Spontaneous symmetry breaking and coherence in two-dimensional electron-hole and exciton systems

Spontaneous symmetry breaking and coherence in two-dimensional electron-hole and exciton systems

S.A. Moskalenko, M.A. Liberman, E.V. Dumanov, and E.S. Moskalenko Institute of Applied Physics of the Academy of Sciences of Moldova, Academic Str. 5, Chisinau, MD2028, Republic of Moldova
Department of Physics, Uppsala University, Box 530, SE-751 21, Uppsala, Sweden
A F Ioffe Physical-Technical Institute, Russian Academy of Sciences, 26 Politekhnicheskaya, 194021 St Petersburg, Russia
August 23, 2019
Abstract

The spontaneous breaking of the continuous symmetries of the two-dimensional (2D) electron-hole systems in a strong perpendicular magnetic field leads to the formation of new ground states and determines the energy spectra of the collective elementary excitations appearing over these ground states. In this review the main attention is given to the electron-hole systems forming coplanar magnetoexcitons in the Bose-Einstein condensation (BEC) ground state with the wave vector , taking into account the excited Landau levels, when the exciton-type elementary excitations coexist with the plasmon-type oscillations. At the same time properties of the two-dimensional electron gas (2DEG) spatially separated as in the case of double quantum wells (DQWs) from the 2D hole gas under conditions of the fractional quantum Hall effect (FQHE) are of great interest because they can influence the quantum states of the coplanar magnetoexcitons when the distance between the DQW layers diminishes. We also consider in this review the bilayer electron systems under conditions of the FQHE with the one half filling factor for each layer and with the total filling factor for two layers equal to unity because the coherence between the electron states in two layers is equivalent to the formation of the quantum Hall excitons (QHExs) in a coherent macroscopic state. This makes it possible to compare the energy spectrum of the collective elementary excitations of the Bose-Einstein condensed QHExs and coplanar magnetoexcitons. The breaking of the global gauge symmetry as well as of the continuous rotational symmetries leads to the formation of the gapless Nambu-Goldstone (NG) modes while the breaking of the local gauge symmetry gives rise to the Higgs phenomenon characterized by the gapped branches of the energy spectrum. These phenomena are equivalent to the emergence of massless and of massive particles, correspondingly, in the relativistic physics. The application of the Nielsen-Chadha theorem establishing the number of the NG modes depending of the number of the broken symmetry operators and the elucidation when the quasi-NG modes appear are demonstrated using as an example related with the BEC of spinor atoms in an optical trap. They have the final aim to better understand the results obtained in the case of the coplanar Bose-Einstein condensed magnetoexcitons. The Higgs phenomenon results in the emergence of the composite particles under the conditions of the FQHE. Their description in terms of the Ginzburg-Landau theory is remembered. The formation of the high density 2D magnetoexcitons and magnetoexciton-polaritons with point quantum vortices attached is suggested. The conditions in which the spontaneous coherence could appear in a system of indirect excitons in a double quantum well structures are discussed. The experimental attempts to achieve these conditions, the main results and the accumulated knowledge are reviewed.

