Spin-1/2 frustrated Heisenberg model show novel ground states due to the quantum fluctulation. To classify the ground states with toprogical invariant, we introduce fractionally quantized Berry phase. The Berry phase for the Heisenberg model is defined by local spin twists in the spin xy plane. As typical examples, we consider plateau states in frustrated magnets, e.g., a multimer ( N -mer) chains. Fractional quantization of the Berry phase comes from rotational symmetry of the twisted cluster. The Z N Berry phases are obtained from the numerical integration with using the ground states calculated by the Lanczos method. As a result, we obtained fractional quantization of Z 2 and Z 4 Berry phases in dimer ( N =2) and tetramer ( N =4) chains respectively.
We study the fractionally quantized
ZN Berry phase,
γN=0,N2π,N4π,…,N2(N−1)π, 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
γN=π(⟨m⟩−1)D, except for the Haldane phase with
γN=0. In addition, we find that a complementary
ZN Berry phase becomes non-zero in the
S=N/2 Haldane phase for
N=2 and 4. Because the exact quantization of the
ZN Berry phases is protected by the translational (or rotational) symmetry along the rung direction, the
ZN Berry phase has the potential to be applied for a wide class of magnetization plateaux in coupled multimer 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 π quantized Berry phases.
Recently by using quantized Berry phases, a prescription for a local characterization of gapped topological insulators is given [1].One requires 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.
The local
ZN quantized Berry phase for the
SU(N) antiferromagnetic Heisenberg spin model is formulated. This quantity, which is a generalization of the local
Z2 Berry phase for SU(2) symmetry, has a direct correspondence to the number of singlet pairs spanning a particular bond, and is effective as a tool to characterize and classify the various symmetry-protected topological phases of one-dimensional
SU(N) spin systems. We extend the path-integral quantum Monte Carlo method for the
Z2 Berry phase in order to calculate the
ZN Berry phase numerically. We demonstrate our method by calculating the
Z4 Berry phase for the bond-alternating
SU(4) antiferromagnetic Heisenberg chain, which is represented by the Young diagram of four columns.
We characterize several phases of gapped spin systems by local order parameters defined by quantized Berry phases [Y. Hatsugai, J. Phys. Soc. Jpn. 75, 123601 (2006)]. This characterization is topologically stable against any small perturbation as long as the energy gap remains finite. The models we pick up are
S=1,2 dimerized Heisenberg chains and
S=2 Heisenberg chains with uniaxial single-ion-type anisotropy. Analytically, we also evaluate the topological local order parameters for the generalized Affleck-Kennedy-Lieb-Tasaki model. The relation between the present Berry phases and the fractionalization in the integer spin chains are discussed as well.
We propose to use quantized Berry phases as local order parameters of gapped quantum liquids, which are invariant under some anti-unitary operation. After presenting a general prescription, the scheme is applied for Heisenberg models with frustrations and two dimensional extended Su-Schrieffer-Heeger models associated with a random dimer covering. In each phases, the quantized Berry phases, as 0 or
π, describe the ground state texture pattern of local singlet bonds and dimer bonds. Also possible applications to large classes of correlated electron systems are discussed.