Physical Review

Electron Correlation in Narrow Energy Bands. II. One Reversed Spin in an Otherwise Fully Aligned Narrow S Band

Laura M. Roth

Published Oct 10, 1969• 59 citations

Abstract

In a previous paper, a new Green's-function decoupling scheme was applied to the Hubbard Hamiltonian, and an improved version of Hubbard's first approximation was obtained. That result did not reduce to the correct low-density limit as obtained by Kanamori. In the present article, the theory is improved for the special case of a single reversed spin in an otherwise fully aligned band, and the improved theory is correct in the low-density limit. Numerical results are presented for the simple cubic lattice. If we define an effective exchange-interaction parameter ${U}_{\mathrm{eff}}$ as the $k=0$ reversed-spin self-energy for $U\ensuremath{\rightarrow}\ensuremath{\infty}$, divided by the number $n\ensuremath{\uparrow}$ of up-spin electrons per site, we find that the present result departs rather rapidly from the Kanamori result as $n\ensuremath{\uparrow}$ is increased, and it is concluded that the Kanamori result overestimates the increase in ${U}_{\mathrm{eff}}$ with $n\ensuremath{\uparrow}$, at least in the present case. For intermediate values of $n\ensuremath{\uparrow}$, the two-pole approximation of the previous article and the present calculation give very similar results for this quantity.

DOI: 10.1103/physrev.186.428

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Electron Correlation in Narrow Energy Bands. I. The Two-Pole Approximation in a Narrow S Band

Laura M. Roth

Physical ReviewAug 10, 1969254 citations

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Three-Body Problem in the One-Dimensional Hubbard Model

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Physical Review

Electron Correlation in Narrow Energy Bands. II. One Reversed Spin in an Otherwise Fully Aligned Narrow S Band

Laura M. Roth

Published Oct 10, 1969• 59 citations

Publisher

Abstract

{U}_{\mathrm{eff}}

as the

k=0

reversed-spin self-energy for

U\ensuremath{\rightarrow}\ensuremath{\infty}

, divided by the number

n\ensuremath{\uparrow}

of up-spin electrons per site, we find that the present result departs rather rapidly from the Kanamori result as

n\ensuremath{\uparrow}

is increased, and it is concluded that the Kanamori result overestimates the increase in

{U}_{\mathrm{eff}}

with

n\ensuremath{\uparrow}

, at least in the present case. For intermediate values of

n\ensuremath{\uparrow}

, the two-pole approximation of the previous article and the present calculation give very similar results for this quantity.

DOI: 10.1103/physrev.186.428

Related papers

Electron Correlation in Narrow Energy Bands. I. The Two-Pole Approximation in a Narrow S Band

Laura M. Roth

An improved version of Hubbard's treatment of correlation in a nondegenerate narrow band is obtained by the use of a new Green's-function decoupling scheme. The resulting one-electron Green's function has two poles on the real axis corresponding to a splitting of the electron bands due to the strong correlations between electrons on the same site. This is the same result that Hubbard obtained, but in our case the poles are shifted and their positions agree with the results of Harris and Lange, who used a moment technique. The theory is applied to a simple cubic lattice with nearest-neighbor interaction, and the lattice is found to be ferromagnetic in the strongly correlated limit for a sufficiently large number of electrons per atom. An examination of the low-density limit shows that the two-pole approximation does not reduce to Kanamori's $T$ -matrix result, a failing which it shares with Hubbard's theory. A method is outlined for improving the theory so as to give the correct low-density result. In addition, the possibility of obtaining a minimum principle for the theory is explored.An improved version of Hubbard's treatment of correlation in a nondegenerate narrow band is obtained by the use of a new Green's-function decoupling scheme. The resulting one-electron Green's function has two poles on the real axis corresponding to a splitting of the electron bands due to the strong correlations between electrons on the same site. This is the same result that Hubbard obtained, but in our case the poles are shifted and their positions agree with the results of Harris and Lange, who used a moment technique. The theory is applied to a simple cubic lattice with nearest-neighbor interaction, and the lattice is found to be ferromagnetic in the strongly correlated limit for a sufficiently large number of electrons per atom. An examination of the low-density limit shows that the two-pole approximation does not reduce to Kanamori's $T$-matrix result, a failing which it shares with Hubbard's theory. A method is outlined for improving the theory so as to give the correct low-density result. In addition, the possibility of obtaining a minimum principle for the theory is explored.