pacs:
71.35.Lk, 67.85.Jk

I Introduction

The collective elementary excitations of the two-dimensional (2D) electron-hole (e-h) systems in a strong perpendicular magnetic field are discussed in the frame of the Bogoliubov theory of quasiaverages [1] taking into account the phenomena related with the spontaneous breaking of the continuous symmetries. The main results in this field have been obtained thanks to the fundamental papers by Goldstone [2], Nambu [3], Higgs [4] and Weinberg [5]. These investigations were influenced by the success of the theory of superconductivity developed originally by Bardin, Cooper and Schriffer [6], refined later by Bogoliubov [1] as well as by the microscopic theory of superfluidity proposed by Bogoliubov [1]. The specific implementation of these concepts and theorems in the case of 2D magnetoexcitons with direct implication of the plasmon-type excitations side-by-side with the exciton-type branches of the energy spectrum is the main topic of the present review. The coplanar electrons and holes in a strong perpendicular magnetic field at low temperatures form the magnetoexcitons, when the Coulomb interaction between electrons and holes lying on the lowest Landau levels (LLLs) plays the main role. However, when the electrons and holes are spatially separated on the different layers of the double quantum well (DQW) the Coulomb e-h interaction diminishes, and the two-dimensional electron gas (2DEG) on one layer and the two-dimensional hole gas (2DHG) on another layer are formed. Their properties under the conditions of the fractional quantum Hall effect (FQHE) can influence the properties of the 2D magnetoexcitons. To the best of our knowledge these aspects of the magnetoexciton physics were not discussed in literature. A short review is given on the Bose-Einstein Condensation (BEC) of the quantum Hall excitons (QHExs) arising in the bilayer electron systems under the conditions of the FQHE at one half filling factor for each layer and the total filling factor equal to unity for both layers. This enables us to compare the phenomenon of the BEC of coplanar magnetoexcitons and of QHExs. Such comparison provides better understanding of the underlying physics and allows to verify accuracy of the made approximations. Because the point vortices play an important role in the understanding of the FQHE the corresponding additional information should be included. The possibility to consider the BEC at as an estimate for the finite temperatures below the Berezinskii-Kosterlitz-Thouless phase transition is suggested. The article is organized as follow. In Section 2 the Bogoliubov theory of the quasiaverages is overviewed. Section 3 is devoted to the Goldstone theorem. The Nambu-Goldstone modes arising under the condition of BEC of the sodium atoms are enumerated in Section 4. The breaking of the local gauge symmetry and the Higgs phenomenon are discussed in Section 5. Section 6 is devoted to the quasi-Nambu-Goldstone modes. In Section 7 the Ginzburg-Landau theory for the FQHE is formulated. The 2D point quantum vortices are described in Section 8. The existence of the statistical gauge vector potential generated by the vortices is considered in Section 9. The BEC of QHExs and the energy spectrum of elementary excitations under these conditions are discussed in Section 10. Section 11 contains the main results concerning the energy spectrum of the exciton and plasmon branches of the collective elementary excitations of the Bose-Einstein condensed coplanar magnetoexcitons.

Ii Bogoliubov’s Theory of Quasiaverages

N.N. Bogoliubov [1] has demonstrated his concept of quasiaverages using the ideal Bose-gas model with the Hamiltonian

(1)

here are the Bose operators of creation and annihilation of particles, and is their chemical potential.
The occupation numbers of the particles are

(2)

where and .
In the normal state, the density of particles in the thermodynamic limit at becomes . At this point, the Bose-Einstein condensation occurs and a finite value of the density of condensed particles appears in the thermodynamic limit

(3)

The operators and asymptotically become numbers, when their commutator

(4)

asymptotically tends to zero and their product is equal to . Then one can write

(5)

On the other hand, the regular averages of the operators and in the Hamiltonian (1) are exactly equal to zero. It is the consequence of the commutativity of the operator and the operator of the total particle number as follows

(6)

As a result, the operators is invariant with respect to the unitary transformation

(7)

with an arbitrary angle . This invariance is called gradient invariance of the first kind or gauge invariance. When does not depend on the coordinate , we have the global gauge invariance and in the case it is called local gauge invariance [2-8] or gauge invariance of the second kind. The invariance (7) implies , which leads to the following average value

Because is an arbitrary angle, there are the selection rules:

(8)

The regular average (8) can also be obtained from the asymptotical expressions (5) if they are integrated over the angle . This apparent contradiction can be resolved if Hamiltonian (1) is supplemented by additional term

(9)

where is the fixed angle and is infinitesimal value.

New Hamiltonian has the form

(10)

It does not conserve the condensate number. Now the regular average values of the operators and over the Hamiltonian differ from zero, i.e., and . The definition of the quasiaverages designated by is the limit of the regular average when tends to zero

(11)

It is important to emphasize that the limit must be effectuated after the thermodynamic limit , . In the thermodynamic limit, is also infinitesimal, and it is possible to choose the ratio of two infinitesimal values and obtaining a finite value

(12)

To calculate the regular average one needs to represent the Hamiltonian (10) in a diagonal form with the aid of the canonical transformation over the amplitudes

(13)

In terms of the new variables the Hamiltonian takes the form

(14)

In the diagonal representation (14), the regular average value exactly equals to zero, while the value is equal to the first term on the right-hand side of formulas (13).

As a result, the quasiaverage is

(15)

It depends on the fixed angle and does not depend on . The spontaneous global gauge symmetry breaking is implied when the phase of the condensate amplitude in Hamiltonian (10) is fixed.

