Ground State Phase Diagram of the $t$-$t^{\prime }$-$J$ model

Abstract

We report results of large scale ground state density matrix renormalization group(DMRG) calculations on $t$-$t^{\prime }$-$J$ cylinders with circumferences 6 and 8. We determine a rough phase diagram which appears to approximate the 2D system. While for many properties, positive and negative $t^{\prime }$ values ($t^{\prime }/t=\pm 0.2$) appear to correspond to electron and hole doped cuprate systems, respectively, the behavior of superconductivity itself shows an inconsistency between the model and the materials. The $t^{\prime }<0$ (hole doped) region shows antiferromagnetism limited to very low doping, stripes more generally, and the familiar Fermi-surface of the hole doped cuprates. However, we find $t^{\prime }<0$ strongly suppresses superconductivity. The $t^{\prime }>0$ (electron doped) region shows the expected circular Fermi pocket of holes around the $(\pi ,\pi )$ point and a broad low-doped region of coexisting antiferromagnetism and $d$-wave pairing with a strong triplet $p$ component at wave-vector $(\pi ,\pi )$ induced by the antiferromagnetism and $d$-wave pairing. The pairing for the electron low-doped system with $t^{\prime }>0$ is strong and unambiguous in the DMRG simulations. At larger doping another broad region with stripes in addition to weaker $d$-wave pairing and striped $p$-wave pairing appears. In a small doping region near $x=0.08$ for $t^{\prime }\sim -0.2$, we find a new type of stripe involving unpaired holes located predominantly on chains spaced three lattice spacings apart. The undoped two-leg ladder regions in between mimic the short-ranged spin correlations seen in two-leg Heisenberg ladders.

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