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arXiv:1808.10138 · Journal of the Physical Society of Japan

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

Isao Maruyama, Shin Miyahara

Published Nov 16, 2018• 8 citations
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Abstract

We study the fractionally quantized $Z_N$ Berry phase, $γ_N=0, {2π\over N}, {4π\over N},\ldots, {2(N-1)π\over N}$, 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 $\langle m\rangle$ and $D$-fold degenerated states has $γ_N=π(\langle m\rangle-1) D$, 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.

DOI
10.7566/jpsj.87.123703

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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.
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In this paper we continue and extend a systematic study of plateaux in magnetization curves of antiferromagnetic Heisenberg spin-1/2 ladders. We first review a bosonic field-theoretical formulation of a single XXZ\mathrm{XXZ}XXZ chain in the presence of a magnetic field, which is then used for an Abelian bosonization analysis of NNN weakly coupled chains. Predictions for the universality classes of the phase transitions at the plateaux boundaries are obtained in addition to a quantization condition for the value of the magnetization on a plateau. These results are complemented by and checked against strong-coupling expansions. Finally, we analyze the strong-coupling effective Hamiltonian for an odd number NNN of cylindrically coupled chains numerically. For N=3N=3N=3 we explicitly observe a spin gap with a massive spinon-type fundamental excitation and obtain indications that this gap probably survives the limit $\stackrel{\ensuremath{\rightarrow}}{N}\ensuremath{\infty}$.
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