Emergent Haldane phase in the S=1 bilinear-biquadratic Heisenberg model on the square lattice

Abstract

Infinite projected entangled pair states simulations of the S = 1 bilinear-biquadratic Heisenberg model on the square lattice reveal an emergent Haldane phase in between the previously predicted antiferromagnetic and 3-sublattice 120 • magnetically ordered phases. This intermediate phase preserves SU(2) spin and translational symmetry but breaks lattice rotational symmetry, and it can be adiabatically connected to the Haldane phase of decoupled S = 1 chains. Our results contradict previous studies which found a direct transition between the two magnetically ordered states.The search for novel states of matter in quantum manybody systems is one of the most active areas in condensed matter physics. A fascinating example is the ground state of the S = 1 antiferromagnetic Heisenberg chain which, unlike the S = 1/2 chain, exhibits an energy gap, exponentially decaying spin-spin correlations, and gapless edge excitations in case of open boundaries. Thanks to Haldane's pioneering work and conjecture that such a gapped state emerges in integer Heisenberg spin chains in general [1,2], this phase has been named after him.The Haldane phase also extends to related S = 1 models, such as weakly-coupled S = 1 Heisenberg chains [3][4][5][6][7] which are realized in several quasi-1D materials [8][9][10][11][12][13], or the S = 1 bilinear-biquadratic Heisenberg (BBH) chain with Hamiltonian H = i,j cos(θ)S i · S j + sin(θ) (S i · S j ) 2 , for θ in between −π/4 and π/4. More recently, the Haldane phase has been understood as a simple example of a symmetry protected topological (SPT) phase [14][15][16][17].In the present work we focus on the BBH model in two dimensions, which has gained much interest in recent years [18][19][20][21][22][23][24][25][26]; firstly, due to its possible connection to the triangular lattice compounds NiGa 2 S 4 [27] and Ba 3 NiSb 2 O 9 [25, 28-30], and secondly, for θ = π/4, the model is equivalent to the SU(3) Heisenberg model which can be experimentally realized using ultra-cold fermionic atoms in optical lattices [31][32][33][34]. The latter has been shown to exhibit 3-sublattice order on the square and triangular lattices [35,36], and an important question concerns the stability of this phase away from the SU(3) symmetric point. Previous studies on the square lattice based on linear flavor-wave theory [24], exact diagonalization [24], and series expansion [26] predicted a direct transition between the AF and the 3-sublattice phase for θ ≈ 0.2π. However, the accurate study of this parameter regime remains very challenging because Quantum Monte Carlo suffers from the negative sign problem.In this paper we show, using state-of-the-art tensor network simulations, that in between the AF and the 3- AF phaseHaldane phase 3-SL 120° phase 0.217(4) 0.189 (2) FIG. 1. Haldane phase emerging in between the antiferromagnetic (AF) and the 3-sublattice 120 • ordered phases.sublattice phase an intermediate quantum paramagnetic phase emerges which preserves translational and SU(2) spin symmetry, but breaks lattice rota...

Ground-state study of the spin-1 bilinear-biquadratic Heisenberg model on the triangular lattice using tensor networks

“…Moreover, in our recent study of the spin-1 BBH model on the square lattice [27,28], we found the occurrence of a quantum paramagnetic phase in between the antiferromagnetic and 120 • magnetically ordered phases, and we were able to show that this phase can be adiabatically connected to the one-dimensional Haldane phase of decoupled spin-1 chains. In addition, we also encountered a partially-magnetic partially-quadrupolar phase in between the antiferroquadrupolar and ferromagnetic phases.…”

“…Inspired by our finding [27] of the one-dimensional Haldane phase extending all the way to the two-dimensional isotropic limit on the square lattice, we searched for signs of a possible quantum paramagnetic phase on the triangular lattice. The simple update results ( Fig.…”

“…More recently, we conducted an iPEPS study of the spin-1 BBH model on the two-dimensional square lattice [27,28], which yielded the occurrence of two additional phases on top of those present in the product ground state phase diagram. In particular, we found that in between the ordinary antiferromagnetic (AFM) and 120 • magnetically ordered (AFM3) phases, a quantum paramagnetic phase arises [27] that is characterized by the fact that it preserves spin-rotation and lattice-translation symmetry while breaking latticerotation symmetry due to energy differences between the x and y-bonds. Moreover, by continuously shrinking all high energy bond couplings to zero, the quantum paramagnetic phase turned out to be adiabatically connected to the Haldane phase of decoupled spin-1 chains, and can thus be viewed as a two-dimensional extension of the latter.…”

“…Motivated by our findings for the corresponding square lattice model [27,28], we also investigate the anisotropic triangular lattice spin-1 BBH model for θ ∈ (−π/4, π/4). We show that, on the triangular lattice, the onedimensional Haldane phase does not reach up to the two-dimensional isotropic model-albeit that it does extend far from the one-dimensional limit.…”

Spin nematics in frustrated spin-dimer systems with bilayer structure

“…7 The emergence of a quasi-one-dimensional Haldane phase in a narrow region between the Néel and AFM3 phases was also reported. 8 The phase diagram for the honeycomb lattice includes the FM, FQ, Néel, and plaquette valence-bond-solid phases. 9 For the triangular lattice, the phase diagram contains the FM, FQ, 120 • -structure antiferromagnetic (120 • -AFM), and AFQ3 phases.…”