Physical ReviewAug 10, 1969254 citations

Correlation Effects on the Spin Susceptibility of Narrow-Band Electrons

M. Kanehisa, Hiroshi Kamimura

To investigate the effects of electron correlation in narrow energy bands, a general formula of static spin susceptibility for interacting electron systems is derived. It is expressed in terms of the one-particle Green's function and its derivative with respect to the average number of the opposite spin electrons, the existence of the latter being the characteristic of electron correlation. The formula is applied to the case of Hubbard's improved approximation (Hubbard III), and gross feature of both the correlation and temperature dependences of the susceptibility is investigated in the insulating and metallic regions. The correlation in the metallic region is reduced for the Hubbard III case so that there occurs no ferromagnetism in a half-filled band.To investigate the effects of electron correlation in narrow energy bands, a general formula of static spin susceptibility for interacting electron systems is derived. It is expressed in terms of the one-particle Green's function and its derivative with respect to the average number of the opposite spin electrons, the existence of the latter being the characteristic of electron correlation. The formula is applied to the case of Hubbard's improved approximation (Hubbard III), and gross feature of both the correlation and temperature dependences of the susceptibility is investigated in the insulating and metallic regions. The correlation in the metallic region is reduced for the Hubbard III case so that there occurs no ferromagnetism in a half-filled band.

Journal of the Physical Society of JapanJan 1, 19737 citations

Three-Body Problem in the One-Dimensional Hubbard Model

Jun-ichi Igarashi

Electron correlation is studied in the special case of a single reversed spin in an otherwise fully aligned band in the one-dimensional Hubbard model for the purpose of examining our previous theory which exactly takes account of the three-body correlation. The importance of the three-body correlation is demonstrated through a comparison of our results with those of exact theories. The ground-state energy and the single particle spectra are in good agreement with McGuire-Lieb-Wu's exact solution and also with exact numerical solutions for finite size systems over a wide range of the intraatomic Coulomb integral and the electron number. The properties of the ground state are studied in detail through the calculation of pair correlation functions.Electron correlation is studied in the special case of a single reversed spin in an otherwise fully aligned band in the one-dimensional Hubbard model for the purpose of examining our previous theory which exactly takes account of the three-body correlation. The importance of the three-body correlation is demonstrated through a comparison of our results with those of exact theories. The ground-state energy and the single particle spectra are in good agreement with McGuire-Lieb-Wu's exact solution and also with exact numerical solutions for finite size systems over a wide range of the intraatomic Coulomb integral and the electron number. The properties of the ground state are studied in detail through the calculation of pair correlation functions.

