Quantized Berry phases for a local characterization of spin liquids in frustrated spin systems

Recently, by using quantized Berry phases, a prescription for a local characterization of gapped topological insulators has been given (Hatsugai 2006 Preprint cond-mat/0603230). It requires that the ground state is gapped and is invariant under some anti-unitary operation. A spin liquid which is realized as a unique ground state of the Heisenberg spin system with frustrations is a typical target system, since pairwise exchange couplings are always time-reversal invariants even with frustrations. As for a generic Heisenberg model with a finite excitation gap, we locally modify the Hamiltonian by a continuous SU(2) twist only at a specific link and define the Berry connection by the derivative. Then the Berry phase evaluated by the entire many-spin wavefunction is used to define the local topological order parameter at the link. We numerically apply this scheme for several spin liquids and show its physical validity. For example, it implies that the Haldane phase of the S = 1 chains is characterized by uniform pi quantized Berry phases.