When the interaction between the particles is taken into account, these differences appear for other amplitudes as well. They give rise to the renormalization of the energy spectrum of the collective elementary excitations. In such a way, the canonical transformation

(16)

introduced for the first time by Bogoliubov [1] in his theory of superfluidity, has a quantum-statistical foundation within the framework of the quasiaverage concept. At the quasiaverage coincides with the average over the quantum-mechanical ground state, which is the coherent macroscopic state [9].

The phenomena related to the spontaneous breaking of the continuous symmetry play an important role in statistical physics.
Some elements of this concept, such as the coherent macroscopic state with a given fixed phase and the displacement canonical transformation of the field operator describing the Bose-Einstein condensate, were introduced by Bogoliubov in the microscopical theory of superfluidity [1] and were generalized in his theory of quasiaverages [1] noted above.
The brief review of the gauge symmetries, their spontaneous breaking, Goldstone and Higgs effects will be presented below following the Ryder’s monograph [7] and Berestetskii’s lectures [8].

Iii Goldstone’s Theorem

Goldstone has considered a simple model of the complex scalar Bose field to demonstrate his main idea. In the classical description the Lagrangian is

(17)

The potential energy has the form

(18)

where is considered as a parameter only, rather than a mass term, is the parameter of self-interaction, whereas the denotations and mean

(19)

The Lagrangian is invariant under the global gauge transformation

(20)

It has the global gauge symmetry. The ground state is obtained by minimizing the potential as follows

(21)

Of interest is the case , when the minima are situated along the ring

(22)

The function is shown in Fig. 1 being plotted against two real components of the fields .

Figure 1: The potential with the minima at and a local maximum at

There is a set of degenerate vacuums related to each other by rotation. The complex scalar field can be expressed in terms of two scalar real fields, such as and , in polar coordinates representation or in the Cartesian decomposition as follows

(23)

The Bogoliubov-type canonical transformation breaking the global gauge symmetry is

(24)

The new particular vacuum state has the average with the particular vanishing vacuum expectation values . It means the selection of one vacuum state with infinitesimal phase . As was pointed in [7], the physical fields are the excitations above the vacuum. They can be realized by performing perturbations about . Expanding the Lagrangian (17) in series of the infinitesimal perturbations , , , and neglecting by the constant terms, we obtain

(25)

or in polar description

(26)

Neglecting by the cubic and quartic terms, we will see that there are the quadratic terms only of the type and , but there is no quadratic terms proportional to and . For real physical problems, for example, for the field theory, the field components and represent massive particles and dispersion laws with energy gap, whereas the field components and represent the massless particles and gapless energy spectrum.
The main Goldstone results can be formulated as follows

(27)

The spontaneous breaking of the global gauge symmetry takes place due to the influence of the quantum fluctuations. They transform the initial field with two massive real components and , and a degenerate ground state with the minima forming a ring into another field with one massive and other massless components, the ground state of which has a well defined phase without initial symmetry.

The elementary excitations above the new ground state changing the value are massive. It costs energy to displace against the restoring forces of the potential . But there are no restoring forces corresponding to displacements along the circular valley formed by initial degenerate vacuums.
Hence, for angular excitations of wavelength, we have as . The dispersion law is and the particles are massless [7]. The particles are known as the Goldstone bosons. This phenomenon is general and takes place in any order of perturbation theory. The spontaneous breaking of a continuous symmetry not only of the type as a global gauge symmetry but also of the type of rotational symmetry entails the existence of massless particles referred to as Goldstone particles or Nambu-Golstone gapless modes. This statement is known as Goldstone theorem. It establishes that there exists a gapless excitation mode when a continuous symmetry is spontaneously broken. The angular excitations are analogous to the spin waves. The latter represent a slow spatial variation of the direction of magnetization without changing of its absolute value. Since the forces in a ferromagnetic are of short range, it requires a very little energy to excite this ground state. So, the frequency of the spin waves has the dispersion law . As was mentioned by Ryder [7], this argument breaks down if there are long-range forces like, for example, the Coulomb force. In this case, we deal with the maxwellian gauge field with local depending on gauge symmetry instead of global gauge symmetry considered above.

After the specific application of the above statement will be demonstrated following Refs. [10-16], where the spinor Bose-Einstein condensates were discussed, we will consider the case of Goldstone field and of the maxwellian field with local gauge symmetry.

