Physical review. B, Condensed matter

Ground-state quantum numbers of the half-filled Hubbard model

Adriana Moreo, Elbio R. Dagotto

Published May 1, 1990• 16 citations

Abstract

Working on a square lattice with an even number of sites ${\mathit{N}}^{2}$ 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 ${\mathit{k}}_{\mathit{x}}$-${\mathit{k}}_{\mathit{y}}$=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.

DOI: 10.1103/physrevb.41.9488

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

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A Weak-Coupling Expansion for the Two-Dimensional Hubbard Model on Square Clusters

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The spin reversal symmetry of the ground state of the half-filled Hubbard model

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Using a method based on Lieb's theorems and SU(2) symmetry for the half-filled Hubbard model on a bipartite lattice with mod A mod = mod B mod =N (where mod A mod ( mod B mod ) is the number of sites in the A (B) sublattice), we show that the ground-state wave function has even (odd) symmetry with respect to the spin reversal operator when N is even (odd) for both U>0 and U<0 cases. These rigorous results hold in all dimensions without the necessity for a periodic boundary condition.

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Physical review. B, Condensed matter

Ground-state quantum numbers of the half-filled Hubbard model

Adriana Moreo, Elbio R. Dagotto

Published May 1, 1990• 16 citations

Publisher

Abstract

Working on a square lattice with an even number of sites

{\mathit{N}}^{2}

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

{\mathit{k}}_{\mathit{x}}

{\mathit{k}}_{\mathit{y}}

=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.

DOI: 10.1103/physrevb.41.9488

Related papers

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

Michele Cini, Gianluca Stefanucci

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 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.

arXiv:cond-mat/0009058Journal of Physics Condensed MatterJan 25, 20012 citations

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The ground state of the two-dimensional Hubbard model on a square lattice is studied in the large-U limit in a half-filled band with a dynamical hole. For a 4\ifmmode\times\else\texttimes\fi{}4 lattice with hopping t=1 and coupling J ${J}_{c}$ \ensuremath{\approxeq}0.075, the ground state has zero momentum (k) and spin 15/2 (ferromagnetic background). For J\ensuremath{\ge} ${J}_{c}$ the ground state is degenerate with nonzero k and spin 1/2, in agreement with recent variational calculations. The nonzero k of the ground state comes from a nontrivial phase in the overlap of the spin states before and after a hole move. For small lattices we show that the effective Hamiltonian of the hole is that of a particle moving in a magnetic field. We also find that many of the features of the weak-coupling region carry over continuously to the strong-coupling region.The ground state of the two-dimensional Hubbard model on a square lattice is studied in the large-U limit in a half-filled band with a dynamical hole. For a 4\ifmmode\times\else\texttimes\fi{}4 lattice with hopping t=1 and coupling J${J}_{c}$\ensuremath{\approxeq}0.075, the ground state has zero momentum (k) and spin 15/2 (ferromagnetic background). For J\ensuremath{\ge}${J}_{c}$ the ground state is degenerate with nonzero k and spin 1/2, in agreement with recent variational calculations. The nonzero k of the ground state comes from a nontrivial phase in the overlap of the spin states before and after a hole move. For small lattices we show that the effective Hamiltonian of the hole is that of a particle moving in a magnetic field. We also find that many of the features of the weak-coupling region carry over continuously to the strong-coupling region.

Physical review. B, Condensed matterOct 1, 198979 citations

A Weak-Coupling Expansion for the Two-Dimensional Hubbard Model on Square Clusters

Barry Friedman

I present a scheme for performing weak-coupling expansions for the Hubbard model on square clusters near half-filling. Using this scheme the 6 × 6 cluster with 2, 3 and 4 holes is considered. I have calculated the ground-state quantum numbers for the 3- and 4-hole cases. In the 3-hole case I find a nonzero momentum near the (±π/2, ±π/2) points, while for the four-hole case the momentum is (0, 0). The calculation of low-lying excited states indicates the opening of a spin gap as one dopes away from half-filling.I present a scheme for performing weak-coupling expansions for the Hubbard model on square clusters near half-filling. Using this scheme the 6 × 6 cluster with 2, 3 and 4 holes is considered. I have calculated the ground-state quantum numbers for the 3- and 4-hole cases. In the 3-hole case I find a nonzero momentum near the (±π/2, ±π/2) points, while for the four-hole case the momentum is (0, 0). The calculation of low-lying excited states indicates the opening of a spin gap as one dopes away from half-filling.

