Structure of Essential Spectra and Discrete Spectrum of the Energy Operator of Six-Electron Systems in the Hubbard Model. First Quintet and First Singlet States ()
1. Introduction
The Hubbard model first appeared in 1963 in the works [1] [2] [3]. The Hubbard model is a simple model of metal was proposed that has become a fundamental model in the theory of strongly correlated electron systems. In that model, a single nondegenerate electron band with a local Coulomb interaction is considered. The model Hamiltonian contains only two parameters: the parameter B of electron hopping from a lattice site to a neighboring site and the parameter U of the on-site Coulomb repulsion of two electrons. In the secondary quantization representation, the Hamiltonian can be written as
(1)
where B is the transfer integral between neighboring sites,
, here
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,
is the spin index,
or
,
and
denote the spin values
and
, and
and
denote Fermi operators of creation and annihilation of an electron with spin
on a site
, here
-dimensional integer valued lattice,
-lattice dimensionality.
The model proposed in [1] [2] [3] 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 [4]. We also recall that the Hubbard model is a particular case of the Shubin-Wonsowsky polaron model [5], which had appeared 30 years before [1] [2] [3]. 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 simplicity and sufficiency of Hamiltonian (1) have made the Hubbard model very popular and effective for describing strongly correlated electron systems.
The Hubbard model well describes the behavior of particles in a periodic potential at sufficiently low temperatures such that all particles are in the lower Bloch band and long-range interactions can be neglected. If the interaction between particles on different sites is taken into account, then the model is often called the extended Hubbard model. In considering electrons in solids, the Hubbard model can be considered a sophisticated version of the model of strongly bound electrons, involving only the electron hopping term in the Hamiltonian. In the case of strong interactions, these two models can give essentially different results. The Hubbard model exactly predicts the existence of so-called Mott insulators, where conductance is absent due to strong repulsion between particles. The Hubbard model is based on the approximation of strongly coupled electrons. In the strong-coupling approximation, electrons initially occupy orbital’s in atoms (lattice sites) and then hop over to other atoms, thus conducting the current. Mathematically, this is represented by the so-called hopping integral. This process can be considered the physical phenomenon underlying the occurrence of electron bands in crystal materials. But the interaction between electrons is not considered in more general band theories. In addition to the hopping integral, which explains the conductance of the material, the Hubbard model contains the so-called on-site repulsion, corresponding to the Coulomb repulsion between electrons. This leads to a competition between the hopping integral, which depends on the mutual position of lattice sites, and the on-site repulsion, which is independent of the atom positions. As a result, the Hubbard model explains the metal-insulator transition in oxides of some transition metals. When such a material is heated, the distance between nearest-neighbor sites increases, the hopping integral decreases, and on-site repulsion becomes dominant.
The Hubbard model is currently one of the most extensively studied multielectron models of metals [6] [7] [8]. Therefore, obtaining exact results for the spectrum and wave functions of the crystal described by the Hubbard model is of great interest. The spectrum and wave functions of the system of two electrons in a crystal described by the Hubbard Hamiltonian were studied in [6]. It is known that two-electron systems can be in two states, triplet and singlet [6] [7] [8]. The work [6] is considered the Hamiltonian of the form
(2)
Here A is the electron energy at a lattice site, B is the transfer integral between neighboring sites,
, where
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,
is the spin index,
or
,
and
denote the spin values
and
, and
and
are the respective electron creation and annihilation operators at a site
. It was proved in [6] that the spectrum of the system Hamiltonian
in the triplet state is purely continuous and coincides with a segment
, where
is the lattice dimensionality, and the operator
of the system in the singlet state, in addition to the continuous spectrum
, has a unique antibound state for some values of the quasimomentum. For the antibound state, correlated motion of the electrons is realized under which the contribution of binary states is large. Because the system is closed, the energy must remain constant and large. This prevents the electrons from being separated by long distances. Next, an essential point is that bound states (sometimes called scattering-type states) do not form below the continuous spectrum. This can be easily understood because the interaction is repulsive. We note that a converse situation is realized for
: below the continuous spectrum, there is a bound state (antibound states are absent) because the electrons are then attracted to one another.
