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arXiv:cond-mat/0606659 · Physical Review B

Finite compressibility in the low-doping region of the two-dimensional t−J model

Massimo Lugas, Leonardo Spanu, Federico Becca, Sandro Sorella

Published Oct 31, 2006• 37 citations
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Abstract

We reexamine the important issue of charge fluctuations in the two-dimensional $t\text{\ensuremath{-}}J$ model by using an improved variational method based on a wave function that contains both antiferromagnetic and $d$-wave superconducting order parameters. In particular, we generalize the wave function introduced some time ago by Bouchaud, Georges, and Lhuillier [J. Phys. (Paris) 49, 553 (1988)] by considering also a long-range spin-spin Jastrow factor, in order to correctly reproduce the small-$q$ behavior of the spin fluctuations. We mainly focus our attention on the physically relevant region $J∕t\ensuremath{\sim}0.4$ and find that, contrary to a previous variational ansatz, this state is stable against phase separation for small hole doping. Moreover, by performing projection Monte Carlo methods based on the so-called fixed-node approach, we obtain clear evidence that the $t\text{\ensuremath{-}}J$ model does not phase separate for $J∕t\ensuremath{\lesssim}0.7$ and that the compressibility remains finite close to the antiferromagnetic insulating state.

DOI
10.1103/physrevb.74.165122

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We reexamine the important issue of charge fluctuations in the two-dimensional t\ensuremath-Jt\text{\ensuremath{-}}Jt\ensuremath-J model by using an improved variational method based on a wave function that contains both antiferromagnetic and ddd-wave superconducting order parameters. In particular, we generalize the wave function introduced some time ago by Bouchaud, Georges, and Lhuillier [J. Phys. (Paris) 49, 553 (1988)] by considering also a long-range spin-spin Jastrow factor, in order to correctly reproduce the small-qqq behavior of the spin fluctuations. We mainly focus our attention on the physically relevant region J∕t\ensuremath∼0.4J∕t\ensuremath{\sim}0.4J∕t\ensuremath∼0.4 and find that, contrary to a previous variational ansatz, this state is stable against phase separation for small hole doping. Moreover, by performing projection Monte Carlo methods based on the so-called fixed-node approach, we obtain clear evidence that the t\ensuremath-Jt\text{\ensuremath{-}}Jt\ensuremath-J model does not phase separate for J∕t\ensuremath≲0.7J∕t\ensuremath{\lesssim}0.7J∕t\ensuremath≲0.7 and that the compressibility remains finite close to the antiferromagnetic insulating state.
The possibility of coexistence of superconductivity (SC) and antiferromagnetic long-range order (AFLRO) of the two-dimensional extended $t\text{\ensuremath{-}}J$ model in the very underdoped region is studied by the variational Monte Carlo (VMC) method. In addition to using previously studied wave functions, a recently proposed wave function generated from the half-filled Mott insulator is used. For hole-doped systems, the phase boundary between AFLRO and ddd-wave SC for the physical parameters, J∕t=0.3J∕t=0.3J∕t=0.3, ${t}^{\ensuremath{'}}∕t=\ensuremath{-}0.3$ and ${t}^{\ensuremath{''}}∕t=0.2$, is located near hole density ${\ensuremath{\delta}}_{c}\ensuremath{\cong}0.06$, and there is no coexistence. The phase transition is first order between these two homogeneous phases at ${\ensuremath{\delta}}_{c}$.
We examine the two-dimensional $t\ensuremath{-}J$ model by using variational approach combined with well established quantum Monte Carlo techniques [S. Sorella et al., Phys. Rev. Lett. 88, 117002 (2002).] that are used to improve systematically the accuracy of the variational ansatz. Contrary to recent density-matrix renormalization group and projected entangled-pair state calculations [P. Corboz et al., Phys. Rev. B 84, 041108(R) (2011).], a uniform phase is found for J/t=0.4J/t=0.4J/t=0.4, even when the calculation is biased with an ansatz that explicitly contains stripe order. Moreover, in the small hole doping regime, that is, $\ensuremath{\delta}\ensuremath{\lesssim}0.1$, our results support the coexistence of antiferromagnetism and superconductivity.
Using computational techniques, it is shown that pairing is a robust property of hole-doped antiferromagnetic insulators. In one dimension and for two-leg ladder systems, a BCS-like variational wave function with long-bond spin singlets and a Jastrow factor provides an accurate representation of the ground state of the t-J model, even though strong quantum fluctuations destroy the off-diagonal superconducting long-range order in this case. However, in two dimensions it is argued-and numerically confirmed using several techniques, especially quantum Monte Carlo-that quantum fluctuations are not strong enough to suppress superconductivity.
We present a systematic study of the phase diagram of the $t\text{\ensuremath{-}}{t}^{\ensuremath{'}}\text{\ensuremath{-}}J$ model by using the Green's function Monte Carlo (GFMC) technique, implemented within the fixed-node (FN) approximation and a wave function that contains both antiferromagnetism and ddd-wave pairing. This enables us to study the interplay between these two kinds of order and compare the GFMC results with the ones obtained by the simple variational approach. By using a generalization of the forward-walking technique, we are able to calculate true FN ground-state expectation values of the pair-pair correlation functions. In the case of ${t}^{\ensuremath{'}}=0$, there is a large region with a coexistence of superconductivity and antiferromagnetism that survives up to ${\ensuremath{\delta}}_{c}\ensuremath{\sim}0.10$ for J∕t=0.2J∕t=0.2J∕t=0.2, and ${\ensuremath{\delta}}_{c}\ensuremath{\sim}0.13$ for J∕t=0.4J∕t=0.4J∕t=0.4. The presence of a finite ${t}^{\ensuremath{'}}∕t<0$ induces a strong suppression of both magnetic (with ${\ensuremath{\delta}}_{c}\ensuremath{\lesssim}0.03$ for J∕t=0.2J∕t=0.2J∕t=0.2 and ${t}^{\ensuremath{'}}∕t=\ensuremath{-}0.2$) and pairing correlations. In particular, the latter ones are depressed both in the low-doping regime and around $\ensuremath{\delta}\ensuremath{\sim}0.25$, where strong size effects are present.
We have studied the $t\ensuremath{-}J$ model using the Green function Monte Carlo technique. We have obtained accurate energies well converged in the thermodynamic limit by performing simulations for up to 242242242 lattice sites. By studying the energy as a function of hole doping we conclude that there is no phase separation in the physical region relevant for high- Tc{T}_{c}Tc​ superconductors. This finding is further supported by the hole-hole correlation function calculation. Remarkably, near the phase separation instability, for ${J}_{c}/t\ensuremath{\sim}0.5$ this function displays enhanced fluctuations at incommensurate wave vectors that scale linearly with the doping, in agreement with experimental findings.