Structure of Essential Spectrum and Discrete Spectrum of the Energy Operator of Three-Magnon Systems in the Isotropic Ferromagnetic Non-Heisenberg Model with Spin One and Nearest-Neighbor Interactions ()
1. Introduction
Two-magnon systems have attracted the attention of many researchers. Probably, such systems were first discussed by Bethe [1] in the context of one-dimensional integer-valued lattices. Bethe proved that no more than one bound state (BS) of the system can exist in the case of one-dimensional isotropic ferromagnet. N. Fukuda and M. Wortis [2] investigated the two-magnon systems in the one-dimensional Heisenberg ferromagnetic model and they have received the confirmation of Bethe results. Worts [3] examined the two-magnon system in a d-dimensional integer-valued lattice for an arbitrary d and proved that in this case, the system has
BSs.
Majumdar [4] investigated the two-magnon system in a one-dimensional Heisenberg ferromagnet with a coupling between nearest and second nearest neighbors for the full quasi-momentum
. He found the spectrum and the BSs of the system numerically. In [5] , such a system was examined for the case of a one-dimensional Heisenberg isotropic ferromagnet with a nearest- and second nearest-neighbor interactions for
and
. The spectrum and the BSs of the system for these values of
were studied with numerical methods. Gochev [6] considered the two-magnon system in a one-dimensional Heisenberg longitudinal ferromagnet with a coupling between nearest and second nearest neighbors for an arbitrary full quasi-momentum. He investigated the spectrum and the BSs of the system analytically.
The two-magnon systems in the anisotropic Heisenberg model with a nearest-neighbor interaction were addressed in [7] . The focus in [8] was on two-magnon systems in a one-dimensional anisotropic Heisenberg ferromagnet with a interaction between nearest and second nearest neighbors. The spectrum and the BSs of such systems were investigated for all values of the full quasi-momentum.
The usual starting point for theoretical studies of magnetically organized matter is the Heisenberg exchange Hamiltonian (with an arbitrary spin s)
(1)
where J is the bilinear exchange interaction parameter for nearest-neighbor atoms;
is the atomic spin operator of the mth node of the ν-dimensional integer-valued lattice
, and
denotes summation over the nearest neighbors. However, the actual isotropic spin exchange Hamiltonian with an arbitrary spin s has the form [9]
(2)
where
are the multipole exchange interaction parameters for nearest-neighbors atoms. Hamiltonian (2) coincides with Hamiltonian (1) only for
, while there are terms with higher powers of
up to
inclusive for
. These terms must be taken into account. Hamiltonian (2) is called the non-Heisenberg Hamiltonian.
Spectrum and BSs of two-magnon system in the non-Heisenberg ferromagnet with the bilinear and biquadratic exchange interactions were studied in works [10] - [17] . The spectrum and the BSs of two-magnon systems in a non-Heisenberg ferromagnet with coupling between nearest neighbors by bilinear and biquadratic interactions were investigated in [10] [11] [12] [13] [14] . Different methods, such as the Green’s function method, the molecular field approximation method, the random phase approximation method, numerical methods, and the use of the creation and annihilation operators through the Holstein-Primakoff transformation, Dyson transformation, Dyson-Maleev transformation, Golghirch transformation, and others, were applied in these works. In [15] [16] , the spectrum and the BSs of this system were investigated for the case of a one-dimensional non-Heisenberg ferromagnet with
and with a coupling between second nearest and third nearest neighbors respectively. The values of the Hamiltonian parameters for which the BSs exist were found, and the energies of these BSs were calculated. In [17] , the spectrum and the BSs of two-magnon system were investigated in a ν-dimensional non-Heisenberg ferromagnet with
and with a coupling between nearest neighbors.
In [18] , the spectrum and the three-magnon BSs of three-magnon systems were investigated in a two-dimensional isotropic and anisotropic Heisenberg ferromagnet in bounded lattice with numerical methods.
In the work [19] [20] investigated the structure of essential spectrum and obtained the lower and upper estimates for the number of three-particle bound states (BS) of the energy operator of two-magnon system in a isotropic Heisenberg and Non-Heisenberg ferromagnet model with impurity in a ν-dimensional lattice
with nearest-neighbor interactions.
