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## Effective Chiral-Spin Hamiltonian for Odd-numbered Coupled Heisenberg Chains (2008)

### BibTeX

@MISC{Subrahmanyam08effectivechiral-spin,

author = {V. Subrahmanyam},

title = {Effective Chiral-Spin Hamiltonian for Odd-numbered Coupled Heisenberg Chains},

year = {2008}

}

### OpenURL

### Abstract

An L× ∞ system of odd number of coupled Heisenberg spin chains is studied using a degenerate perturbation theory, where L is the number of coupled chains. An effective chain Hamiltonian is derived explicitly in terms of two spin half degrees of freedom of a closed chain of L sites, valid in the regime the inter-chain coupling is stronger than the intra-chain coupling. The spin gap has been calculated numerically using the effective Hamiltonian for L = 3,5,7,9 for a finite chain up to ten sites. It is suggested that the ground state of the effective Hamiltonian is correlated, by examining various variational trial states for the effective-spin chain Hamiltonian. 2 There is a growing interest in coupled chains of Hubbard-Heisenberg spin systems, following the experimental realization of coupled arrays of metal-oxide-metal ladder systems [1, 2]. A number of investigations [3, 4, 5, 6, 7] of weakly-coupled Heisenberg spin chains have been carried out, in which the coupling between the chains is weaker than intra-chain coupling, and have provided a strong indication that a system of even number of chains can be understood in terms of a short-range resonating valence bond (RVB) picture, with spin gap and a finite spin-spin correlation length. Whereas a system of odd number of chains is gapless with power-law spin correlations, indicating that it is in the same universality class as a single spin chain. The framework for understanding the difference between even and odd numbered chains is the RVB picture[6, 8], which suggests that an even-numbered coupled chain system can be thought of as an integer spin chain exhibiting a Haldane spin gap, and an odd-numbered coupled system maps onto a half-odd integer spin chain. Below we will construct an effective spin Hamiltonian for odd-numbered coupled system, which is cast explcitily in terms of spin half degrees of freedom. This can be done straightforwardly in the regime the coupling along the chain is weaker than the coupling between the chains, complementary to the earlier studies. We consider the Heisenberg Hamiltonian given as

### Keyphrases

effective chiral-spin hamiltonian odd-numbered coupled heisenberg chain coupled chain intra-chain coupling odd number effective hamiltonian spin half degree spin gap effective spin hamiltonian effective-spin chain hamiltonian closed chain odd-numbered coupled system map even-numbered coupled chain system finite spin-spin correlation length effective chain hamiltonian integer spin chain coupled array rvb picture inter-chain coupling hubbard-heisenberg spin system various variational trial state odd numbered chain heisenberg hamiltonian universality class strong indication single spin chain coupled heisenberg spin chain weakly-coupled heisenberg spin chain power-law spin correlation short-range resonating valence bond half-odd integer spin chain haldane spin gap degenerate perturbation theory odd-numbered coupled system ground state finite chain metal-oxide-metal ladder system experimental realization