J. Phys. A: Math. Gen. 24 (1991)191-202. Printed in the UK Non-analytic finite-size corrections for the Heisenberg chain in a magnetic field with free and twisted boundary conditions H-P Ecklet and C J Hamer Department of Theoretical Physics, University of New South Wales, GPO Box 1, Kensington, NSW 2033, Australia Received 28 August 1990 Abstract. The finite-size energy spectrum of the anisotropic Heisenberg chain in an extemal magnetic field is calculated for free and twisted boundary conditions. As with periodic boundary conditions, it is found that the spectra exhibit nowanalytical telw which do not fit into the form predicted on the basis of conformal invariance unless extra commensurability conditions between the sise of the system and the external field are introduced. Taking these conditions into account some scaling dimensions for associated models are derived. 1. Introduction lllci CUILLqJL U1 CUlllUl'lldl ayLlnlleLry grcauy c.Lllla'lccu LUC u'lu~raaa'rurry U , crlbl- cal two-dimensional classical and (l+l)-dimensional quantum systems (Belavin el n/ 1984). The conformal anomaly e and the scaling dimensions of the primary conformal order parameters classify the system completely (Friedan el a/ 1984). These critical parameters of the bulk system are directly accessible through the finite-size effects of an affiliated system defined on a strip geometry of infinite length but finite width This observation led to numerous studies, both numerical and analytical, of the finite-size effects of critical and conformal invariant systems. Much of the analytical work was concerned with the calculation of the conformal anomaly and scaling dimen- sions of models exactly solvable by the Bethe ansalr (BA) method. The prototype of such models is the XXZ chain of N spins h with various boundary conditions (IIamer 1986, de Vega and Karowski 1987, Woynarovich and Eckle 1987, Woynarovich 1987, Alcarar et al 1987a,b, 1988, Hamer el a/ 1987, Hamer and Batchelor 1988). Conformal invariance predicts a so called tower structure (Cardy 1986) for the spectrum of a one-dimensional quantum system, which is given in the most general form (Bogoliubov el a/ 1987) by 'pL^ ------ L -L. -.L- 1 I_.. _... L,.. .-L .-.. I 11. ..->...L.-,:-- .I ..:L: (E!& e2 s! !98B, P.ff,eck 19%). 2a P,(N',N-)-P,= -(s,+N- N -N-)+2DkF. (1.2) t On leave of absence from lnstitut fiir Theoretische Physik, Universitiit Hannover. Appelstr. 2,3000 Hannover 1, Federd Republic of Germany. 0305-4470/91/010191+ 12103.50 @ 1991 IOP Publishing Ltd 191