The spectrum and BSs of two-magnon systems in a non-Heisenberg ferromagnet with coupling between nearest-neighbors by linear and biquadratic interactions were investigated in [17] .
2. Hamiltonian
In this paper, we investigate the structure of essential spectrum and we obtain the lower and upper estimates for the number of three-magnon bound states of the energy operator of three-magnon system in a isotropic Non-Heisenberg ferromagnet model with spin one and nearest-neighbor interactions in a ν-dimensional lattice
.
In this case the component
of spin operator
take up the values
, i.e.
, or
, or
, where
be the vacuum vector. We consider these case the separately. From the beginning. We consider the case, when the spin component
take up the value 1.
The system Hamiltonian has the form
(3)
acts in the symmetrical Fo’ck space
,
is the atomic spin
operator in the node m,
are the respective the bilinear and biquadratic exchange interaction parameters for nearest-neighbor atoms of the lattice, and
denotes summation over the nearest neighbors. We set
, where
and
are the respective magnon creation and annihilation operators at the site m. Let
be the so-called vacuum vector, which is fully determined by the conditions
and
. The vectors
describe the state of the system of three magnons located at the nodes m, n and l. The vectors
constitute an orthonormal system. We let
denote the Hilbert space spanned by these vectors. It is called the space of three-magnon states of the operator H. The space
is invariant under operator H. We let H3 denote the restriction of the operator H in the space
.
Theorem 1. The space
is invariant with respect to the operator H. The operator H3 is a bounded self-adjoint operator. It generates the bounded self-adjoint operator
, acting in the space
according to the formula
(4)
where
is the Kronecker symbol. The operator H3 acts on the vector
according to the formula
(5)
Proof. The proof is by direct calculation in which we use the well-known commutation relations between the operators
, and
:
, and
.
Lemma 1. The spectra of the operators
and
coincide.
Proof. Because
and
are bounded self-adjoint operators, it follows from the Weyl criterion that there exist a sequence of vectors
such that
,
, and
(6)
where
. On the other hand,
. Here
and
. It follows that
. Consequently,
. Conversely, let
. Again by the Weyl criterion, there then exist a sequence
such that
and
(7)
as
.
Setting
, we have
and
. This, together with Formula (7) and the Weyl criterion, implies that
, and hence
. These two relations imply that
.
We let
denote the Fourier transform:
,
where
is a ν-dimensional torus with the normalized Lebesgue measure
:
. We set
.
Theorem 2. The Fourier transformation transforms the operator
into the bounded self-adjoint operator
acting in the space
according to the formula
. (8)
The following fact is important for further investigating the spectrum of the operator
. Let the full quasi-momentum of the three-magnon system, i.e. sum of quasi-momentum of each three magnons
be fixed. Let
be the space of functions that are quadratically integrable over the manifold
. It is known [21] that the operator
and space
can be expanded into the direct integrals
of the operators
and the spaces
such that the spaces
are invariant with respect to the operators
.
In the isotropic non-Heisenberg Ferromagnet model with spin
, the spectral properties of the considered operator of the energy of three-magnon systems are closely related to those of its two-magnon subsystems (see Formula (11)). We first study the spectrum and BSs of two-magnon systems.
3. Two-Magnon Bound States
The Hamiltoinian of a two-magnon subsystem also has form (3). We let
denote the space of two-magnon states of the operator H. We let H2 denote the restriction of H to the space
.
Theorem 3. The space
is invariant with respect of the operator H. The operator H2 is a bounded self-adjoint operator. It generates the bounded self-adjoint operator
, acting in the space
according to the formula
(9)
The operator H2 acts on the vector
according to the formula
(10)
Lemma 2. The spectra of the operators H2 and
coincide.
Theorem 4. The Fourier transformation transforms the operator
into the bounded self-adjoint operator
acting in the space
according to the formula
(11)
where
Let the full quasi-momentum of the two-magnon system, i.e. sum of quasi-momentum of each two magnons
be fixed. Let
be the space of functions that are quadratically integrable over the manifold
. It is known [21] that the operator
and space
can be expanded into the direct integrals
of the operators
and the spaces
such that the spaces
are invariant with respect to the operators
and the operators
act in the space
according to the formula
(12)
where
and
.
