arXiv:cond-mat/9605178v1 29 May 1996Perturbation study on the spin and charge susceptibilitiesof the two-dimensional Hubbard modelTakashi HottaInstitute for Solid State Physics, University of Tokyo, 7-22-1 Roppongi, Minato-ku, Tokyo 106, JapanSatoshi FujimotoDepartment of Physics, Kyoto University, Kyoto 606, Japan(February 1, 2008)We investigate the spin and charge susceptibilities of the two-dimensional Hubbard model basedupon the perturbative calculation in the strength of correlation U. For U comparable to a barebandwidth, the charge susceptibility decreases near the half-filling as hole-doping approaches zero.This behavior suggesting the precursor of the Mott-Hubbard gap formation cannot be obtainedwithout the vertex corrections beyond the random phase approximation. In the low-temperatureregion, the spin susceptibility deviates from the Curie-Weiss-like law and finally turns to decreasewith the decrease of temperature. This spin-gap-like behavior is originating from the van Hovesingularity in the density of states.I. INTRODUCTIONThe Hubbard model1has been studied extensively asthe fundamental model for strongly correlated electronsystems which exhibit the Mott-Hubbard metal-insulatortransition (MIT)2. Especially, the two-dimensional Hub-bard model (2DHM) has been attracted much attentionin relation to high-temperature superconductors (HTSC)of which normal metallic phase is quite “anomalous” inthe sense that its property seriously deviates from thatpredicted by the conventional Fermi-liquid theory. Thusin this paper, we investigate 2DHM putting stress on thecorrelation effects on the charge and spin susceptibili-ties and grasp the signal of the “anomalous” behavior inthe charge and spin responses near MIT from the weak-coupling perturbation approach.As for the charge degree of freedom, we are interestedin the dependence on hole-doping δ of the charge suscep-tibility. Since the system becomes incompressible nearMIT, it is generally expected that the charge suscep-tibility decreases and eventually vanishes when we ap-proach MIT by putting δ as zero. In fact, such a behav-ior has been obtained in some numerical studies on thebasis of the quantum Monte Carlo (QMC) method.3,4A quite different result, however, has been reported inQMC simulations5: The charge susceptibility divergeslike ∼ 1/δ as δ approaches zero. The difference betweentwo results is not brought about by the scatter of theQMC data among the groups, but is originating fromthe physical interpretation of data. Thus it is desirableto investigate the doping dependence of the charge sus-ceptibility with use of other methods.by using the second-order perturbation theory (SOPT)with respect to the on-site Coulomb interaction U, weinvestigate the tendency of the change of the charge sus-In this paper,ceptibility from that in the non-interacting case due tocorrelation effects.On the other hand, as for an anomalous behavior of thespin response, we would like to note the spin-gap-like be-havior observed in the nuclear spin relaxation rate T−1several HTSC materials.6–14In the higher-temperatureregion than the superconducting transition temperatureTc, (T1T)−1increases in proportional to 1/(T + θ) withthe decrease of T, indicating that (T1T)−1obeys theCurie-Weiss law. Here θ is the Weiss temperature. AsT is decreased, in such materials as La1−xSrxCuO46and YBa2Cu3O7−δ with Tc ∼ 90K,7,8(T1T)−1con-tinues to increase untill there occurs the transition tosuperconducting state.However, in such materials asYBa2Cu3O6.6with Tc∼ 60K,8–10YBa2Cu4O8,11–13andBi2Sr2CaCuO8,14(T1T)−1begins to deviate from theCurie-Weiss law even for T > Tc, and turns to decreasewith the decrease of T. Since this decrease of (T1T)−1seems to be associated with the formation of the spin-singlet state, such a behavior has been frequently called“the spin gap”. Some authors have argued the origin ofthe spin-gap formation such as the spinon pairing15orbi-layer coupling,16but it should be clarified whether itis possible or not to reproduce the spin-gap behavior inthe framework of the Fermi-liquid theory without bi-layercoupling. From the analysis of the normal state of HTSCon the basis of the Fermi-liquid theory in due consider-ation of anti-ferromagnetic (AF) spin fluctuations,17thefollowing relation holds in HTSC:1of(T1T)−1∝ χs(Q,0),(1.1)where χs(Q,0) is the staggered spin susceptibility in thestatic limit with Q = (π,π).calculating χs(Q,0) in SOPT, we make a qualitative in-vestigation on the temperature dependence of (T1T)−1of 2DHM.Thus in this paper, by1
