Journal of Magnetism and Magnetic Materials 310 (2007) 1511–1513 Quantum fluctuations in the competition among spin glass, antiferromagnetism and local pairing superconductivity S.G. Magalhaes à , F.M. Zimmer, C.J. Kipper, E.J. Calegari Laborato´rio de Mecaˆnica Estatı´stica e Teoria da Mate´ria Condensada (PPGFIS-Dep. Fı´sica) UFSM, 97105-900 Santa Maria, RS, Brazil Available online 14 November 2006 Abstract The competition among spin glass (SG), antiferromagnetism (AF) and local pairing superconductivity (PAIR) is studied in a two- sublattice fermionic Ising SG model with a local BCS pairing interaction in the presence of a transverse magnetic field G. The spins in different sublattices interact with Gaussian random couplings with an antiferromagnetic mean. The problem is formulated in a Grassmann path integral formalism. The static ansatz and the replica symmetry are used to obtain the half-filling thermodynamic potential. The results are shown in phase diagrams that exhibit a complex transition line separating the PAIR phase from the others. This line is second order at high temperature which ends in a tricritical point. The presence of G affects deeply the transition lines. r 2006 Elsevier B.V. All rights reserved. PACS: 05.50.+q; 64.60.Cn Keywords: Quantum spin glass; Antiferromagnetism; Superconductivity It is well known that over doping heavy fermions (HF) and high temperature superconductors (HTS) physical systems can present superconductivity (SC), antiferromag- netism (AF) and non-Fermi liquid (NFL) behavior [1,2]. On the other hand, it is recognized that disorder can have an important role in those physical systems. Some theories even claim that it can be the source of NFL behavior [3]. Moreover, disorder can also lead to frustration. In fact, a spin glass (SG) state has been found in some HF and HTS [4] systems together with SC and AF. Nevertheless, there is still relatively little attention given to build theories able to mimic such phase diagrams. Quite recently, a mean field theory [4] has been proposed to study the competition between SG and pair formation in real space (PAIR phase) by a model composed by a random Gaussian fermionic Ising term, a real space BCS pairing interaction with a transverse magnetic field G. However, there is one important shortcoming in this approach which is the lack of an AF solution. In this paper, we generalize the one lattice model introduced in Ref. [4] for two sublattices in order to study the phase boundaries among AF, SG and the PAIR phase. Here, the random Gaussian coupling is only between spins localized in different sublattices (see Ref. [5] and references therein). It is the interlattice random coupling, if compared to Ref. [4], which can produce an AF solution. Quantum fluctuations are introduced as Refs. [4,5], by the presence of an applied transverse field G. Therefore, the Hamiltonian is H ¼ X i a j b J i a j b S z i a S z j b  2G X p X i p S x i p  g N X p X i p j p c y i p " c y i p # c j p # c j p " , ð1Þ where the sums over i p ðj p Þ run over the N sites of each sub- lattice p (p ¼ a or b). The exchange interactions J i a j b are independent random variables that follow Gaussian distributions with mean value 4J 0 =N and variance 32J 2 =N. The spin operators in Eq. (1) are defined as: S z i p ¼ 1 2 ½ ^ n i p "  ^ n i p #  and S x i p ¼ 1 2 ½c y i p " c i p # þ c y i p " c i p #  where ^ n i p s gives the fermions number at site i p with spin projection s ¼" or #c y i p s and c i p s are the fermion creation and annihilation operators, respectively. The problem is formulated within a path integral formalism with Grassmann fields. The configurational ARTICLE IN PRESS www.elsevier.com/locate/jmmm 0304-8853/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2006.10.655 à Corresponding author. Tel.: +55 553 2208862; fax: +55 553 2208032. E-mail address: ggarcia@ccne.ufsm.br (S.G. Magalhaes).