HIDDEN Z(2)XZ(2) SYMMETRY-BREAKING AND THE HALDANE PHASE IN THE S=1/2 QUANTUM SPIN CHAIN WITH BOND ALTERNATION

We study the S = 1/2 spin chain with the Hamiltonian H = SIGMA(j=1)L-1 {sigma-2j(x)sigma-2j+1x+sigma-2j(y)sigma-2j+1y+lambda-sigma-2j(z)sigma-2j+1} + beta-SIGMA(j=1)L(1-sigma-2j-1.sigma-2j), where sigma(i) = (sigma(i)x, sigma(i)y, sigma(i)z) are the Pauli matrices. We find that a nonlocal unitary transformation reveals the hidden Z2 X Z2 symmetry of the system. It has been argued that a similar hidden Z2 X Z2 Symmetry of the S = 1 chain is fully broken when and only when the system exhibits the Haldane pp. We prove that the present system exhibits both an excitation gap and a full breaking of the hidden Z2 X Z2 symmetry in a range of the parameter space including the line beta = 0, lambda > -1. We argue that the range with such properties indeed extends to the limit beta --> infinity in which the present model reduces to the S = 1 spin chain. This observation provides support of Hida's conclusion that the Haldane gap in the S = 1 chain is continuously connected to the gap in the decoupled S = 1/2 system with beta = 0.