Iv Bogoliubov’s Excitations and the Nambu-Goldstone Modes

The above formulated theorems can be illustrated using the specific example of the Bose-Einstein condensed sodium atoms Na in an optical-dipole trap following the investigations of Murata, Saito and Ueda [10] on the one side and of Uchino, Kobayashi and Ueda [11] on the other side. There are numerous publications on this subject among which should be mentioned [12-17]. The sodium atom Na has spin of the hyperfine interaction and obey the Bose statistics. Resultant spin of the interacting bosons with is which takes the values . The contact hard-core interaction constant is characterized by s-wave scattering length , which is not zero for when two atomic spins form a singlet, and for , when they form a quintuplet. The constant and enter the combinations and which determine the Hamiltonian. The description of the atomic Bose gas in an optical-dipole trap is possible in the plane-wave representation due to the homogeneity and the translational symmetry of the system. It means that the components of the Bose field operator can be represented in the form:

(28)

where is the volume of the system, and is the annihilation operator with the wave vector and the magnetic quantum number , which in the case takes three values 1, 0, -1. The spinor Bose-Einstein condensates were realized experimentally by the MIT group [12] for different spin combinations using the sodium atoms Na in a hyperfine spin states in a magnetic trap and then transforming them to the optical-dipole trap formed by the single infrared laser. The Bose-Einstein condensates were found to be long-lived. Some arguments concerning the metastable long-lived states were formulated. The states may appear if the energy barriers exist, which prevent the system from direct evolving toward its ground states. If the thermal energy needed to overcome these barriers is not available, the metastable state may be long-lived and these events are commonly encountered. Even the Bose-Einstein condensates in the dilute atomic gases can also be formed due to the metastability. Moreover, in the gases with attractive interactions the Bose-Einstein condensates may be metastable against the collapse just due to the energy barriers [12]. Bellow we will discuss the Bogoliubov-type collective elementary excitations arising over the metastable long-lived ground states of the spinor-type Bose-Einstein condensates (BEC-tes) following Ref.[10, 11], so as to demonstrate the formation of the Nambu-Goldstone modes.

The Hamiltonian considered in [10] is given by formulas (3) and (4), and has the form

(29)

Here the following designations were used

(30)

The repeated indexes mean summation over 1,0,-1. The symbol denotes the normal ordering of the operators. The coefficient is the sum of the linear Zeeman energy and of the Lagrangian multiplier, which is introduced to set the total magnetization in the direction to a prescribed value. This magnetization is conserved due to the axisymmetry of the system in a magnetic field. q is the quadratic Zeeman effect energy, which is positive in the case of spin for Na and Rb atoms. The spin-spin interaction is of ferromagnetic-type with for Rb atoms and is antiferromagnetic-type with for Na atoms [10]. Taking into account that in many experimental situations the linear Zeeman effect can be ignored and the quadratic Zeeman effect term can be manipulated experimentally, in [11] both cases of positive and negative at were investigated for spin and spin Bose-Einstein condensates (BECs). We restrict ourselves to review some spinor phases with spin discussed in [11] so as to demonstrate the relations between the Nambu-Goldstone(NG) modes of the Bogoliubov energy spectra and the spontaneous breaking of the continuous symmetries. The description of the excitations is presented in Refs. [10, 11] in the number-conserving variant of the Bogoliubov theory [1]. There is no need to introduce the chemical potential as a Lagrangian multiplier in order to adjust the particle number to a prescribed value. The BEC takes place on a superposition state involving the single-particle states with wave vector and different magnetic quantum numbers

(31)

The order parameter has a vector form and consists of three components: . The vacuum state means the absence of the atoms. The ground state wave function of the BEC-ed atoms is given by the formula (8) of Ref. [11]

(32)

In the mean-field approximation the operators , are replaced by the numbers , where is the number of the condensed atoms. After this substitution, the initial Hamiltonian loses its global gauge symmetry and does not commute with the operator . The order parameters are chosen so that to minimize the expectation value of the new Hamiltonian and its ground state and to satisfy the normalization condition . To keep the order parameter of each phase unchanged it is necessary to specify the combination of the gauge transformation and spin rotations [11]. This program was carried out in [18-21].

The initial Hamiltonian (29) in the absence of the external magnetic field has the symmetry representing the global gauge symmetry and the spin-rotation symmetry . The generators of these symmetries are referred to as symmetry generators and have the form

(33)

Unlike the symmetry group with three generators , and , the symmetry group has only one generator which describes the spin rotation around the axis and looks as follows:

(34)

In the presence of an external magnetic field, the symmetry of the Hamiltonian is . The breaking of the continuous symmetry means the breaking of their generators. The number of the broken generators (BG) is denoted as . There are 4 generators in the case of symmetry and two in the case of symmetry.