It is known that the continuous spectrum of the operator
does not depend on the functions
and consists of the intervals
,
where
and
.
Definition 1. The eigenfunction
of the operator
corresponding to the eigenvalue
is called the bound state (BS) of the operator
with quasi-momentum
, and the quantity
is called the energy of this BS.
We consider the operator
, acting in the space
according to the formula
This operator is totally continuous in the space
for values of
.
Let
, where
and
In these formulas,
Lemma 3. A number
be an eigenvalue of the operator
if and only if it is a zero of the function
, i.e.
.
Proof. In the case under consideration, the equation for the eigenvalues is an integral equation with a degenerate kernel. It is therefore equivalent to a system of linear homogeneous algebraic equations. It is known that such a system has a nontrivial solution if and only if its determinant is equal to zero. In this case, the determinant of this linear homogeneous algebraic system is equal to function
.
Theorem 5. Let
and
be arbitrary. Then the operator
has two BSs
and
(not taking the order of the energy degeneration into account) with the energy values
,
, and
is degenerate
times, while
is not degenerate,
, for all
, i.e. the energy values of these BSs lie below the continuous spectrum domain of the operator
.
Proof. If
, then
, and
Solving the equation
, we prove the theorem.
Let
.
Theorem 6. Let
and
. Then the operator
has only one BS
with the energy value
, and this energy level is degenerate
times. In addition, if
, then
, and if
, then
. When
, this BS vanishes because it is incorporated into the continuous spectrum.
Proof. The proof of this theorem is based on the equality
with
and also on the corresponding form of the determinant
.
From Theorem 5 and 6 and later is obviously, what the spectrum of the Hamiltonian
by different value of
differ from one another.
In the case where
, the change of the energy spectrum of the operator
is described by the following theorems.
Theorem 7.
1) Let
and
or
.
a) If respectively
or
, then the operator
has two BSs
and
with the corresponding energy values
.
b) If respectively
or
, then the operator
has only one BS
with the energy value
, and
.
2) Let
and
or
.
a) If respectively
or
, then the operator
has only one BS
with the energy value
.
b) If
, then the operator
has no BS. Above,
, and
.
3) Let
and
or
.
a) If respectively
or
, then the operator
has two BSs
and
with the corresponding energy values
, and
.
b) If respectively
or
, then the operator
has three BSs
,
, and
with the corresponding energy values
, and
.
4) Let
and
or
.
a) If respectively
or
, then the operator
has two BSs
and
with the corresponding energy values
.
b) If respectively
or
, then the operator
has only BS
with the energy value
. In this case, the second BS vanishes because it is incorporated into the continuous spectrum.
5) Let
and
. Then the operator
has only one BS
with the energy value
.
6) Let
and
. Then the operator
has two BSs
and
with the corresponding energy values
, and
.
In the case where
and
, the change of the energy spectrum is described by the following theorem.
Theorem 8.
1) If
and
, then the operator
has two BSs
and
with the corresponding energy values
.
2) If
and
, then the operator
has only one BS
with the energy value
.
3) If
and
, then the operator
has two BSs
and
with the corresponding energy values
and
.
4) If
and
, then the operator
has two BSs
and
with the corresponding energy values
.
5) If
and
, then the operator
has no BS.
6) If
and
, then the operator
has only one BS
with the energy value
.
A sketch proof of Theorems 7-8 is given below. In the case under consideration, the equation for the eigenvalues is an integral equation with a degenerate kernel. It is therefore equivalent to a system of linear homogeneous algebraic equations. It is known that such a system has a nontrivial solution if and only if its determinant is equal to zero. In this case, the equation
it therefore equivalent to the equation stating that the determinant of the system is zero. Expressing all integrals in the equation
, through the integral
, we find that the equation
is equivalent to the equation
(13)
Because
is a continuous function for
and
, the function
is an increasing function of z for
. Moreover,
as
,
as
,
as
and
as
. Analysis of Equation (13) outside the set
, leads to the proof of Theorems 7-8.
