We question the validity of the finite-size-correction method based on conformal field theories. It is shown that the method leads to incorrect critical exponents on some of the correlation functions for a one-dimensional free fermion model. The model exhibits the same critical behaviour as in the well-known square-lattice Ising model that is believed to be conformally invariant.

Journal of the Physical Society of JapanOct 15, 19932 citations

Finite-size corrections to the energy spectra of gapless one-dimensional systems in the presence of boundaries

Yifan Liu, Haruki Shimizu, Atsushi Ueda +1 more

arXiv:2405.06891SciPost PhysicsOct 2, 20246 citations

arXiv:hep-th/9606137 · Annals of Physics

Modular Invariance, Lattice Field Theories, and Finite Size Corrections

C. Nash, Denjoe O’Connor

Published Apr 1, 1999• 17 citations

arXiv HTML PDF

DOI: 10.1006/aphy.1998.5868

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We present the finite-size scaling theory of one-dimensional quantum critical systems in the presence of boundaries. While the finite-size spectrum in the conformal limit, namely of a conformal field theory with conformally invariant boundary conditions, is related to the dimensions of boundary operators by Cardy, the actual spectra of lattice models are affected by both bulk and boundary perturbations and contain non-universal boundary energies. We obtain a general expression of the finite-size energy levels in the presence of bulk and boundary perturbations. In particular, a generic boundary perturbation related to the energy-momentum tensor gives rise to a renormalization of the effective system size. We verify our field-theory formulation by comparing the results with the exact solution of the critical transverse-field Ising chain and with accurate numerical results on the critical three-state Potts chain obtained by Density-Matrix Renormalization Group.We present the finite-size scaling theory of one-dimensional quantum critical systems in the presence of boundaries. While the finite-size spectrum in the conformal limit, namely of a conformal field theory with conformally invariant boundary conditions, is related to the dimensions of boundary operators by Cardy, the actual spectra of lattice models are affected by both bulk and boundary perturbations and contain non-universal boundary energies. We obtain a general expression of the finite-size energy levels in the presence of bulk and boundary perturbations. In particular, a generic boundary perturbation related to the energy-momentum tensor gives rise to a renormalization of the effective system size. We verify our field-theory formulation by comparing the results with the exact solution of the critical transverse-field Ising chain and with accurate numerical results on the critical three-state Potts chain obtained by Density-Matrix Renormalization Group.

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We find the exact finite size and lattice corrections, to the partition function

Z

, of a continuous spin Gaussian model on a toroidal triangular lattice, with periods

{L}_{0}

and

{L}_{1}

. The spins carry a representation of the fundamental group of the torus labeled by

{u}_{0}

and

{u}_{1}

and a mass

m

. Summing ${Z}^{\ensuremath{-}1/2}$ over

{u}_{i}

gives the corresponding result for the Ising model. The limits $m\ensuremath{\rightarrow}0$ and ${u}_{i}\ensuremath{\rightarrow}0$ do not commute. With

m\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0

and

{u}_{i}

nonzero the model exhibits a vortex critical phase. In the continuum limit, for arbitrary

m

, the finite size corrections to $\ensuremath{-}\mathrm{ln}Z$ are modular invariant. In the limit ${L}_{1}\ensuremath{\rightarrow}\ensuremath{\infty}$ the ``cylinder charge''

c({u}_{0}{,m}^{2}{L}_{0}^{2})

ranges nonmonotonically from $2[{1+6u}_{0}({u}_{0}\ensuremath{-}1)]$ for

m\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0

to zero for $m\ensuremath{\rightarrow}\ensuremath{\infty}$.