The phase transition of the spinor Bose gas from the normal state to the Bose-Einstein condensed state was introduced mathematically into Hamiltonian (29) using the Bogoliubov displacement canonical transformation, when the single-particle creation and annihilation operators with the given wave vector , for example, , were substituted by the macroscopically numbers describing the condensate formation. The different superpositions of the single-particle states determine the structure of the finally established spinor phases [11]. Nielsen and Chadha [17] formulated the theorem which establishes the relation between the number of the Nambu-Goldstone modes, which must be present between the amount of the collective elementary excitations, which appear over the ground state of the system if it is formed as a result of the spontaneous breaking of the continuous symmetries. The number of NG modes of the first type with linear (odd) dispersion law in the limit of long wavelengths denoted as being accounted once, and the number of the NG modes of the second type with quadratic (even) dispersion law at small wave vectors, being accounted twice give rise to the expression , which is equal to or greater than the number of the broken symmetry generators. The theorem [17] says

(35)

The theorem has been verified in [11] for multiple examples of the spin and spin Bose-Einstein condensate phases. In the case of spin nematic phases, the special Bogoliubov modes that have linear dispersion relation but do not belong to the NG modes were revealed. The Bogoliubov theory of the spin and spin Bose-Einstein condensates (BECs) in the presence of the quadratic Zeeman effect was developed by Uchino, Kobayashi and Ueda [11] taking into account the Lee, Huang, Yang (LHY) corrections to the ground state energy, pressure, sound velocity and quantum depletion of the condensate. Many phases that can be realized experimentally were discussed to examine their stability against the quantum fluctuations and the quadratic Zeeman effect. The relations between the numbers of the NG modes and of the broken symmetry generators were verified. A brief review of the results concerning the spin phases of [11] is presented below so as to demonstrate, using these examples, the relations between the Bogoliubov excitations and the Nambu-Goldstone modes.

The first example is the ferromagnetic phase with , and the vector order parameter

(36)

The modes with and are already diagonalized, whereas the mode is diagonalized by the standard Bogoliubov transformation. The Bogoliubov spectrum is given by formulas (33) and (34) of Ref. [11]

(37)

The mode is massless. In the absence of a magnetic field, when , the mode is also massless with the quadratic dispersion law. The initial symmetry of the Hamiltonian before the phase transition is , whereas the final, remaining symmetry after the process of BEC is the symmetry of the ferromagnetic i.e. . From the four initial symmetry generators , , and remains only the generator of the symmetry. The generators and were broken by the ferromagnet phase, whereas the gauge symmetry operator was broken by the Bogoliubov displacement transformation. The number of the broken generators , , is three, i.e., . In this case , and , being equal to . The equality takes place. In the presence of an external magnetic field, with , the initial symmetry before the phase transition is with two generators and , whereas after the BEC and the ferromagnetic phase formation the remained symmetry is . Only one symmetry generator was broken. It means , and . The equality also takes place.

The condition to be hold is required for the Bogoliubov mode to be stable. It ensures the mechanical stability of the mean-field ground state. Otherwise, the compressibility would not be positive definite and the system would become unstable against collapse. In the case , and the state would undergo the Landau instability for the and modes with quadratic spectra and the dynamical instability for the mode with a linear spectrum (36) of Ref.[11].
There are two polar phases. One with the parameters

(38)

and the other with the parameters

(39)

These two polar phases have two spinor configurations which are degenerate at and connect other by transformation. However, for nonzero the degeneracy is lifted and they should be considered as different phases. This is because the phase has a remaining symmetry , whereas the phase is not invariant under any continuous transformation. The number of NG modes is different in each phase and the low-energy behavior is also different. Following formulas (40)-(42) of [11] the density fluctuation operator and the spin fluctuation operators and were introduced

(40)

Their Bogoliubov energy spectra are

(41)

In the presence of an external magnetic field, the initial symmetry is , whereas after the BEC and the formation of the phase with the remaining symmetry is also . Only the symmetry and its generator were broken during the phase transition. It means we have in this case , and . The equality holds. Density mode is massless because the gauge symmetry is spontaneously broken in the mean-field ground state, while the transverse magnetization modes and are massive for non zero , since the rotational degeneracies about the x and y axes do not exist being lifted by the external magnetic field. In the limit of infinitesimal nevertheless nonzero, the transverse magnetization modes and become massless. It occurs because before the BEC in the absence of an external magnetic field the symmetry of the spinor Bose gas is , whereas after the phase transition it can be considered as a remaining symmetry . The generators , , were broken, whereas the generator remained. In this case we have , and the equality looks as .