The energy spectrum in the case where
for the full quasi momenta of the form
is described below. It is easy to see that if the parameters
, and
satisfy the conditions of Theorems 7-8, the statements of the theorems are true. Only one additional BS
appears, whose energy value is
, moreover
, if
. If
, the operator
has no additional BS.
The proof of this statements is based on the fact that if
and
then the function
has the form
(14)
where
The equation
is therefore equivalent to the equation
(15)
and
(16)
It is easy to see that Equation (15) has a unique solution
if
; if
, this solution satisfies the condition
. If
, Equation (15) has no solution. Expressing the integrals in Equation (16) through the integral
, we obtain an equation of the form
where
and
. In this case,
. In turn, for
, the latter equation is equivalent to the equation of the form
(17)
Analyzing Equation (17) outside the set
and taking into account that the function
is monotonic for
, we obtain statements similar to the statements in Theorems 7-8.
For all other quasi momenta
, there exist sets
, of the parameters
, and
such that in every set
the operator
has exactly j BSs (taking the energy degeneration order into account) with the corresponding energy values
.
Indeed, in this case and for
, the function
has the form
,
where
, and
,
;
,
,
,
,
,
.
In these formulas,
,
,
,
.
Expressing all integrals in the equation
through
and rearranging algebraically, we reduce the latter equation to the form
(18)
where
is a fifth-order polynomial in z, and
is a lower-order polynomial in z. Analyzing Equation (18) outside the set
and taking into account that the function
with
is monotonic, we can easily verify that the equation has no more than five solutions outside the set
.
We now consider the case of
. Let the full quasi-momentum have the form
. If the parameters
and
satisfy the conditions in Theorems 7-8, then statements similar to those in the theorems are true. Only one additional BS
appears, whose energy value is
. This energy level is twice degenerate and
, if
. This additional BS vanishes when
because it is incorporated into the continuous spectrum.
To prove this, we note that in this case, the function
has the form
where
Therefore the equation
is equivalent to the equations
(19)
and
(20)
It is easy see that Equation (19) has a unique double solution
if
and
, if
. Expressing all integrals in Equation (20) through
, we obtain the equation
(21)
where
, and
. Here
. If
, Equation (21) is, in turn, equivalent to the equation
(22)
Analyzing Equation (22) outside the set
and taking into account that the function
for
, is monotonic, we prove the statements made above.
If
, the system has no more than seven BSs (taking the energy degeneration order into account), and there exist sets
, of the parameters
, and
such that in every set
the system has exactly k BSs. The energy values of these BSs lie outside the set
. When passing from one of these sets to another, either some additional BSs of the operator
appear or some existing BSs vanish.
In this case, the function
has the form
, where
and
,
;
,
,
;
,
;
,
,
;
. In these formulas
,
,
.
,
.
Expressing all integrals in the equation
through
and rearranging algebraically, we reduce this equation to the form
where
is a seventh-order polynomial in z, and
is a lower-order polynomial in
Therefore, this equation has no more than seven solutions outside the set
.
For an arbitrary
and
, if the parameters
and
satisfy the conditions in Theorems 7-8, statements similar to those in the theorems are true. In this situation, the operator
with
has only one additional BS. The energy z of this additional BS is degenerate
times. Moreover,
, if
. For all other values of the full quasi-momentum
of the system, the operator
has no more than
BSs (taking the energy degeneration order into account) with the energy values lying outside the set
.
The proof of these statements is based on finding zeros of the determinants
of the operators. Expressing all integrals in
through
, we can bring the equation
to the form
(23)
where
is a
th-order polynomial in z and
is also a polynomial in z whose order (with respect to
) is lower. Analysis of Equation (23) outside the set
leads to the proof of the statements made above.
Theorem 9. Let
and
be an arbitrary number. Then the operator
has no more than one BS, and the corresponding energy level
is not degenerate.
Proof. If
, the relations
,
hold. Using the determinant
and solving the corresponding equation, we obtain the statement in Theorem 9.
