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Physical review. B, Condensed matter

Variational Monte Carlo study of incommensurate antiferromagnetic phases in the two-dimensional Hubbard model

Thierry Giamarchi, C. Lhuillier

Published Dec 1, 1990• 78 citations
Publisher

Abstract

We use a variational Monte Carlo technique to study the ground state of the two-dimensional Hubbard model on a square lattice. We study the stability of the usual commensurate antiferromagnetic phase against the formation of domain walls for various system sizes, band filling, and Hubbard repulsion. An instability towards diagonal domain walls is found for the values of the Hubbard repulsion studied (U=4-10t). We compute the condensation energy of the holes in walls. Such an incommensurate antiferromagnetic phase is much stabler than any other solution. Whether paramagnetic or purely superconducting, but superconductivity is found still to coexist with incommensurate antiferromagnetism. We carefully compare our results with those of the Hartree-Fock model and shed some light on the limitations of the Hartree-Fock solutions.

DOI
10.1103/physrevb.42.10641

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We use a variational Monte Carlo technique to study the ground state of the two-dimensional Hubbard model on a square lattice. We study the stability of the usual commensurate antiferromagnetic phase against the formation of domain walls for various system sizes, band filling, and Hubbard repulsion. An instability towards diagonal domain walls is found for the values of the Hubbard repulsion studied (U=4-10t). We compute the condensation energy of the holes in walls. Such an incommensurate antiferromagnetic phase is much stabler than any other solution. Whether paramagnetic or purely superconducting, but superconductivity is found still to coexist with incommensurate antiferromagnetism. We carefully compare our results with those of the Hartree-Fock model and shed some light on the limitations of the Hartree-Fock solutions.
We use a variational Monte Carlo technique to study the ground state of the two-dimensional Hubbard and t-J models on a square lattice. We use a trial wave function that allows a continuous description of the paramagnetic, antiferromagnetic, and superconducting phases, as well as the coexistence of these phases, with no a priori constraint on double occupancy. The phase diagram of both models is given for intermediate coupling (U=10t). We show that the apparent discrepancies between the two models, that appeared in previous variational Monte Carlo studies, are not present when this sufficiently good variational wave function is used. The two models are in good qualitative agreement. We find that except at half-filling a pure antiferromagnetic phase does not exist but is always in coexistence with a d-wave superconducting phase, and is followed by a pure d-wave superconducting phase at doping up to \ensuremath{\delta}\ensuremath{\sim}0.3. The staggered magnetization and the superconducting gap are measured, and a comparison with other analytical or numerical results is made.
The magnetic states occurring in the two-dimensional Hubbard model in the vicinity of half-filling of the conduction band are studied in the case of weak correlations, using Hartree-Fock theory. It is shown that at zero temperature the commensurate antiferromagnetic state is unstable against domain-wall formation for arbitrarily small deviations from half-filling. The resulting incommensurate antiferromagnet remains initially insulating. The predictions of the model for the polarization and modulation wave vector of the incommensurate antiferromagnet are discussed.
The fixed-node Monte Carlo method is extended to lattice fermion models. This method replaces the problem of finding the ground state of a many-fermion system by an effective eigenvalue problem of finding the lowest-energy wave function in a given region of the configuration space. It has previously only been applied to fermions moving in continuous space. The discreteness of the configuration space causes the algorithm to differ from that of the continuum. The method is tested against a known limiting case where exact results are available for comparison. Good agreement is found. Using the fixed-node Monte Carlo method, we study the domain-wall phase in the ground state of the two-dimensional Hubbard model. The existence of a domain-wall phase which has domains of antiferromagnetic phases separated by walls of holes is recently suggested by inhomogeneous Hartree-Fock (HF) and variational Monte Carlo (VMC) calculations. Large improvements of the energies are found. The domain walls are broader than those obtained by the HF and VMC calculations.
We investigate the ground-state phase diagram of the two-dimensional Hubbard model based on the optimization variational Monte Carlo method. We use a wave function that is an off-diagonal type given as ψ=exp⁡(−λK)PGψ0ψ=\exp(-λK)P_Gψ_0ψ=exp(−λK)PG​ψ0​, where ψ0ψ_0ψ0​ is a one-particle state, PGP_GPG​ is the Gutzwiller operator, KKK is the kinetic operator, and λλλ is a variational parameter. The many-body effect plays an important role as an origin of spin correlation and superconductivity in correlated electron systems. We examine the competition between the antiferromagnetic state and superconducting state by varying the Coulomb repulsion UUU, the band parameter t′t't′ and the electron density nen_ene​. We show a phase diagram that includes superconducting and antiferromagnetic phases and that t′=0t'=0t′=0 is most favorable for superconductivity.
A variational ground state of the repulsive Hubbard model on a square lattice is investigated numerically for an intermediate coupling strength (U=8t)(U=8t)(U=8t) and for moderate sizes (from $6\ifmmode\times\else\texttimes\fi{}6$ to $10\ifmmode\times\else\texttimes\fi{}10$). Our ansatz is superior to other widely used variational wave functions. The results for order parameters and correlation functions provide insight into the antiferromagnetic state at half-filling as well as strong evidence for a superconducting phase away from half-filling.
Using a variational Monte Carlo method, the interplay between antiferromagnetism and superconductivity is studied for the two- dimensional Hubbard model with a diagonal transfer t’. It is found with a refined function that the optimized state is sensitive to the values of model parameters, U/t, t’/t, and δ (doping rate), and becomes a coexisting state or a pure superconducting (SC) state. SC properties in the coexisting state also undergo considerable changes around the crossover point Uco(~10t) from a fragile BCS type for U < Uco co to an unconventional one for U>Uco, in which pairs are formed through local exchange or doublon-holon processes, but supercurrent flows separately with the doped holes.