Quantum liquid in antiferromagnetic chains: A stochastic geometric approach to the Haldane gap

The S=1 quantum antiferromagnetic chain with the Hamiltonian H= ${\mathcal{J}}_{\mathit{i}}$ ${\mathrm{S}}_{\mathit{i}}^{\mathit{x}}$ ${\mathit{S}}_{\mathit{i}+1\mathrm{}}^{\mathit{x}}$+${\mathit{S}}_{\mathit{i}}^{\mathit{y}}$ ${\mathit{S}}_{\mathit{i}+1\mathrm{}}^{\mathit{y}}$+λ${\mathit{S}}_{\mathit{i}}^{\mathit{z}}$ ${\mathit{S}}_{\mathit{i}+1\mathrm{}}^{\mathit{z}}$+D(${\mathit{S}}_{\mathit{i}}^{\mathit{z}}$ ${)}^2$ is studied. By developing a (path-integral-type) stochastic geometric representation and using the ideas of percolation, we find that the Haldane phase with a unique disordered ground state and a gap can be regarded as ‘‘liquid’’. The large-D phase and the Néel ordered phase are identified as ‘‘gas’’ and ‘‘solid’’, respectively. We introduce a new order parameter that distinguishes the Haldane phase from other disordered phases such as the large-D phase or the dimer phase.