4. Structure of Essential Spectrum and Discrete Spectrum of Three-Magnon Systems
We first determine the structure of the essential spectrum of the three-magnon system and then estimate the number of three-magnon BSs in this system. Comparing Formulas (8) and (11) and using tensor products of Hilbert spaces and tensor products of operators in Hilbert spaces [22] , we can verify that the operator
can be represented in the form
(24)
where I is the unit operator in the space
, and
and
and
are the energy operator of two-magnon systems, and
are finite-dimensional operator,
,
,
,
.
The spectrum of
, where A and B are densely defined bounded linear operators, was studied in [23] [24] [25] . Explicit formulas expressing
and
in terms of
and
were given in those papers:
. It is clear that
.
Note that, what the problems of finite-rank perturbations for the compact operators be considered in the work [26] [27] [28] .
The following theorems describe the structure of the essential spectrum of
.
Theorem 10. If
and
be arbitrary. Then the essential spectrum of the operator
consists of the set of five points:
, and the inequality
holds for the number N of three-magnon BSs.
Proof. It can be seen from Theorem 5, that for
the operator
has exactly two BSs
and
(not taking the order of the energy degeneration into account) with the energy values
,
, while the continuous spectrum of the operator
is consists of one point
, therefore the essential spectrum of operator
is consists of points
,
,
,
,
, i.e.
. The operator
has a eigenvalues to equally
,
,
, and
. The operator
is a finite rank operator, to rank is equal to
. Consequently, in this case the number of three-magnon BS
.
Theorem 11. Let
and
. Then the essential spectrum of the operator
consists of the set of three points:
, and the inequality
holds for the number N of three-magnon BSs.
Proof. It can be seen from Theorem 6, that for
and
the operator
has a unique BS
with the energy value
, and this energy level is ν-fold degenerate. The continuous spectrum of the operator
is consists of point
, therefore the essential spectrum of the operator
consists of points
,
,
, i.e.
. The operator
has a eigenvalues equal to
. The operator
has a finite rank operator, with rank to equal to
. Consequently, in this case, the number of three-magnon BS
.
We let
,
,
,
,
,
.
Theorem 12. Let
,
and
,
,
or
,
,
.
a) If
,
,
or
,
,
, then the essential spectrum of the operator
consists of the union of five intervals:
and the inequality
holds for the number N of three-magnon BSs.
b) If
,
,
or
,
,
, then the essential spectrum of the operator
consists of the union of three intervals:
, and the inequality
holds for the number of three-magnon BSs N.
c) If
,
,
or
,
,
or
,
,
or
,
,
, then the essential spectrum of the operator
consists of the union of four intervals:
, and the inequality
holds for the number of three-magnon BSs N.
Theorem 13. Let
and
,
,
or
,
,
.
a) If
or
, and
or
, and
or
, respectively, then the essential spectrum of the operator
consists of the union of three intervals:
, and the inequality
holds for the number of three-magnon BSs N.
b) If
and
and
, then the essential spectrum of the operator
consists of single interval:
, and the inequality
holds for the number of three-magnon BSs N. Here
.
c) If
or
or
, then the essential spectrum of the operator
consists of the union of two intervals:
, and the inequality
holds for the number of three-magnon BSs N.
Theorem 14. Let
and
,
,
or
,
,
.
a) If
,
,
or
,
,
, then the essential spectrum of the operator
consists of the union of five intervals:
, and the inequality
holds for the number of three-magnon BSs N.
b) If
,
,
or
,
,
, then the essential spectrum of the operator
consists of the union of seven intervals:
, and the inequality
holds for the number of three-magnon BSs N.
c) If
,
,
or
,
,
or
,
,
, then the essential spectrum of the operator
consists of the union of six intervals:
, and the inequality
holds for the number of three-magnon BSs N.
Theorem 15. Let
and
or
a) If
,
, or
,
, then the essential spectrum of the operator
consists of the union of five intervals:
, and the inequality
holds for the number of three-magnon BSs N.
b) If
,
or
,
, then the essential spectrum of the operator
consists of the union of three intervals:
, and the inequality
holds for the number of three-magnon BSs N.
c) If
,
, or
,
, then the essential spectrum of the operator
consists of the union of four intervals:
, and the inequality
holds for the number of three-magnon BSs N.
d) If
,
or
,
, then the essential spectrum of the operator
consists of the union of four intervals:
, and the inequality
holds for the number of three-magnon BSs N.
