ISSN 1063-7761, Journal of Experimental and Theoretical Physics, 2010, Vol. 111, No. 4, pp. 635–644. © Pleiades Publishing, Inc., 2010. Original Russian Text © S.V. Nikolaev, S.G. Ovchinnikov, 2010, published in Zhurnal Éksperimental’noі i Teoreticheskoі Fiziki, 2010, Vol. 138, No. 4, pp. 717–728. 635 1. INTRODUCTION It has become clear in recent decades that correla- tion effects play a leading role in the formation of physical properties of a large group of materials with strong electron correlations (in particular, high-tem- perature superconductors). The quasi-two-dimen- sional structure of HTSC cuprates is also an important factor determining the properties of these materials. The most promising model for theoretical study of quasi-two-dimensional systems with strong electron correlations is the two-dimensional (2D) Hubbard model [1]. In spite of its apparent simplicity, this model helps to reveal general fundamental properties of these materials. An important constraint in a detailed theoretical description of properties of strongly correlated systems is the fact that the kinetic and potential energies are of the same order of magni- tude. This means that perturbation theory is applicable neither in the weak coupling limit nor in the strong coupling limit. However, it is well known that an anal- ysis of 2D systems based on the theories of dynamic mean field, which are most advanced theories in the Hubbard model [2–4], is impossible due to the disre- gard of spatial correlations of particles in this theory. A large number of recent publications devoted to the development of a new class of theories can be classified as quantum cluster theories [5]. The very idea of divid- ing the entire lattice into clusters and to take into account the interaction in a cluster and the interaction between clusters in perturbation theory was presented long ago [6–8]. Later, a number of theories were developed based on this principle, e.g., the cluster dynamic mean field theory (CDMFT) [9], dynamic cluster approximation (DCA) [10], and cluster pertur- bation theory (CPT) [11]. An analogous approach was used in [12] for describing plaquette deformation of a 2D quantum magnet. In these theories, the first step is the division of the entire lattice into cluster and taking into account intr- acluster interactions exactly. For this purpose, the Lanczos method [13, 14] of exact diagonalization was mainly used. It is shown below that this approach is not correct without an appropriate control of the number of states in the Hilbert state and may lead to incorrect results. For this reason, the exact complete diagonalization method taking into account all excited states in each subspace of the Hilbert state of local cluster states is used in our study. The second step in constructing cluster theories is the allowance for interactions between clusters. At this stage, all quantum cluster theories differ significantly. For example, in CDMFT and DCA, a self-consistent approach is used to take into account the interaction ELECTRONIC PROPERTIES OF SOLID Cluster Perturbation Theory in Hubbard Model Exactly Taking into Account the Short-Range Magnetic Order in 2 × 2 Cluster S. V. Nikolaev a,b, * and S. G. Ovchinnikov a,c a Kirenskii Institute of Physics, Siberian Branch, Russian Academy of Sciences, Krasnoyarsk, 660036 Russia b Dostoevsky State University, Omsk, 644077 Russia c Siberian Federal University, Krasnoyarsk, 660041 Russia *e-mail: 25sergeyn@mail.ru Received December 17, 2009 Abstract—The cluster perturbation theory is presented in the 2D Hubbard model constructed using X oper- ators in the Hubbard-I approximation. The short-range magnetic order is taken into account by dividing the entire lattice into individual 2 × 2 clusters and solving the eigenvalue problem in an individual cluster using exact diagonalization taking into account all excited levels. The case of half-filling taking into account jumps between nearest neighbors is considered. As a result of numerical solution, a shadow zone is discovered in the quasiparticle spectrum. It is also found that a gap in the density of states in the quasiparticle spectrum at zero temperature exists for indefinitely small values of Coulomb repulsion parameter U and increases with this parameter. It is found that the presence of this gap in the spectrum is due to the formation of a short-range antiferromagnetic order. An analysis of the temperature evolution of the density of states shows that the metal–insulator transition occurs continuously. The existence of two characteristic energy scales at finite temperatures is demonstrated, the larger scale is associated with the formation of a pseudogap in the vicinity of the Fermi level, and the smaller scale is associated with the metal–insulator transition temperature. A peak in the density of states at the Fermi level, which is predicted in the dynamic mean field theory in the vicinity of the metal–insulator transition, is not observed. DOI: 10.1134/S1063776110100146