For the first band, the spectrum is independent of the parameter U of the on-site Coulomb interaction of two electrons and corresponds to the energy of two noninteracting electrons, being exactly equal to the triplet band. The second band is determined by Coulomb interaction to a much greater degree: both the amplitudes and the energy of two electrons depend on U, and the band itself disappears as
and increases without bound as
. 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 [9]. In the three-electron systems are exists quartet state, and two type doublet states. The quartet state corresponds to the free motion of three electrons over the lattice with the basic functions
. In the work [9] is proved that the essential spectrum of the system in a quartet state consists of a single segment and the three-electron bound state or the three-electron antibound state is absent. The doublet state corresponds to the basic functions
, and
. If
and
, then the essential spectrum of the system of first doublet state operator
is exactly the union of three segments and the discrete spectrum of
consists of a single point, i.e., in the system exists unique antibound state. In the two-dimensional case, we have the analogous results. In the three-dimensional case, or the essential spectrum of the system in the first doublet state operator
is the union of three segments and the discrete spectrum of operator
consists of a single point, i.e., in the system exists only one antibound state, or the essential spectrum of the system in the first doublet state operator
is the union of two segments and the discrete spectrum of the operator
is empty, or the essential spectrum of the system in the first doublet state operator
is consists of a single segment, and discrete spectrum is empty, i.e., in the system the antibound state is absent. In the one-dimensional case, the essential spectrum of the operator
of second doublet state is the union of three segments, and the discrete spectrum of operator
consists of no more than one point. In the two-dimensional case, we have analogous results. In the three-dimensional case, or the essential spectrum of the system in the second doublet state operator
is the union of three segments and the discrete spectrum of operator
consists of no more than one point, i.e., in the system exists no more than one antibound state, or the essential spectrum of the system in the second doublet state operator
is the union of two segments and the discrete spectrum of the operator
is empty, or the essential spectrum of the system in the second doublet state operator
is consists of a single segment, and discrete spectrum is empty, i.e., in the system the antibound state is absent.
The spectrum of the energy operator of system of four electrons in a crystal described by the Hubbard Hamiltonian in the triplet state was studied in [10]. In the four-electron systems are exists quintet state, and three type triplet states, and two type singlet states. The triplet state corresponds to the basic functions
,
,
.
If
and
, then the essential spectrum of the system first triplet state operator
is exactly the union of two segments and the discrete spectrum of operator
is empty. In the two-dimensional case, we have the analogous results. In the three-dimensional case, the essential spectrum of the system first triplet-state operator
is the union of two segments and the discrete spectrum of operator
is empty, or the essential spectrum of the system first triplet-state operator
is single segment and the discrete spectrum of operator
is empty. If
and
, then the essential spectrum of the system second triplet state operator
is exactly the union of three segments and the discrete spectrum of operator
is consists no more than one point. In the two-dimensional case, we have the analogous results. In the three-dimensional case, the essential spectrum of the system second triplet-state operator
is the union of three segments and the discrete spectrum of the operator
is consists no more than one point, or the essential spectrum of the system second triplet-state operator
is the union of two segments and the discrete spectrum of the system second triplet state operator
is empty, or the essential spectrum of the system second triplet-state operator
is consists of single segment and the discrete spectrum of the operator
is empty.
If
and
, the essential spectrum of the system third triplet-state operator
is exactly the union of three segments and the discrete spectrum of the operator
is consists no more than one point. In two-dimensional case, we have analogous results. In the three-dimensional case, the essential spectrum of the system third triplet-state operator
is the union of three segments, and the discrete spectrum of the operator
is consists no more than one point or the essential spectrum of the system third triplet-state operator
is the union of two segments, and the discrete spectrum of the operator
is empty, or the essential spectrum of the system third triplet-state operator
is consists of single segment, and the discrete spectrum of the operator
is empty. We see that there are three triplet states, and they have different origins.
The spectrum of the energy operator of four-electron systems in the Hubbard model in the quintet, and singlet states were studied in [11]. The quintet state corresponds to the free motion of four electrons over the lattice with the basic functions
. In the work [11] proved, that the spectrum of the system in a quintet state is purely continuous and coincides with the segment
, and the four-electron bound states or the four-electron antibound states is absent. The singlet state corresponds to the basic functions
,
, and these two singlet states have different origins.
