Finite-size study of the one-dimensional spin-(1/2) dimerized Heisenberg chain

Abstract

We present an analysis based on the extrapolation of finite-size results to the infinite onedimensional spin-2 dimerized isotropic Heisenberg system for the whole range of the dimerization parameter 5 (I 5 I &1) at zero temperature. This system undergoes a transition at 5=0, and a gap opens in the spectrum of elementary excitations. The exponent v, which characterizes the opening of the gap, is estimated with use of the finite-size results (with size up to N =18). We investigate two finite-size-scaling hypotheses, assuming a pure power-1aw behavior [case (1)], or taking into account logarithmic corrections [case (2)]. To estimate v, we use the derivative cf the reciprocal of the gap, the derivative of the gap, and the Callan-Symanzik function. We show that the first of these is less affected by finite-size corrections than are the other two. Using it, we have obtained, in case (1), v=0. 71+0.01, in agreement with previous estimates, and in case (2), v=0. 668+0.001, in very good agreement with the value v= -, conjectured by den Nijs. We also show that, far from criticality, the ground-state energy per site may be described by the form I 5 I" with x =1.34+0.02. However, results for the derivative of this quantity show a different functional dependence upon 5, at least for 5 & 0.4. In fact, both the values of the ground-state energy and its derivative agree with the third-order perturbation theory of Harris, with agreement improving as 5 approaches 1.

Self-Consistent-Field Study of the Spin-1/2 Dimerized Antiferromagnetic Heisenberg Chain

“…The slopes of the straight lines are 4 3 and 2 3 , respectively, which are compared with 1.36 and 0.668 from finite-size calculation [6,7] and 1.251 and 0.667 from DMRG [12]. We conclude that the power law relation between the ground-state energy or the energy gap and dimerization is the asymptotic behavior in the limit of small d.…”

“…The discussion will focus on the specific nature of the behavior of both E 0 and D near the uniform limit, which is important for the spinPeierls theory [7]. Let us calculate analytically the asymptotic behavior for d 3 0.…”

“…In the thermodynamic limit (N 3 I) and near criticality (d % 0), the ground state energy per site E 0 d, and the gap Dd, in the excitation spectrum yield the pure power-law behavior predicted by Cross and Fisher [10] (that is, E 0 d À E 0 0=J $ jdj 2n and Dd $ jdj n with n 2=3). Many authors have attempted to calculate the exponent n by various methods, such as the extrapolation of finite-size results [6,7], the bosonization method [8], the real-space renormalization-group (RG) method [11], and the density-matrix renormalization group (DMRG) [12], the molecular-field approach [13], the perturbation expansion [14,15], and so on.…”

Excitation spectra of structurally dimerized and spin-Peierls chains in a magnetic field

“…The dimerization parameter δ is a constant for structurally dimerized chains, whereas δ ∝ K −3/2 for spin-Peierls systems. 7,11,12 The structurally dimerized chain can be viewed as a limiting case of Eq.…”

Quantum entanglement in the one-dimensional spin-orbitalSU(2)⊗XXZmodel

“…It is obvious that H AO and H FO stand for the same (translational invariant) spin Hamiltonian, and H FO will have lower ground state energy when x > − S j · S j+1 AF ≃ 0.4431. The dimerised AF Heisenberg chain (3.13) related to spin-Peierls state cannot be solved trivially, with the exception of the free-dimer limit (y = −0.25) and the uniform Heisenberg limit (y = −∞) [82], One finds the ground state energy per site ε ∞ (δ) of a pure dimerised spin chain [83],…”