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arXiv:cond-mat/0009058 · Journal of Physics Condensed Matter

Antiferromagnetism of the two-dimensional Hubbard model at half-filling: the analytic ground state for weak coupling

Michele Cini, Gianluca Stefanucci

Published Jan 25, 2001• 2 citations
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Abstract

We introduce a local formalism to deal with the Hubbard model on an N × N square lattice (for even N ) in terms of eigenstates of number operators, having well defined point symmetry. For U →0, the low-lying shells of the kinetic energy are filled in the ground state. At half-filling, using the 2 N -2 one-body states of the partially occupied shell hf , we build a set of degenerate unperturbed ground states with S z = 0 which are then resolved by the Hubbard interaction Ŵ = U ∑ r r ↑ r ↓ . We study the many-body eigenstates in hf of the kinetic energy with vanishing eigenvalue of the Hubbard repulsion ( W = 0 states). In the S z = 0 sector, this is an N -times-degenerate multiplet. From the singlet component one obtains the ground state of the Hubbard model for U = 0 + , which is unique, in agreement with a theorem of Lieb. The wave function demonstrates an antiferromagnetic order, a lattice step translation being equivalent to a spin flip. We show that the total momentum vanishes, while the point symmetry is s or d for even or odd N /2, respectively.

DOI
10.1088/0953-8984/13/6/308

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We introduce a local formalism to deal with the Hubbard model on an N × N square lattice (for even N ) in terms of eigenstates of number operators, having well defined point symmetry. For U →0, the low-lying shells of the kinetic energy are filled in the ground state. At half-filling, using the 2 N -2 one-body states of the partially occupied shell hf , we build a set of degenerate unperturbed ground states with S z = 0 which are then resolved by the Hubbard interaction Ŵ = U ∑ r r ↑ r ↓ . We study the many-body eigenstates in hf of the kinetic energy with vanishing eigenvalue of the Hubbard repulsion ( W = 0 states). In the S z = 0 sector, this is an N -times-degenerate multiplet. From the singlet component one obtains the ground state of the Hubbard model for U = 0 + , which is unique, in agreement with a theorem of Lieb. The wave function demonstrates an antiferromagnetic order, a lattice step translation being equivalent to a spin flip. We show that the total momentum vanishes, while the point symmetry is s or d for even or odd N /2, respectively.
We investigate the ground state (T = 0 K) of the one-dimensional symmetrical (n = 1) Hubbard model formalized in terms of the system of integral equations, which we previously obtained using the method of the generating functional of Green’s functions with the subsequent Legendre transformation. In a wide range of variations in the parameter of Coulomb interaction U, the following characteristics of the system have been calculated: the electron density of states, the electron band spectrum, the number of doubly occupied lattice sites, the localized magnetic moment, the correlator of the square of the longitudinal component of spin at a site,<S Z 2>, and the internal energy of the system. It has been shown that, for all U > 0, the model yields two solutions, i.e., an antiferromagnetic insulator and a paramagnetic insulator, in which there are no single-electron quasi-particles at the Fermi level. The energy of the paramagnetic solution in the region of U < 1.1 is considerably less than that of the antiferromagnetic solution for the case of U > 1.1, we have the opposite situation.
Working on a square lattice with an even number of sites N2{\mathit{N}}^{2}N2 and assuming a smooth connection between strong and weak coupling, we show that the ground state of the two-dimensional half-filled Hubbard model has momentum kx{\mathit{k}}_{\mathit{x}}kx​-ky{\mathit{k}}_{\mathit{y}}ky​=0, it is even under reflections, and transforms as an s wave under rotations when N/2 is even and as a d wave when N/2 is odd.
As a prototype model of antiferromagnetism, we propose a repulsive Hubbard Hamiltonian defined on a graph $\ensuremath{\Lambda}=\mathcal{A}\ensuremath{\cup}\mathcal{B}$ with $\mathcal{A}\ensuremath{\cap}\mathcal{B}=\ensuremath{\emptyset},$ and bonds connecting any element of A\mathcal{A}A with all the elements of B.\mathcal{B}.B. Since all the hopping matrix elements associated with each bond are equal, the model is invariant under an arbitrary permutation of the A\mathcal{A}A sites and/or of the B\mathcal{B}B sites. This is the Hubbard model defined on the so-called (NA,NB){(N}_{A}{,N}_{B})(NA​,NB​) complete bipartite graph, NA(NB){N}_{A}{(N}_{B})NA​(NB​) being the number of elements in A(B).\mathcal{A}(\mathcal{B}).A(B). In this paper, we analytically find the exact ground state for NA=NB=N{N}_{A}{=N}_{B}=NNA​=NB​=N at half filling for any N; the repulsion has a maximum at a critical N-dependent value of the on-site Hubbard U. The wave function and the energy of the unique, singlet ground state assume a particularly elegant form for $\stackrel{\ensuremath{\rightarrow}}{N}\ensuremath{\infty}.$ We also calculate the spin-spin correlation function and show that the ground state exhibits an antiferromagnetic order for any nonzero U even in the thermodynamic limit. We are aware of no previous explicit analytic example of an antiferromagnetic ground state in a Hubbard-like model of itinerant electrons. The kinetic term induces nontrivial correlations among the particles, and an antiparallel spin configuration in the two sublattices becomes energetically favored at zero temperature. On the other hand, if the thermodynamic limit is taken and then zero temperature is approached, a paramagnetic behavior results. The thermodynamic limit does not commute with the zero-temperature limit, and this fact can be made explicit by the analytic solutions.
In the first part of this paper we explore a configuration-interaction approach to the two-dimensional Hubbard model near half filling. To reproduce the result of degenerate second-order perturbation theory it is necessary to include Fock states having kinetic energy within 4{ital t} of the lowest kinetic-energy states. Using the ground state from the largest configuration-interaction calculation, we calculate the extended {ital s}- and {ital d}-wave superconducting correlation functions as a function of distance. There are indications of a nondecaying long-distance tail in the {ital d}-wave correlation function, that is, {ital d}-wave superconductivity in the ground state of the repulsive Hubbard model. In the second part of the paper, we reexamine our effective Hamiltonian for the repulsive Hubbard model near half filling. Simple considerations lead us to a more intuitively plausible form consistent with our previous results and spin-rotational invariance.
In the first part of this paper we explore a configuration-interaction approach to the two-dimensional Hubbard model near half filling. To reproduce the result of degenerate second-order perturbation theory it is necessary to include Fock states having kinetic energy within 4t of the lowest kinetic-energy states. Using the ground state from the largest configuration-interaction calculation, we calculate the extended s- and d-wave superconducting correlation functions as a function of distance. There are indications of a nondecaying long-distance tail in the d-wave correlation function, that is, d-wave superconductivity in the ground state of the repulsive Hubbard model. In the second part of the paper, we reexamine our effective Hamiltonian for the repulsive Hubbard model near half filling. Simple considerations lead us to a more intuitively plausible form consistent with our previous results and spin-rotational invariance.