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arXiv:cond-mat/9707108 · Physical review. B, Condensed matter

Quantum Monte Carlo simulations of infinitely strongly correlated fermions

Michael Brunner, A. Muramatsu

Published Oct 15, 1998• 14 citations
arXivHTMLPDF

Abstract

Numerical simulations of the two-dimensional $t\ensuremath{-}J$ model in the limit $J/t\ensuremath{\ll}1$ are performed for rather large systems $(N\ensuremath{\sim}10\ifmmode\times\else\texttimes\fi{}10)$ using a world-line loop algorithm. It is shown that in the one-hole case with $J=0,$ very low temperatures $(\ensuremath{\beta}t\ensuremath{\sim}2500)$ are necessary in order to reach Nagaoka's state. Full polarization becomes unstable for $J\ensuremath{\sim}{10}^{\ensuremath{-}4}t$ towards partial polarization up to $J/t\ensuremath{\lesssim}0.01.$ $J/t\ensuremath{\gtrsim}0.05$ leads to minimal spin. The two-hole case shows enhanced total spin up to the lowest attainable temperatures $(\ensuremath{\beta}t=150),$ well below those reached by other finite-temperature methods.

DOI
10.1103/physrevb.58.r10100

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Numerical simulations of the two-dimensional t\ensuremath−Jt\ensuremath{-}Jt\ensuremath−J model in the limit J/t\ensuremath≪1J/t\ensuremath{\ll}1J/t\ensuremath≪1 are performed for rather large systems (N\ensuremath∼10\ifmmode×\else\texttimes\fi10)(N\ensuremath{\sim}10\ifmmode\times\else\texttimes\fi{}10)(N\ensuremath∼10\ifmmode×\else\texttimes\fi10) using a world-line loop algorithm. It is shown that in the one-hole case with J=0,J=0,J=0, very low temperatures (\ensuremathβt\ensuremath∼2500)(\ensuremath{\beta}t\ensuremath{\sim}2500)(\ensuremathβt\ensuremath∼2500) are necessary in order to reach Nagaoka's state. Full polarization becomes unstable for J\ensuremath∼10\ensuremath−4tJ\ensuremath{\sim}{10}^{\ensuremath{-}4}tJ\ensuremath∼10\ensuremath−4t towards partial polarization up to J/t\ensuremath≲0.01.J/t\ensuremath{\lesssim}0.01.J/t\ensuremath≲0.01. J/t\ensuremath≳0.05J/t\ensuremath{\gtrsim}0.05J/t\ensuremath≳0.05 leads to minimal spin. The two-hole case shows enhanced total spin up to the lowest attainable temperatures (\ensuremathβt=150),(\ensuremath{\beta}t=150),(\ensuremathβt=150), well below those reached by other finite-temperature methods.
We present different numerical calculations based on variational quantum Monte Carlo simulations supporting a ferromagnetic ground state for finite and small hole densities in the two-dimensional infinite-U Hubbard model. Moreover, by studying the energies of different total spin sectors, these calculations strongly suggest that the paramagnetic phase is unstable against a phase with a partial polarization for large hole densities delta approximately 0.40 with evidence for a second-order transition to the paramagnetic large doping phase.
Two-dimensional Hubbard lattices with two or three holes are investigated as a function of UUU in the large-UUU limit. In the so-called Nagaoka limit (one-hole system at infinite UUU), it is known that the Hubbard model exhibits a ferromagnetic ground state. Here, by means of exact full configuration interaction quantum Monte Carlo simulations applied to periodic lattices up to 24 sites, we compute spin-spin correlation functions as a function of increasing UUU. The correlation functions clearly demonstrate the onset of ferromagnetic domains, centered on individual holes. The overall total spin of the wave functions remains the lowest possible (0 or 12\frac{1}{2}21​, depending on the number of holes). The ferromagnetic domains appear at interaction strengths comparable to the critical interaction strengths of the Nagaoka transition in finite systems with strictly one hole. The existence of such ferromagnetic domains is the signature of Nagaoka physics in Hubbard systems with a small (but greater than 1) number of holes.
We investigate Nagaoka ferromagnetism in the two-dimensional Hubbard model with one hole using the spin-adapted [SU(2)\text{SU}(2)SU(2) conserving] full configuration interaction quantum Monte Carlo method. This methodology gives us access to the ground-state energies of all possible spin states SSS of finite Hubbard lattices, here obtained for lattices up to 26 sites for various interaction strengths (UUU). The critical interaction strength, Uc{U}_{c}Uc​, at which the Nagaoka transition occurs is determined for each lattice and is found to be proportional to the lattice size for the larger lattices. Below Uc{U}_{c}Uc​, the overall ground states are found to favour the minimal total spin (S=12S=\frac{1}{2}S=21​), and no intermediate spin state is found to be the overall ground state on lattices larger than 16 sites. However, at Uc{U}_{c}Uc​, the energies of all the spin states are found to be nearly degenerate, implying that large fluctuations in total spin can be expected in the vicinity of the Nagaoka transition.
The Nagaoka $(U=\ensuremath{\infty})$ limit of the Hubbard model on a square lattice is mapped onto the itinerant-localized Kondo model at infinitely strong coupling. Such a model is well suited to perform quantum Monte Carlo (QMC) simulations to compute spin correlation functions. For periodic boundary conditions, this model is shown to exhibit no short-range ferromagnetic (FM) spin correlations at any doping $\ensuremath{\delta}\ensuremath{\ge}0.01$ and at finite temperature T=0.1t.T=0.1t.T=0.1t. Our simulations give no indication that there is a tendency towards ferromagnetic ordering in the ground state, with more than one hole. Employing on the other hand the open boundary conditions (or mixed boundary conditions) may result in the qualitatively different results for the thermodynamic limit depending on the way one chooses to approach this limit. These observations imply that the relevant thermodynamic limit remains unclear.
Ferromagnetic domains in the large-UUU Hubbard model with a few holes: an FCIQMC study
Two-dimensional Hubbard lattices with two or three holes are investigated as a function of UUU in the large-UUU limit. In the so-called Nagaoka limit (one-hole system at infinite UUU), it is known that the Hubbard model exhibits a ferromagnetic ground-state. Here, by means of exact FCIQMC simulations applied to periodic lattices up to 24 sites, we compute spin-spin correlation functions as a function of increasing UUU. The correlation functions clearly demonstrate the onset of ferromagnetic domains, centred on individual holes. The overall total spin of the wavefunctions remain the lowest possible (0, or 12\frac{1}{2}21​, depending on the number of holes). The ferromagnetic domains appear at interaction strengths comparable to the critical interaction strengths of the Nagaoka transition in finite systems with strictly one hole. The existence of such ferromagnetic domains is the signature of Nagaoka physics in Hubbard systems with a small (but greater than one) number of holes.
Using an original Quantum Monte Carlo algorithm, we study the thermodynamical properties of a single hole in the two-dimensional infinite-U Hubbard model at finite temperature. We investigate the energy and the spin correlators as a function of an external orbital magnetic field. This field is found to destroy the Nagaoka ferromagnetism and to induce chirality in the spin background. The applied field is partially screened by a fictitious magnetic field coming from the chirality. Our algorithm allows us to reach a temperature low enough to discuss the ground state properties of the model.