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arXiv:cond-mat/9506026 · Physical review. B, Condensed matter

Semiclassical description of spin ladders

David Sénéchal

Published Dec 1, 1995• 53 citations
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Abstract

The Heisenberg spin ladder is studied in the semiclassical limit, via a mapping to the nonlinear \ensuremath{\sigma} model. Different treatments are needed if the interchain coupling K is small, intermediate, or large. For intermediate coupling a single nonlinear \ensuremath{\sigma} model is used for the ladder. It predicts a spin gap for all nonzero values of K if the sum s+s\ifmmode \tilde{}\else \~{}\fi{} of the spins of the two chains is an integer, and no gap otherwise. For small K, a better treatment proceeds by coupling two nonlinear \ensuremath{\sigma} models, one for each chain. For integer s=s\ifmmode \tilde{}\else \~{}\fi{}, the saddle point approximation predicts a sharp drop in the gap as K increases from zero. A Monte Carlo simulation of a spin-1 ladder is presented which supports the analytical results.

DOI
10.1103/physrevb.52.15319

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The Heisenberg spin ladder is studied in the semiclassical limit, via a mapping to the nonlinear \ensuremath{\sigma} model. Different treatments are needed if the interchain coupling K is small, intermediate, or large. For intermediate coupling a single nonlinear \ensuremath{\sigma} model is used for the ladder. It predicts a spin gap for all nonzero values of K if the sum s+s\ifmmode \tilde{}\else ~{}\fi{} of the spins of the two chains is an integer, and no gap otherwise. For small K, a better treatment proceeds by coupling two nonlinear \ensuremath{\sigma} models, one for each chain. For integer s=s\ifmmode \tilde{}\else ~{}\fi{}, the saddle point approximation predicts a sharp drop in the gap as K increases from zero. A Monte Carlo simulation of a spin-1 ladder is presented which supports the analytical results.
The well known Haldane map from spin chains into the O(3) nonlinear sigma model is generalized to the case of spin ladders. This map allows us to explain the different qualitative behaviour between even and odd ladders, in exactly the same way as it explains the difference between integer and half-integer spin chains. Namely, for even ladders the topological term in the sigma model action is absent, while for odd ladders the parameter, which multiplies the topological term, is equal to , where S is the spin of the ladder. Hence even ladders should have a dynamically generated spin gap, while odd ladders with half-integer spin should stay gapless and physically equivalent to a perturbed Wess - Zumino - Witten model in the infrared regime. We also derive some consequences from the dependence of the sigma model coupling constant on the ladder Heisenberg couplings constants.
Heisenberg antiferromagnetic spin ``ladders'' (two coupled spin chains) are low-dimensional magnetic systems which for S=1/2 interpolate between half-integer-spin chains, when the chains are decoupled, and effective integer-spin one-dimensional chains in the strong-coupling limit. The spin-1/2 ladder may be realized in nature by vanadyl pyrophosphate, (VO)2{)}_{2})2​P2{\mathrm{P}}_{2}P2​O7{\mathrm{O}}_{7}O7​. In this paper we apply strong-coupling perturbation theory, spin-wave theory, Lanczos techniques, and a Monte Carlo method to determine the ground-state energy and the low-lying excitation spectrum of the ladder. We find evidence of a nonzero spin gap for all interchain couplings ${\mathit{J}}_{\mathrm{\ensuremath{\perp}}}$>0. A band of spin-triplet excitations above the gap is also analyzed. These excitations are unusual for an antiferromagnet, since their long-wavelength dispersion relation behaves as (k-k0{\mathit{k}}_{0}k0​)2{)}^{2})2 (in the strong-coupling limit ${\mathit{J}}_{\mathrm{\ensuremath{\perp}}}$\ensuremath{\gg}J, where J is the in-chain antiferromagnetic coupling). Their band is folded, with a minimum energy at k0{\mathit{k}}_{0}k0​=\ensuremath{\pi}, and a maximum between k1{\mathit{k}}_{1}k1​=\ensuremath{\pi}/2 (for ${\mathit{J}}_{\mathrm{\ensuremath{\perp}}}$=0) and 0 (for ${\mathit{J}}_{\mathrm{\ensuremath{\perp}}}$=\ensuremath{\infty}). We also give numerical results for the dynamical structure factor S(q,\ensuremath{\omega}), which can be determined in neutron scattering experiments. Finally, possible experimental techniques for studying the excitation spectrum are discussed.
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.
The spin S = 1 Heisenberg chain is an interesting object. First of all, as we have already discussed in Part III, spin chains with integer and half-integer spins have very different low energy properties. The results of Chapter 16 suggest that chains with integer spin have a spectral gap which is generated dynamically. The fact that there are many experimental realizations of quasi-one-dimensional antiferromagnets with S = 1 makes the situation even more interesting. The most well studied compound is Ni(C 2 H 8 N 2 ) 2 NO 2 (ClO 4 ), abbreviated as NENP. Localized spins belong to magnetic Ni ions and all the other ingredients are necessary just to arrange them into well separated chains. The experiments show that there is indeed a spectral gap (Aijro et al. , 1989; Renard et al. , 1987; see Fig. 36.1). This gap is not very small in comparison with the bandwidth, but one can still hope that the continuum description is good enough. The Monte Carlo simulations give for the spin S = 1 Heisenberg chain the ratio of the correlation length to the lattice spacing ξ/ a = 6.2 (Nomura, 1989) which can still be treated as a large number. There is another intriguing fact about the S =1 Heisenberg chain which makes it attractive for a theorist. It is the presence of a hidden, so-called topological order. This feature puts this theory in a much broader context of spin liquids , magnetically disordered systems with a hidden order. Spin ladder The spin S = 1 Heisenberg chain can be described as a limiting case of two ferromagnetically coupled S = 1/2 chains. The problem of coupled chains (so-called spin ladders ) is interesting in its own right.
The antiferromagnetic Heisenberg model on a chain with nearest- and next-nearest-neighbor couplings is mapped onto the SO(3) nonlinear sigma model in the continuum limit. In one spatial dimension this model is always in its disordered phase and a gap opens to excited states. The latter form a doubly degenerate spin-1 branch at all orders in 1/N. We argue that this feature should be present in the spin-1 Heisenberg model itself. Exact diagonalizations are used to support this claim. The inapplicability of this model to half-integer spin chains is discussed.
We study a model of two weakly coupled isotropic spin-1/2 Heisenberg chains with an antiferromagnetic coupling along the chains (spin ladder). It is shown that the system always has a spectral gap and the lower lying excitations are triplets. For the case of identical chains the model in the continuous limit is shown to be equivalent to four decoupled noncritical Ising models with the Z2{\mathit{Z}}_{2}Z2​\ifmmode\times\else\texttimes\fi{}SU(2) symmetry. For this case we obtain the exact expressions for asymptotics of spin-spin correlation functions. It is shown that when the chains have different exchange integrals J1{\mathit{J}}_{1}J1​\ensuremath{\gg}J2{\mathit{J}}_{2}J2​ the spectrum at low energies is described by the O(3)-nonlinear \ensuremath{\sigma} model. We discuss the topological order parameter related to the gap formation and give a detailed description of the dynamical magnetic susceptibility. \textcopyright{} 1996 The American Physical Society.