Spectra of the energy operator of four-electron systems in the triplete state in the Hubbard model

We investigate spectral properties of a four-electron system in the Hubbard Model framework in the v— dimensional lattice Zv. We prove that the essential spectrum of the system in a quintet state consists of a single segment and the four-electron bound state or four-electron anti-bound state is absent. We show that the essential spectrum of the system in a triplete states is the union of at most three segments. We also prove that four-electron bound states or a four-electron anti-bound state exists in triplete states. We prove that in the system exists three triplete states and their spectrum are different.


Introduction
In the early 1970s, three papers [2,3,8], where a simple model of a metal was proposed that has bocome a fundamental model in the theory of strongly correlated electron systems. Note that these papers appeared almost simultaneously and independently. In that model, a single nondegenerate electron band with a local Coulomb interaction was considered. The model Hamiltonian contains only two parameters: the matrix element t of electron hopping from a lattice site to a neighboring site and the parameter U of the one-site Coulomb repulsion of two-electrons. In the secondary quantization representation, the Hamiltonian can be written as follows where a + m,γ and a m,γ denote Fermi operators of creation and annihilation of an electron with spin γ on a site m and the summation over τ means summation over the nearest neighbors on the lattice.
The model proposed in [3,2,8] was called the Hubbard model after John Hubbard, who made a fundamental contribution to studying the statistical mechanics of that system, although the local form of Coulomb interaction was first introduced for an impurity model in a metal by Anderson [1]. We also recall that the Hubbard model is a particular case of the Shubin-Wonsowsky polaron model [15], which had appeared 30 years before [3,2,8]. In the Shubin-Wonsowsky model, along with the on-site Coulomb interaction, the interaction of electrons on neighboring sites is also taken into account.
The Hubbard model is an approximation used in solid state physics to describe the transition between conducting and insulating states. It is the simplest model describing particle interaction on a lattice. Its Hamiltonian contains only two terms: the kinetic term corresponding to the the band itself disappears as U → 0 and increases without bound as U → ∞. The second band largely corresponds to a one-particle state, namely, the motion of the doublet, i.e., two-electron bound states.
The spectrum and wave functions of the system of three electrons in a crystal described by the Hubbard Hamiltonian were studied in [16].
Here, we consider the energy operator of four-electron systems in the Hubbard model and describe the structure of the essential and discrete spectra of the system for quintet and triplet states. The Here, A is the electron energy at a lattice site, B is the transfer integral between neighboring sites (we assume that B > 0 for convenience), τ = ±e j , j = 1, 2, ..., ν, where e j are unit mutually orthogonal vectors, which means that summation is taken over the nearest neighbors, U is the parameter of the on-site Coulomb interaction of two electrons, γ is the spin index, γ =↑ or γ =↓, ↑ and ↓ denote the spin values 1 2 and − 1 2 , and a + m,γ and a m,γ are the respective electron creation and annihilation operators at a site m ∈ Z ν .
The energy of the system depends on its total spin S. In the case of a saturated ferromagnetic state (S = Ne 2 , where N e is the number of electrons in the system), the solution of the problem is exact and trivial for any admissible number of electrons N e . In that case, the system is an ideal Fermi gas of electrons with the same direction of the spin projections.
Along with the Hamiltonian, the N e electron system is characterized by the total spin S, Hamiltonian (2) commutes with all components of the total spin operator S = (S + , S − , S z ), and the structure of eigenfunctions and eigenvalues of the system therefore depends on S. The Hamiltonian H acts in the antisymmetric Fock space H as .