If
and
, then the essential spectrum of the system of first singlet-state operator
is exactly the union of three segments and the discrete spectrum of the operator
is consists only one point. In the two-dimensional case, we have the analogous results. In the three-dimensional case, the essential spectrum of the system first singlet-state operator
is the union of three segments and the discrete spectrum of the operator
is consists only one point, or the essential spectrum of the system of first singlet-state operator
is the union of two segment and the discrete spectrum of the operator
is empty, or the essential spectrum of the system of first singlet-state operator
is consists of single segment and the discrete spectrum of operator
is empty. If
and
, then the essential spectrum of the system of second singlet-state operator
is exactly the union of three segments and the discrete spectrum of operator
is consists only one point. In two-dimensional case, we have the analogous results. In the three-dimensional case, the essential spectrum of the system second singlet-state operator
is the union of three segments and the discrete spectrum of the operator
is consists only one point, or the essential spectrum of the system of second singlet-state operator
is the union of two segment and the discrete spectrum of the operator
is empty, or the essential spectrum of the system of second singlet-state operator
is consists of single segment and the discrete spectrum of operator
is empty.
The structure of essential spectra and discrete spectrum of the energy operator of five-electron systems in the Hubbard model in the fifth doublet state were studied in [12].
In the five-electron systems exists the sextet state, and five type doublet states, and four type quartet states.
The structure of essential spectrum and discrete spectra of the energy operator of five-electron systems in the Hubbard model in the doublet state were investigated in [13] and [14].
The structure of essential spectra and discrete spectrum of the energy operator of five electron systems in the Hubbard model in a sextet and quartet states were studied in [15]. The spectrum of the energy operator of two-electron systems in the impurity Hubbard model in the triplet and singlet state was studied in the work [16].
The spectrum of the energy operator of three-electron systems in the impurity Hubbard model in the second doublet state was studied [17]. The structure of essential spectra and discrete spectrum of three-electron systems in the impurity Hubbard model in the Quartet state were studied in [18]. The structure of essential spectra and discrete spectrum of four-electron systems in the impurity Hubbard model in the first triplet state were studied in [19].
Consequently, in previous works, the spectrum of the energy operator of two, three, four and five electronic systems in the Hubbard model was studied, and the spectrum of the energy operator of two and three electronic systems and four-electron systems for first triplet state in the Impurity Hubbard model was investigated. Naturally, similar problems should be considered for the energy operator of six electronic systems in the Hubbard model.
2. Hamiltonian of the System
We consider the energy operator of six-electron systems in the Hubbard model and describe the structure of the essential spectra and discrete spectrum of the system for octet state, and first quintet and first singlet states in the lattice. The Hamiltonian of the chosen model has the form (2).
In the six electron systems has a octet state, and quintet states, and triplet states, and singlet states. The energy of the system depends on its total spin S. Along with the Hamiltonian, the
electron system is characterized by the total spin S,
,
,
.
Hamiltonian (2) commutes with all components of the total spin operator
, and the structure of eigenfunctions and eigenvalues of the system therefore depends on S. The Hamiltonian H acts in the antisymmetric Fock space
.
3. Six-Electron Octet State in the Hubbard Model
Let
be the vacuum vector in the space
. The octet state corresponds to the free motion of six electrons over the lattice with the basic functions
. The subspace
, corresponding to the octet state is the set of all vectors of the form
,
, where
is the subspace of antisymmetric functions in the space
. We denote by
the restriction of operator H to the subspace
.
Theorem 1. The subspace
is invariant under the operator H, and the restriction
of operator H to the subspace
is a bounded self-adjoint operator. It generates a bounded self-adjoint operator
acting in the space
as
(3)
The operator
acts on a vector
as
(4)
Proof. We act with the Hamiltonian H on vectors
using the standard anticommutation relations between electron creation and annihilation operators at lattice sites,
,
, and also take into account that
, where
is the zero element of
. This yields the statement of the theorem.
Lemma 1. The spectra of the operators
and
coincide.
Proof. Because the operators
and
are bounded self-adjoint operators, it follows that if
, then the Weyl criterion (see [20], chapter VII, paragraph 3, pp. 262-263) implies that there is a sequence
such that
and
. We set
. Then
as
, where
. It follows that
. Consequently,
.
Conversely, let
. Then, by the Weyl criterion, there is a sequence
such that
and
. Setting
,
, we conclude that
and
as
. This means that
and hence
. These two relations imply
.
We call the operator
the six-electron octet state operator in the Hubbard model.
Let
be the Fourier transform, where
is the
-dimensional torus endowed with the normalized Lebesgue measure
, i.e.
.
