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arXiv:cond-mat/9809040 · Physical review. B, Condensed matter

Correlated wave functions and the absence of long-range order in numerical studies of the Hubbard model

M. Guerrero, Gerardo Ortíz, James E. Gubernatis

Published Jan 15, 1999• 22 citations
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Abstract

We present a formulation of the constrained path Monte Carlo method for fermions that uses trial wave functions that include many-body effects. This formulation allows us to implement a whole family of generalized mean-field states as constraints. As an example, we calculated superconducting pairing correlation functions for the two-dimensional repulsive Hubbard model using a BCS trial state as the constraint. We compared the results with the case where a free-electron trial wave function is used. We found that the correlation functions are independent of which state is used as the constraint, which reaffirms the results previously found by Zhang et al. [Phys. Rev. Lett. 78, 4486 (1997)] regarding the suppression of long-range pairing correlations as the system size increases.

DOI
10.1103/physrevb.59.1706

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We present a formulation of the constrained path Monte Carlo method for fermions that uses trial wave functions that include many-body effects. This formulation allows us to implement a whole family of generalized mean-field states as constraints. As an example, we calculated superconducting pairing correlation functions for the two-dimensional repulsive Hubbard model using a BCS trial state as the constraint. We compared the results with the case where a free-electron trial wave function is used. We found that the correlation functions are independent of which state is used as the constraint, which reaffirms the results previously found by Zhang et al. [Phys. Rev. Lett. 78, 4486 (1997)] regarding the suppression of long-range pairing correlations as the system size increases.
We introduce an efficient and numerically stable technique to make use of a BCS trial wave function in the computation of correlation functions of strongly correlated quantum fermion systems. The technique is applicable to any projection approach involving paths of independent-fermion propagators, for example, in mean-field or auxiliary-field quantum Monte Carlo (AFQMC) calculations. Within AFQMC, in the absence of the sign problem, the methodology allows the use of a BCS reference state which can greatly reduce the required imaginary time of projection and improves Monte Carlo sampling efficiency and statistical accuracy for systems where pairing correlations are important. When the sign problem is present, the approach provides a powerful generalization of the constrained-path AFQMC technique which usually uses Slater determinant trial wave functions. As a demonstration of the capability of the methodology, we present benchmark results for the attractive Hubbard model, both spin balanced (no sign problem) and with a finite spin polarization (with sign problem).
Ground state properties of the Hubbard model are of fundamental importance to understand the mechanism of unconventional superconductivity in the high-T_c cuprates and other materials. One of the most powerful numerical methods for strongly interacting models is quantum Monte Carlo, which however faces a fundamental limitation, the Fermion sign problem. The sign problem can be mitigated using approximate methods such as Constrained Path Monte Carlo, where the sign of the overlap with a trial wavefunction is constrained to remain positive. Beyond the constraint, additional approximations must be made in order to measure observables for operators that do not commute with the Hamiltonian. We examine critically the most commonly used approximation, back propagation, as well as a recently proposed constraint release measurement technique. In comparisons with a variety of systems that can be solved numerically exactly by other methods, we find that back propagation tends to underestimate long-range superconducting pair-pair correlations. Except in special cases such as one dimension, back propagation introduces an uncontrolled approximation even when the exact trial wavefunction is used. The constraint release technique can provide accurate results, with the disadvantages that it much more computationally expensive and reintroduces the sign problem.
We study the effects of disorder on long-range antiferromagnetic correlations in the half-filled, two-dimensional, repulsive Hubbard model at T=0.T=0.T=0. A mean field approach is first employed to gain a qualitative picture of the physics and to guide our choice for a trial wave function in a constrained path quantum Monte Carlo (CPQMC) method that allows for a more accurate treatment of correlations. Within the mean field calculation, we observe both Anderson and Mott insulating antiferromagnetic phases. There are transitions to a paramagnet only for relatively weak coupling, U<2tU<2tU<2t in the case of bond disorder, and U<4tU<4tU<4t in the case of on-site disorder. Using ground-state CPQMC we demonstrate that this mean field approach significantly overestimates magnetic order. For U=4t,U=4t,U=4t, we find a critical bond disorder of ${V}_{c}\ensuremath{\approx}(1.6\ifmmode\pm\else\textpm\fi{}0.4)t$ even though within mean field theory no paramagnetic phase is found for this value of the interaction. In the site disordered case, we find a critical disorder of ${V}_{c}\ensuremath{\approx}(5.0\ifmmode\pm\else\textpm\fi{}0.5)t$ at U=4t.U=4t.U=4t.
The d-wave pair-pair correlation of the two-dimensional Hubbard model is investigated by the power-Lanczos method. A Jastrow-BCS type wave function is used in our variational study. By assuming a monotonic behavior of the long-range pairing correlation under Lanczos iteration, we found there is no long-range pairing order in the intermediate interaction region (U ≃ 10), which is consistent with the former result by the constrained-path Monte Carlo method. We also present numerical results obtained by the exact diagonalization of the 4×4 lattice in supporting our conclusion. PACS numbers: 74.20.-z, 71.18.+y I.
A linked cluster expansion of the Gutzwiller variational wave function is used to study the one- and two-dimensional Hubbard model; in parallel the variational results are compared with quantum Monte Carlo simulations on finite-size lattices. Detailed results are presented for the magnetic and pairing correlations as a function of wave vector, electron density, and interaction strength U. There is no sign for superconducting long-range order although both the d-wave and the extended s-wave pairing fluctuations are substantially enhanced as U increases.
We present the results of a quantum Monte Carlo study of the extended sss and the ${d}_{{x}^{2}{\ensuremath{-}y}^{2}}$ pairing correlation functions for the two-dimensional Hubbard model, computed with the constrained-path Monte Carlo method. For small lattice sizes and weak interactions, we find that the ${d}_{{x}^{2}{\ensuremath{-}y}^{2}}$ pairing correlations are stronger than the extended sss pairing correlations and are positive when the pair separation exceeds several lattice constants. As the system size or the interaction strength increases, the magnitude of the long-range part of both correlation functions vanishes.