In the polar phase with the parameters (39) the density and spin fluctuation operators were introduced by formulas (57)-(59) of Ref. [11]

(42)

with the Bogoliubov energy spectra described by formulas (65)-(67) [11]:

(43)

At in contrast to the case one of the spin fluctuation mode becomes massless. The initial symmetry of the system is . It has the symmetry generators and . They are completely broken during the phase transition. After the phase transition and the phase formation there are not any symmetry generators. The number of the broken generator is 2 ( ), whereas the numbers and are 2 and 0, respectively. As in the previous cases, the equality occurs in the Nielsen and Chadha rule. For the Bogoliubov spectra to be real the condition , and must be satisfied, otherwise, the state will be dynamically unstable.

Side by side with the spinor-type three-dimensional (3D) atomic Bose-Einstein condensates in the optical traps, we will discuss also the case of the Bose-Einstein condensation of the two-dimensional (2D) magnetoexcitons in semiconductors [22-25]. The collective elementary excitations under these conditions were investigated in [26-31] and will be described in Section 11. As was shown above, the spontaneous symmetry breaking yields Nambu-Goldstone modes, which play a crucial role in determining low-energy behavior of various systems [5, 32-38]. Side by side with the global gauge symmetry the local symmetry does exist.

V Spontaneous breaking of the local gauge symmetry and the Higgs phenomenon

The interaction of the electrons with the electromagnetic field can be described introducing into the Lagrangian the kinetic momentum operators instead of canonical ones what is equivalent to introduce the covariant derivatives instead of the differential ones . They are determined in Ref. [8] as

(44)

where and are the scalar and vector potentials of the electromagnetic field (EMF). Below we will use also the denotations of Ref.[7]

(45)

The Lagrangian of the free EMF has the form [7]

(46)

being expressed through the antisymmetric tensors and . They are determined as four-dimensional curls of and .

(47)

The full Lagrangian of the electrons and EMF reads [7]

As before is a parameter so that in the case and in the absence of the EMF vacuum values are determined by the formula (22).
The invariance of the Lagrangian (48) under the transformation [8]

(49)

in the presence of the EMF can be achieved only under the concomitant transformation of its potential in the form [8]

(50)

Indeed in this case the Lagrangian (48) remains invariant [7] as follows

(51)

Introducing the gauge transformation of the field function (24) and expanding the Lagrangian in power series on the small physical fields and we obtain the constant, quadratic, cubic and quartic terms. The quadratic part looks as [7]

(52)

The second term is proportional to . It indicates that the photon becomes massive. The scalar field is also a massive one. The field takes part in the mixed term and can be eliminated by the supplementary gauge transformation (50). Following Ref [7] the Lagrangian (52) can be presented in the form

(53)

It contains two fields only: the photon with longitudinal component and spin 1 and field with spin 0. They are both massive. The field , which in the case of spontaneous breaking of the global symmetry became massless forming a Goldstone boson, in this case disappeared. The photon became massive. This phenomenon is called the Higgs phenomenon [7].

One possible illustration of the described above effect will be considered below following the paper by Halperin, Lee and Read [39]. They considered the two-dimensional (2D) system of spinless electrons under the conditions of the quantum Hall effect. Then the Hamiltonian consists of the kinetic energy operator

(54)

with 2D electrons with the mass and the charge situated in a uniform external perpendicular magnetic field B with the vector potential . The potential energy operator depends on the Coulomb interaction between the electrons. The creation and annihilation operators , obey to the Fermi statistics as was the case of Ref.[39], but we will consider following the Ref.[40] a more general case including also the Bose statistics

(55)

The signs correspond to the Fermi and Bose statistics. In Ref.[39] the new ”quasiparticle” operators , were introduced by the relations

(56)

with an integer number m and with the phase operator

(57)

It depends on the angle between the vector and the in-plane axis x being determined by the formula

(58)

and by the density operator

(59)

These operators have the properties

(60)