ISSN 0021-3640, JETP Letters, 2012, Vol. 96, No. 3, pp. 171–175. © Pleiades Publishing, Inc., 2012. Original Russian Text © M.A. Timirgazin, A.K. Arzhnikov, V.Yu. Irkhin, 2012, published in Pis’ma v Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki, 2012, Vol. 96, No. 3, pp. 184–188. 171 1. INTRODUCTION Metal–insulator transitions [1] have been inten- sively studied starting from the 1940s until now. How- ever, the quantitative description of the metal–insula- tor transitions and sometimes the qualitative under- standing of the physical phenomena determining these transitions in definite systems still remain unsatisfac- tory [2]. According to the classification of [2], metal–insu- lator transitions are separated into two types: filling control transitions, which occur during the change in the electron concentration, and bandwidth control transitions at the fixed concentration equal to one electron per site (half-filling), caused by a change in the interaction parameters. In addition, several types of transitions can be distinguished in this classification depending on the magnetic state of metal, dielectric, and possible superconducting phase. Usually the metal–insulator transitions are consid- ered in the context of the electron correlations, the simplest model for the description of which is the Hubbard model. Hubbard [3] considered the metal– insulator transition in the paramagnetic phase, which is caused by the appearance of the correlation gap and is continuous. Later, the description of such a transi- tion was improved by the use of the many-electron operator approach [4]. Recently, the limit of the infi- nite dimension of the space d has been used in the description of the Mott–Hubbard transitions, which is considered within the dynamic mean field theory [5], in particular, using the numerical renormalization group method [6]. The possibility of the phase transi- tions of the first order has been considered. The antiferromagnetic ordering is important for the description of metal–insulator transitions in the ground state of the Hubbard model. If the bare elec- tron spectrum  k satisfies the nesting condition  k + Q = – k (e.g., for simple lattices with nearest-neighbor hopping), the exponentially small dielectric gap of the Slater type (antiferromagnetic subbands) appears at any weak interaction U between electrons. When the nesting condition is violated (e.g., when next-nearest- neighbor hopping with the energy scale W ' is taken into account), the metal–insulator transition occurs at a finite interaction. It was shown in [7] that it is a transition of the first order (earlier this conclusion was made for the case of the long-range Coulomb interac- tion). In the case of small W ', the comparison of the energies of the antiferromagnetic insulator and para- magnetic metal gives for the transition point (1) where W and ρ are the bare values of the band width and the density of states on the Fermi level, respec- tively. The possible scenarios of the transition to the insulator antiferromagnetic phase from the metal paramagnetic or metal antiferromagnetic phase were considered in [7]. However, to the best of our knowl- edge, no further calculations for definite lattices were performed. The main aim of our studies is the description of the metal–insulator transitions of the Slater type, i.e., transitions related to the appearance of the gap in the energy spectrum of electronic states as a result of the magnetic ordering. Thus, we will consider transitions from the magnetic ordered metal state to the magnetic ordered dielectric one disregarding strong correla- 1 / U c  ρ W/ W ' ( ) , ln Metal–Insulator Transition in the Hubbard Model with Incommensurate Magnetic Structures M. A. Timirgazin a, b , A. K. Arzhnikov a, b , and V. Yu. Irkhin a, b a Physical-Technical Institute, Ural Branch, Russian Academy of Sciences, Izhevsk, 426000 Russia b Institute of Metal Physics, Ural Branch, Russian Academy of Sciences, Izhevsk, 426000 Russia e-mail: timirgazin@gmail.com Received June 15, 2012 The metal–insulator transition for the square, simple cubic, and body centered cubic lattices has been studied within the Hubbard model at half-filling taking into account nearest- and next-nearest-neighbor electron hopping. Both staggered antiferromagnetic and incommensurate magnetic states (spin-spiral wave) have been considered. The inclusion of the latter states for the three-dimensional lattices does not change the gen- eral pattern of the metal–insulator transition, but opens the fundamentally new possibility of the metal–insu- lator transition of the first order between the magnetically ordered states for the square lattice. DOI: 10.1134/S002136401215012X