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Journal of Physics Condensed Matter

The spin-1/2 Heisenberg spin ladder with bond alternation

Kiyohito Totsuka, Masuo Suzuki

Published Jul 24, 1995• 38 citations
Publisher

Abstract

The spin-1/2 Heisenberg spin ladder model is studied by via a perturbation calculation and via bosonization. We mainly focus on the case of ferromagnetic interchain coupling in relation to the Haldane problem. Using the GKO coset construction of the conformal field theory, we bosonize the system in terms of the SU(2) (spin) sector and a residual Ising sector. The interchain interaction generically drives the system to strong-coupling massive phases (the Haldane phase or the dimerized phase). However, it is suggested that the criticality appears for particular choices of the couplings.

DOI
10.1088/0953-8984/7/30/012

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The spin-1/2 Heisenberg spin ladder model is studied by via a perturbation calculation and via bosonization. We mainly focus on the case of ferromagnetic interchain coupling in relation to the Haldane problem. Using the GKO coset construction of the conformal field theory, we bosonize the system in terms of the SU(2) (spin) sector and a residual Ising sector. The interchain interaction generically drives the system to strong-coupling massive phases (the Haldane phase or the dimerized phase). However, it is suggested that the criticality appears for particular choices of the couplings.
We construct a field-theoretic description of two coupled spin-1 Heisenberg chains, starting with the known representation of a single spin-1 chain in terms of Majorana fermions (or Ising models). After reexamining the bosonization rules for two Ising models, taking particular care of order and disorder operators, we obtain a bosonic description of the spin-1 ladder. From renormalization-group and mean-field arguments, we conclude that, for a small interchain coupling, the spin-1 ladder is approximately described by three decoupled, two-frequency sine-Gordon models. We then predict that, starting with decoupled chains, the spin gap decreases linearly with interchain coupling, in both the ferromagnetic and antiferromagnetic directions. Finally, we discuss the possibility of an incommensurate phase in the spin-1 zigzag chain.
We introduce a set of discrete modular transformations and in order to study the relationships between the different phases of the Heisenberg ladders obtained with all possible exchange coupling constants. For the two-legged ladder we show that the resonating valence bond (RVB) phase is invariant under the transformation, while the Haldane phase is invariant under . These two phases are related by . Moreover, there is a `mixed' phase, that is invariant under , and which under becomes the RVB phase, while under becomes the Haldane phase. For odd ladders there exists only the transformation which, for strong coupling, maps the effective antiferromagnetic spin chain onto the spin chain. Our work is based on a combination of approximate methods such as bosonization, perturbation theory and the nonlinear sigma model, adapted to the different regimes of coupling constants.
Using the density matrix renormalization group method, we determine the phase diagram of a frustrated bond-alternating S = 1/2 Heisenberg ladder with ferro-antiferromagnetic couplings at zero temperature. With the interactions between spins along the rungs set, we identify three spin-gapped phases (the Haldane phase, the singlet phase and the dimer phase) in the whole parameter range. The analysis of our data shows that two-leg spin bond-alternating ladders have a rich phase diagram if both rung and diagonal couplings are taken into account.
We investigate an S =(1/2) ladder model with a ferromagnetic interchain coupling (-λ) and an antiferromagnetic intrachain coupling based on the bosonization method, which reduces to the S =1 antiferromagnetic Heisenberg model in the limit λ→∞. We argue with the aid of Wilson's renormalization group method that the ground state is the Haldane state with an excitation gap for λ>0, that the gap opens at λ=0 exponentially and that the critical index of the coherence length is unity (ν=1) on the critical line between the antiferromagnetic and the Haldane phases. Furthermore, it is found that the string order parameter proposed by den Nijs and Rommelse is expressed by a local order parameter in the bosonization method.
The critical properties of the S=1/2S=1/2S=1/2 Heisenberg two-leg ladders are investigated in a magnetic field. Combining the exact diagonalization method and the finite-size-scaling analysis based on conformal field theory, we calculate the critical exponents of spin correlation functions numerically. For a strong interchain coupling, magnetization dependence of the critical exponents shows characteristic behavior depending on the sign of the interchain coupling. We also calculate the critical exponents for the S=1/2S=1/2S=1/2 Heisenberg two-leg ladder with a diagonal interaction, which is thought of as a model Hamiltonian of the organic spin ladder compound ${\mathrm{Cu}}_{2}(1,4\ensuremath{-}\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{z}\mathrm{a}\mathrm{c}\mathrm{y}\mathrm{c}\mathrm{l}\mathrm{o}\mathrm{h}\mathrm{e}\mathrm{p}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{e}{)}_{2}{\mathrm{Cl}}_{4}.$ Numerical results are compared with experimental results of temperature dependence of the NMR relaxation rate 1/T1.{1/T}_{1}.1/T1​.
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.