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npj Quantum Materials

Stable diagonal stripes in the t–J model at nh = 1/8 doping from fPEPS calculations

Shaojun Dong, Chao Wang, Yong‐Jian Han, Chao Yang, Lixin He

Published May 8, 2020• 11 citations
Publisher

Abstract

Abstract We investigate the two-dimensional t–J model at a hole doping of $${\bar{n}}_{{\rm{h}}}=1/8$$ 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.

DOI
10.1038/s41535-020-0226-4

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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.
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 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.
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 ${\ensuremath{\rho}}_{l}\ensuremath{\sim}0.75\text{--}0.8$.
We studied the ground state of the two-dimensional Hubbard model near half filling on the square lattice. Using Variational Monte Carlo method, we show that the diagonal stripe state of bond-centered type, in which the domain wall is located between two sites, becomes more stable than the vertical stripe state when the doping hole density x is as low as x ≃0.06. The inverse of stripe periodicity δ and the hole density x are observed to have the relationship δ∼ x (δ< x ) for the bond-centered diagonal stripe state (the vertical stripe state) in the light-doping region. These results are in good agreement with elastic neutron scattering experiments in La 2- x Sr x CuO 4 .
We study the competition between stripe states with different periods and a uniform ddd-wave superconducting state in the extended two-dimensional Hubbard model at 1/8 hole doping using infinite projected entangled-pair states (iPEPS). With increasing strength of negative next-nearest neighbor hopping ${t}^{\ensuremath{'}}$, the preferred period of the stripe decreases. For the values of ${t}^{\ensuremath{'}}$ predicted for cuprate high-Tc{T}_{c}Tc​ superconductors, we find stripes with a period 4 in the charge order, in agreement with experiments. Superconductivity in the period 4 stripe is suppressed at 1/81/81/8 doping. Only at larger doping, $0.18\ensuremath{\lesssim}\ensuremath{\delta}<0.25$, the period 4 stripe exhibits coexisting ddd-wave superconducting order. The uniform ddd-wave state is only favored for sufficiently large positive ${t}^{\ensuremath{'}}$.
We demonstrate the use of finite-size fermionic projected entangled pair states, in conjunction with variational Monte Carlo, to perform accurate simulations of the ground state of the 2D Hubbard model. Using bond dimensions of up to D=28, we show that we can surpass state-of-the-art density matrix renormalization group energies that use up to m=32 000 SU(2) multiplets on eight-leg ladders. We further apply our methodology to 10×16, 12×16, and 16×16 lattices at 1/8 hole doping and observe the dimensional crossover between stripe orientations. Our Letter shows the power of finite-size fermionic tensor networks to resolve the physics of the 2D Hubbard model and related problems.