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arXiv:0807.2896 · Journal of Physics Conference Series

Quantized Berry phases of a Spin-1/2 frustrated two-leg ladder with four-spin exchange

Isao Maruyama, Takaaki Hirano, Yasuhiro Hatsugai

Published Jan 1, 2009• 4 citations
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Abstract

A spin-1/2 frustrated two-leg ladder with four-spin exchange interaction is studied by quantized Berry phases. We found that the Berry phase successfully characterizes the Haldane phase in addition to the rung-singlet phase, and the dominant vector-chirality phase. The Hamiltonian of the Haldane phase is topologically identical to the S=1 antiferromagnetic Heisenberg chain. Decoupled models connected to the dominant vector-chirality phase revealed that the local object identified by the non-trivial ($π$) Berry phase is the direct product of two diagonal singlets.

DOI
10.1088/1742-6596/145/1/012052

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A spin-1/2 frustrated two-leg ladder with four-spin exchange interaction is studied by quantized Berry phases. We found that the Berry phase successfully characterizes the Haldane phase in addition to the rung-singlet phase, and the dominant vector-chirality phase. The Hamiltonian of the Haldane phase is topologically identical to the S=1 antiferromagnetic Heisenberg chain. Decoupled models connected to the dominant vector-chirality phase revealed that the local object identified by the non-trivial (πππ) Berry phase is the direct product of two diagonal singlets.
A spin-12\frac{1}{2}21​ two-leg ladder with four-spin ring exchange is studied by quantized Berry phases, used as local-order parameters. Reflecting local objects, nontrivial $(\ensuremath{\pi})$ Berry phase is founded on a rung for the rung-singlet phase and on a plaquette for the vector-chiral phase. Since the quantized Berry phase is topologically invariant for gapped systems with the time-reversal symmetry, topologically identical models can be obtained by the adiabatic modification. The rung-singlet phase is adiabatically connected to a decoupled rung-singlet model and the vector-chiral phase is connected to a decoupled vector-chiral model. Decoupled models reveal that the local objects are a local singlet and a plaquette singlet, respectively.
We study the fractionally quantized ZNZ_NZN​ Berry phase, γN=0,2πN,4πN,…,2(N−1)πNγ_N=0, {2π\over N}, {4π\over N},\ldots, {2(N-1)π\over N}γN​=0,N2π​,N4π​,…,N2(N−1)π​, to characterize local NNN-mer spin structures at magnetization plateaux in spin-1/2 Heisenberg multimer (NNN-mer) models, i.e., highly frustrated NNN-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 NNN types of Berry phases, which characterize magnetization-plateau phases, appear in a magnetic phase diagram when N=2N=2N=2 and 444. We show that magnetization plateau with magnetization ⟨m⟩\langle m\rangle⟨m⟩ and DDD-fold degenerated states has γN=π(⟨m⟩−1)Dγ_N=π(\langle m\rangle-1) DγN​=π(⟨m⟩−1)D, except for the Haldane phase with γN=0γ_N=0γN​=0. In addition, we find that a complementary ZNZ_NZN​ Berry phase becomes non-zero in the S=N/2S=N/2S=N/2 Haldane phase for N=2N=2N=2 and 4. Because the exact quantization of the ZNZ_NZN​ Berry phases is protected by the translational (or rotational) symmetry along the rung direction, the ZNZ_NZN​ Berry phase has the potential to be applied for a wide class of magnetization plateaux in coupled multimer systems.
We present the phase diagram of the S=1/2S=1/2S=1/2 Heisenberg model on the two leg ladder with cyclic four-spin exchange, determined by a combination of exact diagonalization and density-matrix renormalization group techniques. We find six different phases and regimes: the rung singlet phase, a ferromagnetic phase, two symmetry broken phases with staggered dimers and staggered scalar chiralities, and a gapped region with dominant vector chirality, or collinear spin correlations. We localize the phase transitions and investigate their nature.
A spin-1/2 Heisenberg model with a chirality-chirality interaction (CCI) on a two-leg ladder provides a minimal setup to explore an interplay between spin and chirality degrees of freedom. This model is potentially relevant for understanding the effects of a four-spin ring exchange and a phonon-mediated interaction on a ladder. By applying the Abelian bosonization and two types of spin-chirality duality, we derive a comprehensive ground-state phase diagram that includes previously unexplored cases of a ferromagnetic rung exchange J⊥<0J_\perp<0J⊥​<0 or a negative CCI Kch<0K_{\rm ch}<0Kch​<0. The phase diagram consists of the rung singlet (RS) phase, the Haldane phase, and three non-magnetic ordered phases. Moreover, the RS and Haldane phases consist of several regions with different dominant correlations in terms of spin, dimer, or chirality. We perform numerical simulations to confirm the predicted phase structure and critical properties.
We investigate a frustrated four-leg spin-1/2 ladder using density matrix renormalization group calculations. The uniform system displays three regimes: short-range antiferromagnetic legs, short-range ferromagnetic legs, and an effective spin-2 Heisenberg chain, separated by a crossover and a first-order transition. The spin-2 regime is confirmed through its finite string order parameter, edge-localized excitations, and excellent agreement with a projected Sr = 2 effective Hamiltonian. Recasting the model as two frustrated two-leg ladders coupled by rung and diagonal interactions, we track how the trivial and Haldane phases of an isolated ladder evolve as interladder couplings are introduced. The resulting phase diagrams reveal crossover and first-order lines whose locations are captured by the spin-2 projection and show how singlet- and triplet-dominated regimes reorganize when two ladders merge into a four-leg structure, clarifying the emergence of effective spin-1 versusspin-2 behavior.
The magnetization M(h)M(h)M(h) is used to identify three singlet quantum phases of the ladder with isotropic exchange interactions. The Dimer phase with frustrated F exchanges in rungs and legs has a first-order M(h)M(h)M(h) transition at 000 K from singlet to ferromagnetic at the saturation field hsh_shs​. The Haldane-DAF phase with strong F exchange in rungs and net AF exchange between rungs has continuous M(h)M(h)M(h) and is adiabatically connected to the S=1S = 1S=1 Heisenberg AF chain. The AF phase with strong F exchange in legs and net AF exchange between legs has continuous M(h)M(h)M(h) and is adiabatically connected to the spin-1/2 J1−J2J_1-J_2J1​−J2​ model with J1>0J_1 > 0J1​>0 and J2<0J_2 < 0J2​<0. All three singlet phases have finite gaps to the lowest triplet state.