The one-dimensional quantum Heisenberg model with random \ifmmode\pm\else\textpm\fi{}J bonds is studied for S=1/2 and S=1. The specific heat and the zero-field susceptibility are calculated by using high-temperature series expansions and the quantum transfer matrix method. The susceptibility shows a Curie-like temperature dependence at low temperatures as well as at high temperatures. The numerical results for the specific heat suggest that there are anomalously many low-lying excitations. The qualitative nature of these excitations is discussed based on the exact diagonalization of finite-size systems.The one-dimensional quantum Heisenberg model with random \ifmmode\pm\else\textpm\fi{}J bonds is studied for S=1/2 and S=1. The specific heat and the zero-field susceptibility are calculated by using high-temperature series expansions and the quantum transfer matrix method. The susceptibility shows a Curie-like temperature dependence at low temperatures as well as at high temperatures. The numerical results for the specific heat suggest that there are anomalously many low-lying excitations. The qualitative nature of these excitations is discussed based on the exact diagonalization of finite-size systems.

arXiv:cond-mat/9507083Physical review. B, Condensed matterDec 1, 199518 citations

Specific heat and susceptibility of the Heisenberg chain with finite concentration of impurities

P. Schlottmann

Journal of Magnetism and Magnetic MaterialsFeb 1, 19951 citation

Classical heisenberg model at zero temperature

Mariya V Shcherbina

Theoretical and Mathematical PhysicsOct 1, 19895 citations

The high-temperature susceptibility of the classical Heisenberg model in four dimensions

Sati McKenzie, Cyril Domb, D L Hunter

The high-temperature susceptibility of the four-dimensional classical Heisenberg model is studied by the method of series expansions. High-temperature series are presented to order K 9 for the hyper face centred (HFC) lattice, to order K 11 for the hyper body centred cubic (HBCC) lattice and to order K 12 for the hyper simple cubic (HSC) lattice. The last three coefficients for the HSC lattice and all the coefficients for the other two lattices are new. The series are analysed for singularities of the form t -1 mod In t mod p , predicted by the renormalisation group theory (t=1-K c /K, where K is the high-temperature expansion variable J/kT). Fairly good convergence is obtained for p approximately=0.45 for all three lattices, in agreement with renormalisation group calculations.The high-temperature susceptibility of the four-dimensional classical Heisenberg model is studied by the method of series expansions. High-temperature series are presented to order K 9 for the hyper face centred (HFC) lattice, to order K 11 for the hyper body centred cubic (HBCC) lattice and to order K 12 for the hyper simple cubic (HSC) lattice. The last three coefficients for the HSC lattice and all the coefficients for the other two lattices are new. The series are analysed for singularities of the form t -1 mod In t mod p , predicted by the renormalisation group theory (t=1-K c /K, where K is the high-temperature expansion variable J/kT). Fairly good convergence is obtained for p approximately=0.45 for all three lattices, in agreement with renormalisation group calculations.

