Yao Shenand Fu-Lin Zhang

1 School of Criminal Investigation,People's Public Security University of China,Beijing 100038,China

2 Department of Physics,School of Science,Tianjin University,Tianjin 300072,China

Abstract Gentile statistics describes fractional statistical systems in the occupation number representation.Anyon statistics researches those systems in the winding number representation.Both of them are intermediate statistics between Bose–Einstein and Fermi–Dirac statistics.The second quantization of Gentile statistics shows a lot of advantages.According to the symmetry requirement of the wave function and the property of braiding,we give the general construction of transformation between anyon and Gentile statistics.In other words,we introduce the second quantization form of anyons in an easier way.This construction is a correspondence between two fractional statistics and gives a new description of anyon.Basic relations of second quantization operators,the coherent state and Berry phase are also discussed.

Keywords:gentile statistics,anyon,fractional statistics,computational method

1.Introduction

In nature,basic particles are divided into two categories:bosons and fermions.Bosons obey Bose–Einstein statistics.Fermions follow Fermi–Dirac statistics.The symmetry requests the wave functions of bosons are symmetric,while they are anti-symmetric for fermions.Regarding many body system,when we exchange two bosons,the wave function is invariant.When two fermions are exchanged,the wave function of the many body system gets a phase π(?1).There could be only one fermion in single quantum state at most,while the number of boson in one state is not limited.In these decades,researchers found some intermediate cases between Bose and Fermi statistics called intermediate statistics or fractional statistics.Anyon and Gentile statistics are two typical and important cases.

Wilczek pointed out that braiding two different anyons gave the wave function an additional phase2π αk,where k is the winding number and α is statistical parameter[1,2].α could be fraction and depends on the type of anyon.The model of anyons is called flux tube model.This model could be interpreted as a point charged vortex.Analogue to the electromagnetic field,but the charge and the vector potential are both fictitious for anyons.The first Kitaev model is a honeycomb spin lattice model.Considering the strong correlation,it is simplified into a quadrilateral spin lattice model.For example,according to the first Kitaev model(1/2 anyon)[3–5],anyon has four superselection sectors:e(electric charge),m(magnetic vortex),ε=e×mand vacuum.Moving an anyon around another,which belongs to a different superselection sector except vacuum,gives the wave function the phase π(?1).

In 1940,Gentile proposed an intermediate statistics named after him[6].The maximum occupation number of Gentile statistics is limited to a finite number n[7–9].The states of Gentile statistics are represented by the practical occupation number,which are very convenient to operate.Gentile statistics has advantages in dealing with many body problems.The spins of fermions are half-integer.When the number of the fermions is even in the system,the case becomes complicated.If the distances of the fermions are very close,they can be regarded as bosons.But if their distances are not close enough,we need Gentile statistics to describe the system.

Because of these properties,the representations of anyon and Gentile statistics are called the winding number representation and the occupation number representation.They describe intermediate statistics in different aspects.One of the correspondences between these two representations has been discussed in[10].That is a special algebra we constructed in 2010[10].More and more real anyon systems and Gentile systems are observed these decades[11–15].Fractional statistics are applied in many hot fields of physics.For instance,topological quantum computation[16–18],quantum phase[19–21],quantum information[22,23],solving many-body interacting problem approximately using quantum field method[24–27],and quantum materials[28,29].

It has to be noticed that the wave function is also invariant when one fermion goes around another.In another word,the wave function of fermion returns to itself for braiding one circle.Exchanging two particles is half a circle which gives the fermionic wave function the phase factor?1.Bosons and fermions must be the ultimate limits of anyons and Gentile particles.As for 1/2 anyon,one circle corresponds to additional phase factor?1,and half circle is a complex numberi.In quantum mechanics,all quantities can be observed are real.This limitation is inconvenient.Fortunately,the group of anyon is braiding group.Therefore,it's advisable that we can weaken the limitation of fermionic symmetry to give a general construction of transformation between anyon and Gentile statistics.In this case,the limits of anyons and Gentile particles are bosons and fermions after adjustment.

This paper is organized as following:in section 2,we construct a general form of transformation between anyon and Gentile statistics.In section 3,we give the results of the second quantization operators of anyons in the winding number representation.In sections 4 and 5,the coherent state and Berry phase of anyon are discussed respectively.Finally,in section 6,the main results are concluded and further work is discussed.

2.The construction of transformation between two representations

Gentile and anyon statistics are both intermediate statistics.The maximum occupation number of Gentile particles in one state is a finite number n.The state of Gentile statistics is represented to be∣ν〉n,where ν is the practical particle number in one state.n is the maximum occupation number,which is an integer ranging from one to infinity.Thus the representation of Gentile statistics is called the occupation number representation.n→∞and n=1 correspond to Bose and Fermi statistics.It is very convenient to describe systems in the occupation number representation.After the second quantization,the basic operator relation is

where a,b,a?,b?are annihilation and creation operators of Gentile system and a=b*.a=b*means that the space of the states we considered consists of two parts:the normal state space and its complex conjugate space.If we denote the space ofa?, aas V andb?, basV*,the space of the entire system can be written as V?V*.It has been proved that only a?, ais not enough to represent the whole system just like the Holstein–Primakoff representation which is a modified bosonic realization[7,8].An additional constraint must be introduced.If we want to express the system with no constraint,the complex conjugate space is needed[8].The commutation relation returns to boson and fermion when the maximum occupation number n→∞and n=1.When n→∞,the phase factor is exp (i2π/(n +1))=1.When n=1,we have exp (i2π/(n +1))=-1.In these two limits,the space of the states returns to V.The particle number operator is constructed as

