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arXiv:cond-mat/0107021 · Physical review. B, Condensed matter

Universal relation between the dispersion curve and the ground-state correlation length in one-dimensional antiferromagnetic quantum spin systems

Kouichi Okunishi, Yuko Akutsu, Noriko Akutsu, Takao Yamamoto

Published Aug 23, 2001• 15 citations
arXivHTMLPDF

Abstract

We discuss a universal relation $\ensuremath{\varepsilon}(i\ensuremath{\kappa})=0$ with $\mathrm{Re}\ensuremath{\kappa}=1/\ensuremath{\xi}$ in 1D quantum spin systems with an excitation gap, where $\ensuremath{\varepsilon}(k)$ is the dispersion curve of the low-energy excitation and $\ensuremath{\xi}$ is the correlation length of the ground state. We first discuss this relation for integrable models such as the Ising model in a transverse filed and the $\mathrm{XYZ}$ model. We secondly make a derivation of the relation for general cases, in connection with the equilibrium crystal shape in the corresponding 2D classical system. We finally verify the relation for the $S=1$ bilinear-biquadratic spin chain and the $S=1/2$ zigzag spin ladder numerically.

DOI
10.1103/physrevb.64.104432

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Quantum spin chains and the conformal anomaly

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Finite-size scaling studies of massive one-dimensional lattice models

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Spin-1 biquadratic chain: Magnetization curve (abstract)

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One New Method to Obtain the Correlation Langth of Solvable Models

