[Paper Review] Z_3 symmetry-protected topological phases in the SU(3) AKLT model

Takahiro Morimoto, Hiroshi Ueda+2 more|arXiv (Cornell University)|Sep 5, 2014

Quantum Chromodynamics and Particle Interactions4 citations

TL;DR

This paper constructs Z₃ symmetry-protected topological (SPT) phases in one-dimensional SU(3) spin systems using matrix product states (MPS) with 3×3 matrices, derived from a nontrivial cocycle in H²(Z₃×Z₃, U(1)) ≅ ℤ₃. It introduces an SU(3) AKLT Hamiltonian with bilinear and biquadratic terms that realizes a nontrivial Z₃ SPT phase across a range of coupling ratios, including the SU(3) AKLT point and the limit with vanishing biquadratic coupling, confirmed via iDMRG and string order parameters.

ABSTRACT

We study $\mathbb{Z}_3$ symmetry-protected topological (SPT) phases in one-dimensional spin systems with $Z_3 imes Z_3$ symmetry. We construct ground-state wave functions of the matrix product form for nontrivial $\mathbb{Z}_3$ phases and their parent Hamiltonian from a cocycle of the group cohomology $H^2(Z_3 imes Z_3,U(1))$. 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($N$) case, the SU($N$) AKLT Hamiltonian, is also presented which realizes nontrivial $\mathbb{Z}_N$ 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 $θ$ controlling the ratio of the bilinear and biquadratic coupling constants. The nontrivial $\mathbb{Z}_3$ SPT phase is found for a range of the parameter $θ$ including the point of vanishing biquadratic term ($θ=0$) as well as the SU(3) AKLT point [$θ=\arctan(2/9)$]. A continuous phase transition to the SU(3) dimer phase takes place at $θ\approx -0.027π$, with a central charge $c\approx3.2$. For SU(3) symmetric cases we define string order parameters for the $\mathbb{Z}_3$ 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.

Motivation & Objective

• To generalize the AKLT construction to higher-rank SU(N) spin chains with Z_N×Z_N symmetry protection.
• To identify and characterize Z₃ SPT phases in one-dimensional spin systems with SU(3) symmetry.
• To construct explicit matrix product state wave functions and parent Hamiltonians for nontrivial Z₃ SPT phases using group cohomology.
• To establish string order parameters for Z₃ SPT phases analogous to the Haldane phase.
• To determine the ground-state phase diagram of the SU(3) bilinear-biquadratic model using iDMRG and identify phase transitions.

Proposed method

• Constructs Z₃ SPT wave functions using matrix product states (MPS) with 3×3 matrices derived from a nontrivial cocycle in H²(Z₃×Z₃, U(1)) ≅ ℤ₃.
• Defines the SU(3) AKLT Hamiltonian as a bilinear-biquadratic model in the adjoint representation of su(3), using traceless generators t^a normalized as tr(t^a t^b) = (1/2)δ_ab.
• Uses the infinite-size density matrix renormalization group (iDMRG) to compute the ground-state phase diagram as a function of the coupling ratio parameter θ.
• Introduces string order parameters for Z₃ SPT phases analogous to the Haldane phase, using non-local transformations to detect hidden order.
• Derives the transfer matrix M = ∑_m A^m ⊗ (A^m)* and proves its eigenvalues include 1/N² and (N²−1)/N² for N=3, confirming topological order.
• Generalizes the construction to SU(N) AKLT models, showing nontrivial Z_N SPT phases for arbitrary N via the same MPS and Hamiltonian framework.

Experimental results

Research questions

• RQ1 Can Z₃ symmetry-protected topological phases be realized in one-dimensional SU(3) spin chains with Z₃×Z₃ symmetry?
• RQ2 How can matrix product states with 3×3 matrices be used to construct nontrivial Z₃ SPT phases protected by group cohomology?
• RQ3 What is the ground-state phase diagram of the SU(3) bilinear-biquadratic model as a function of the coupling ratio θ?
• RQ4 Does the SU(3) AKLT point and the limit with vanishing biquadratic coupling both belong to the same nontrivial Z₃ SPT phase?
• RQ5 Can string order parameters be defined to detect Z₃ SPT order in SU(3) chains, similar to the Haldane phase?

Key findings

• The nontrivial Z₃ SPT phase is realized for a range of coupling parameters θ, including both the SU(3) AKLT point (θ = arctan(2/9)) and the limit with vanishing biquadratic coupling (θ = 0).
• A continuous phase transition to an SU(3) dimer phase occurs at θ ≈ −0.027π with central charge c ≈ 3.2, indicating critical behavior.
• The SU(3) AKLT Hamiltonian is constructed as a bilinear-biquadratic model in the adjoint representation, with terms involving su(3) generators in the adjoint representation.
• The ground state is a valence bond solid formed by quark-antiquark singlets between neighboring sites, with virtual degrees of freedom transforming under the fundamental and conjugate representations of SU(3).
• String order parameters are defined for the Z₃ SPT phase, analogous to the Haldane phase, and serve as order parameters for the hidden symmetry breaking.
• The SU(N) AKLT model is generalized to realize nontrivial Z_N SPT phases for arbitrary N, with the parent Hamiltonian constructed from group cohomology data in H²(Z_N×Z_N, U(1)) ≅ ℤ_N.