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arXiv:cond-mat/9602055 · Physical review. B, Condensed matter

Quantum spin ladders atT=0 and at high temperatures studied by series expansions

J. Oitmaa, Rajiv R. P. Singh, Weihong Zheng

Published Jul 1, 1996• 62 citations
arXivHTMLPDF

Abstract

We have carried out extensive series studies, at T=0 and at high temperatures, of two-chain and three-chain spin-half ladder systems with antiferromagnetic intrachain and both antiferromagnetic and ferromagnetic interchain couplings. Our results confirm the existence of a gap in the two-chain Heisenberg ladders for all nonzero values of the interchain couplings. Complete dispersion relations for the spin-wave excitations are computed. For three-chain systems, our results are consistent with a gapless spectrum. We also calculate the uniform magnetic susceptibility and specific heat as a function of temperature. We find that as T\ensuremath{\rightarrow}0, for the two-chain system the uniform susceptibility goes rapidly to zero, whereas for the three-chain system it approaches a finite value. These results are compared in detail with previous studies of finite systems. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physrevb.54.1009

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We have carried out extensive series studies, at T=0 and at high temperatures, of two-chain and three-chain spin-half ladder systems with antiferromagnetic intrachain and both antiferromagnetic and ferromagnetic interchain couplings. Our results confirm the existence of a gap in the two-chain Heisenberg ladders for all nonzero values of the interchain couplings. Complete dispersion relations for the spin-wave excitations are computed. For three-chain systems, our results are consistent with a gapless spectrum. We also calculate the uniform magnetic susceptibility and specific heat as a function of temperature. We find that as T\ensuremath{\rightarrow}0, for the two-chain system the uniform susceptibility goes rapidly to zero, whereas for the three-chain system it approaches a finite value. These results are compared in detail with previous studies of finite systems. \textcopyright{} 1996 The American Physical Society.
The properties of antiferromagnetic Heisenberg S= 1/2 ladders with two, three and four chains are expanded in the ratio of the intrachain and interchain coupling constants. A simple mapping procedure is introduced to relate the four- and two-chain ladders, which keeps to moderate values of the expansion parameters. A second-order calculation of the spin gap to the lowest triplet excitation in the two- and four-chain ladders is found to be quite accurate, even at the isotropic point where the couplings are equal. Similar expansions and mapping procedures are presented for the three-chain ladders, which are in the same universality class as single chains.
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.
Abstract In this paper we consider ladders with two, three, and four coupled Ising spin chains, characterized by interchain and intrachain couplings, to study in detail their magnetic behaviour for different ratios of the interaction constants and different values of magnetic field, by using a transfer matrix method and a complex algorithm, realizing the diagonalization and the differentiation. We have proposed a combined algorithm to realize also the generation of the elements of the transfer matrix for more complicated ladder configurations. The problem of Ising ladders is straightforward with a crossover between a 2D high‐temperature region and a 1D low‐temperature region when the 2D correlation length equals the width. To emphasize the fact that there is no finite magnetization in absence of the magnetic field, we study also analytically this case for the ladder with two coupled Ising chains. In presence of the magnetic field, for very weak magnetic fields (of the order 10−8, in reduced units) the magnetization goes from the value one to the value zero in a narrow interval of nonzero temperatures, where also the susceptibility exhibits a very high peak. This narrow interval, increasing the number of chains, is displaced towards higher temperatures. Also, in the case of interchain antiferromagnetic couplings, the formation of interchain spin pairs becomes clear.
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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.
A full description of the previously proposed method of calculating low temperature specific heat and susceptibility of quantum spin chains are given. By the use of internal energy as the expansion variable the domain of convergence of the high temperature series is remarkably extended. The low temperature properties are obtained by continuing it in a natural way to polynomials in internal energy, whose form is suggested by the spin wave theory. The method is checked in the S =1/2 XY model and is then applied to the Heisenberg model of S =1/2, 1, 3/2 and 2. The results are satisfactory in ferromagnets and reasonable in the S =1/2 antiferromagnet. In antiferromagnets of S ≧1 the results suggest inadequacy of our polynomial, which is, presumably, due to the existence of an energy gap.