arXiv:1104.5463 · Physical Review B

Stripes in the two-dimensional t - J model with infinite projected entangled-pair states

Philippe Corboz, Steven R. White, Guifré Vidal, Matthias Troyer

Published Jul 29, 2011• 211 citations

Abstract

We simulate the $t$-$J$ 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 ${\ensuremath{\rho}}_{l}\ensuremath{\sim}0.75\text{--}0.8$.

DOI: 10.1103/physrevb.84.041108

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arXiv:1104.5463 · Physical Review B

Stripes in the two-dimensional t - J model with infinite projected entangled-pair states

Philippe Corboz, Steven R. White, Guifré Vidal, Matthias Troyer

Published Jul 29, 2011• 211 citations

arXiv HTML PDF

Abstract

We simulate the

t

J

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

DOI: 10.1103/physrevb.84.041108

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

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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.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.

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We develop an improved variant of $U(1)$ -symmetric infinite projected entangled-pair states (iPEPS) ansatz to investigate the ground-state phase diagram of the spin- $1/2$ square ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model. In order to improve the accuracy of the ansatz, we discuss a simple strategy to select automatically relevant symmetric sectors and also introduce an optimization method to treat second-neighbor interactions more efficiently. We show that variational ground-state energies of the model obtained by the $U(1)$ -symmetric iPEPS ansatz (for a fixed bond dimension $D)$ set a better upper bound, improving previous tensor-network-based results. By studying the finite- $D$ scaling of the magnetically order parameter, we find a N\'eel phase for ${J}_{2}/{J}_{1}<0.53$ . For $0.53<{J}_{2}/{J}_{1}<0.61$ , a nonmagnetic columnar valence bond solid (VBS) state is established as observed by the pattern of local bond energy. The divergent behavior of correlation length $\ensuremath{\xi}\ensuremath{\sim}{D}^{1.2}$ and vanishing order parameters are consistent with a deconfined N\'eel-to-VBS transition at ${J}_{2}^{{c}_{1}}/{J}_{1}=0.530(5)$ , where estimated critical anomalous exponents are ${\ensuremath{\eta}}_{s}\ensuremath{\sim}0.6$ and ${\ensuremath{\eta}}_{d}\ensuremath{\sim}1.9$ for spin and dimer correlations, respectively. We show that the associated VBS order parameter monotonically increases with ${J}_{2}/{J}_{1}$ and finally a first-order quantum phase transition takes place at ${J}_{2}^{{c}_{2}}/{J}_{1}=0.610(2)$ to the conventional Stripe phase. We compare our results with earlier DMRG and PEPS studies and suggest future directions for resolving remaining issues.We develop an improved variant of $U(1)$-symmetric infinite projected entangled-pair states (iPEPS) ansatz to investigate the ground-state phase diagram of the spin-$1/2$ square ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model. In order to improve the accuracy of the ansatz, we discuss a simple strategy to select automatically relevant symmetric sectors and also introduce an optimization method to treat second-neighbor interactions more efficiently. We show that variational ground-state energies of the model obtained by the $U(1)$-symmetric iPEPS ansatz (for a fixed bond dimension $D)$ set a better upper bound, improving previous tensor-network-based results. By studying the finite-$D$ scaling of the magnetically order parameter, we find a N\'eel phase for ${J}_{2}/{J}_{1}<0.53$. For $0.53<{J}_{2}/{J}_{1}<0.61$, a nonmagnetic columnar valence bond solid (VBS) state is established as observed by the pattern of local bond energy. The divergent behavior of correlation length $\ensuremath{\xi}\ensuremath{\sim}{D}^{1.2}$ and vanishing order parameters are consistent with a deconfined N\'eel-to-VBS transition at ${J}_{2}^{{c}_{1}}/{J}_{1}=0.530(5)$, where estimated critical anomalous exponents are ${\ensuremath{\eta}}_{s}\ensuremath{\sim}0.6$ and ${\ensuremath{\eta}}_{d}\ensuremath{\sim}1.9$ for spin and dimer correlations, respectively. We show that the associated VBS order parameter monotonically increases with ${J}_{2}/{J}_{1}$ and finally a first-order quantum phase transition takes place at ${J}_{2}^{{c}_{2}}/{J}_{1}=0.610(2)$ to the conventional Stripe phase. We compare our results with earlier DMRG and PEPS studies and suggest future directions for resolving remaining issues.

arXiv:1711.07584Physical review. B./Physical review. BMay 8, 201887 citations

We simulate the

t

J

\ensuremath{\pi}

{\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

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.

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.

We study the spin-1 Heisenberg model on the square lattice with the antiferromagnetic nearest-neighbor

{J}_{1}

and the next-nearest-neighbor

{J}_{2}

couplings by using the infinite projected entangled-pair state (iPEPS) ansatz and the density matrix renormalization group (DMRG) calculation. The iPEPS simulation, which studies the model directly in the thermodynamic limit, finds a crossing of the ground state from the N\'eel magnetic state to the stripe magnetic state at ${J}_{2}/{J}_{1}\ensuremath{\simeq}0.549$, showing a direct phase transition. In the finite-size DMRG calculation on the cylinder geometry up to the cylinder width

{L}_{y}=10

, we find that around the same critical point the N\'eel and the stripe orders are strongly suppressed, which implies the absence of an intermediate phase. Both calculations identify that the stripe order comes with a first-order transition at ${J}_{2}/{J}_{1}\ensuremath{\simeq}0.549$. Our results indicate that unlike the spin-1/2 ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ square model, quantum fluctuations in the spin-1 model may not be strong enough to stabilize an intermediate nonmagnetic phase.

We develop an improved variant of

U(1)

-symmetric infinite projected entangled-pair states (iPEPS) ansatz to investigate the ground-state phase diagram of the spin-

1/2

square ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model. In order to improve the accuracy of the ansatz, we discuss a simple strategy to select automatically relevant symmetric sectors and also introduce an optimization method to treat second-neighbor interactions more efficiently. We show that variational ground-state energies of the model obtained by the

U(1)

-symmetric iPEPS ansatz (for a fixed bond dimension

D)

set a better upper bound, improving previous tensor-network-based results. By studying the finite-

D

scaling of the magnetically order parameter, we find a N\'eel phase for

{J}_{2}/{J}_{1}<0.53

. For

0.53<{J}_{2}/{J}_{1}<0.61

, a nonmagnetic columnar valence bond solid (VBS) state is established as observed by the pattern of local bond energy. The divergent behavior of correlation length $\ensuremath{\xi}\ensuremath{\sim}{D}^{1.2}$ and vanishing order parameters are consistent with a deconfined N\'eel-to-VBS transition at

{J}_{2}^{{c}_{1}}/{J}_{1}=0.530(5)

, where estimated critical anomalous exponents are ${\ensuremath{\eta}}_{s}\ensuremath{\sim}0.6$ and ${\ensuremath{\eta}}_{d}\ensuremath{\sim}1.9$ for spin and dimer correlations, respectively. We show that the associated VBS order parameter monotonically increases with

{J}_{2}/{J}_{1}

and finally a first-order quantum phase transition takes place at

{J}_{2}^{{c}_{2}}/{J}_{1}=0.610(2)

to the conventional Stripe phase. We compare our results with earlier DMRG and PEPS studies and suggest future directions for resolving remaining issues.