This equation tells us how many particles are there in one state.The eigenvalue of the particle number operator is the practical number of particles in one state.The second quantization of quantum field can be expressed in the language of the creation and annihilation operators.For bosons,the exchange of two particles gives nothing(+1),and the commutation relation reads[a ,a?]=aa?-a?a =1.For fermions,the exchange of two particles gives?1 which means{a ,a?}=aa?-(-1)a?a =1.Comparing these two commutation relations with the basic commutation relation of Gentile statistics,we get the phase of exchanging Gentile particles ei2π(n+1).So the commutation relation equation(1)tells us that there will be a phase factor ei2π(n+1)when two Gentile particles are exchanged.With regard to anyon,two different anyons braid for one circle gives the phase factor ei2πkα.We want to relate these two phase factors to each other.At the same time,bosons and fermions are their limits.This is quite natural for bosons.As we mentioned above,there is a problem for fermions.Braiding two fermions for one circle gives the wave function nothing while exchanging them gives?1.If we follow this requirement strictly,for example,exchanging two different 1/2 anyons will create a complex numberi.However,the difficulty lies in the phase factor of complex number could not be observed or detected in quantum mechanics,because the observable quantities are averages〈ψ∣ψ〉.Considering these,we have to weaken the limitation of symmetry.Because of the braiding group of anyons,we assume that the wave functions of anyons go back to themselves after braiding them for g circles.Our basic assumption goes to

ν is the practical particle number in one state andν∈N.So we have the change of the winding numberThe relation of two phase factors between two representations is

or

3.Basic relations of the second quantization operators

The second quantization form has the advantage in describing the creation and annihilation of particles.In this part,we transform the basic relations of the second quantization operators in the occupation number representation to the winding number representation to give anyon another description.

According to equations(3)and(5),the main operator relation becomes

The states in the occupation number representation are∣ν〉n.In the winding number representation,the states can be expressed as

where

a?creates a particle each time and b annihilates one.And

We also construct the particle number operatorNas

where

Now,let us consider the more general operator relations for arbitrary operators:u,v,w…The intermediate statistical bracket?n bracket is substituted by[u,v]α,g(see equation(6)).We give the famous Jacobi-like identities as examples,

4.The coherent states

The eigenstate of annihilation operator is called the coherent state[30].The coherent state of harmonic oscillator is very important in physics[31].In Gentile statistics,the coherent state is the eigenstate of annihilation operator b.In our construction,the coherent state is expressed as∣χ〉α,gin the winding number representation,so we have another expression:

where the eigenvalue χ is a Grassmann number.Grassmann numbers do not commute with the states.They satisfy

here we take

as an example and we also have

Under these assumptions,we can construct the coherent state as

where M is the normalization constant,the coefficients read

As forb?,b,there will be the same relations.

5.Berry phase

Berry phase is a very important concept in differential geometry.The appearance of Berry phase is a topological effect.The source is that the spherical surface is not homeomorphic to plane.Under an adiabatic and unitary transformation,the eigenstate of Gentile system could obtain a phase factor exp (iην)whereηνis an angle which depends on the practical particle number,and this factor is Berry phase.We assume the unitary transformation is denoted byU(θ),where θ is some kind of angle.Operator A under this unitary transformation becomeU(θ)AU?(θ).In the occupation number representation of Gentile statistics,we adopt

We have

According to equation(3),we relateexp (iην)to the phase factor of anyon in the winding number representation,equation(27)becomes

The positive and negative value ofηνcorrespond to two closed trajectories in two opposite directions.

To show the non-uniqueness of the construction,here we introduce another instruction,supposeand

Then we get a restrict of the winding number

The positive and negative value of k means braiding clockwise and counterclockwise.For instance,when n=3 and g=2,we haveν=0,1,2,3 andα=1/6.So the restrict of k isk=6ν-9=-9,-3,3,9.The difference is that equation(28)is a direct usage of equation(3),and equation(29)forces those two phase factors equal to each other to give a restrict of the winding number.

6.Conclusion and discussion

Anyon and Gentile statistics are two typical intermediate statistics beyond Bose–Einstein and Fermi–Dirac statistics.Bose and Fermi statistics are their ultimate limit conditions.When two different kinds of anyons are braiding,the wave function of the system could obtain an additional phase factor which depends on the statistical parameter and the winding number.The maximum occupation number of Gentile statistics is neither 1 nor∞,but a finite number n.After the second quantization,exchanging two Gentile particles also gives the wave function a phase factor which depends on n.The second quantization form is very convenient to create,annihilate particles and gives special properties of the system.So it is worth researching the second quantization form of anyons.But this is not easy.The algebra of braiding group of anyon,which is called Hopf algebra,is complicated.But the braiding property of anyon shows a possible correspondence between anyon and Gentile statistics.The symmetry requirement must be weakened to make the construction selfconsistent.From this point of view,we introduce a general construction of transformation between anyon and Gentile statistics to change the results of Gentile statistics in the winding number representation of anyons.In other words,we give the second quantization form of anyons indirectly.This paper is a new description of anyon.The coherent state and Berry phase are also discussed.Further researches of anyon properties are proceeding.

Acknowledgments

The research was supported by the Fundamental Research Funds for the Central Universities Grant No.2020JKF306 and NSFC Grant No.11675119.