Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:cond-mat/9907284 · Journal of Physics A Mathematical and General

Logarithmic corrections in quantum impurity problems

Ian K. Affleck, Shaojin Qin

Published Oct 28, 1999• 35 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/0305-4470/32/45/301

Related papers

Logarithmic corrections to bulk and surface criticality in a three-dimensional quantum Heisenberg antiferromagnet

Xuyang Liang, Xiao-Chuan Wu, Zenan Liu +3 more

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.

arXiv:2607.12029•Jul 13, 2026•0 citations

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

K. Nomura

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.

Physical review. B, Condensed matter•Dec 1, 1993•38 citations

Temperature-dependent logarithmic corrections in the spin-1/2 Heisenberg chain

Victor Barzykin

We obtain the logarithmic corrections to the dynamic response function and nuclear magnetic resonance (NMR) T 1 - and T 2 G -rates in the spin-1/2 antiferromagnetic Heisenberg chain using the perturbative renormalization group for the leading irrelevant operator. The result is compared with NMR experiments on Sr 2 CuO 3 .

arXiv:cond-mat/9904250•Journal of Physics Condensed Matter•Feb 17, 2000•15 citations

Logarithmic corrections to the ground-state energy of finite spin−12 antiferromagnetic chains due to ferromagnetic end bonds

Jizhong Lou, Jianhui Dai, Shaojin Qin +2 more

We analyze the logarithmic corrections due to ferromagnetic impurity ending bonds of open spin-1/2 antiferromagnetic chains, using the density matrix renormalization group technique. A universal finite-size scaling $\ensuremath{\sim}1/(L\mathrm{ln}L)$ for impurity contributions in the quasidegenerate ground-state energy is demonstrated for a zigzag spin-1/2 chain at the critical next-nearest-neighbor coupling and the standard Heisenberg spin-1/2 chain, in the long-chain limit. Using an exact solution for the latter case it is argued that one can extract the impurity contributions to the entropy and specific heat from the scaling analysis. It is also shown that a pure spin-3/2 open Heisenberg chain belongs to the same universality class.

arXiv:cond-mat/0003071•Physical review. B, Condensed matter•Oct 1, 2000•12 citations

The Ising model in a random boundary field

John Cardy

Random magnetic fields acting on the boundary of a system are relevant if eta /sub ///(1. For the case of the two-dimensional Ising model they are marginal, giving rise to calculable logarithmic corrections to boundary quantities near the bulk critical point. These considerations may be important in the study of finite-size effects in the critical behaviour of adsorbed systems.

Journal of Physics A Mathematical and General•Nov 21, 1991•24 citations

Three loop renormalization group for a marginally perturbed SU(2) WZW model

Chigak Itoi

Employing a simple calculation method obtained by M.-H. Kato, we calculate the three loop renormalization group in the $su(2)$ coset conformal field theory with a slightly relevant perturbation and the $su(2)$ Wess-Zumino-Witten model with a particular invariant marginal perturbation. Zamolodchikov's $c$-theorem, exact data of the perturbation operator and a known exact form of the operator product coefficient enable us to calculate the beta function, the gamma function and the $c$-function to three loop order. This result gives the logarithmic finite size correction to the ground state energy and the low temperature behavior of the specific heat in the Heisenberg antiferromagnetic chain with high accuracy. We describe the consistency with results obtained by several authors on the basis of its exact solvability. We discuss an experiment of the specific heat and the suceptibility recently observed.

arXiv:cond-mat/9804075•Apr 7, 1998•1 citation
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 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 analyze the logarithmic corrections due to ferromagnetic impurity ending bonds of open spin-1/2 antiferromagnetic chains, using the density matrix renormalization group technique. A universal finite-size scaling $\ensuremath{\sim}1/(L\mathrm{ln}L)$ for impurity contributions in the quasidegenerate ground-state energy is demonstrated for a zigzag spin-1/2 chain at the critical next-nearest-neighbor coupling and the standard Heisenberg spin-1/2 chain, in the long-chain limit. Using an exact solution for the latter case it is argued that one can extract the impurity contributions to the entropy and specific heat from the scaling analysis. It is also shown that a pure spin-3/2 open Heisenberg chain belongs to the same universality class.
Employing a simple calculation method obtained by M.-H. Kato, we calculate the three loop renormalization group in the su(2)su(2)su(2) coset conformal field theory with a slightly relevant perturbation and the su(2)su(2)su(2) Wess-Zumino-Witten model with a particular invariant marginal perturbation. Zamolodchikov's ccc-theorem, exact data of the perturbation operator and a known exact form of the operator product coefficient enable us to calculate the beta function, the gamma function and the ccc-function to three loop order. This result gives the logarithmic finite size correction to the ground state energy and the low temperature behavior of the specific heat in the Heisenberg antiferromagnetic chain with high accuracy. We describe the consistency with results obtained by several authors on the basis of its exact solvability. We discuss an experiment of the specific heat and the suceptibility recently observed.