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arXiv:cond-mat/9802035 · Physical review. B, Condensed matter

Magnetization plateaux in N -leg spin ladders

D. C. Cabra, A. Honecker, Pierre Pujol

Published Sep 1, 1998• 227 citations
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Abstract

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 $\mathrm{XXZ}$ chain in the presence of a magnetic field, which is then used for an Abelian bosonization analysis of $N$ 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 $N$ of cylindrically coupled chains numerically. For $N=3$ we explicitly observe a spin gap with a massive spinon-type fundamental excitation and obtain indications that this gap probably survives the limit $\stackrel{\ensuremath{\rightarrow}}{N}\ensuremath{\infty}$.

DOI
10.1103/physrevb.58.6241

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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 N\ensuremath→\ensuremath∞\stackrel{\ensuremath{\rightarrow}}{N}\ensuremath{\infty}N\ensuremath→​\ensuremath∞.
We analyze the phase diagram of a system of spin-1/2 Heisenberg antiferromagnetic chains interacting through a zig-zag coupling, also called zig-zag ladders. Using bosonization techniques we study how a spin-gap or more generally plateaux in magnetization curves arise in different situations. While for coupled XXZ chains, one has to deal with a recently discovered chiral perturbation, the coupling term which is present for normal ladders is restored by an external magnetic field, dimerization or the presence of charge carriers. We then proceed with a numerical investigation of the phase diagram of two coupled Heisenberg chains in the presence of a magnetic field. Unusual behaviour is found for ferromagnetic coupled antiferromagnetic chains. Finally, for three (and more) legs one can choose different inequivalent types of coupling between the chains. We find that the three-leg ladder can exhibit a spin-gap and/or non-trivial plateaux in the magnetization curve whose appearance strongly depends on the choice of coupling.
Magnetization processes of spin- 12\frac{1}{2}21​ Heisenberg ladders are studied using strong-coupling expansions, numerical diagonalization of finite systems, and a bosonization approach. We find that the magnetization exhibits plateaux as a function of the applied field at certain rational fractions of the saturation value. Our main focus is on ladders with three legs where plateaux with magnetization one third of the saturation value are shown to exist.
The experimental realization of various spin ladder systems has prompted their detailed theoretical investigations. Here we study the evolution of ground-state magnetization with an external magnetic field for two different antiferromagnetic systems: a three-legged spin-1/21/21/2 ladder, and a two-legged spin-1/21/21/2 ladder with an additional diagonal interaction. The finite system density-matrix renormalization-group method is employed for numerical studies of the three-chain system, and an effective low-energy Hamiltonian is used in the limit of strong interchain coupling to study the two- and three-chain systems. The three-chain system has a magnetization plateau at one-third of the saturation magnetization. The two-chain system has a plateau at zero magnetization due to a gap above the singlet ground state. It also has a plateau at half of the saturation magnetization for a certain range of values of the couplings. We study the regions of transitions between plateaus numerically and analytically, and find that they are described, at first order in a strong-coupling expansion, by an XXZ\mathrm{XXZ}XXZ spin-1/21/21/2 chain in a magnetic field; the second-order terms give corrections to the XXZ\mathrm{XXZ}XXZ model. We also study numerically some low-temperature properties of the three-chain system, such as the magnetization, magnetic susceptibility and specific heat.
We have studied the ground state phase diagram of a spin ladder with alternating rung exchange J⊥n=J⊥[1+(−1)nδ]J^{n}_{\perp} = J_{\perp}[1 + (-1)^{n} δ]J⊥n​=J⊥​[1+(−1)nδ] in a magnetic filed, in the limit where the rung coupling is dominant. In this limit the model is mapped onto an XXZXXZXXZ Heisenberg chain in a uniform and staggered longitudinal magnetic fields, where the amplitude of the staggered field is ∼δ\sim δ∼δ. We have shown that the magnetization curve of the system exhibits a plateau at magnetization equal to the half of the saturation value. The width of a plateau scales as δνδ^νδν, where ν=4/5ν=4/5ν=4/5 in the case of ladder with isotropic antiferromagnetic legs and ν=2ν=2ν=2 in the case of ladder with isotropic ferromagnetic legs. We have calculated four critical fields (Hc1±H^{\pm}_{c1}Hc1±​ and Hc2±H^{\pm}_{c2}Hc2±​) corresponding to transitions between different magnetic phases of the system. We have shown that these transitions belong to the universality class of the commensurate-incommensurate transition.
We study numerically the formation of magnetization plateaux with the Lanczos method in two-leg ladders with mixed spins of magnitudes (S1,S2)=(1,1/2){(S}_{1}{,S}_{2})=(1,1/2)(S1​,S2​)=(1,1/2) located at alternating positions along the ladder and with dimerization $\ensuremath{\gamma}.$ For interchain coupling ${J}^{\ensuremath{'}}>0$ and $\ensuremath{\gamma}=0,$ we find normalized plateaux at m=1/3m=1/3m=1/3 starting at zero field and m=1m=1m=1 (saturation), while when $\ensuremath{\gamma}\ensuremath{\ne}0$ is columnar, another extra plateau at m=2/3m=2/3m=2/3 shows up. For ${J}^{\ensuremath{'}}<0,$ when $\ensuremath{\gamma}<{\ensuremath{\gamma}}_{c}{(J}^{\ensuremath{'}})$ we find no plateau while for $\ensuremath{\gamma}>{\ensuremath{\gamma}}_{c}{(J}^{\ensuremath{'}})$ we find four plateaux at m=0,1/3,2/3,1.m=0,1/3,2/3,1.m=0,1/3,2/3,1. We also apply several approximate analytical methods (spin-wave theory, low-energy effective Hamiltonians, and bosonization) to understand these findings and to conjecture the behavior of ferrimagnetic ladders with a larger number of legs.
We examine the properties of the Bethe ansatz solvable two- and three-leg spin-SSS ladders. These models include Heisenberg rung interactions of arbitrary strength and thus capture the physics of the spin-SSS Heisenberg ladders for strong rung coupling. The discrete values derived for the magnetization plateaux are seen to fit with the general prediction based on the Lieb-Schultz-Mattis theorem. We examine the magnetic phase diagram of the spin-1 ladder in detail and find an extended magnetization plateau at the fractional value 〈M〉=12〈M〉=\frac{1}{2}〈M〉=21​ in agreement with the experimental observation for the organic polyradical spin-1 ladder compound BIP-TENO.