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arXiv:cond-mat/0405420 · Physical Review B

Computer-aided perturbation theory by cumulants: Dimerized and frustrated spin- 12 chain

Steffen Sykora, A. Hübsch, K. W. Becker

Published Aug 17, 2004• 3 citations
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Abstract

This paper demonstrates that a computer-aided perturbation theory can easily be realized by use of a cumulant approach. In contrast to a recent alternative formulation on the basis of Wegner's flow equation method the present approach can be applied to systems with arbitrary Hilbert space. In particular an equidistant spectrum of the unperturbed part of the Hamiltonian is not needed. The method is illustrated in detail for dimerized and frustrated spin $\frac{1}{2}$ chains for which the ground state energy is calculated up to seventh order perturbation theory.

DOI
10.1103/physrevb.70.054408

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This paper demonstrates that a computer-aided perturbation theory can easily be realized by use of a cumulant approach. In contrast to a recent alternative formulation on the basis of Wegner's flow equation method the present approach can be applied to systems with arbitrary Hilbert space. In particular an equidistant spectrum of the unperturbed part of the Hamiltonian is not needed. The method is illustrated in detail for dimerized and frustrated spin 12\frac{1}{2}21​ chains for which the ground state energy is calculated up to seventh order perturbation theory.
The real-space renormalization group for the study of periodic systems goes through the reduction of the Hilbert space to products of lowest eigenstates of identical blocks. Instead of working with the Hamiltonian, it is possible to estimate effective interactions between blocks from the spectrum of dimers or trimers of blocks, according to the effective Hamiltonian theory of Bloch. Tests on a series of spin problems [cohesive energy of the one-dimensional (1D) chain and of the 2D square lattice, excitation energies of the dimerized 1D spin chain, and behavior of the frustrated chain in the low frustration regime] show the potentiality of the method.
In this paper we present results for the ground state and low-lying excitations of the S=1/2S=1/2S=1/2 alternating Heisenberg antiferromagnetic chain. Our more conventional techniques include perturbation theory about the dimer limit and numerical diagonalization of systems of up to 28 spins. An application of multiple precision numerical diagonalization allows us to determine analytical perturbation series to high order; the results found using this approach include ninth-order perturbation series for the ground state energy and one magnon gap. We also determine the fifth-order dispersion relation and third-order exclusive neutron scattering structure factor for one-magnon modes and numerical and analytical binding energies of S=0S=0S=0 and S=1S=1S=1 two-magnon bound states.
A zero-temperature renormalization group (RG) approach is applied to the one-dimensional, spin-1/2 antiferromagnetic Heisenberg dimerized (alternating) chain. Specifically, the ground state energy and lowest-lying spectral excitations are examined. The calculation indicates the existence of a gap in the spectrum of the dimerized chain which vanishes only in the limit of a uniform spin chain, in contrast to a recent Green’s function approach. The RG results are in reasonable agreement with numerical extrapolations on the exact eigenvalue spectrum of finite chains of up to 12 spins. Both methods are compared with several other approximate treatments of the Heisenberg system, and tested by comparison with exact results for the spin-1/2 XY dimerized chain.
In conformal field theory, key properties of spin-12\frac{1}{2}21​ chains, such as the ground state energy per site and the excitation gap, scale with dimerization $\ensuremath{\delta}$ as ${\ensuremath{\delta}}^{\ensuremath{\alpha}}$ with known exponents $\ensuremath{\alpha}$ and logarithmic corrections. The logarithmic corrections vanish in a spin chain with nearest (J=1)(J=1)(J=1) and next-nearest-neighbor interactions (J2)({J}_{2})(J2​), for J2c=0.2411{J}_{2\mathrm{c}}=0.2411J2c​=0.2411. Density-matrix renormalization group analysis of a frustrated spin chain with no logarithmic corrections yields the field theoretic values of $\ensuremath{\alpha}$, and the scaling relation is valid up to the physically realized range, $\ensuremath{\delta}\ensuremath{\sim}0.1$. However, chains with logarithmic corrections (J2<0.2411J)({J}_{2}<0.2411\phantom{\rule{0.3em}{0ex}}\mathrm{J})(J2​<0.2411J) are more accurately fit by simple power laws with different exponents for physically realized dimerizations. We show the exponents decreasing from approximately 3∕43∕43∕4 to 2∕32∕32∕3 for the spin gap and from approximately 3∕23∕23∕2 to 4∕34∕34∕3 for the energy per site, and error bars in the exponent also decrease as J2{J}_{2}J2​ approaches to J2c{J}_{2\mathrm{c}}J2c​.