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Physical review. B, Condensed matter

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

Shi‐Jie Xiong

Published Jan 1, 1994• 0 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.49.589

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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.
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 JJc{J}_{c}Jc​\ensuremath{\approxeq}0.075, the ground state has zero momentum (k) and spin 15/2 (ferromagnetic background). For J\ensuremath{\ge}Jc{J}_{c}Jc​ 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.
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.
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.
The coupled-cluster method has recently proved to be efficient in calculating the ground state of strongly interacting spins or fermions on lattices. We apply it to the ground state of one hole in the half-filled Hubbard model on a two-dimensional square lattice. In a wide range for the interaction 10\ensuremath{\lesssim}U/t\ensuremath{\lesssim}80, the ground-state energy of a hole is fitted by a simple function \ensuremath{\epsilon}/t\ensuremath{\approxeq}-3.40+8.9(t/U)0.70{)}^{0.70})0.70, in good agreement with the results obtained through exact diagonalizations on small clusters. The effective mass of a hole m∗{\mathit{m}}^{\mathrm{*}}m∗=8t/W (W represents the bandwidth) is proportional to U/T (m∗{\mathit{m}}^{\mathrm{*}}m∗\ensuremath{\approxeq}0.845U/t) for 5\ensuremath{\lesssim}U/t\ensuremath{\lesssim}20 and saturates to a constant value m∗{\mathit{m}}^{\mathrm{*}}m∗\ensuremath{\approxeq}84 for large U/t.
A variational ground state of the repulsive Hubbard model on a square lattice is investigated numerically for an intermediate coupling strength (U=8t)(U=8t)(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.
The ground-state wave function and energy are calculated for the one-band Hubbard Hamiltonian on a 4\ifmmode\times\else\texttimes\fi{}4 cluster of the square lattice up to first and second order of the on-site Coulomb repulsion U, respectively. Changes induced by a deviation of the number of electrons from half-filling (one electron per lattice site) are closely followed. Charge-charge and spin-spin correlation functions are directly obtained from the ground-state wave function and show the existence of antiferromagnetic correlations that extend over the whole cluster at half-filling. Correlations among sites separated by more than a lattice site diminish very rapidly as holes are introduced into the cluster. The analysis is extended to the negative-U case. Ground-state properties change qualitatively.