We set
. In the quasimomentum representation, the operator
acts in the Hilbert space
, where
is the subspace of antisymmetric functions in
.
Theorem 2. The Fourier transform of operator
is an operator
acting in the space
be the formula
where
and
is the subspace of antisymmetric functions in
.
Proof. The proof is by direct calculation in which we use the Fourier transformation in formula (3).
The spectrum of operator
is a purely continuous and coincide with the interval
.
4. Structure of the Essential Spectrum and Discrete Spectrum of Operator First Quintet State
The first quintet state corresponding to the free motion of six electrons over lattice and their interactions with the basic functions
(5)
The subspace
, corresponding to the first quintet state is the set of all vectors of the form
(6)
where
is the subspace of antisymmetric functions in the space
. We denote by
the restriction of operator H to the subspace
.
Theorem 3. The subspace
is invariant under the operator H, and the restriction
of operator H to the subspace
is a bounded self-adjoint operator. It generates a bounded self-adjoint operator
acting in the space
as
(7)
The operator
acts on a vector
as
(8)
Lemma 2. The spectra of the operators
and
coincide.
The proof of the lemma 2 similarly to proof the lemma 1.
We set
. In the quasimomentum representation, the operator
acts in the Hilbert space
, where
is the subspace of antisymmetric functions in
.
Theorem 4. The Fourier transform of operator
is an operator
, acting in the space
be the formula
(9)
where
, and
is the subspace of antisymmetric functions in
.
Proof. The proof is by direct calculation in which we use the Fourier transformation in formula (7).
Using tensor products of Hilbert spaces and tensor products of operators in Hilbert spaces [21], and taking into account that the function
is an antisymmetric function, we can verify that the operator
can be represented in the form
(10)
where
(11)
here
,
(12)
here
,
(13)
here
, and I is the unit operator in space
.
Consequently, we must investigate the spectrum of the operators
,
, and
.
Let the total quasimomentum of the two-electron system
be fixed. We let
denote the space of functions that are square integrable on the manifold
. It is known [22] that the operator
and the space
can be decomposed into a direct integral
,
of operators
and spaces
, such that the spaces
are invariant under the operators
and each operator
acts in
according to the formula
, where
. Therefore, the function
is an antisymmetric function of parameters
, and all parameters
a changed in the
-dimensional torus
, because all integral’s
are equal.
First, we investigate the spectrum of the operator
.
It is known that the continuous spectrum of
is independent of the parameter U and consists of the intervals
.
Definition 1. The eigenfunction
of the operator
corresponding to an eigenvalue
is called a bound state (BS) (antibound state (ABS)) of
with the quasi momentum
, and the quantity
is called the energy of this state.
We consider the operator
acting the space
according to the formula
(14)
It is a completely continuous operator in
for
.
We set
(15)
Lemma 3. A number
is an eigenvalue of the operator
if and only if it is a zero of the function
, i.e.,
.
Proof. Let the number
be an eigenvalue of the operator
, and
be the corresponding eigenfunction, i.e.,
(16)
Let
. Then
(17)
i.e., the number
is an eigenvalue of the operator
. It then follows that
.
Now let
be a zero of the function
, i.e.,
. It follows from the Fredholm theorem than the homogeneous equation
has a nontrivial solution. This means that the number
is an eigenvalue of the operator
.
We consider the one-dimensional case.
Theorem 5.
a) Let
and
, then for all values of parameters of the Hamiltonian, the operator
has a unique eigenvalue
, that is below the continuous spectrum of the operator
, i.e.,
.
b) Let
and
, then for all values of parameters of the Hamiltonian, the operator
has a unique eigenvalue
, that is above the continuous spectrum of the operator
, i.e.,
.
Proof. If
, then in the one-dimensional case, the function
decreases monotonically outside the continuous spectrum domain of the operator
, i.e., in the intervals
and
. For
the function
decreases from 1 to
,
as
,
as
. Therefore, below the value
, the function
has a single zero at the point
. For
, and
, the function
decreases from
to 1,
as
,
as
. Therefore, above the value
, the function
cannot vanish. If
, and
, the function
increases from 1 to
,
as
,
as
. Therefore, belove the value
, the function
cannot vanish. For
, and
, the function
increases from
to 1,
as
,
as
. Therefore, above the value
, the function
vanishes on a single point
.
In two-dimensional case, we have analogously results. We now consider the three-dimensional case. Let
, and
. We denote
, and
.