Journal of Physics A Mathematical and GeneralDec 1, 19824 citations

Classical Heisenberg Model

G S Joyce

Exact expressions for the partition function, spin pair correlation function, and susceptibility of the onedimensional isotropic classical Heisenberg model are obtained in zero external field with cyclic boundary conditions. It is shown that the methods used to derive these results enable the partition functions and susceptibilities of finite clusters of interacting classical spins to be evaluated in terms of the $3n\ensuremath{-}j$ symbols of Wigner. Exact results in one dimension are also obtained for the partition function and susceptibility of a "planar" classical Heisenberg model. In this model the spin vectors interact via a Heisenberg coupling but each spin vector is restricted to lie in a plane.The anisotropic classical Heisenberg model described by the Hamiltonian $\mathcal{H}=\ensuremath{-}\ensuremath{\Sigma}\stackrel{}{(\mathrm{ij})}2({{J}_{\mathrm{ij}}}^{x}{{s}_{i}}^{x}{{s}_{j}}^{x}+{{J}_{\mathrm{ij}}}^{y}{{s}_{i}}^{y}{{s}_{j}}^{y}+{{J}_{\mathrm{ij}}}^{z}{{s}_{i}}^{z}{{s}_{j}}^{z})\ensuremath{-}mH\ensuremath{\Sigma}\stackrel{N}{j=1}{{s}_{i}}^{z},$ where ${{s}_{i}}^{x}$ , ${{s}_{i}}^{y}$ , and ${{s}_{i}}^{z}$ are components of the unit vector ${\mathbf{s}}_{i}$ , is also considered. A perturbation series for the zero-field free energy of the anisotropic model in one dimension with nearest-neighbor interactions ${{J}_{\mathrm{ij}}}^{x}={{J}_{\mathrm{ij}}}^{y}=J$ and ${{J}_{\mathrm{ij}}}^{z}=\ensuremath{\gamma}J$ is developed in powers of $\ensuremath{\gamma}\ensuremath{-}1$ using the isotropic model as the unperturbed system. Detailed calculations are performed to third order in $\ensuremath{\gamma}\ensuremath{-}1$. It is found that the perturbation series for the energy per spin breaks down as $T\ensuremath{\rightarrow}0$. A high-temperature series expansion for the anisotropic model, which is valid for a general interaction potential and lattice, is derived by generalizing the methods developed by Horwitz and Callen for the Ising model. This series is rearranged to give a simplified diagram expansion. Finally, a practical technique for calculating the high-temperature series expansions of the zero-field free energy and susceptibility of the isotropic classical Heisenberg model is presented.Exact expressions for the partition function, spin pair correlation function, and susceptibility of the onedimensional isotropic classical Heisenberg model are obtained in zero external field with cyclic boundary conditions. It is shown that the methods used to derive these results enable the partition functions and susceptibilities of finite clusters of interacting classical spins to be evaluated in terms of the $3n\ensuremath{-}j$ symbols of Wigner. Exact results in one dimension are also obtained for the partition function and susceptibility of a "planar" classical Heisenberg model. In this model the spin vectors interact via a Heisenberg coupling but each spin vector is restricted to lie in a plane.The anisotropic classical Heisenberg model described by the Hamiltonian $\mathcal{H}=\ensuremath{-}\ensuremath{\Sigma}\stackrel{}{(\mathrm{ij})}2({{J}_{\mathrm{ij}}}^{x}{{s}_{i}}^{x}{{s}_{j}}^{x}+{{J}_{\mathrm{ij}}}^{y}{{s}_{i}}^{y}{{s}_{j}}^{y}+{{J}_{\mathrm{ij}}}^{z}{{s}_{i}}^{z}{{s}_{j}}^{z})\ensuremath{-}mH\ensuremath{\Sigma}\stackrel{N}{j=1}{{s}_{i}}^{z},$ where ${{s}_{i}}^{x}$, ${{s}_{i}}^{y}$, and ${{s}_{i}}^{z}$ are components of the unit vector ${\mathbf{s}}_{i}$, is also considered. A perturbation series for the zero-field free energy of the anisotropic model in one dimension with nearest-neighbor interactions ${{J}_{\mathrm{ij}}}^{x}={{J}_{\mathrm{ij}}}^{y}=J$ and ${{J}_{\mathrm{ij}}}^{z}=\ensuremath{\gamma}J$ is developed in powers of $\ensuremath{\gamma}\ensuremath{-}1$ using the isotropic model as the unperturbed system. Detailed calculations are performed to third order in $\ensuremath{\gamma}\ensuremath{-}1$. It is found that the perturbation series for the energy per spin breaks down as $T\ensuremath{\rightarrow}0$. A high-temperature series expansion for the anisotropic model, which is valid for a general interaction potential and lattice, is derived by generalizing the methods developed by Horwitz and Callen for the Ising model. This series is rearranged to give a simplified diagram expansion. Finally, a practical technique for calculating the high-temperature series expansions of the zero-field free energy and susceptibility of the isotropic classical Heisenberg model is presented.

Physical ReviewMar 10, 1967196 citations

New method of deriving high temperature expansions for the Heisenberg model

Cyril Domb, David W. Wood

Physics LettersJan 1, 196421 citations

{{s}_{i}}^{x}

{{s}_{i}}^{y}

, and

{{s}_{i}}^{z}

are components of the unit vector

{\mathbf{s}}_{i}

, is also considered. A perturbation series for the zero-field free energy of the anisotropic model in one dimension with nearest-neighbor interactions

{{J}_{\mathrm{ij}}}^{x}={{J}_{\mathrm{ij}}}^{y}=J

and ${{J}_{\mathrm{ij}}}^{z}=\ensuremath{\gamma}J$ is developed in powers of $\ensuremath{\gamma}\ensuremath{-}1$ using the isotropic model as the unperturbed system. Detailed calculations are performed to third order in $\ensuremath{\gamma}\ensuremath{-}1$. It is found that the perturbation series for the energy per spin breaks down as $T\ensuremath{\rightarrow}0$. A high-temperature series expansion for the anisotropic model, which is valid for a general interaction potential and lattice, is derived by generalizing the methods developed by Horwitz and Callen for the Ising model. This series is rearranged to give a simplified diagram expansion. Finally, a practical technique for calculating the high-temperature series expansions of the zero-field free energy and susceptibility of the isotropic classical Heisenberg model is presented.