Fractionally Quantized Berry Phases of Magnetization Plateaux in Spin-1/2 Heisenberg Multimer Chains

We study the fractionally quantized Z_N Berry phase, γ_N = 0,{2π}/{N},{4π}/{N}, \ldots, {2(N-1)π}/{N}, to characterize local N-mer spin structures at magnetization plateaux in spin-1/2 Heisenberg multimer (N-mer) models, i.e., highly frustrated N-leg ladder models, which are generalizations of an orthogonal dimer chain and have exact ground states in the strong multimer coupling region. We demonstrate that all N types of Berry phases, which characterize magnetization-plateau phases, appear in a magnetic phase diagram when N = 2 and 4. We show that magnetization plateau with magnetization <m> and D-fold degenerated states has γ_N = π(<m> - 1)D, except for the Haldane phase with γ_N = 0. In addition, we find that a complementary Z_N Berry phase becomes non-zero in the S = N/2 Haldane phase for N = 2 and 4. Because the exact quantization of the Z_N Berry phases is protected by the translational (or rotational) symmetry along the rung direction, the Z_N Berry phase has the potential to be applied for a wide class of magnetization plateaux in coupled multimer systems.