Nonperturbative solutions to the Hubbard model are found by using the functional derivative method. A system of closed equations is obtained for the Green's functions of the Hubbard model. Exact expressions for the self-energy are derived which involve only first order functional derivatives. A scheme is proposed for systematically calculating self-energy corrections. We apply the solution to finite rings of two and four lattice sites and compare to the results of numerical calculations on these systems.Nonperturbative solutions to the Hubbard model are found by using the functional derivative method. A system of closed equations is obtained for the Green's functions of the Hubbard model. Exact expressions for the self-energy are derived which involve only first order functional derivatives. A scheme is proposed for systematically calculating self-energy corrections. We apply the solution to finite rings of two and four lattice sites and compare to the results of numerical calculations on these systems.

Jan 1, 19870 citations

Non-perturbative solutions to the Hubbard Model

Henry L. Neal

Springer eBooksNov 13, 20050 citations

Nonperturbative approach to the attractive Hubbard model

S. Allen, A.–M. S. Tremblay

A nonperturbative approach to the single-band attractive Hubbard model is presented in the general context of functional-derivative approaches to many-body theories. As in previous work on the repulsive model, the first step is based on a local-field-type ansatz, on enforcement of the Pauli principle and a number of crucial sumrules. The Mermin-Wagner theorem in two dimensions is automatically satisfied. At this level, two-particle self-consistency has been achieved. In the second step of the approximation, an improved expression for the self-energy is obtained by using the results of the first step in an exact expression for the self-energy, where the high- and low-frequency behaviors appear separately. The result is a cooperon-like formula. The required vertex corrections are included in this self-energy expression, as required by the absence of a Migdal theorem for this problem. Other approaches to the attractive Hubbard model are critically compared. Physical consequences of the present approach and agreement with Monte Carlo simulations are demonstrated in the accompanying paper (following this one).A nonperturbative approach to the single-band attractive Hubbard model is presented in the general context of functional-derivative approaches to many-body theories. As in previous work on the repulsive model, the first step is based on a local-field-type ansatz, on enforcement of the Pauli principle and a number of crucial sumrules. The Mermin-Wagner theorem in two dimensions is automatically satisfied. At this level, two-particle self-consistency has been achieved. In the second step of the approximation, an improved expression for the self-energy is obtained by using the results of the first step in an exact expression for the self-energy, where the high- and low-frequency behaviors appear separately. The result is a cooperon-like formula. The required vertex corrections are included in this self-energy expression, as required by the absence of a Migdal theorem for this problem. Other approaches to the attractive Hubbard model are critically compared. Physical consequences of the present approach and agreement with Monte Carlo simulations are demonstrated in the accompanying paper (following this one).

arXiv:cond-mat/0010039Physical review. B, Condensed matterJul 31, 200142 citations

Local dynamics and strong correlation physics: One- and two-dimensional half-filled Hubbard models