Europhysics Letters (EPL)Mar 1, 19918 citations

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Journal of Physics Condensed MatterFeb 14, 19940 citations

Exact ground state of the Hubbard model on a double-layer square lattice

Shi‐Jie Xiong

A group of analytical wave functions is confirmed to be the ground state of a Hubbard model on a double-layer square lattice in the case of half-filling and in the case with one or two doped holes for a range of parameters. The method is based on the variational principle, and no approximations are applied. In the case of half-filling, the ground state is a product of local pairs. In the case of one or two holes, the doped holes are freely moving in the above background. In both cases there exists a long-range order of the evenness and oddness with respect to the layer parity.A group of analytical wave functions is confirmed to be the ground state of a Hubbard model on a double-layer square lattice in the case of half-filling and in the case with one or two doped holes for a range of parameters. The method is based on the variational principle, and no approximations are applied. In the case of half-filling, the ground state is a product of local pairs. In the case of one or two holes, the doped holes are freely moving in the above background. In both cases there exists a long-range order of the evenness and oddness with respect to the layer parity.

Physical review. B, Condensed matterJan 1, 19940 citations

Superconductivity and antiferromagnetism in the two-dimensional Hubbard model: A variational study

David Eichenberger, D. Baeriswyl

A variational ground state of the repulsive Hubbard model on a square lattice is investigated numerically for an intermediate coupling strength $(U=8t)$ and for moderate sizes (from $6\ifmmode\times\else\texttimes\fi{}6$ to $10\ifmmode\times\else\texttimes\fi{}10$). Our ansatz is superior to other widely used variational wave functions. The results for order parameters and correlation functions provide insight into the antiferromagnetic state at half-filling as well as strong evidence for a superconducting phase away from half-filling.A variational ground state of the repulsive Hubbard model on a square lattice is investigated numerically for an intermediate coupling strength $(U=8t)$ and for moderate sizes (from $6\ifmmode\times\else\texttimes\fi{}6$ to $10\ifmmode\times\else\texttimes\fi{}10$). Our ansatz is superior to other widely used variational wave functions. The results for order parameters and correlation functions provide insight into the antiferromagnetic state at half-filling as well as strong evidence for a superconducting phase away from half-filling.

arXiv:0708.2795Physical Review BNov 30, 200771 citations

Working on a square lattice with an even number of sites

{\mathit{N}}^{2}

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

{\mathit{k}}_{\mathit{x}}

{\mathit{k}}_{\mathit{y}}

=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.

{J}_{c}

\ensuremath{\approxeq}0.075, the ground state has zero momentum (k) and spin 15/2 (ferromagnetic background). For J\ensuremath{\ge}

{J}_{c}

the ground state is degenerate with nonzero k and spin 1/2, in agreement with recent variational calculations. The nonzero k of the ground state comes from a nontrivial phase in the overlap of the spin states before and after a hole move. For small lattices we show that the effective Hamiltonian of the hole is that of a particle moving in a magnetic field. We also find that many of the features of the weak-coupling region carry over continuously to the strong-coupling region.

A variational ground state of the repulsive Hubbard model on a square lattice is investigated numerically for an intermediate coupling strength

(U=8t)

and for moderate sizes (from $6\ifmmode\times\else\texttimes\fi{}6$ to $10\ifmmode\times\else\texttimes\fi{}10$). Our ansatz is superior to other widely used variational wave functions. The results for order parameters and correlation functions provide insight into the antiferromagnetic state at half-filling as well as strong evidence for a superconducting phase away from half-filling.