An effective quantum field theory of the 2D Hubbard model on a square lattice near half-filling is presented and studied. This effective model describes so-called nodal and antinodal fermions, and it is derived from the lattice model using a certain partial continuum limit. It is shown that the nodal fermions can be bosonized, which leads to spin-charge separation and a 2D analogue of a Wess-Zumino-Witten model. A bosonization formula for the nodal fermion field operator is obtained, and an exactly solvable model of interacting 2D fermions is identified. Different ways of treating the antinodal fermions are also proposed.
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
3He, UBe3, 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 describe a low-temperature approach to the two-dimensional half-filled Hubbard model which allows us to study both antiferromagnetism and single-particle properties. This approach ignores amplitude fluctuations of the antiferromagnetic (AF) order parameter and is valid below a crossover temperature
TX which marks the onset of AF short-range order. Directional fluctuations (spin waves) are described by a nonlinear sigma model $(\mathrm{NL}\ensuremath{\sigma}\mathrm{M})$ that we derive from the Hubbard model. The parameters of the $\mathrm{NL}\ensuremath{\sigma}\mathrm{M}$---the spin stiffness and spin-wave velocity---are calculated as a function of the Coulomb repulsion U. The $\mathrm{NL}\ensuremath{\sigma}\mathrm{M}$ is solved by a saddle-point approximation within the
CP1 representation where the N\'eel field is parametrized by two Schwinger bosons. At zero temperature, there is always Bose condensation of the Schwinger bosons, which signals AF long-range order for any value of the Coulomb repulsion. At finite temperature, the AF long-range order is suppressed (in agreement with the Mermin-Wagner theorem), but the AF correlation length remains exponentially large. In the
CP1 representation, the fermion field is naturally expressed as the product of a Schwinger boson and a pseudofermion whose spin is quantized along the (fluctuating) N\'eel field. This allows us to write the fermion Green's function as the product (in direct space) of the Schwinger boson propagator (which is derived from the $\mathrm{NL}\ensuremath{\sigma}\mathrm{M})$ and the pseudofermion propagator. At zero temperature and weak coupling, our results are typical of a Slater antiferromagnet. The AF gap is exponentially small; there are well-defined Bogoliubov quasiparticles (QP's) (carrying most of the spectral weight) coexisting with a high-energy incoherent excitation background. As U increases, the Slater antiferromagnet progressively becomes a Mott-Heisenberg antiferromagnet. The Bogoliubov bands evolve into Mott-Hubbard bands separated by a large AF gap. A significant fraction of spectral weight is transferred from the Bogoliubov QP's to incoherent excitations. At finite temperature, there is a metal-insulator transition between a pseudogap phase at weak coupling and a Mott-Hubbard insulator at strong coupling. Finally, we point out that our results straightforwardly translate to the half-filled attractive Hubbard model, where the $\mathbf{q}=(\ensuremath{\pi},\ensuremath{\pi})$ charge and
q=0 pairing fluctuations combine to form an order parameter with SO(3) symmetry.
The ground-state energy E and momentum distribution n k of the electrons are expanded from the atomic limit for the half-filled Hubbard model. The coefficients of expansion are represented by the spin correlation functions of the spin 1 / 2 Heisenberg model. Using the spin-wave theory, approximate values of coefficients are calculated for the square lattice and the simple cubic lattice. In one dimension, the theory shows a good agreement with the exact solution. An effective spin Hamiltonian for the half-filled Hubbard model with arbitrary hopping integrals is obtained up to the fifth order. It is shown that the fourth term contains four spin interactions.
A detailed quantitative study of the physical properties of the infinite-dimensional Hubbard model at half filling is presented. The method makes use of an exact mapping onto a single-impurity model supplemented by a self-consistency condition. This coupled problem is solved numerically. Results for thermodynamic quantities (specific heat, entropy, . . .), one-particle spectral properties, and magnetic properties (response to a uniform magnetic field) are presented and discussed. The nature of the Mott-Hubbard metal-insulator transition found in this model is investigated. A numerical solution of the mean-field equations inside the antiferromagnetic phase is also reported.
The Hubbard model including nearest neighbor interaction is studied at T=0 on a d-dimensional hypercubic lattice (d\ensuremath{\gg}1) close to half filling. For the model in d=\ensuremath{\infty} we derive the exact result that the ground state at weak coupling is phase separated. Results for lower dimensions are then derived in a 1/d expansion. To obtain these results we first consider possible second order transitions. One then finds that the broken-symmetry phase near half filling is incommensurate. However, the corresponding ground state has negative compressibility and is hence thermodynamically unstable. A Maxwell construction is used to construct the actual phase separated ground state, which consists of homogeneous lower-density and antiferromagnetic or charge density wave higher-density regions. It is shown that both the doping level below which phase separation occurs and the order parameter differ from the corresponding Hartree results by a renormalization factor q of order unity. This renormalization factor q is calculated systematically up to O(1/d) in a 1/d expansion and turns out to be identical to the renormalization factor previously calculated for the low-temperature thermodynamics at half filling. \textcopyright{} 1996 The American Physical Society.