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We discuss a universal relation \ensuremathε(i\ensuremathκ)=0\ensuremath{\varepsilon}(i\ensuremath{\kappa})=0\ensuremathε(i\ensuremathκ)=0 with Re\ensuremathκ=1/\ensuremathξ\mathrm{Re}\ensuremath{\kappa}=1/\ensuremath{\xi}Re\ensuremathκ=1/\ensuremathξ in 1D quantum spin systems with an excitation gap, where \ensuremathε(k)\ensuremath{\varepsilon}(k)\ensuremathε(k) is the dispersion curve of the low-energy excitation and \ensuremathξ\ensuremath{\xi}\ensuremathξ is the correlation length of the ground state. We first discuss this relation for integrable models such as the Ising model in a transverse filed and the XYZ\mathrm{XYZ}XYZ model. We secondly make a derivation of the relation for general cases, in connection with the equilibrium crystal shape in the corresponding 2D classical system. We finally verify the relation for the S=1S=1S=1 bilinear-biquadratic spin chain and the S=1/2S=1/2S=1/2 zigzag spin ladder numerically.
The conformal anomaly c determines the universality class of a model system in statistical mechanics. The value of c characterizes both 2D classical models and their 1D quantum counterparts. The conformal anomaly may therefore be determined numerically for quantum spin chains using the relation: E0(N)≂E0(∞)−(NΔE/12)c(1/N2), where E0 (N) is the ground-state energy of an N-spin finite system, E0 (∞) is the ground-state energy in the thermodynamic limit, and ΔE is the energy gap between the ground state at k=0 and the first excited state of the dispersion curve at k=2π/N. The numerical approach is highly successful when tested on the integrable s= 1/2 Heisenberg antiferromagnetic XXZ chain and the integrable s=1 SU(2) model. The method gives c=1 to within 2% accuracy for the s=1 and (3)/(2) XY chains, placing them in the universality class of the 2D XY model. The result c=1 (2% accuracy) is obtained for the s= (3)/(2) Heisenberg antiferromagnetic chain, in agreement with the Haldane prediction. The s=1 pure antiferromagnetic biquadratic chain and the s=1 XXZ model with uniaxial anisotropy in the vicinity of the critical point Δ=Δ2 ∼1.15 −1.18 have also been studied.
In this paper we propose an efficient routine to carry out finite size scaling studies of one-dimensional massive lattice models. Unlike massless systems, for which the lattice and continuum models yield the same results for low-lying excitations due to the infinite correlation length, the massive continuum models can at best approximate their lattice counterparts because of the intrinsic length scale $\ensuremath{\xi}\ensuremath{\sim}{\ensuremath{\Delta}}^{\ensuremath{-}1},$ where $\ensuremath{\Delta}$ is the mass gap. On examples of antiferromagnetic gapped spin-1/2 XXZ\mathrm{XXZ}XXZ chains we show explicitly that several relations between physical quantities like the mass gap, spin-wave velocity v,v,v, and the correlation length \ensuremath{\xi}, derived from the continuum models deviate significantly from the numerical results obtained for finite chains, when $\ensuremath{\xi}$ is comparable with the lattice spacing a0.{a}_{0}.a0​. On the other hand, we find if the dispersion for elementary excitations is modified from the ``Lorentz invariant'' form $\sqrt{{\ensuremath{\Delta}}^{2}{+v}^{2}{p}^{2}}$ to $\sqrt{{\ensuremath{\Delta}}^{2}{+v}^{2}{\mathrm{sin}}^{2}p},$ the spin-wave velocity vvv becomes almost size independent, and the appropriately defined correlation length using that dispersion agrees very well with numerical results. In fact, this follows from the Bethe ansatz solution for the spin-1/2 XXZ\mathrm{XXZ}XXZ chains. Based on our previous experience and these considerations we propose scaling equations to obtain the ground-state energy, mass gap, correlation length, spin-wave velocity, and scattering length between the massive elementary excitations. Only three low energy levels are needed in this method. Although the method is illustrated on the gapped XXZ\mathrm{XXZ}XXZ quantum spins, it is applicable to all one-dimensional lattice models with massive relativistic low energy dispersions. To substantiate our statement we show explicitly that the numerical data for the Ising model in a transverse magnetic field fully agree with the finite-size scaling based on the correlation length newly defined from the exact analytic solution.
The antiferromagnetic (AFM) s=1 chain with biquadratic exchange coupling has recently been the subject of considerable attention, and some controversy. Theoretical arguments by Affleck that the model has dimerized character associated with a small spectral excitation gap have not received unanimous support from numerical calculations. For finite systems the field-dependent dispersion spectra are interesting and informative concerning the nature of the model. The excitations divide into two classes having very different character, corresponding to SzT even and SzT odd. Reflection symmetry about k=π/2 is apparent if states with a given total spin ST, rather than SzT, are plotted as a function of wavevector k, consistent with the Affleck prediction of dimerized character. The high degeneracy of the eigenstates reflects the existence of new symmetry operators which (a) shift the SzT value, (b) change wave vector k to π/2−k, and (c) shift the total spin quantum ST, of the energy eigenstates. Parkinson has recently observed that finite N states with SzT even and SzT≥2 map into states of the finite N s= 1/2 XXZ model at a special value of uniaxial anisotropy. Barber and Batchelor (unpublished) have shown the existence of an analytic mapping between states of the s=1 biquadratic chain and the 2D 9-state Potts model. They further show that all states of the Parkinson s= 1/2 XXZ model map into states of the s=1 biquadratic model. Hence, they find the ground state and first excited state energies exactly, and verify the Affleck picture. Their approach falls short of establishing complete integrability, however. Numerical studies on finite chains indicate that the T=0 magnetization curve is determined by the lowest-energy states for given SzT even, which are states of the s= 1/2 XXZ chain. Hence the magnetization curve of the s=1 AFM biquadratic chain is the same as the analytically known magnetization curve of the s= 1/2 XXZ chain!
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.
Some measurements have shown that the second-order exchange interaction is non-negligible in ferromagnetic compounds whose microscopic interactions are described by means of half-odd integer quantum spins. In these spin systems the ground state is either ferromagnetic or antiferromagnetic when the bilinear exchange interaction is dominant. Instead, in ferromagnetic systems characterized by bilinear and biquadratic exchange interactions of comparable magnitude, the energy minimum occurs when spins are in a canting ground-state. To this aim, a one-dimensional (1D) quantum spin chain and a two-dimensional (2D) lattice of quantum spins subjected to periodic boundary conditions are modeled via the generalized quantum Heisenberg Hamiltonian containing, in addition to the isotropic and short-range bilinear exchange interaction of the Heisenberg type, a second-order interaction, the isotropic and short-range biquadratic exchange interaction between nearest-neighbors quantum spins. For these 1D and 2D quantum systems a generalization of the Mermin–Wagner–Hohenberg theorem (also known as Mermin–Wagner–Berezinksii or Coleman theorem) is given. It is demonstrated, by means of quantum statistical arguments, based on Bogoliubov’s inequality, that, at any finite temperature, (1) there is absence of long-range order and that (2) the law governing the vanishing of the order parameter is the same as in the bilinear case for both 1D and 2D quantum ferromagnetic systems. The physical implications of the absence of a spontaneous spin symmetry breaking in 1D spin chains and 2D spin lattices modeled via a generalized quantum Heisenberg Hamiltonian are discussed.