Quintet state
Let φ 0 be the vacuum vector in the space H as . The quintet state corresponds to the free motion of four electrons over the lattice with the basis functions The subspace H q 2 , corresponding to the quintet state is the set of all vectors of the form where l as 2 is the subspace of antisymmetric functions in the space l 2 ((Z ν ) 4 ).
Proof. We act with the Hamiltonian H on vectors ψ ∈ H q 2 using the standard anticommutation relations between electron creation and annihilation operators at lattice sites, {a m,γ , a + n,β } = δ m,n δ γ,β , {a m,γ , a n,β } = {a + m,γ , a + n,β } = θ, and also take into account that a m,γ φ 0 = θ, where θ is the zero element of H q 2 . This yields the statement of the theorem.
Lemma 2.2. The spectra of the operators H q and H q 2 coincide. The proof follows by using the Weyl criterion [13]. We call H q as the four-electron quintet operator.
We let F denote the Fourier transform: In the quasi-momentum representation, the operator H q acts in the where L as 2 is the subspace of antisymmetric functions in L 2 ((T ν ) 4 ) and h(x; y; z; t) = 4A + 2B (cos x i + cos y i + cos z i + cos t i ).
It is obvious that the spectrum of H q is purely continuous and coincides with the value set of the function h(x; y; z; t), i.e. with the set Therefore, the quintet state spectrum is independent of the Coulomb interaction parameter U and is the set of energies of four noninteracting electrons moving in the crystal. This result is totally natural because the quintet state cannot contain states with two electrons at a site. Hence, in the quintet state, the spectrum of four-electron systems can be evaluated exactly and is purely continuous. The spectral problem that we consider here is a particular case of the problem of finding the spectrum of a system of N noninteracting electrons in a crystal lattice.
By Hunds rule, the minimum-energy state in an N − electron system is the state where all spins are directed upward, i.e., the state ↑↑ ... ↑ . By the Pauli exclusion principle, this state cannot contain states with two electrons at one site. In this case, the spectrum of the system is independent of the Coulomb interaction parameter U and is the band energy of N noninteracting electrons moving in the crystal. The spectrum of the system is then purely continuous.

First triplet state
In the system there exist three triplet states. The triplet state corresponds to the basis functions We see that there are three such states, and they have different origins. The subspace k H t 1 , k = 1, 2, 3 corresponding to the triplet states in the set of all vectors of the forms ψ = ∑ m,n,p,r f (m, n, p, r) k t 1 m,n,p,r , k = 1, 2, 3, f ∈ l as 2 , where l as 2 is the subspace of antisymmetric functions in the space l 2 ((Z ν ) 4 ).
The restriction k H t 1 = H/k H t 1 , k = 1, 2, 3 of H to the subspace k H t 1 is a bounded self-adjoint operator. where δ k,j is the Kronecker symbol. The operator 1 H t 1 acts on a vector ψ ∈ 1 H t 1 as Proof. The proof of the theorem can be obtained from the explicit form of the action of H on vectors Ψ ∈ 1 H t 1 using the standard anti-commutation relations between electron creation and annihilation operators.
In the quasi-momentum representation, the operator 1 H t 1 acts in the Hilbert space L as 2 ((T ν ) 4 as where L as 2 is the subspace of antisymmetric functions in L 2 ((T ν ) 4 , and h(λ, µ, γ, θ) has a form (5).
Using that the function f (λ, µ, γ, θ) is the antisymmetrical function, we verify that the operator 1 H t 1 can be represented in the form where I is the unit operator in the two-electron space, ( and ( We must therefore investigate the spectrum and bound states of the operators H 1 and H 2 , respectively. Let the total quasimomentum of the two-electron system λ + θ = Λ 1 be fixed. We let L 2 (Γ Λ ) denote the space of functions that are square integrable on the manifold Let < H ′ , (.) > be a separable Hilbert space, and < M, µ > be a measurable space with σ− finite measure µ. The By H we denote a linear space of all weakly measurable mappings f : for all ψ ∈ H ′ . In this case A is called decomposable and will be written as The following result is a well-known theorem about the description of decomposable operators as direct integral.

µ > is separable measurable spaces whit σ− finite measure and H ′ is separable. Let A be an algebra of decomposable operators with layers, which multiple to the identity operator. In this case, A ∈ L(H ′ ) is decomposable if and only if A is commuted whit every operators from A.
Since It is known [12,14] that the operator H 1 and the space such that the spaces H 2Λ 1 are then invariant under the operators H 1Λ 1 and each operator H 1Λ 1 It is known that the continuous spectrum of H 1Λ 1 is independent of U and coincides with the segment [m Λ 1 , Then the homogeneous equation has a nontrivial solution. It then follows that the number z = z 0 is an eigenvalue of H 1 .
It is clear that the continuous spectrum of H 1 coincides with the segment We consider the one-dimensional case. Proof. In the one-dimensional case, the function ∆ 1 Λ 1 (z) increases monotonically outside the continuous spectrum domain of the singlet-state operator H 1 . For z < m 1 Λ 1 , the function ∆ 1 Λ 1 (z) increases from 1 to +∞, and one has We now consider the two-dimensional case. Proof. In the two-dimensional case, the function ∆ 2 Λ 1 (z) increases monotonically outside the continuous spectrum domain of H 1 . For z < m 2 Λ 1 , the function ∆ 2 Λ 1 (z) increases from 1 to +∞, In the three-dimensional case, the operator H 1 for some values of the parameters in the Hamiltonian has a unique eigenvalue z, such that z > M 3 Λ 1 . We consider the Watson integral [18] Because the measure ν is normalized,

