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Journal of Physics A Mathematical and General

Non-analytic finite-size corrections in the one-dimensional Bose gas and Heisenberg chain

Ferenc Woynarovich, Hans-Peter Eckle, Trong Tuong Truong

Published Sep 21, 1989• 100 citations
Publisher

Abstract

The energy spectra of the one-dimensional Bose gas and Heisenberg chain in an external field (chemical potential in the case of the Bose gas and magnetic field in the case of the Heisenberg chain) are calculated. It is found that, at general values of the external field, the spectra are not of the form expected on the basis of conformal invariance. The conformal structure can be recovered only if some extra conditions are imposed on the size of the system and the external field. It is argued that these additional conditions must be satisfied when taking the continuum limit in order to arrive at a conformally invariant theory.

DOI
10.1088/0305-4470/22/18/035

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The energy spectra of the one-dimensional Bose gas and Heisenberg chain in an external field (chemical potential in the case of the Bose gas and magnetic field in the case of the Heisenberg chain) are calculated. It is found that, at general values of the external field, the spectra are not of the form expected on the basis of conformal invariance. The conformal structure can be recovered only if some extra conditions are imposed on the size of the system and the external field. It is argued that these additional conditions must be satisfied when taking the continuum limit in order to arrive at a conformally invariant theory.
The finite-size effects in the spectrum of a Hubbard chain are obtained for both the repulsive and attractive cases. It is shown that the finite-size corrections-similar to the case of a Heisenberg chain or Bose gas-are nonanalytic unless some conditions are imposed on the chemical potential, magnetic field and chain length. If these conditions are met, the spectrum shows a similar tower structure as expected in conformal theories, although the model in general is not conformally invariant. In the special case when the two fermi velocities are equal, the model is conformally invariant with c=2, the indices are similar to the Gaussian form and there are four marginal operators.
The finite-size energy spectrum of the anisotropic Heisenberg chain in an external magnetic field is calculated for free and twisted boundary conditions. As with periodic boundary conditions, it is found that the spectra exhibit non-analytical terms which do not fit into the form predicted on the basis of conformal invariance unless extra commensurability conditions between the size of the system and the external field are introduced. Taking these conditions into account some scaling dimensions for associated models are derived.
The finite-size effects in the spectrum of a Hubbard chain are obtained for both the repulsive and attractive cases. It is shown that the finite-size corrections-similar to the case of a Heisenberg chain or Bose gas-are nonanalytic unless some conditions are imposed on the chemical potential, magnetic field and chain length. If these conditions are met, the spectrum shows a similar tower structure as expected in conformal theories, although the model in general is not conformally invariant. In the special case when the two fermi velocities are equal, the model is conformally invariant with c=2, the indices are similar to the Gaussian form and there are four marginal operators.
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.