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

The S=1 quantum antiferromagnetic chain with the Hamiltonian H= Jᵢ Sᵢˣ{S}ᵢ₊₁ˣ$+${S}ᵢʸSᵢ₊₁ʸ+{λ}Sᵢᶻ Sᵢ₊₁ᶻ+D(Sᵢᶻ{)}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 N\'eel 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.