W .
Otherwise, the operator H 1 has no eigenvalue.
Proof. In the three-dimensional case, Hence, the function ∆ 3 Λ 1 (z) cannot vanish below the continuous spectrum. Above the continuous spectrum, Inspecting the equation ∆ 3 Λ 1 (z) = 0 above the continuous spectrum domain of the operator H 1 , we obtain the statement of the theorem.
We note that the converse situation is realized for U < 0. We must therefore investigate the spectrum of the operator H 2 . ( ]. Now, using the obtained results and representation (9), we describe the structure of the essential spectrum and the discrete spectrum of the operator 1 H 1 t . The spectrum of the operator A ⊗ I + I ⊗ B, where A and B are densely defined bounded linear operators, was studied in [5], [6], [7]. Explicit formulas were given there that express the essential spectrum σ ess (A ⊗ I + I ⊗ B) and the discrete spectrum σ disc (A

It is clear that σ(A
Theorem 3.7. At ν = 1 and U > 0 the essential spectrum of the system first triplet-state operator 1 H 1 t is exactly the union of two segments: Proof. It follows from representation (9) that and in the one-dimensional case, the continuous spectrum of H 1 is , and the discrete spectrum of H 2 is empty. Therefore, the essential spectrum of the system first triplet-state operator 1 H 1 t is the union of two segments, and the first triplet-state operator 1 H 1 t has no eigenvalue.
The following theorem is proved totally similarly to Theorem 3.6.

W .
Then the essential spectrum of the system first triplet-state operator 1 H 1 t is the union of two . The discrete spectrum of 1 H 1 t is empty. Here and hereafter, z is an eigenvalue of H 1 .

Let
Then the essential spectrum of the system first triplet-state operator 1 H 1 t is the segment , and the discrete spectrum is empty.
Proof. The proof uses representation (9) and the results for the spectra of H 1 and H 2 in the case ν = 3.

Second triplet state
where δ k,j is the Kronecker symbol. The operator 2 H t 1 acts on a vector ψ ∈ 2 H t 1 as Proof. The proof of the theorem follows from the explicit form of the action of the Hamiltonian H on vectors ψ ∈ 2 H t 1 using standard anticommutation relations for electron creation and annihilation operators.
In the quasimomentum representation, the operator 2 H t 1 acts in the Hilbert space L as 2 ((T ν ) 4 as where L as 2 is the subspace of antisymmetric functions in L 2 ((T ν ) 4 , and h(λ, µ, γ, θ) has form (5). Taking into account that the function f (λ, µ, γ, θ) is antisymmetric, we can rewrite formula (12) We verify that the operator 2 H t 1 can be represented in the form where ( We must therefore investigate the spectrum and bound states of the operators H 3 , and H 4 , and H 5 . Let the total quasimomentum of the two-electron system λ + γ = Λ 3 be fixed. We let L 2 (Γ Λ 3 ) denote the space of functions that are square integrable on the manifold That the operator H 3 and the space H 2 ≡ L 2 ((T ν ) 2 ) can be decomposed into a direct integral of operators H 3Λ 3 and spaces H 2Λ 3 = L 2 (Γ Λ 3 ), such that the spaces H 2Λ 3 are then invariant under the operators H 3Λ 3 and each operator H 3Λ 3 It is known that the continuous spectrum of H 3Λ 3 is independent of U and coincides with the segment [m Λ 3 ,

Lemma 4.2. The number z = z 0 not belonging to the continuous spectrum of the operator H 3 is an eigenvalue of that operator if and only if it is a zero of the function
The proof of the statements in the lemma 4.2 is similar to the proof of lemma 3.2.
It is clear that the continuous spectrum of H 3 coincides with the segment We consider the one-dimensional case.  Proof. In the one-dimensional case, the function ∆ 1 Λ 3 (z) decreases monotonically outside the continuous spectrum domain of the operator H 3 . For z < m 1 Λ 3 , the function ∆ 1 Λ 3 (z) decreases from 1 to −∞, as z → +∞. Therefore, above the value M 1 Λ 3 , ∆ 1 Λ 3 (z) cannot vanish. We now consider the two-dimensional case. Proof. In the two-dimensional case, the function ∆ 2 Λ 3 (z) decreases monotonically outside the continuous spectrum domain of H 3 . For z < m 2 Λ 3 , the function ∆ 2 Λ 1 (z) decreases from 1 to −∞, In the three-dimensional case, the operator H 3 for some values of the parameters in the Hamiltonian has a unique eigenvalue z, such that z < m 3 Λ 3 . Theorem 4.5. At ν = 3 and U > 0 and the total quasimomentum Λ 3 of the system have the form

