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arXiv:cond-mat/9610168 · Physical Review Letters

Magnetization Plateaus in Spin Chains: “Haldane Gap” for Half-Integer Spins

Masaki Oshikawa, Masanori Yamanaka, Ian K. Affleck

Published Mar 10, 1997• 710 citations
arXivHTMLPDF

Abstract

We discuss zero-temperature quantum spin chains in a uniform magnetic field, with axial symmetry. For integer or half-integer spin, $S$, the magnetization curve can have plateaus and we argue that the magnetization per site $m$ is topologically quantized as $n(S\ensuremath{-}m)\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\mathrm{integer}$ at the plateaus, where $n$ is the period of the ground state. We also discuss conditions for the presence of the plateau at those quantized values. For $S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}3/2$ and $m\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/2$, we study several models and find two distinct types of massive phases at the plateau. One of them is argued to be a ``Haldane gap phase'' for half-integer $S$.

DOI
10.1103/physrevlett.78.1984

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We discuss zero-temperature quantum spin chains in a uniform magnetic field, with axial symmetry. For integer or half-integer spin, SSS, the magnetization curve can have plateaus and we argue that the magnetization per site mmm is topologically quantized as n(S\ensuremath−m)=integern(S\ensuremath{-}m)\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\mathrm{integer}n(S\ensuremath−m)=integer at the plateaus, where nnn is the period of the ground state. We also discuss conditions for the presence of the plateau at those quantized values. For S=3/2S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}3/2S=3/2 and m=1/2m\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/2m=1/2, we study several models and find two distinct types of massive phases at the plateau. One of them is argued to be a ``Haldane gap phase'' for half-integer SSS.
Plateaus can be observed in the zero-temperature magnetization curve of quantum spin systems at rational values of the magnetization. In one dimension, the appearance of a plateau is controlled by a quantization condition for the magnetization which involves the length of the local spin and the volume of a translational unit cell of the ground state. We discuss examples of geometrically frustrated quantum spin systems with large (in general unbounded) periodicities of spontaneous breaking of translational symmetry in the ground state. In two dimensions, we discuss the square, triangular and Kagomé lattices using exact diagonalization (ED) for up to N = 40 sites. For the spin-1/2 XXZ model on the triangular lattice we study the nature and stability region of a plateau at one third of the saturation magnetization. The Kagomé lattice gives rise to particularly rich behaviour with several plateaus in the magnetization curve and a jump due to local magnon excitations just below saturation.
We unveil nontrivial topological properties of zero-temperature magnetization plateau states in periodically modulated quantum spin chains under a uniform magnetic field. As positions of plateaus are uniquely determined by the modulation period of exchange couplings, we find that the topologically nontrivial plateau state can be characterized by a nonzero integer Chern number and has nontrivial edge excitations. Our study clarifies the topological origin of the well-known phenomena of quantized magnetization plateaus in one-dimensional quantum spin systems and relates the plateau state to the correlated topological insulator.
Hubbard chains with periodically modulated coupling constants in a magnetic field exhibit gaps at zero temperature in their magnetic and charge excitations in a variety of situations. In addition to fully gapped situations (plateau in the magnetization curve and charge gap), we have shown [Phys. Lett. A 268, 418 (2000)] that plateaus also appear in the presence of massless modes, leading to a plateau with a magnetization m whose value depends continuously on the filling n. Here we detail and extend the arguments leading to such doping-dependent magnetization plateaus. First we analyze the low-lying excitations using Abelian bosonization. We compute the susceptibility and show that due to the constraint of fixed n, it vanishes at low temperatures (thus leading to a magnetization plateau) even in the presence of one massless mode. Next we study correlation functions and show that one component of the superconducting order parameter develops quasi-long-range order on a doping-dependent magnetization plateau. We then use perturbation theory in the on-site repulsion U to compute the width of these plateaus up to first order in U. Finally, we compute ground state phase diagrams and correlation functions by Lanczos diagonalization of finite clusters, confirming the presence of doping-dependent plateaus and their special properties.
The bulk electric polarization works as a nonlocal order parameter that characterizes topological quantum matters. Motivated by a recent paper [H. Watanabe et al., Phys. Rev. B 103, 134430 (2021)], we discuss magnetic analogs of the bulk polarization in one-dimensional quantum spin systems, that is, quantized magnetizations on the edges of one-dimensional quantum spin systems. The edge magnetization shares the topological origin with the fractional edge state of the topological odd-spin Haldane phases. Despite this topological origin, the edge magnetization can also appear in topologically trivial quantum phases. We develop straightforward field theoretical arguments that explain the characteristic properties of the edge magnetization. The field theory shows that a U(1) spin-rotation symmetry and a site-centered or bond-centered inversion symmetry protect the quantization of the edge magnetization. We proceed to discussions that quantum phases on nonzero magnetization plateaus can also have the quantized edge magnetization that deviates from the magnetization density in bulk. We demonstrate that the quantized edge magnetization distinguishes two quantum phases on a magnetization plateau separated by a quantum critical point. The edge magnetization exhibits an abrupt stepwise change from zero to 1/21/21/2 at the quantum critical point because the quantum phase transition occurs in the presence of the symmetries protecting the quantization of the edge magnetization. We also show that the quantized edge magnetization can result from the spontaneous ferrimagnetic order.
A symmetry-protected topologically ordered phase is a short-range entangled state, for which some imposed symmetry prohibits the adiabatic deformation into a trivial state which lacks entanglement. In this paper we argue that magnetization plateau states of one-dimensional antiferromagnets which satisfy the conditions $S\ensuremath{-}m\ensuremath{\in}$ odd integer, where SSS is the spin quantum number and mmm the magnetization per site, can be identified as symmetry-protected topological states if an inversion symmetry about the link center is present. This assertion is reached by mapping the antiferromagnet into a nonlinear sigma model type effective field theory containing a novel Berry phase term (a total derivative term) with a coefficient proportional to the quantity $S\ensuremath{-}m$, and then analyzing the topological structure of the ground state wave functional which is inherited from the latter term. A boson-vortex duality transformation is employed to examine the topological stability of the ground state in the absence/presence of a perturbation violating link-center inversion symmetry. Our prediction based on field theories is verified by means of a numerical study of the entanglement spectra of actual spin chains, which we find to exhibit twofold degeneracies when the aforementioned condition is met. We complete this study with a rigorous analysis using matrix product states.