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arXiv:2307.03653 · Europhysics Letters (EPL)

Spinor bosons realization of the SU(3) Haldane phase with adjoint representation

Junjun Xu

Published Jan 1, 2026• 0 citations
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Abstract

Abstract The Haldane phase with local SU (3) adjoint representation constitutes the simplest non-trivial symmetry-protected topological phases in the SU Heisenberg spin chains. In this paper, we propose to realize this phase by a two-species spinor Bose gas, with each species labeling the quark or antiquark states of SU (3) symmetry. In the strong-coupling limit, we determine the ground-state phase diagram, and identify a quantum phase transition from the Haldane phase to a dimer phase. We show how to characterize the Haldane phase through its edge excitations. We also explain the physics at the dimer phase, by constructing an explicit ground-state ansatz at the dimer point.

DOI
10.1209/0295-5075/ae33eb

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Abstract The Haldane phase with local SU (3) adjoint representation constitutes the simplest non-trivial symmetry-protected topological phases in the SU Heisenberg spin chains. In this paper, we propose to realize this phase by a two-species spinor Bose gas, with each species labeling the quark or antiquark states of SU (3) symmetry. In the strong-coupling limit, we determine the ground-state phase diagram, and identify a quantum phase transition from the Haldane phase to a dimer phase. We show how to characterize the Haldane phase through its edge excitations. We also explain the physics at the dimer phase, by constructing an explicit ground-state ansatz at the dimer point.
We give a bosonic realization of the SU(3) antiferromagnetic Heisenberg (AFH) chain in the alternating conjugate representation, and study its phase diagram as a function of staggered interactions and anisotropy along the T3T^3T3 and T8T^8T8 directions. Unlike the SU(2) case, we observe a chiral-reversed quantum phase transition, where each competing phase is adiabatically connected to one of the chiral Haldane phases predicted in the SU(3) AFH chain with local adjoint representation. In the vicinity of the Heisenberg point, we identify a symmetry-protected topological state that appears at the first excited energy level. We also study the spontaneous Z3\mathbb{Z}_3Z3​ symmetry breaking of the system, and provide a variational wavefunction that captures the transition from the topological phase to the trivial phase. Finally, we propose an experimental realization of our bosonic model by two spin-1/2 bosons in an optical lattice.
Gapped quantum spin chains with symmetry PSU(N)=SU(N)/ZN\text{PSU}(N)=\text{SU}(N)/{\mathbb{Z}}_{N}PSU(N)=SU(N)/ZN​ are known to possess NNN distinct symmetry protected topological phases. Besides the trivial phase, there are $N\ensuremath{-}1$ Haldane phases which are distinguished by the occurrence of massless boundary spins. We give an expression for a universal AKLT Hamiltonian for an arbitrary symmetry group, including the case where inversion symmetry is broken. Using a diagrammatic method, we construct Hamiltonians for two of the topological classes in SU(N)\text{SU}(N)SU(N) that arise when the physical spin is in the adjoint representation. Since our results are valid for all NNN, we also discuss the large-NNN limit.
Abstract Topology in quantum many-body systems has profoundly changed our understanding of quantum phases of matter. The model that has played an instrumental role in elucidating these effects is the antiferromagnetic spin-1 Haldane chain 1,2 . Its ground state is a disordered state, with symmetry-protected fourfold-degenerate edge states due to fractional spin excitations. In the bulk, it is characterized by vanishing two-point spin correlations, gapped excitations and a characteristic non-local order parameter 3,4 . More recently it has been understood that the Haldane chain forms a specific example of a more general classification scheme of symmetry-protected topological phases of matter, which is based on ideas connected to quantum information and entanglement 5–7 . Here, we realize a finite-temperature version of such a topological Haldane phase with Fermi–Hubbard ladders in an ultracold-atom quantum simulator. We directly reveal both edge and bulk properties of the system through the use of single-site and particle-resolved measurements, as well as non-local correlation functions. Continuously changing the Hubbard interaction strength of the system enables us to investigate the robustness of the phase to charge (density) fluctuations far from the regime of the Heisenberg model, using a novel correlator.
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.
Chiral Haldane phases are examples of one-dimensional topological states of matter which are protected by the projective SU(N)\mathrm{SU}(N)SU(N) group (or its subgroup ${\mathbb{Z}}_{N}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{N})$ with N>2N>2N>2. The unique feature of these symmetry-protected topological (SPT) phases is that they are accompanied by inversion-symmetry breaking and the emergence of different left and right edge states which transform, for instance, respectively in the fundamental $(\mathbit{N})$ and antifundamental $(\overline{\mathbit{N}})$ representations of SU(N)\mathrm{SU}(N)SU(N). We show, by means of complementary analytical and numerical approaches, that these chiral SPT phases as well as the nonchiral ones are realized as the ground states of a generalized two-leg SU(N)\mathrm{SU}(N)SU(N) spin ladder in which the spins in the first chain transform in $\mathbit{N}$ and the second in $\overline{\mathbit{N}}$. In particular, we map out the phase diagram for N=3N=3N=3 and 4 to show that all the possible symmetry-protected topological phases with projective SU(N)\mathrm{SU}(N)SU(N) symmetry appear in this simple ladder model.
The nature of symmetry-protected topological phases of Heisenberg spin chains in totally symmetric representations of rank NNN of the SU(N\mathrm{SU}(NSU(N) group is investigated through a Majorana fermion study starting from an integrable point. The latter approach generalizes the one pioneered by Tsvelik [Phys. Rev. B 42, 10499 (1990)] to describe the low-energy properties of the Haldane phase of the spin-1 Heisenberg chain from three massive Majorana fermions. We find, for all NNN's, the emergence of a nondegenerate gapped phase with edge states whose topological protection depends on the parity of NNN. Whereas for NNN odd, there is no such protection, the phase with even NNN is shown to be topologically protected. We find that the phase belongs to the same topological class as the phase with edge states living in self-conjugate fully antisymmetric representation of the SU(N)\mathrm{SU}(N)SU(N) group.