Theorem 16. Let
and
. Then the essential spectrum of the operator
consists of the union of three intervals:
, and the inequality
holds for the number of three-magnon BSs N.
Theorem 17. Let
and
. Then the essential spectrum of the operator
consists of the union of five intervals:
, and the inequality
holds for the number of three-magnon BSs N.
If
and
and
has the form
,
, and
, then the essential spectrum of the operator
is investigated to analogously to one-dimensional case.
In the essential spectrum of the operator
is appear only two additional intervals and corresponding estimation for the number of three-magnon BSs, in the case of, when the operator
and
and
has a correspondingly, correspondingly to equal to number a and b and c, BSs, that the estimation
changed to the estimation
.
For arbitrary values
and
and
and
, the essential spectrum of the operator
is consists of the union of no more that
intervals, and the relation
, holds for the number of three-magnon BSs N, where
.
Theorem 18. If
and the number
be arbitrary. Then the essential spectrum of the operator
consists of the union of no more then three intervals:
, and the inequality
holds for the number of three-magnon BSs N.
The cases, when
or
or
investigated the similarly. Analogously is investigated essential spectrum and discrete spectrum of the operator
for the other cases.
Obviously, that the case, when the of spin component
take the value −1 coincide, with cases when spin component
take the value 1.
5. Case, When
We already say,what the spin component
can by take on a value −1, 0, 1, i.e. can by
or
or
.
Now we consider the case, when the of spin component
take value 0.
Hamiltonian of the system also has the form (3) and acts in the symmetrical Fo'ck space
. In this case the vacuum vector
uniquely determined by the conditions:
.
Theorem 19. The space
is invariant with respect of the operator H. The operator H3 is a bounded self-adjoint operator. It generates the bounded
self-adjoint operator
, acting in the space
according to the formula
(25)
where
is the Kronecker symbol. The operator H3 acts on the vector
according to the formula
(26)
Theorem 20. The Fourier transformation transforms the operator
into the bounded self-adjoint operator
, acting in the space
according to the formula
(27)
where
The spectral properties of the considered energy operator of three-magnon systems in the isotropic ferromagnetic non-Heisenberg model are closely related to those of its two-magnon subsystems. We first study the spectrum and bound states of two-magnon subsystems.
Theorem 21. The space
is invariant with respect of the operator H. The operator H2 is a bounded self-adjoint operator. It generates the bounded self-adjoint operator
, acting in the space
according to the formula
(28)
The operator H2 acts on the vector
according to the formula
(29)
Theorem 22. The Fourier transformation transforms the operator
into the bounded self-adjoint operator
, acting in the space
according to the formula
(30)
where
Let the full quasi-momentum of the system
be fixed. Let
be the space of functions that are quadratically integrable over the manifold
. It is known [21] that the operator
and space
can be expanded into the direct integrals
of the operators
and the spaces
such that the spaces
are invariant with respect to the operators
and the operators
act in the space
according to the formula
(31)
where
and
.
Theorem 23. Let full quasi-momentum of the system
by arbitrary. Then the operator
has a unique BS
with the energy value
and it is ν-fold degenerated.
Let the full quasi-momentum of the system
be fixed. Then the operator
and space
can be expanded into the direct integrals
.
We now determine the structure of the essential spectrum of the three-magnon system and then estimate the number of three-magnon BSs in this system. Comparing Formulas (27) and (30) and using tensor products of Hilbert spaces and tensor products of operators in Hilbert spaces [22] , we can verify that the operator
can be represented in the form
(32)
where I is the unit operator in the space
, and
are finite-dimensional operator (see (27)).
Theorem 24. Let full quasi-momentum of the system
by arbitrary. Then the essential spectrum of the operator
consists of the three points: 0,
and
, where
and
and
is a eigenvalue of the operators
and
and
, correspondingly, and the inequality
holds for the number of three-magnon BSs N.
The finding results shown the structure of essential spectra and discrete spectrum of three-magnon system, in the cases, when the component
of spin is receive the value 1 and 0, is strongly different one another.