W .
Otherwise, the operator H 3 has no eigenvalue.
Proof. In the three-dimensional case, Hence, the function ∆ 3 Λ 3 (z) has a single zero below the continuous spectrum, if U > 12B cos Λ 0 3 2 W . Above the continuous spectrum, Inspecting the equation ∆ 3 Λ 3 (z) = 0 above the continuous spectrum domain of the operator H 3 , we obtain the statement of the theorem.  12 We note that the converse situation is realized for U < 0. The spectrum of operator ( H 4 f )(µ, γ) = U ∫ T ν f (s, µ+γ −s)ds is purely discrete and consists is a single point z = U, for arbitrary value ν.
Now we investigated the spectrum of the operator We let Λ 4 = µ + θ, and Λ 5 = γ + θ. It is known that the continuous spectrum of H 5 is independent of U and coincides with the We let L 2 (Γ Λ 4 ) denote the space of functions that are square integrable on the manifold That the operator H 5 and the space H 2 ≡ L 2 ((T ν ) 2 ) can be decomposed into a direct integral of operators H 5Λ 4 and spaces H 2Λ 4 = L 2 (Γ Λ 4 ), such that the spaces H 2Λ 4 are then invariant under the operators H 5Λ 4 and each operator H 5Λ 4

Lemma 4.6. The number z = z 0 not belonging to the continuous spectrum of the operator H 4 is an eigenvalue of that operator if and only if it is a zero of the function
Proof. The proof of the statements in the lemma 4.6 is similar to the proof of Lemma 3.2.
We consider the one-dimensional case. Proof. In the one-dimensional case, the function ∆ 1 Λ 5 (z) increases monotonically outside the continuous spectrum domain of the operator H 5 . For z < m 1 Λ 4 , the function ∆ 1 Λ 5 (z) increases from 1 to +∞, as z → +∞. Therefore, above the value M 1 Λ 4 , the function ∆ 1 Λ 5 (z) has a single zero at the point We now consider the two-dimensional case. Proof. In the two-dimensional case, the function ∆ 2 Λ 5 (z) increases monotonically outside the continuous spectrum domain of H 5 . For z < m 2 Λ 4 , the function ∆ 2 Λ 5 (z) increases from 1 to +∞, In the three-dimensional case, the operator H 5 for some values of the parameters in the Hamiltonian has a unique eigenvalue z, such that z > M 3 Λ 4 . Theorem 4.9. At ν = 3 and U > 0 and the total quasimomentum Λ 5 of the system have the form Otherwise, the operator H 5 has no eigenvalue.
Proof. In the three-dimensional case, We note that the converse situation is realized for U < 0. The spectrum of operator ( H 7 f )(µ, γ) = U ∫ T ν f (s, µ+γ −s)ds is purely discrete and consists is a single point z = U, for arbitrary value ν. Now we investigated the spectrum of the operator We let Λ 4 = µ + θ, and Λ 5 = γ + θ. It is known that the continuous spectrum of H 8 is independent of U and coincides with the segment [m ν We let L 2 (Γ Λ 5 ) denote the space of functions that are square integrable on the manifold That the operator H 8 and the space H 2 ≡ L 2 ((T ν ) 2 ) can be decomposed into a direct integral of operators H 8Λ 5 and spaces H 2Λ 5 = L 2 (Γ Λ 5 ), such that the spaces H 2Λ 5 are then invariant under the operators H 8Λ 5 and each operator H 8Λ 5 Lemma 5.6. The number z = z 0 not belonging to the continuous spectrum of the operator H 8 is an eigenvalue of that operator if and only if it is a zero of the function ∆ ν Λ 5 (z), i.e., ∆ ν Λ 5 (z 0 ) = 0.
Proof. The proof of the statements in the lemma 5.6 is similar to the proof of lemma 3.2.

W .
Otherwise, the operator H 8 has no eigenvalue.
Proof. In the three-dimensional case,