Journal of the Physical Society of JapanJan 15, 198539 citations

Dynamical correlation-hole approach to the Hubbard model

M. C. Refolio, J. M. López Sancho, J. Rubio +1 more

The Hubbard I decoupling approximation [J. Hubbard, Proc. R. Soc. London Ser. A 276, 238 (1963)] is extended by introducing a k-dependent self-energy which describes a mobile correlation hole of opposite spin propagating along with the electron in its way around the lattice. The theory, despite its simplicity, (1) becomes exact in the strong-coupling limit; and (2) reproduces with great accuracy the ground-state energy, the double occupancy, the on-site correlation functions, and the effective hopping for all the values of the Hubbard Coulomb repulsion U. It also gives reasonable spectral functions leading to (3) a single-particle Green function showing the well-known spectral-weight transfer between the Hubbard bands as soon as the electron density deviates from half-filling, and (4) a momentum distribution as well as magnetic and charge structure factors in qualitative agreement with quantum Monte Carlo simulations, slave-boson calculations, and conserving approximations like fluctuation exchange and parquet. On the other hand, as in most decoupling schemes, liftime effects and the near-Fermi-level additional structure characteristic of strongly correlated electron systems are missing.The Hubbard I decoupling approximation [J. Hubbard, Proc. R. Soc. London Ser. A 276, 238 (1963)] is extended by introducing a k-dependent self-energy which describes a mobile correlation hole of opposite spin propagating along with the electron in its way around the lattice. The theory, despite its simplicity, (1) becomes exact in the strong-coupling limit; and (2) reproduces with great accuracy the ground-state energy, the double occupancy, the on-site correlation functions, and the effective hopping for all the values of the Hubbard Coulomb repulsion U. It also gives reasonable spectral functions leading to (3) a single-particle Green function showing the well-known spectral-weight transfer between the Hubbard bands as soon as the electron density deviates from half-filling, and (4) a momentum distribution as well as magnetic and charge structure factors in qualitative agreement with quantum Monte Carlo simulations, slave-boson calculations, and conserving approximations like fluctuation exchange and parquet. On the other hand, as in most decoupling schemes, liftime effects and the near-Fermi-level additional structure characteristic of strongly correlated electron systems are missing.

Physical review. B, Condensed matterFeb 15, 19991 citation

Density of states for a structurally disordered Hubbard model

K. L. Liu, B. Bergersen

We study the density of states for a structurally random Hubbard model with a half-filled band. Calculations are carried out with the hopping integral t(R) obtained from a hydrogenic 1S wave function and with an on-site Coulomb repulsion [Formula: see text]. The equation of motion for the retarded single-particle Green's function is treated in the Hubbard III decoupling approximation. The configurational averaging for the Green's function is simplified using the approximation of Kikuchi. Coefficients in the locator–interactor expansion are calculated analytically up to seventh order in t. Several approximants for this formal series are investigated, including Padé approximant, continued fraction approximation, and quadratic Padé approximant. The last two approximants yield densities of state in satisfactory agreement with the result of Aoki and Kamimura in the high concentration region and our results are expected to represent an improvement for intermediate densities. However, for very low densities the Hubbard III scheme will underestimate the effect of spin correlations and our previous treatment in the U = ∞ limit would be more appropriate for the lower Hubbard band.We study the density of states for a structurally random Hubbard model with a half-filled band. Calculations are carried out with the hopping integral t(R) obtained from a hydrogenic 1S wave function and with an on-site Coulomb repulsion [Formula: see text]. The equation of motion for the retarded single-particle Green's function is treated in the Hubbard III decoupling approximation. The configurational averaging for the Green's function is simplified using the approximation of Kikuchi. Coefficients in the locator–interactor expansion are calculated analytically up to seventh order in t. Several approximants for this formal series are investigated, including Padé approximant, continued fraction approximation, and quadratic Padé approximant. The last two approximants yield densities of state in satisfactory agreement with the result of Aoki and Kamimura in the high concentration region and our results are expected to represent an improvement for intermediate densities. However, for very low densities the Hubbard III scheme will underestimate the effect of spin correlations and our previous treatment in the U = ∞ limit would be more appropriate for the lower Hubbard band.

