SPT) states, of which the Haldane phase of the spin-1 Heisenberg antiferromagnetic chain is a preeminent example, we provide direct evidence that the Q1D Haldane phase is indeed a nontrivial SPT state.

DOI
10.1142/s0217984914300178

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We review the basic properties of the Haldane phase in spin-1 Heisenberg antiferromagnetic chains, including its persistence in quasi-one-dimensional (Q1D) geometries. Using large-scale numerical simulations, we map out the phase diagram for a realistic model applicable to experimental Haldane compounds. We also investigate the effect of different chain coupling geometries and confirm a general mean-field universality of the critical coupling times the coordination number of the lattice. Inspired by recent developments in the characterization of symmetry protected topological (SPT) states, of which the Haldane phase of the spin-1 Heisenberg antiferromagnetic chain is a preeminent example, we provide direct evidence that the Q1D Haldane phase is indeed a nontrivial SPT state.
We consider a quasi-one-dimensional system of spin-1 Heisenberg antiferromagnetic chains in two-dimensional and three-dimensional hypercubic lattices with interchain coupling J and uniaxial single-ion anisotropy D. Using large-scale numerical simulations, we map out the J-D phase diagram and investigate the low-lying excitations of the Haldane phase in the J≪1 limit. We also provide direct evidence that the Haldane phase remains a nontrivial symmetry-protected topological state for small but finite J.
The nonordinary surface transitions at the (2+12+12+1)-dimensional quantum critical point precluded in the corresponding classical critical point have been found recently. The mechanism of such behavior that is only found in dimerized Heisenberg models to date is still under debate. To illuminate the role of symmetry-protected topological (SPT) phases in inducing such nonordinary behaviors, we study a system on a two-dimensional square lattice consisting of interacting spin-1 Haldane chains, which has a genuine SPT phase---the Haldane phase---at weak interchain interactions and a quantum critical point, belonging to the classical three-dimensional (3D) O(3) universality class, to the N\'eel phase. Different from previously studied models, there is no dimerization in the current model. Cutting the system along the chain direction or perpendicular to the chain direction exposes two different surfaces. Using unbiased quantum Monte Carlo simulations, we find that the two different types of surfaces show completely different surface critical behaviors at the bulk critical point, resulting from different surface states in the SPT phase. For the system with surfaces along the chain direction, the surface critical behavior is of ordinary type of the bulk 3D O(3) critical point, while for the surfaces perpendicular to the chain direction, the surface critical behavior is nonordinary, consistent with nonordinary transitions found in dimerized Heisenberg models. Our numerical results demonstrate that the gapless surface state in the gapped SPT phase together with the gapless mode of the critical point is a pure quantum scenario that leads to the nonordinary transition.
Using stochastic series expansion quantum Monte Carlo and density matrix renormalization group methods, we investigate the ground-state phase diagram of the S=1/2S=1/2S=1/2 Heisenberg model on the two-dimensional square-hexagon-octagon (SHO) lattice. The model incorporates nearest-neighbor interactions J1J_1J1​ (intrahexagon interaction) and J2J_2J2​ (interhexagon), as well as a selected third-neighbor interaction J3J_3J3​ along the xxx direction. We identify five distinct phases in the parameter regime 0<λ1=J2/J1<4,0<λ2=J3/J1<40<λ_1=J_2/J_1<4, 0<λ_2=J_3/J_1<40<λ1​=J2​/J1​<4,0<λ2​=J3​/J1​<4: a Néel antiferromagentic phase, two dimer phases (orthogonal and ladder staggered dimers), a hexagon singlet phase, and notably a Haldane-like symmetry-protected topological (SPT) phase. The topological nature of the Haldane-like phase is confirmed by the degeneracy of the ground-state energy under open boundary conditions and the twofold degeneracy of the entanglement spectrum. Phase boundaries are accurately determined using finite-size scaling of the spin stiffness and Binder cumulant. Data collapse analysis reveals that all transitions from nonmagnetic phases to the antiferromagnetic phase belong to the three-dimensional O(3)O(3)O(3) Heisenberg universality class. In addition, we investigate the robustness of the SPT phase to other interactions, such as those that could arise in experimental materials. Our work establishes a comprehensive theoretical framework for understanding magnetic and topological phases on the SHO lattice.
We study the quenching of the Haldane gap in quasi-one-dimensional systems of weakly coupled spin-1 antiferromagnetic Heisenberg chains. The critical interchain coupling Jc required to stabilize long range magnetic order can be accurately determined from large scale quantum Monte Carlo calculations. Several different geometries of coupled chains are studied, illustrating the dependence of Jc on the coordination of chains. For bipartite geometries, ferromagnetically coupled chains yield similar magnitudes for Jc.
We study the quenching of the Haldane gap in quasi-one-dimensional systems of weakly coupled spin-1 antiferromagnetic Heisenberg chains. The critical interchain coupling Jc required to stabilize long range magnetic order can be accurately determined from large scale quantum Monte Carlo calculations. Several different geometries of coupled chains are studied, illustrating the dependence of Jc on the coordination of chains. For bipartite geometries, ferromagnetically coupled chains yield similar magnitudes for Jc.
The ground state of the S=1 antiferromagnetic Heisenberg chain belongs to the Haldane phase--a well-known example of the symmetry-protected topological phase. A staggered field applied to the S=1 antiferromagnetic chain breaks all the symmetries that protect the Haldane phase as a topological phase, reducing it to a trivial phase. That is, the Haldane phase is then connected adiabatically to an antiferromagnetic product state. Nevertheless, as long as the symmetry under site-centered inversion combined with a spin rotation is preserved, the phase is still distinct from another trivial phase. We demonstrate the existence of such distinct symmetry-protected trivial phases using a field-theoretical approach and numerical calculations. Furthermore, a general proof and a nonlocal order parameter are given in terms of a matrix-product state formulation.