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COMPARATIVE STUDY ON THE HUBBARD MODEL ON ANTIFERROMAGNETIC INSULATORS USING QUANTUM ELECTRON SIMULATION TOOLBOX (QUEST)

Tantasso Sabo, Micheal Oboh

Published Jan 1, 2013• 0 citations
OpenAlex

Abstract

Magnetic property of a 2D Hubbard model is inferred from the result of Monte Carlo simulation. Lattice size of 4x4, and temperature down to T = 0.25 where studied. The plots of ρ versus µ exhibit some of the fundamental physics of the Hubbard model, namely the “Mott insulating gap”. At T = 0 the chemical potential has the property that µ = ρ

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Magnetic property of a 2D Hubbard model is inferred from the result of Monte Carlo simulation. Lattice size of 4x4, and temperature down to T = 0.25 where studied. The plots of ρ versus µ exhibit some of the fundamental physics of the Hubbard model, namely the “Mott insulating gap”. At T = 0 the chemical potential has the property that µ = ρ
We use quantum Monte Carlo methods to determine T=0T\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0T=0 Green functions, $G(\stackrel{\ensuremath{\rightarrow}}{r},\ensuremath{\omega})$, on lattices up to $16\ifmmode\times\else\texttimes\fi{}16$ for the 2D Hubbard model at U/t=4U/t\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}4U/t=4. For chemical potentials $\ensuremath{\mu}$ within the Hubbard gap $|\ensuremath{\mu}|<{\ensuremath{\mu}}_{c}$ and at long distances $\stackrel{\ensuremath{\rightarrow}}{r}$, $G(\stackrel{\ensuremath{\rightarrow}}{r},\ensuremath{\omega}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{\mu})\ensuremath{\sim}{e}^{\ensuremath{-}|\stackrel{\ensuremath{\rightarrow}}{r}|/{\ensuremath{\xi}}_{l}}$ with critical behavior ${\ensuremath{\xi}}_{l}\ensuremath{\sim}|\ensuremath{\mu}\ensuremath{-}{\ensuremath{\mu}}_{c}{|}^{\ensuremath{-}\ensuremath{\nu}}$, $\ensuremath{\nu}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.26\ifmmode\pm\else\textpm\fi{}0.05$. This result stands in agreement with the assumption of hyperscaling with correlation exponent $\ensuremath{\nu}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/4$ and dynamical exponent z=4z\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}4z=4. In contrast, the generic band insulator as well as the metal-insulator transition in the 1D Hubbard model are characterized by $\ensuremath{\nu}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/2$ and z=2z\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2z=2.
The Hubbard model has been widely used to describe magnetism due to itinerant electrons. However, except for the one-dimensional case, no exact solution of the model exists. We have studied the two-dimensional Hubbard model on a square lattice using a Monte Carlo simulation technique. Simulations were performed on lattices of up to 8×8 in spatial size. We will discuss results for thermodynamic properties and correlation functions as a function of temperature and interaction strength. For the half-filled band case, our results indicate that the system exhibits long-range antiferromagnetic order in the ground state for any nonzero interaction, although substantially reduced from the classical Néel state due to both charge and spin fluctuations. For the non-half-filled band case, our results suggest that no long-range magnetic order exists. We also don’t find any indication of short-ranged ferromagnetic correlations in the parameter range studied. We have also studied the predictions of various approximate treatments for this model, particularly the static approximation and the coherent potential approximation (CPA). By comparison with the Monte Carlo results, we discuss the validity of these schemes. Finally, we discuss the effect of changing the band structure on the magnetic properties of the system.
The Quantum Monte-Carlo simulations of the two-dimensional Hubbard model are presented for the half filling. The method based on the direct-space proposed by Suzuki and al., and Hirsch and al. was used. The states generated by this method are basis states in occupation number representation built with Wannier states localised on each site of the square array. The configurations of fermions can be observed on the real 2D array. An antiferromagnetic factor is defined and calculated for each temperature. The curves of energy, specific heat, conducivity and antiferromagnetic factor are presented for different values of the repulsive coulombian on site interaction U. There is a metal-insulator transition at low temperature for small values of U. This transition corresponds with a paramagnetic-ferromagnetic first order transition. Indeed, for these interaction values, the energy curves show a gap which is a characteristic of a first order transition. An hysteresis phenomenon appears on the conductivity curves. There is a behaviour change for U/t=3.5. For the values U>3.5 there is ferromagnetic-paramagnetic change without observable effect on the energy and the specific heat. The metal-insulator transition does not exist any more, the conductivity stays very small. Isotherms of the physical quantities versus U/t show a transition which seems to be the metal-insulator Mott transition. These results allow to draw a phase diagram with two first order transition lines.
Magnetic properties of the two-dimensional Hubbard-model are inferred from results of Monte Carlo simulations. Lattice sizes up to 8\ifmmode\times\else\texttimes\fi{}8 and temperatures down to T=t20T=\frac{t}{20}T=20t​ (t=hoppingt=\mathrm{hopping}t=hopping) were studied. The half-filled system is found to exhibit antiferromagnetic long-range order for all values of the Coulomb repulsion UUU. The low-temperature magnetic properties are found to be well described by spin-wave theory with renormalized local moment and spin-wave velocity. Numerical evidence presented suggests that when doped the system loses the long-range order immediately away from half filling.
The Quantum Monte Carlo simulations of the ionic Hubbard model on a two-dimensional square lattice at half filling were performed. The method based on the direct-space, proposed by Suzuki and al., Hirsch and al., was used. Cycles of increasing and decreasing values of the Coulomb interaction $ U $ were performed for fixed temperature ($ kT=0.01 $). Results indicate that, at low temperature, the two insulator phases are separated by a metallic phase for weak to intermediate values of the staggered potential $ Δ$. For large Coulomb repulsion the system is in a Mott insulator with an antiferromagnetism order. On increasing and decreasing the Coulomb interaction $ U $ the metal-Mott insulator transition shows an hysteresis phenomenon while the metal-band insulator transition is continue. For large $ Δ$ it seems that the metallic region shrinks to a single metallic point. However, the band insulator to the Mott insulator transition is not direct for the studied model. A phase diagram is drawn for the temperature $ kT=0.01 $. For $ Δ=0.5 $ cycles of increasing and decreasing temperature were programmed for different values of the Coulomb interaction $ U $ . A behaviour change appears for $ U\simeq 1.75 $. This suggests that a crossover line divides the metallic region of the phase diagram.
The ground state of the 2D (square lattice) finite-size Hubbard model is studied at half filling with a newly developed Quantum Monte Carlo method. We have studied square cells from 18 to 242 sites for U=3 with high statistical accuracy to perform a reliable size scaling analysis. The results confirm that the system is an antiferromagnetic insulator. The finite size scaling results are discussed in detail, focusing on the behaviour of magnetization, and most particularly of momentum distribution.