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

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 n͑S 2 m͒ 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 3͞2 and m 1͞2, 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. [S0031-9007(97)02624-0] PACS numbers: 75.10.JmOne-dimensional antiferromagnets are expected not to have long-range magnetic order in general. It was argued by Haldane [1], in 1983, that for integer, but not half-integer spin, S, there is a gap to the excited states. In the presence of a magnetic field, the S 1͞2 Heisenberg antiferromagnetic (AF) chain remains gapless from zero field up to the saturation field, where the ground state is fully polarized [2]. For integer S, the gap persists up to a critical field, equal to the gap, where Bose condensation of magnons occurs [3]. The S 1 Heisenberg AF chain is known to be gapless from the critical field up to the saturation field [4]. Recently Hida observed that an S 1͞2 antiferromagnetic chain with period 3 exchange coupling shows a plateau in the magnetization curve at magnetization per site m 1͞6 (1͞3 of the full magnetization) [5]. Related works on bond-alternating chains have also been reported [6-9] including experimental observation [10].In this Letter, we consider the zero-temperature behavior of general quantum spin chains, including chains with periodic structures, in a uniform magnetic field pointing along the direction of the axial symmetry (z axis) (i.e., the total S z is conserved). We argue that, in quantum spin chains, there is a phenomenon which is strikingly analogous to the quantum Hall effect-topological quantization of a physical quantity under a changing magnetic field [11]. We first consider an extension of the Lieb-Schultz-Mattis (LSM) theorem [12] to the case with an applied field. This indicates that translationally invariant spin chains in an applied field can be gapful without breaking translation symmetry, only when the magnetization per spin, m, obeys S 2 m integer. We expect such gapped phases to correspond to plateaus at these quantized values of m. "Fractional quantization" can also occur, if accompanied by (explicit or spontaneous) breaking of the translational symmetry. The generalized LSM theorem does not prove the presence of the plateau, however. Thus we construct a corresponding argument using Abelian bosonization, which is in complete agreement with the generalized LSM theorem, and also gives a condition for the presence of the plateau. As simplest examples, we study translationally invariant S 3͞2 chains at m 1͞2. We present numerical diagonalization and density matrix renormalization group (DMRG) [13] calculations, which demonstrate the existen...

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