Theorem 6. Let
.
a) If
, and
, then the operator
has a unique eigenvalue
, that is below the continuous spectrum of
, i.e.,
.
b) If
, and
, then the operator
has no eigenvalue, that is below the continuous spectrum of
.
c) If
, and
, then the operator
has a unique eigenvalue
, that is above the continuous spectrum of
, i.e.,
.
d). If
, and
, then the operator
has no eigenvalue, that is above the continuous spectrum of
.
Now, we consider the operator
, and investigated the spectra this operator. We set
(18)
It is known the continuous spectrum of the operator
, is consists of the segment
.
Theorem 7. a) Let
and
, then for all values of parameters of the Hamiltonian, the operator
has a unique eigenvalue
, that is above the continuous spectrum of
, i.e.,
.
b) Let
and
, then for all values of parameters of the Hamiltonian, the operator
has a unique eigenvalue
, that is below the continuous spectrum of
, i.e.,
.
The proof of Theorem 7 are similarly the proof of Theorem 5.
We denote
, and
.
Theorem 8. Let
.
a) If
, and
, then the operator
has a unique eigenvalue
, that is above the continuous spectrum of
, i.e.,
.
b) If
, and
, then the operator
has no eigenvalue, that is above the continuous spectrum of
.
c) If
, and
, then the operator
has a unique eigenvalue
, that is below the continuous spectrum of
, i.e.,
.
d) If
, and
, then the operator
has no eigenvalue, that is below the continuous spectrum of
.
Now, we consider the operator
, and investigated the spectra this operator. We set
(19)
It is known the continuous spectrum of the operator
, is consists of the segment
.
Theorem 9. a) Let
and
, then for all values of parameters of the Hamiltonian, the operator
has a unique eigenvalue
, that is below the continuous spectrum of
, i.e.,
.
b) Let
and
, then for all values of parameters of the Hamiltonian, the operator
has a unique eigenvalue
, that is above the continuous spectrum of
, i.e.,
.
In the two-dimensional case, we have analogously results.
We consider the three-dimensional case.
We denote
, and
.
Theorem 10. Let
.
a) If
, and
, then the operator
has a unique eigenvalue
that is below the continuous spectrum of
, i.e.,
.
b) If
, and
, then the operator
has no eigenvalue, that is below the continuous spectrum of
.
c) If
, and
, then the operator
has a unique eigenvalue
that is above the continuous spectrum of
, i.e.,
.
d) If
, and
, then the operator
has no eigenvalue, that is above the continuous spectrum of
.
Now, using the obtained results and representation (10), we describe the structure of essential spectrum and discrete spectrum of the energy operator of six electron systems in the Hubbard model in the first quintet state.
Theorem 11. Let
. Then
A) If
and
, then the essential spectrum of the operator
is consists of the union of seven segments:
and the discrete spectrum of the operator
is consists of no more one eigenvalue:
, or
. Here and hereafter
,
,
,
,
,
, and
,
,
.
B) If
, and
, then the essential spectrum of the operator
is consists of the union of seven segments:
and the discrete spectrum of the operator
is consists of no more one eigenvalue:
, or
. Here
,
,
.
Proof. A) It follows from representation (10), and from Theorems 5 and 7 and 9, that in one-dimensional case, if
, then the continuous spectrum of the operator
is consists from
and discrete spectrum of the operator
is consists of unique eigenvalue
, the continuous spectrum of the operator
is consists from
, and discrete spectrum of the operator
is consists of unique eigenvalue
, and the continuous spectrum of the operator
is consists from
, and discrete spectrum of the operator
is consists of unique eigenvalue
. Therefore, the essential spectrum of the operator
is consists from of the union of seven segments:
and discrete spectrum of the operator
is consists is the no more then one eigenvalues:
, or
. These is given to the proof of statement A) from Theorem 11.
The statements B) from Theorem 11. are proved similarly.
In the two-dimensional case, we have the analogously results.
Now, we consider the three-dimensional case.
Theorem 12. Let
. Then
A) If
and
, and
, and
, and
, or
, and
, and
, or
, and
, and
, then the essential spectrum of the operator
is consists of the union of seven segments:
and the discrete spectrum of the operator
is consists of no more one eigenvalue:
, or
. Here and hereafter
,
,
,
,
,
, and
,
, and
are the eigenvalues, correspondingly, the operator’s
,
and
.
