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Journal of Physics A Mathematical and General

Finite-size effects in a non-half-filled Hubbard chain

Ferenc Woynarovich

Published Oct 7, 1989• 170 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/22/19/017

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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 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 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 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.
In this chapter we want to refine the analysis of the ground state and low-lying excitations of the Hubbard model in the phases with gapless modes, i.e. phases II, IV and V discussed in Chapters 6, 7, by taking account of corrections which are important when considering Hubbard chains of finite length L . For the generic case, i.e. away from half-filling in a magnetic field, the finite-size corrections to the spectrum of the Hubbard model have been calculated by F. Woynarovich [487]. These results are the basis for our discussion in the following Chapter 9 of the asymptotic behaviour of correlation functions within the conformal approach [6, 51, 62, 75] and thereby will allow us to make contact with Haldane's Luttinger liquid approach for the description of one-dimensional strongly correlated electron systems [189–192] Generic case – the repulsive Hubbard model in a magnetic field To investigate how the thermodynamic limit is approached we have to take into account finite-size corrections in our previous derivation of integral equations from Takahashi's equations. This analysis has to be performed separately for each of the phases with gapless excitations identified before. From a technical point of view the most complex situation is found in phase IV – the partially filled, partially magnetized band with two massless modes. The finite-size scaling behaviour in the phases with a single gapless mode can be studied using the same techniques and we will point out the differences to the ‘generic’ case studied in this section later in this chapter.
The finite-size effects in the spectrum of the integrable XXZ Heisenberg chain with arbitrary spin-S are studied analytically on the basis of the string hypothesis for bound states and numerically by solving the associated Bethe ansatz equations. The finite-size corrections to the spectrum and to the low-temperature behaviour of the free energy are found to be similar to the universal predictions for conformally invariant theories, although the model is, in general, not Lorentz invariant, since it can have an arbitrary number of branches for low-energy excitations with different Fermi velocities depending on the value of S and the anisotropy parameter gamma .
The finite-size corrections to the ground state and the energy of the low magnetisation (S<<N) states as a function of the size N are calculated analytically for the one-dimensional half-filled Hubbard model with on-site repulsion (U>O). It is found that the contribution of the charge degrees of freedom is negligible, while the contribution of the spin degrees is the same as that in the one-dimensional isotropic Heisenberg model. The analytical results are compared to numerical ones obtained for the chain lengths up to N=512.