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arXiv:1111.5158 · Physical Review B

Absence of static stripes in the two-dimensional t − J model determined using an accurate and systematic quantum Monte Carlo approach

Wen-Jun Hu, Federico Becca, Sandro Sorella

Published Feb 28, 2012• 40 citations
arXivHTMLPDF

Abstract

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 $J/t=0.4$, 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.

DOI
10.1103/physrevb.85.081110

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We examine the two-dimensional t\ensuremath−Jt\ensuremath{-}Jt\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, \ensuremathδ\ensuremath≲0.1\ensuremath{\delta}\ensuremath{\lesssim}0.1\ensuremathδ\ensuremath≲0.1, our results support the coexistence of antiferromagnetism and superconductivity.
We simulate the ttt-JJJ model in two dimensions by means of infinite projected entangled-pair states (iPEPS) generalized to arbitrary unit cells, finding results similar to those previously obtained by the density-matrix renormalization group (DMRG) for wide ladders. In particular, we show that states exhibiting stripes, that is, a unidirectional modulation of hole-density and antiferromagnetic order with a $\ensuremath{\pi}$-phase shift between adjacent stripes, have a lower variational energy than uniform phases predicted by variational and fixed-node Monte Carlo simulations. For a fixed unit-cell size the energy per hole is minimized for a hole density ${\ensuremath{\rho}}_{l}\ensuremath{\sim}0.5$ per unit length of a stripe. The superconducting order parameter is maximal around ${\ensuremath{\rho}}_{l}\ensuremath{\sim}0.75\text{--}0.8$.
Abstract We investigate the two-dimensional t–J model at a hole doping of nˉh=1/8{\bar{n}}_{{\rm{h}}}=1/8nˉh​=1/8 n ¯ h = 1 / 8 using recently developed high accuracy fermionic projected entangled pair states method. By applying the stochastic gradient descent method combined with the Monte Carlo sampling technique, we obtain the ground state hole energy Ehole = −1.621 for J/t = 0.4. We show that the ground state has stable diagonal stripes instead of vertical stripes with a width of 4 unit cells, and stripe filling ρl = 0.5. We further show that the long-range superconductivity order is suppressed at this point.
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 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 $J∕t\ensuremath{\sim}0.4$ 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.
Stable diagonal stripes in the t-J model at nˉh\bar{n}_hnˉh​=1/8 doping from fPEPS calculations
We investigate the 2D t-J model at a hole doping of nˉh\bar{n}_hnˉh​=1/8 using recently developed high accuracy fermionic projected entangled pair states(fPEPS) method. By applying stochastic gradient descent method combined with Monte Carlo sampling technique, we obtain the ground state hole energy EholeE_{\rm hole}Ehole​=-1.6186 for J/tJ/tJ/t=0.4. We show that the ground state has stable diagonal stripes instead of vertical stripes with width of 4 unit cells, and stripe filling ρlρ_lρl​=0.5. We further show that the long range superconductivity order is suppressed at this point.
We investigate the ground-state properties of two strongly-correlated systems: the two-dimensional t−Jt-Jt−J model that can be used to study the high-temperature superconducting phase in the doped antiferromagnet, and the frustrated N\'eel antiferromagnet described by the J1−J2J_{1}-J_{2}J1​−J2​ Heisenberg model on the square lattice, which is widely considered as the prototype model for spin frustration. In this thesis, we apply to these two systems state-of-the-art quantum Monte Carlo techniques, including the variational and the Green's function Monte Carlo with the fixed-node approximation. Few Lanczos steps are used to systematically improve the accuracy of the trial wave functions. By introducing a suitable regularization scheme for the variational calculations with few Lanczos steps, stable and controllable simulations can be performed up to very large cluster sizes with very good accuracy. In the two-dimensional t−Jt{-}Jt−J model at J/t=0.4J/t=0.4J/t=0.4, we show that the accuracy of the Gutzwiller-projected variational state (containing dx2−y2d_{x^2-y^2}dx2−y2​ pairing) can be improved much by few Lanczos steps; in addition, the fixed-node Monte Carlo with these systematically improvable trial wave functions gives results that are comparable with the best accurate DMRG ones. Our main outcome is that the ground state is homogeneous and no evidence of stripes is detected around the doping δ=1/8\delta=1/8δ=1/8. Indeed, our best approximation to the ground state does not show any tendency towards charge inhomogeneity. Furthermore, our results show that a uniform state containing superconductivity and antiferromagnetism is stabilized at low hole doping, i.e., δ≲0.1\delta \lesssim 0.1δ≲0.1. In the J1−J2J_{1}-J_{2}J1​−J2​ Heisenberg model on the square lattice, we use the projected mean-field state that is built from Abrikosov fermions having a Z2Z_2Z2​ gauge structure and four Dirac points in the spinon spectrum. No spin or dimer order is found in the strongly frustrated regime and our calculations imply that a spin liquid phase may faithfully represent the exact ground state around J2/J1=0.5J_{2}/J_{1}=0.5J2​/J1​=0.5. The few Lanczos step technique is used to systematically improve the accuracy of the variational states both for the ground state and for few relevant low-energy excitations. This procedure allows us to estimate, in a valuable and systemic way, the spin gaps within thermodynamical limit and to show a solid evidence of an unconventional gapless excitation spectrum in the strongly frustrated regime, i.e., J2/J1≃0.5J_{2}/J_{1}\simeq 0.5J2​/J1​≃0.5. In particular, we found gapless triplet excitations at momenta (π,0)(\pi,0)(π,0) and (0,π)(0,\pi)(0,π), which are compatible with the presence of four Dirac points at momenta (±π2,±π2)(\pm \frac{\pi}{2},\pm \frac{\pi}{2})(±2π​,±2π​) in the spinon spectrum.
The long-studied Hubbard model is one of the simplest models of copper-oxide superconductors. However, the connection between the model and the experimental phase diagram is still under debate, in particular regarding the existence and extent of the ddd-wave superconducting phase. Recent rapid progress in improving the accuracy of numerical solvers has opened a way to answer this question reliably. Here, we study the hole-doping concentration ($\ensuremath{\delta}$) dependence of the Hubbard model in the ground states on a square lattice at strong coupling U/t=10U/t=10U/t=10, for the on-site interaction UUU and the transfer ttt, using a variational Monte Carlo method. The method, which combines tensor network and Lanczos methods on top of Pfaffian wave functions, reveals a rich phase diagram, in which many orders compete severely and degenerate within the energy range of 0.01t0.01t0.01t. We have identified distinct phases including a uniform ddd-wave superconducting phase for $0.17\ensuremath{\lesssim}\ensuremath{\delta}\ensuremath{\lesssim}0.22$ and a stripe charge/spin ordered phase for $\ensuremath{\delta}\ensuremath{\lesssim}0.17$ with the stripe period depending on $\ensuremath{\delta}$, together with presumable spatially coexisting antiferromagnetic and stripe order for $\ensuremath{\delta}\ensuremath{\lesssim}0.07$ and coexisting stripe and ddd-wave superconductivity for $0.07\ensuremath{\lesssim}\ensuremath{\delta}\ensuremath{\lesssim}0.17$. The present, improved method revealed a wider region of a charge uniform superconducting phase than the previous studies and shows a qualitative similarity to the phase diagram of the cuprate superconductors. The superconducting order parameter is largest at doping of around $\ensuremath{\delta}=0.17$ in the ground state, which undergoes phase transitions from an inhomogeneous to a uniform state.