B) If
, and
,
, and
, or
, and
, or
, and
, then the essential spectrum of the operator
is consists of the union of four segments:
, or
, or
, and the discrete spectrum of the operator
is empty set:
.
C) If
, and
, and
, and
, and
, or
, and
, and
, and or
, and
, and
, then the essential spectrum of the operator
is consists of the union of two segments:
or
or
and the discrete spectrum of the operator
is empty:
.
D) If
, and
, and
, and
, and
, or
, and
,
, and
, or
, and
,
, then the essential spectrum of the operator
is consists of single segments:
, and the discrete spectrum of the operator
is empty set:
.
Theorem 13. Let
, then
A) If
and
, and
, and
, and
, or
, and
, or
, and
, and
, and
, or
, and
, or
,
, and
, and
, or
, and
, then the essential spectrum of the operator
is consists of the union of seven segments:
and the discrete spectrum of the operator
is consists of no more one eigenvalue:
, or
. Here and hereafter
,
,
,
,
,
, and
,
, and
are the eigenvalues, correspondingly, the operator’s
,
and
.
B). If
, and
,
, and
, and
, or
,
, and
, and
or
, and
, and
, and
, or
, and
, and
, and
, or
, and
, and
, and
, or
,
, and
, and
, then the essential spectrum of the operator
is consists of the union of four segments:
or
or
and the discrete spectrum of the operator
is empty set:
.
C). If
, and
, and
,
, and
or
, and
, and
, and
, or
, and
, and
, and
, or
and
, and
and
, or
and
, and
and
or
and
, and
and
, then the essential spectrum of the operator
is consists of the union of two segments:
or
or
and the discrete spectrum of the operator
is empty:
.
D) If
, and
, and
, and
, and
, or
, or
, and
, and
, and
, or
, and
, or
and
, and
, and
, or
and
, then the essential spectrum of the operator
is consists of single segment:
, and the discrete spectrum of the operator
is empty set:
.
We now consider the three-dimensional case, while as
. Then the continuous spectrum of the operator
is consists of the segment:
. We consider the Watson integral [23]
.
Theorem 14. Let
and
. Then
A) If
, and
, then the operator
has a unique eigenvalue
the below of the continuous spectrum of the operator
. If
, and
. then the operator
has no eigenvalue of the below the continuous spectrum of the operator
.
B) If
, and
, then the operator
has a unique eigenvalue
the above of the continuous spectrum of the operator
. If
, and
, then the operator
has no eigenvalue of the above the continuous spectrum of the operator
.
We now consider the three-dimensional case, while as
. Then the continuous spectrum of the operator
is consists of the segment:
.
Theorem 15. Let
and
. Then
A) If
, and
, then the operator
has a unique eigenvalue
the above of the continuous spectrum of the operator
. If
, and
, then the operator
has no eigenvalue of the below the continuous spectrum of the operator
.
B) If
, and
, then the operator
has a unique eigenvalue
the below of the continuous spectrum of the operator
. If
, and
, then the operator
has no eigenvalue of the below the continuous spectrum of the operator
.
We now consider the three-dimensional case, while as
. Then the continuous spectrum of the operator
is consists of the segment:
.
Theorem 16. Let
and
. Then
A) If
, and
, then the operator
has a unique eigenvalue
the below of the continuous spectrum of the operator
. If
, and
, then the operator
has no eigenvalue of the below the continuous spectrum of the operator
.
B) If
, and
, then the operator
has a unique eigenvalue
the above of the continuous spectrum of the operator
. If
, and
, then the operator
has no eigenvalue of the above the continuous spectrum of the operator
.
Theorem 17. Let
and
,
, and
. Then
A) If
, and
, and
, and
, or
, or
,
,
, and
, or
, or
,
,
,
, or
, then the essential spectrum of the operator
is consists of the union of seven intervals:
and discrete spectrum of the operator
is consists of no more then one eigenvalue:
, or
. Here and hereafter
,
,
,
,
,
, and
,
, and
are the eigenvalues of the operators
, correspondingly.
B) If
,
,
, and
, or
,
,
, and
, or
,
,
, and
, or
,
,
, and
, or
,
,
, and
, or
,
,
, and
, then the essential spectrum of the operator
is consists of the union of four intervals:
, or
, or
, and discrete spectrum of the operator
is empty set:
.
