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Physical review. B, Condensed matter

Logarithmic corrections of the one-dimensionalS=1/2 Heisenberg antiferromagnet

K. Nomura

Published Dec 1, 1993• 38 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.48.16814

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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 study the Heisenberg chain with frustration and alternation which is expressed by , both analytically and numerically. We focus on the case, where there is no marginal operator which brings about the logarithmic corrections in various quantities. By using the bosonization method, we calculate the energy gap, the change in the ground-state energy due to the alternation (so-called energy gain), the spin correlation and the string correlation (not only their exponents but also their amplitudes), and compare them with the results of the numerical diagonalization for finite systems. We point out the existence of the logarithmic correction in the energy gain despite the absence of the marginal operator. Taking into account this logarithmic correction, we can obtain a reasonable hyperscaling relation between the critical exponents of the energy gap and an energy gain from the numerical data.
The effect of a bulk marginal operator on boundary critical phenomena in two space-time dimensions is considered. The particular case of an open S = ½ antiferromagnetic Heisenberg chain, corresponding to a Wess-Zumino-Witten nonlinear model, is solved. In this case, the required renormalization group coefficient is associated with a novel operator product expansion in which three operators approach the same point. Resulting logarithmic corrections occurring in finite-size calculations and nuclear magnetic resonance experiments are discussed.
At the bulk upper critical dimension, marginally irrelevant interactions generate multiplicative logarithmic corrections to mean-field scaling. While these corrections are well understood for bulk observables, their consequences for boundary criticality, particularly for finite-size scaling, remain much less explored. Here we combine large-scale quantum Monte Carlo simulations with boundary renormalization-group analysis to study a (3 + 1)D O(3) quantum critical point. After verifying the known logarithmically modified bulk finite-size scaling, including the correlation-length scaling governed by the logarithmic finite-size exponent \hat{\coppa}, we tune the surface coupling to identify ordinary, special, and extraordinary boundary regimes. For the ordinary and special transitions, we derive logarithmic correction exponents and \hat{\coppa}-dependent finite-size scaling forms for boundary correlations, including results that have not been systematically established before. These predictions are quantitatively supported by Monte Carlo data. In the extraordinary regime, we find long-range surface magnetic order and a logarithmically enhanced surface-bulk correlation.
We present an analysis based on the extrapolation of finite-size results to the infinite one-dimensional spin-(1/2) dimerized isotropic Heisenberg system for the whole range of the dimerization parameter \ensuremath{\delta} (\ensuremath{\Vert}\ensuremath{\delta}\ensuremath{\Vert}\ensuremath{\le}1) at zero temperature. This system undergoes a transition at \ensuremath{\delta}=0, and a gap opens in the spectrum of elementary excitations. The exponent \ensuremath{\nu}, 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-law behavior [case (1)], or taking into account logarithmic corrections [case (2)]. To estimate \ensuremath{\nu}, we use the derivative of 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), \ensuremath{\nu}=0.71\ifmmode\pm\else\textpm\fi{}0.01, in agreement with previous estimates, and in case (2), \ensuremath{\nu}=0.668\ifmmode\pm\else\textpm\fi{}0.001, in very good agreement with the value \ensuremath{\nu}=(2/3) conjectured by den Nijs. We also show that, far from criticality, the ground-state energy per site may be described by the form \ensuremath{\Vert}\ensuremath{\delta}${\ensuremath{\Vert}}^{x}$ with x=1.34\ifmmode\pm\else\textpm\fi{}0.02. However, results for the derivative of this quantity show a different functional dependence upon \ensuremath{\delta}, at least for \ensuremath{\delta}\ensuremath{\gtrsim}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 \ensuremath{\delta} approaches 1.
The spin-1/2 alternating Heisenberg model with X Y -like anisotropy is studied by numerical methods. This model exhibits critical behavior in the ground state as the bond-alternation parameter δ approaches 0. The critical exponents characterizing this ground-state critical point are estimated. The results agree well with those of the analytical calculation of Nakano and Fukuyama. A logarithmic correction of the type predicted by Black and Emery is found to be consistent with numerical data in the isotropic Heisenberg model but not in anisotropic models.