Canadian Journal of PhysicsJan 1, 19813 citations

Perturbational treatment of correlation effects in the Hubbard model

G. Bulk, Rainer J. Jelitto

In the general frame of a modified many-body perturbation theory, the single-band Hubbard model is treated in the limit of weak electron-electron interaction. In this modification the decomposition of the Hamiltonian into an effective free system and an effective perturbation is rearranged iteratively in each order of the formalism, resulting in a successive reduction of the interaction part. In comparison with conventional perturbation techniques, this means an extended range of validity. Imposing thermodynamic self-consistency, we calculate the wave-vector-dependent electronic self-energy up to second order and analyze the influence of the Coulomb correlations. Several well-known physical properties of the Hubbard model are recovered in characteristic single-particle quantities. Therefore, the effect of the band occupation n and the electron-electron repulsion U on spectral densities, band structures, densities of states, magnetization, etc. is investigated in detail. Some of the most remarkable results are: first, a splitting of the free Bloch band into two quasiparticle subbands when the coupling strength reaches moderate values. These subbands are linked by a small damping-induced background in the density of states and are separated roughly by an energy of the order of U. In addition, the upper subband is shifted to higher energies in the spectrum proportionally to U while the lower one is nearly fixed. Second, we found a paramagnetic-ferromagnetic phase diagram that has a critical coupling strength ${U}_{c}$ and exhibits strong indications of a lower critical occupation number ${n}_{c}$ . There is no spontaneous magnetization in the system for parameters smaller than these values. Compared with the Stoner model, one thus obtains striking improvements. In many cases our conclusions for the weak-coupling limit are qualitatively in good agreement with well-known results in the strong-correlation regime.In the general frame of a modified many-body perturbation theory, the single-band Hubbard model is treated in the limit of weak electron-electron interaction. In this modification the decomposition of the Hamiltonian into an effective free system and an effective perturbation is rearranged iteratively in each order of the formalism, resulting in a successive reduction of the interaction part. In comparison with conventional perturbation techniques, this means an extended range of validity. Imposing thermodynamic self-consistency, we calculate the wave-vector-dependent electronic self-energy up to second order and analyze the influence of the Coulomb correlations. Several well-known physical properties of the Hubbard model are recovered in characteristic single-particle quantities. Therefore, the effect of the band occupation n and the electron-electron repulsion U on spectral densities, band structures, densities of states, magnetization, etc. is investigated in detail. Some of the most remarkable results are: first, a splitting of the free Bloch band into two quasiparticle subbands when the coupling strength reaches moderate values. These subbands are linked by a small damping-induced background in the density of states and are separated roughly by an energy of the order of U. In addition, the upper subband is shifted to higher energies in the spectrum proportionally to U while the lower one is nearly fixed. Second, we found a paramagnetic-ferromagnetic phase diagram that has a critical coupling strength ${U}_{c}$ and exhibits strong indications of a lower critical occupation number ${n}_{c}$. There is no spontaneous magnetization in the system for parameters smaller than these values. Compared with the Stoner model, one thus obtains striking improvements. In many cases our conclusions for the weak-coupling limit are qualitatively in good agreement with well-known results in the strong-correlation regime.

Physical review. B, Condensed matterJan 1, 199034 citations

{U}_{\mathrm{eff}}

as the

k=0

reversed-spin self-energy for

U\ensuremath{\rightarrow}\ensuremath{\infty}

, divided by the number

n\ensuremath{\uparrow}

of up-spin electrons per site, we find that the present result departs rather rapidly from the Kanamori result as

n\ensuremath{\uparrow}

is increased, and it is concluded that the Kanamori result overestimates the increase in

{U}_{\mathrm{eff}}

with

n\ensuremath{\uparrow}

, at least in the present case. For intermediate values of

n\ensuremath{\uparrow}

, the two-pole approximation of the previous article and the present calculation give very similar results for this quantity.

T

-matrix result, a failing which it shares with Hubbard's theory. A method is outlined for improving the theory so as to give the correct low-density result. In addition, the possibility of obtaining a minimum principle for the theory is explored.

{U}_{c}

and exhibits strong indications of a lower critical occupation number

{n}_{c}

. There is no spontaneous magnetization in the system for parameters smaller than these values. Compared with the Stoner model, one thus obtains striking improvements. In many cases our conclusions for the weak-coupling limit are qualitatively in good agreement with well-known results in the strong-correlation regime.