Tudor D. Stanescu, Philip Phillips

We report on a nonperturbative approach to the one-dimensional (1D) and two-dimensional (2D) Hubbard models that is capable of recovering both strong- $(U\ensuremath{\gg}t)$ and weak-coupling $(U\ensuremath{\ll}t)$ limits, with U the on-site Coulomb repulsion and t the kinetic energy. Dynamical corrections to the electron self-energy in the single particle Green function are explicitly included by expanding in terms of the 16 eigenstates that characterize two nearest neighbor sites. We first show that even when U is much smaller than the bandwidth, the Mott-Hubbard gap never closes at half filling in both 1D and 2D. Consequently, the Hubbard model at half filling is always in the strong-coupling nonperturbative regime. For both large and small U, we find that the population of nearest-neighbor singlet states approaches a value of order unity as $\stackrel{\ensuremath{\rightarrow}}{T}0$ as would be expected for antiferromagnetic order. We also find that the double occupancy is a smooth monotonic function of U and approaches the anticipated noninteracting limit of $1/4$ as $\stackrel{\ensuremath{\rightarrow}}{U}0$ and vanishes as $\stackrel{\ensuremath{\rightarrow}}{U}\ensuremath{\infty}.$ Finally, we compute the heat capacity $[C(T,U)]$ for both 1D and 2D. Our results for 1D at moderate to high temperatures are in quantitative agreement with those of the exact Bethe ansatz solution, differing by no more than $1%.$ In addition, we find that in 2D, the $C(T,U)$ curves vs T for different values of U exhibit a universal crossing point at two characteristic temperatures $T\ensuremath{\approx}1.7t\ifmmode\pm\else\textpm\fi{}0.1t$ and $T\ensuremath{\approx}0.4\ifmmode\pm\else\textpm\fi{}0.1t$ as is seen universally in Hubbard models and experimentally in a wide range of strongly correlated systems such as ${}^{3}\mathrm{He},$ ${\mathrm{UBe}}_{3},$ and ${\mathrm{CeCu}}_{6\ensuremath{-}x}{\mathrm{Al}}_{x}.$ The success of this method in recovering well-established results that stem fundamentally from the Coulomb interaction suggests that local dynamics are at the heart of the physics of strongly correlated systems.We report on a nonperturbative approach to the one-dimensional (1D) and two-dimensional (2D) Hubbard models that is capable of recovering both strong- $(U\ensuremath{\gg}t)$ and weak-coupling $(U\ensuremath{\ll}t)$ limits, with U the on-site Coulomb repulsion and t the kinetic energy. Dynamical corrections to the electron self-energy in the single particle Green function are explicitly included by expanding in terms of the 16 eigenstates that characterize two nearest neighbor sites. We first show that even when U is much smaller than the bandwidth, the Mott-Hubbard gap never closes at half filling in both 1D and 2D. Consequently, the Hubbard model at half filling is always in the strong-coupling nonperturbative regime. For both large and small U, we find that the population of nearest-neighbor singlet states approaches a value of order unity as $\stackrel{\ensuremath{\rightarrow}}{T}0$ as would be expected for antiferromagnetic order. We also find that the double occupancy is a smooth monotonic function of U and approaches the anticipated noninteracting limit of $1/4$ as $\stackrel{\ensuremath{\rightarrow}}{U}0$ and vanishes as $\stackrel{\ensuremath{\rightarrow}}{U}\ensuremath{\infty}.$ Finally, we compute the heat capacity $[C(T,U)]$ for both 1D and 2D. Our results for 1D at moderate to high temperatures are in quantitative agreement with those of the exact Bethe ansatz solution, differing by no more than $1%.$ In addition, we find that in 2D, the $C(T,U)$ curves vs T for different values of U exhibit a universal crossing point at two characteristic temperatures $T\ensuremath{\approx}1.7t\ifmmode\pm\else\textpm\fi{}0.1t$ and $T\ensuremath{\approx}0.4\ifmmode\pm\else\textpm\fi{}0.1t$ as is seen universally in Hubbard models and experimentally in a wide range of strongly correlated systems such as ${}^{3}\mathrm{He},$ ${\mathrm{UBe}}_{3},$ and ${\mathrm{CeCu}}_{6\ensuremath{-}x}{\mathrm{Al}}_{x}.$ The success of this method in recovering well-established results that stem fundamentally from the Coulomb interaction suggests that local dynamics are at the heart of the physics of strongly correlated systems.

arXiv:cond-mat/0104478Physical review. B, Condensed matterNov 28, 200117 citations

Non-perturbative ground-state of a generalized Hubbard model

Oracio Navarro

Czechoslovak Journal of PhysicsApr 1, 19961 citation

1/4

as $\stackrel{\ensuremath{\rightarrow}}{U}0$ and vanishes as $\stackrel{\ensuremath{\rightarrow}}{U}\ensuremath{\infty}.$ Finally, we compute the heat capacity

[C(T,U)]

for both 1D and 2D. Our results for 1D at moderate to high temperatures are in quantitative agreement with those of the exact Bethe ansatz solution, differing by no more than

1%.

In addition, we find that in 2D, the

C(T,U)

curves vs T for different values of U exhibit a universal crossing point at two characteristic temperatures $T\ensuremath{\approx}1.7t\ifmmode\pm\else\textpm\fi{}0.1t$ and $T\ensuremath{\approx}0.4\ifmmode\pm\else\textpm\fi{}0.1t$ as is seen universally in Hubbard models and experimentally in a wide range of strongly correlated systems such as

{}^{3}\mathrm{He},

{\mathrm{UBe}}_{3},

and ${\mathrm{CeCu}}_{6\ensuremath{-}x}{\mathrm{Al}}_{x}.$ The success of this method in recovering well-established results that stem fundamentally from the Coulomb interaction suggests that local dynamics are at the heart of the physics of strongly correlated systems.