C) If
,
,
, and
,
,
,
, and
, or
,
,
, and
, or
,
,
, and
, or
,
, and
, and
, or
,
, and
, then the essential spectrum of the operator
is consists of the union of two intervals:
, or
, or
, and discrete spectrum of the operator
is empty set:
.
D) If
,
,
, and
, or
, or
,
, and
, and
, or
, or
,
,
, and
, or
, then the essential spectrum of the operator
is consists of unique interval:
, and discrete spectrum of the operator
is empty set:
.
Theorem 18. Let
and
,
, and
. Then
A) If
,
,
, and
, or
, or
,
,
, and
, or
, or
, and
,
, and
, or
, then the essential spectrum of the operator
is consists of the union of seven intervals:
and discrete spectrum of the operator
is consists of no more then one eigenvalue:
, or
. Here and hereafter
,
,
,
,
,
, and
,
, and
are the eigenvalues of the operators
, correspondingly.
B) If
, and
,
and
, or
,
,
and
, or
,
,
and
, or
,
,
, and
, or
,
,
, and
, or
,
,
, and
, then the essential spectrum of the operator
is consists of the union of four intervals:
, or
, or
, and discrete spectrum of the operator
is empty set:
.
C) If
, and
,
and
, or
,
,
and
, or
,
,
and
, or
,
,
, and
, or
,
,
, and
, or
,
,
, and
, then the essential spectrum of the operator
is consists of the union of two intervals:
, or
, or
, and discrete spectrum of the operator
is empty set:
.
D) If
, and
,
and
, or
,
,
and
, or
,
,
and
, or
,
,
, and
, or
,
,
, and
, or
,
,
, and
, then the essential spectrum of the operator
is consists of the unique interval:
, and discrete spectrum of the operator
is empty set:
.
5. Six-Electron First Singlet State in the Hubbard Model
Let
be the vacuum vector in the space
. The first singlet state corresponds to the free motion of six electrons over the lattice and their interactions with the basic functions
. The subspace
, corresponding to the first singlet state is the set of all vectors of the form
,
where
is the subspace of antisymmetric functions in the space
.
Theorem 19. The subspace
is invariant under the operator H, and the restriction
of operator H to the subspace
is a bounded self-adjoint operator. It generates a bounded self-adjoint operator
acting in the space
as
(20)
The operator
acts on a vector
as
Lemma 4. The spectra of the operators
and
coincide.
We call the operator
the six-electron first singlet state operator.
Let
be the Fourier transform, where
is the
-dimensional torus endowed with the normalized Lebesgue measure
, i.e.
.
We set
. In the quasimomentum representation, the operator
acts in the Hilbert space
, where
is the subspace of antisymmetric functions in
.
Theorem 20. The Fourier transform of operator
is an bounded self-adjoint operator
acting in the space
be the formula
(21)
where
.
The proof Theorem 20, is straightforward of (20) using the Fourier transformation.
Taking into account that the function
is antisymmetric, and using tensor products of Hilbert spaces and tensor products of operators in Hilbert spaces, we can verify that the operator
can be represented in the form
(22)
where
and I is the unit operator in the two-electron space
.
Consequently, we must investigate the spectrum of the operators
,
, and
.
Let the total quasimomentum of the two-electron system
be fixed. We let
denote the space of functions that are square integrable on the manifold
. It is known that the operator
and the space
can be decomposed into a direct integral
,
of operators
and spaces
, such that the spaces
are invariant under the operators
and each operator
acts in
according to the formula
, where
.
It is known that the continuous spectrum of operator
is independent of the parameter U and consists of the intervals
.
We set
.
Lemma 5. A number
is an eigenvalue of the operator
if and only if it is a zero of the function
, i.e.,
.
The proof the Lemma 5 is analogously to the proof of Lemma 3 in the these work.
We consider the one-dimensional case.
Theorem 21. a) If
and
, then for all values of parameters of the Hamiltonian, the operator
has a unique eigenvalue
, that is below the continuous spectrum of
, i.e.,
.
b) If
and
, then for all values of parameters of the Hamiltonian, the operator
has a unique eigenvalue
, that is above the continuous spectrum of
, i.e.,
.
The proof of Theorem 21 are similarly the proof of Theorem 5.
In the two-dimensional case, we have analogous results.
We consider the Watson integral. Because the measure
is normalized,
Let
, and
.
