QUANTUM LIQUID IN ANTIFERROMAGNETIC CHAINS - A STOCHASTIC GEOMETRIC APPROACH TO THE HALDANE GAP

The S = 1 quantum antiferromagnetic chain with the Hamiltonian H = SIGMA-iS(i)xS(i)x+1 + S(i)yS(i)y+1 + lambda-S(i)zS(i)z+1 + D(S(i)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 Neel 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.