On the Application of the Non Linear Sigma Model to Spin Chains and Spin Ladders

Abstract

We review the non linear sigma model approach (NLSM) to spin chains and spin ladders, presenting new results. The generalization of the Haldane's map to ladders in the Hamiltonian approach, give rise to different values of the $\theta$ parameter depending on the spin S, the number of legs $n_{\ell }$ and the choice of blocks needed to built up the NLSM fields. For rectangular blocks we obtain $\theta =0$ or $2\pi S$ depending on wether $n_{\ell }$, is even or odd, while for diagonal blocks we obtain $\theta =2\pi S{n}_{\ell }$. Both results agree modulo $2\pi$, and yield the same prediction, namely that even ( resp. odd) ladders are gapped (resp. gapless). For even legged ladders we show that the spin gap collapses exponentially with $n_{\ell }$ and we propose a finite size correction to the gap formula recently derived by Chakravarty using the 2+1 NSLM, which gives a good fit of numerical results. We show the existence of a Haldane phase in the two legged ladder using diagonal blocks and finally we consider the phase diagram of dimerized ladders.

25 pages, Latex, 7 figures in postscript files, Proc. of the 1996 El Escorial Summer School on "Strongly Correlated Magnetic and Superconducting Systems". Some more references are added

References in corpus (10)

• Surprises on the Way from 1D to 2D Quantum Magnets: the Novel Ladder Materials
• Antiferromagnetic spin ladders: crossover between spin S = 1/2 and S = 1 chains
• A Monte Carlo Study of Correlations in Quantum Spin Ladders
• Equivalence of the antiferromagnetic Heisenberg ladder to a single $S=1$ chain
• The Non Linear Sigma Model and Spin Ladders
• Susceptibility and Low Temperature Thermodynamics of Spin-1/2 Heisenberg Ladders
• Dimensional Crossover in Quantum Antiferromagnets
• Semiclassical description of spin ladders
• Low-Energy Properties of Antiferromagnetic Spin-1/2 Heisenberg Ladders with an Odd Number of Legs
• Hidden Order and Dimerization Transition in $S=2$ Chains

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• From asymptotic freedom to $\theta$ vacua: Qubit embeddings of the O(3) nonlinear $\sigma$ model
• Self-conjugate representation SU(3) chains
• Soft supersymmetry breaking in the nonlinear sigma model