Physica B 334 (2003) 451–455 Spin gap in doped dimerized chain near half-filling Xuefan Jiang a,b, *, G.Y. Guo a a Department of Physics, National Taiwan University, Taipei 106, Taiwan b Department of Physics, Changshu College, Changshu 215500, Jiangsu, China Received 11 January 2003; accepted 2 April 2003 Abstract By applying a variational approximation to the bosonized Hamiltonian, the spin gap in a doped spontaneously dimerized spin chain is investigated analytically based on the tJJ 0 model in one dimension in the region of small hole dopingandsmall-J limit.Itisfoundthatthespingapiseasilydestroyedbyholedoping.Thereisacriticalvalueofthe holedensityatwhichthesystemundergoesacontinuouscrossoverfromafrustratedquantumliquidwithaspingapto a Tomonaga–Luttinger liquid without a spin gap. Our results are consistent with that from the exact diagonalization method and can also explain some experimental behaviors in the high-T c materials. r 2003 Elsevier Science B.V. All rights reserved. PACS: 75.10.Jm; 74.72.h; 74.20.Mn Keywords: Spin gap; Doping; Dimerization The discovery of quasi-one-dimensional (1D) cuprates has stimulated the study of 1D strongly correlated electron systems. Although high-T C cuprates are at least two-dimensional (2D) sys- tems, it is also interesting to investigate the 1D model. Since it is relatively easy to explore, 2D strongly correlated systems could share some properties of the 1D case [1]. On the other hand, frustration of exchange interactions in spin and electron systems may bring exotic ground state thatcannotbeunderstoodfromtheclassicalpoint of view. Among them, a 1D antiferromagnetic Heisenberg spin system with first- and second- neighborexchangeinteractionsisknowntohavea dimer ground state and to have a spin excitation gap [2–8]. Therefore, an interesting question is how the dimer character of the ground state and the spin gap changes when holes are doped. The effectsoffrustrationinspinwasconsideredtobea candidate causing a spin gap in the itinerant electron systems by the numerical diagonalization whereafewholesaredopedinthehalf-filledband [5]. The tJ model is one of the simplest models for studying strongly correlated electron systems and is a most important problem for high-T C super- conductivity. Interacting 1D electron systems generally behave as Tomonaga–Luttinger (TL) liquids [9,10] in which the correlation functions have power-law decays with exponents that depend on the interaction strength. Using exact diagonalizationmethod,thephasediagramforthe *Corresponding author. Tel.: 0086253596273; fax: 008651252773486. E-mail address: jxfjxf@jlonline.com (X. Jiang). 0921-4526/03/$-see front matter r 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0921-4526(03)00176-5