† Corresponding author. E-mail:
roject supported by the National Natural Science Foundation of China (Grant No. 11674059) and Natural Science Foundation of Fujian Province, China (Grant Nos. 2016J01008 and 2016J01009).
We consider the construction of exact eigenstates of the two-dimensional Fermi–Hubbard model defined on an L × L lattice with a periodic condition. Based on the characteristics of Slater determinants, several methods are introduced to construct exact eigenstates of the model. The eigenstates constructed are independent of the on-site electron interaction and some of them can also represent exact eigenstates of the two-dimensional Bose–Hubbard model.
The two-dimensional Fermi–Hubbard model is important when attempting to understand strongly correlated electronic systems.[1–5] In the model, the electrons can hop from one site to its nearest neighbors and there will be an interaction between two electrons when they are on the same site. The competition between the electron hopping and the on-site electron interaction makes it hard to find exact eigenstates of the model.[6–8] In 1990, Yang and Zhang showed that many new exact eigenstates of the model could be constructed from the known ferromagnetic eigenstates based on the SO4 symmetry of the model.[9] There are also other methods to construct exact eigenstates of the model.[10] Finding out more exact eigenstates of this model not only is an intellectual challenge, but also can increase our understanding of the model especially when the SO4 symmetry is considered.[9,11] In this paper, we will present some new methods to construct more exact eigenstates of the model by connecting the problem with the Slater determinant. The eigenstates we constructed are independent of the on-site electron interaction, i.e., they are eigenstates for any value of the on-site electron interaction strength, which were ever believed to have been all figured out by Yang and Zhang and no new such eigenstates could be constructed.[9]
The two-dimensional Fermi–Hubbard model we considered is defined on a periodic two-dimensional L × L lattice with the Hamiltonian
When the on-site electron interaction strength U is zero, it is easy to find exact eigenstates of the model Hamiltonian H. By introducing the following annihilation operators in momentum space:
In the following, we will give some methods to construct exact eigenstates of the Hamiltonian H for nonzero U. To reach this goal, we consider the following quantum state with j spin-up electrons and (n−j) spin-down electrons:
The simplest case may be g(
An enormous number of eigenstates of H can be constructed through Eq. (
To give some new eigenstates of H, in this case we assume
(i) Assume
Since
(ii) Assume
In the above two situations, we have given two methods to construct the eigenstate of the Hamiltonian H by choosing proper vectors
We have considered the construction of exact eigenstates of the two-dimensional Fermi–Hubbard model. One may wonder whether our methods are also applicable for constructing exact eigenstates of the two-dimensional Bose–Hubbard model, where the operators obey the Bose commutation relations and many particles can be on the same site. Since our constructions prevent double occupation on the same site and do not use the Fermi commutation relations, the eigenstates we constructed for the Fermi–Hubbard model will also be eigenstates for the Bose–Hubbard model when the Bose commutation relations do not make the states zero. For example, the state |Ψ⟩ in Eq. (
The number of electrons in our constructed eigenstates cannot reach or be close to the Hall filling, i.e., L2 electrons in the L × L lattice, which is believed to have a rich physics.[12,13] However, Yang has shown that applying the η† pair operator on a known eigenstate can obtain a new eigenstate with two more electrons.[14] Therefore, we can apply the η† pair operator many times on our constructed eigenstates to obtain new eigenstates with the number of electrons reaching or being close to the Hall filling. The eigenstates obtained in this way are not the ground state, but their corresponding eigenvalues can give some upper bound for the ground state energy and the knowing of these excited states can reduce the size of the space to numerically search the ground state, as the ground state is orthogonal to all excited states.
In summary, we have given some methods to construct exact eigenstates of the two-dimensional Fermi–Hubbard model. The basic idea is to prevent two electrons from being on the same site through the Slater determinant, therefore the eigenstates constructed are independent of the on-site interaction strength U. The finding that there are many U-independent eigenstates itself is of interest, and the connection with the Slater determinant can give some inspiration for understanding the two-dimensional Fermi–Hubbard model.
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