It is known that the continuous spectrum of operator
is independent of U and coincides with the segment
.
Theorem 22. a) Let
and
, and the total quasimomentum
of the system have the form
. Then the operator
has a unique eigenvalue
, if
, that is below the continuous spectrum of operator
. Otherwise, the operator
has no eigenvalue, that is below the continuous spectrum of operator
.
b) Let
and
, and the total quasimomentum
of the system have the form
. Then the operator
has a unique eigenvalue
, if
, that is above the continuous spectrum of operator
. Otherwise, the operator
has no eigenvalue, that is above the continuous spectrum of operator
.
We let
. We now investigated the spectrum of the operator
, i.e., the operator
It is known that the continuous spectrum of the operator
is independent of U and coincides with the segment
.
We set
.
Theorem 23. a) If
and
, then for all values of parameters of the Hamiltonian, the operator
has a unique eigenvalue
, that is above the continuous spectrum of
, i.e.,
.
b) If
and
, then for all values of parameters of the Hamiltonian, the operator
has a unique eigenvalue
, that is below the continuous spectrum of
, i.e.,
.
Now, we consider three-dimensional case. Let
and
.
It is known that the continuous spectrum of
is independent of U and coincides with the segment
.
Theorem 24. a). Let
and
, and the total quasimomentum
of the system have the form
. Then the operator
has a unique eigenvalue
, if
, that is above the continuous spectrum of operator
. Otherwise, the operator
has no eigenvalue, that is above the continuous spectrum of operator
.
b). Let
and
, and the total quasimomentum
of the system have the form
. Then the operator
has a unique eigenvalue
, if
, that is below the continuous spectrum of operator
. Otherwise, the operator
has no eigenvalue, that is above the continuous spectrum of operator
.
Let
. We now investigated the spectra of operator
, i.e., the operator
It is known that the continuous spectrum of the operator
is independent of U and coincides with the segment
.
Denote
.
Theorem 25. a). If
and
, and for all values of parameters of the Hamiltonian, the operator
has a unique eigenvalue
, that is below the continuous spectrum of operator
, i.e.,
.
b) If
and
, and for all values of parameters of the Hamiltonian, the operator
has a unique eigenvalue
, that is above the continuous spectrum of operator
, i.e.,
.
Let
and
.
It is known that the continuous spectrum of
is independent of
and coincides with the segment
.
Theorem 26. a). Let
and
, and the total quasimomentum
of the system have the form
. Then the operator
has a unique eigenvalue
, if
, that is below the continuous spectrum of operator
. Otherwise, the operator
has no eigenvalue, that is below the continuous spectrum of operator
.
b) Let
and
, and the total quasimomentum
of the system have the form
. Then the operator
has a unique eigenvalue
, if
, that is above the continuous spectrum of operator
. Otherwise, the operator
has no eigenvalue, that is above the continuous spectrum of operator
.
We now using the obtaining results and the representation (22), we can describe the structure of essential spectrum and discrete spectrum of the operator of first six-electron singlet state operator
:
Theorem 27. If
and
, then the essential spectrum of the system first six-electron singlet state operator
is exactly the union of seven segments:
The discrete spectrum of operator
consists of no more than one point:
, or
.
Here and hereafter
,
,
,
,
,
,
,
,
.
The proof of Theorem 27 are similarly the proof of Theorem 11.
Theorem 28. If
and
, then the essential spectrum of the system first six-electron singlet state operator
is exactly the union of seven segments:
The discrete spectrum of operator
is consists of no more than one point:
, or
.
Here
,
,
.
Let
, and
, and
, and
.
Now, using the obtained results and representation (22), we describe the structure of the essential spectrum and the discrete spectrum of the system first six-electron singlet state operator
:
Let
and
, and
, and
.
Theorem 29. The following statements hold:
a) Let
, and
,
, and
, or
, or
, and
,
, and
, or
, or
, and
, and
, and
, or
. Then the essential spectrum of the system first six-electron singlet state operator
is consists of the union of seven segments:
The discrete spectrum of the operator
is consists of no more one point:
, or
.
Here and hereafter
,
,
,
,
,
, and
is an eigenvalue of the operator
, and
is an eigenvalue of the operator
, and
is an eigenvalue of the operator
.
b) Let
, and
,
,
, or
, and
, and
, and
, or
, and
, and
, and
, or
, and
, and
, and
, or
, and