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arXiv:cond-mat/9602026 · Physical review. B, Condensed matter

Numerical evidence for multiplicative logarithmic corrections from marginal operators

Sebastian Eggert

Published Oct 1, 1996• 266 citations
arXivHTMLPDF

Abstract

Field theory calculations predict multiplicative logarithmic corrections to correlation functions from marginally irrelevant operators. However, for the numerically most suitable model, the spin-\textonehalf{} chain, these corrections have been controversial. In this paper, the spin-spin correlation function of the antiferromagnetic spin-\textonehalf{} chain is calculated numerically in the presence of a next-nearest-neighbor coupling ${J}_{2}$ for chains of up to 32 sites. By varying the coupling strength ${J}_{2}$ we can control the effect of the marginal operator, and our results unambiguously confirm the field theory predictions. The critical value at which the marginal operator vanishes has been determined to be at $\frac{{J}_{2}}{J}=0.241167\ifmmode\pm\else\textpm\fi{}0.000005$.

DOI
10.1103/physrevb.54.r9612

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There has recently been considerable interest in the numerical calculation of critical exponents for quantum spin chains with general spin s, stimulated by a remarkable prediction by Haldane and conformally invariant field theoretic calculations. In several instances numerical calculations have differed considerably from theoretical predictions, with misleading results. These discrepancies have generally been attributed to the presence of marginal operators resulting in logarithmic corrections which slow numerical convergence. Such phenomena seem to be ubiquitous in the area of quantum spin chains. However, until recently, no calculation was available for estimating quantitatively the expected shifts in exponent values obtained by numerical means. The purpose of this work is to call attention to such a calculation1 for the integrable Takhtajan–Babujian family of quantum spin chains, and make specific comparisons with a variety of spin chain exponents obtained numerically.2,3 Agreement with the analytic formulas is extremely good in all cases. This is an indication that finite-size scaling is a powerful tool for extracting critical exponents even in models with logarithmic corrections, provided that these corrections are taken into account in lowest order using the method of Ref. 1.
It is well known that the free electron Kondo problem can be described by a one-dimensional (1D) model because only the s-wave part of the electronic wavefunction is affected by the Kondo coupling. Moreover, since only the spin degrees of freedom are involved in the Kondo interaction, and due to spin–charge separation in 1D, the universal low energy long distance physics of the Kondo model also arises when a magnetic impurity is coupled to the end of a gapless antiferromagnetic J 1 –J 2 spin- chain, where J 1 ( J 2 ) is the (next) nearest neighbor coupling. Experimental realizations of such spin chain models are possible and, using various analytical and numerical techniques, we present a detailed and quantitative comparison between the usual free electron Kondo model and such spin chain versions of the Kondo problem. For the gapless J 1 –J 2 spin chain two cases are studied, with zero next nearest neighbor coupling, J 2 = 0, and with a critical second-neighbor coupling, J 2 = J 2 c . We first focus on the spin chain impurity model with a critical second-neighbor antiferromagnetic exchange where a bulk marginal coupling present in the spin chain model for J 2 < J 2 c vanishes . There, the usual Kondo physics is recovered in the spin chain model in the low energy regime (up to negligible corrections, dropping as powers of inverse length or energy). At J 2 c the spin chain model is not exactly solvable and we demonstrate the equivalence to the Kondo problem by comparing density matrix renormalization group calculations on the frustrated spin chain model with exact Bethe ansatz calculations of the electronic Kondo problem. We then analyze the nearest neighbor model ( J 2 = 0) where a new kind of Kondo effect occurs due to the presence of the bulk marginal coupling. This marginal coupling slightly alters the β function for the Kondo coupling, leading to a slower variation of the Kondo temperature, T K , with the bare Kondo coupling. In their exact Bethe ansatz solution of this spin chain impurity model ( J 2 = 0), Frahm and Zvyagin noted this relation as well as the connection to the Kondo problem. Here, by numerically solving the Bethe ansatz equations we provide further evidence for the connection to Kondo physics and in addition we present low temperature quantum Monte Carlo results for the impurity susceptibility that further support this connection.
We investigate the effect of the marginal operator on the critical exponents of the one-dimensional S=1/2 Heisenberg model, using the relation between the critical exponents and the energy gaps of the finite system. The energy gaps behave as \ensuremath{\Delta}E\ensuremath{\propto}1/L with the logarithmic corrections from the marginally irrelevant operator. The numerical results obtained with the Bethe ansatz are well explained by the two-loop renormalization of the marginal coupling. Thus the discrepancies between the one-loop renormalization prediction and numerical results are resolved, though it is found that the nonuniversal constant of the logarithmic correction in the ground state does not agree with the asymptotic expansion of the Bethe ansatz.
We use the density-matrix renormalization-group technique developed by White to calculate the spin correlation functions 〈Sn+lz{\mathit{S}}_{\mathit{n}+\mathit{l}}^{\mathit{z}}Sn+lz​ Snz{\mathit{S}}_{\mathit{n}}^{\mathit{z}}Snz​〉=(-1)l{)}^{\mathit{l}})l\ensuremath{\omega}(l,N) for isotropic Heisenberg rings up to N=70 sites. The correlation functions for large l and N are found to obey the scaling relation \ensuremath{\omega}(l,N)=\ensuremath{\omega}(l,\ensuremath{\infty})${\mathit{f}}_{\mathit{XY}}^{\mathrm{\ensuremath{\alpha}}}$(l/N) proposed by Kaplan et al., which is used to determine \ensuremath{\omega}(l,\ensuremath{\infty}). The asymptotic correlation function \ensuremath{\omega}(l,\ensuremath{\infty}) and the magnetic structure factor S(q=\ensuremath{\pi}) show logarithmic corrections consistent with \ensuremath{\omega}(l,\ensuremath{\infty})\ensuremath{\sim}a\ensuremath{\surd}lncl /l, where c is related to the cut-off dependent coupling constant geff{\mathit{g}}_{\mathrm{eff}}geff​(l0{\mathit{l}}_{0}l0​)=1/ln(cl0{\mathit{cl}}_{0}cl0​), as predicted by field theoretical treatments.
Exact asymptotic expressions of the uniform parts of the two-point correlation functions of bilinear spin operators in the Heisenberg antiferromagnetic spin-12\frac{1}{2}21​ chain are obtained. Apart from the algebraic decay, the logarithmic contribution is identified, and the numerical prefactor is determined. We also confirm numerically the multiplicative logarithmic correction of the staggered part of the bilinear spin operators $\ensuremath{\langle}\ensuremath{\langle}{S}_{0}^{a}{S}_{1}^{a}{S}_{r}^{b}{S}_{r+1}^{b}\ensuremath{\rangle}\ensuremath{\rangle}={(\ensuremath{-}1)}^{r}d/(r{ln}^{\frac{3}{2}}r)+(3{\ensuremath{\delta}}_{a,b}\ensuremath{-}1){ln}^{2}r/(12{\ensuremath{\pi}}^{4}{r}^{4})$, and estimate the numerical prefactor as $d\ensuremath{\simeq}0.067$. The relevance of our results for ground-state fidelity susceptibility at the Berezinskii-Kosterlitz-Thouless quantum phase transition points in one-dimensional systems is discussed at the end of our work.