arXiv:cond-mat/0605226v1 [cond-mat.str-el] 9 May 2006 / Journal of Magnetism and Magnetic Materials 0 (2008) 1–0 1 Checkerboard order in the t–J model on the square lattice D. Poilblanc a,c,d,1 , C. Weber b , F. Mila c , M. Sigrist d a Laboratoire de Physique Th´ eorique UMR 5152, C.N.R.S. & Universit´ e de Toulouse, F-31062 Toulouse, France b Institut Romand de Recherche Num´ erique en Physique des Mat´ eriaux (IRRMA), PPH-Ecublens, CH-1015 Lausanne, Switzerland c Institute of Theoretical Physics, Ecole Polytechnique F´ ed´ erale de Lausanne, BSP 720, CH-1015 Lausanne, Switzerland d Theoretische Physik, ETH-H¨onggerberg, 8093 Z¨ urich, Switzerland 1. Introduction With constantly improving resolution of experimental techniques, novel features in the global phase diagram of high-T c cuprate superconductors have emerged. One of the most striking is the observation by scanning tunnelling microscopy (STM), in the pseudogap phase of underdoped Bi 2 Sr 2 CaCu 2 O 8+δ [1] and Ca 2−x Na x CuO 2 Cl 2 single crystals [2], of a form of local electronic ordering, with a spatial period close to four lattice spacings. These observations raise important theoretical questions about the relevance of such structures in the framework of strongly correlated models. Here, we analyze the stability of new inhomogeneous phases [3,4,5] which can compete in certain conditions with the d-wave superconducting RVB state [6]. Both mean-field [3] (MF) and numerical Variational Monte Carlo results [5](VMC) will be summarized here. 2. Renormalized Mean-field Theory We describe the doped antiferromagnet by a t−J model, H = −t  〈ij 〉σ (c † i,σ c j,σ + h.c.)+ J  〈ij 〉 S i · S j . (1) First, we replace the local constraints of no doubly occupied sites by statistical Gutzwiller weights (see below) and use a mean-field decoupling in the particle-hole channel to obtain a self-consistent renormal- ized MF hamiltonian [7], H MF = − t  〈ij 〉σ g t ij (c † i,σ c j,σ + h.c.) − 3 4 J  〈ij 〉σ g J ij (χ ji c † i,σ c j,σ + h.c. −|χ ij | 2 ) , (2) where χ ji = 〈c † j,σ c i,σ 〉 and where we explicitly assume an inhomogeneous solution and hence inhomoge- neous Gutzwiller factors, g J ij = 4(1 − x i )(1 − x j ) (1 − x 2 i )(1 − x 2 j ) − 8x i x j |χ ij | 2 + 16|χ ij | 4 , (3) g t ij =[ 4x i x j (1 − x i )(1 − x j ) (1 − x 2 i )(1 − x 2 j ) + 8(1 − x i x j )|χ ij | 2 + 16|χ ij | 4 ] 1 2 ,