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Real Gentile Statistical Systems: N-Annulenes Yao Shen, and Yu-Lin Zhao J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.8b05121 • Publication Date (Web): 13 Jul 2018 Downloaded from http://pubs.acs.org on July 17, 2018
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The Journal of Physical Chemistry
Real Gentile Statistical Systems: N-Annulenes Yao Shen1, ∗ and Yu-Lin Zhao2 1
School of Criminal Science and Technology, People’s Public Security University of China, Beijing 100038, China 2 Department of Physics, Tianjin University, Tianjin 300350, China Gentile statistics, which is famous for its advantages in dealing with composite particle systems, is a kind of fractional statistics. These years, researchers are concentrating on finding real systems that obey Gentile statistics. Five years ago, we discovered that the cyclic hydrocarbon polyenes called N-annulenes in H¨ uckel model had certain correspondence with Gentile oscillators. According to their rotation and dihedral symmetry, these two systems had the same energy levels and partition functions. In this paper, we give further discussions. We discuss the transformations of wave functions between N-annulenes and Gentile oscillators, the creation and annihilation operators in site picture of N-annulenes, the coherent state and the mathematical proofs of the intermediate commutation relations of operators. All our works prove that N-annulenes in H¨ uckel model are real Gentile statistical systems and offer a new algebraic method to deal with the problems of electron-electron and electron-phonon interactions.
I.
INTRODUCTION
Bosons and fermions are two types of well-known elementary particles in nature. Bosons whose wave functions are symmetric obey Bose-Einstein statistics, , whereas fermions whose wave functions are antisymmetric obey Fermi-Dirac statistics. Bosons and fermions are distinguished by the maximum occupation number(MON), which goes towards infinity by bosons, while by fermions the number equals one. The MON of bosons is infinity while that of fermions is one. Gentile statistics [1–4] and anyons statistics [5–8] are both fractional statistics. The characteristics of Gentile statistics and anyons are MON and wave function respectively. The MON of Gentile statistics is a finite number n. Bosons and fermions are two limits of Gentile statistics. When n equal to ∞, Gentile statistics becomes Bose-Einstein statistics. When n equal to 1, Gentile statistics becomes Fermi-Dirac statistics. According to the topology property, the braiding of two different anyons gives the wave function an additional phase. MON and wave function describe the fractional statistics from two different angles, and they could transform to each other [9]. The pictures of Gentile statistics and anyons are called the occupation number picture and the winding number picture. Gentile statistics is very convenient to deal with composite particle systems, especially fermions that are not approaching each other close enough to constitute bosons [10–14]. In paper [15], we showed the algebra of Gentile statistics. Fractional statistics is usually simulated by nuclear spin systems [16] and photon systems [17]. It can bring marvelous effects such as the fractional quantum Hall effect [18–20], the simulation of quantum computation [21–24], fractional excitations in cold atomic gases [25] and Fermi gas superfluid [26]. In 2013, we pointed out that the energy spectrum and partition functions of
∗ Corresponding
N-annulenes in H¨ uckel model could correspond to Gentile oscillators [27]. We then concluded that conjugate polymers (N-annulenes) offered a possible way to realize fractional statistics [28, 29]. In this paper, we research the details of the correspondence between N-annulenes in H¨ uckel model and Gentile oscillator, including the transformations of wave functions, the creation and annihilation operators, the basic operator relation of n-bracket and the coherent state. In the end, we conclude that N-annulenes in H¨ uckel model are accurate Gentile statistical systems. This paper is organized as follows. In Sec. II, The transformations of wave functions between Gentile occupation number picture and site picture of N-annulenes are constructed. In Sec. III, we show how the creation and annihilation operators act on the states of N-annulenes, express the operators in site picture and verify the basic operator relation of n-bracket. Finally, a possible construction of the coherent state of N-annulenes is discussed in Sec. IV.
II.
TRANSFORMATIONS OF WAVE FUNCTIONS
The cyclic hydrocarbon polyenes described by CN HN are called N-annulenes. They are monocyclic unsaturated hydrocarbons of organic molecules. H¨ uckel model is the simplest but very effective model to deal with these planar conjugated molecules. To be specific, N-annulenes are divided into two topological classes-H¨ uckel annulenes and M¨obius annulenes. All pz orbitals of H¨ uckel annulenes are parallel. The pz orbitals of M¨obius annulenes flip π. N-annulenes are usually described by regular polygons. When the N-annulene systems have N sites, the Hamiltonian can be written as
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H± = H0 ± β(|N i h1| + |1i hN |),
(1)
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where H± denote H¨ uckel annulenes and M¨obius annulenes respectively, and H0 is the linear polyene part H0 = β
N −1 X
(|ji hj + 1| + |j + 1i hj|),
(2)
j=1
where β is the resonance integral and j is the site number. The wave function of these two topological classes are 1 |ψj i = √ N
III.
OPERATORS IN SITE PICTURE OF N-ANNULENES
For Gentile oscillator, the creation and annihilation operators are denoted by a† and b. They are not Hermitian conjugate unless n → ∞ or n = 1. The basic operator relation is b, a† n = 1,
N X
e
−i 2jπs N
|si ,
(3)
s=1
(7)
where 2π
[u, υ]n = uυ − ei n+1 υu.
where ( 0, ±1, ±2, . . . , ± N 2−1 , j= 0, ±1, ±2, . . . , ± N2 − 1 , N2 ,
odd N even N
(4)
and |si is the site state. Odd N describe H¨ uckel annulenes and even N are M¨obius annulenes [30]. In our paper [27], we pointed out that N-annulenes and Gentile oscillators had the same energy spectrum and partition functions. Thus we predicted N-annulenes were fractional statistical systems (Gentile oscillators). In this paper, we want to give the specific transformation relations of wave functions between N-annulenes and Gentile oscillators. In other words, we express Gentile oscillators in site picture of N-annulenes. In the maximum occupation number picture of Gentile oscillators, the states are written by |νin where ν is the real occupation number, and n is the maximum occupation number (0 ≤ ν ≤ n). We assume the relation N = n + 1 and ( 2j − 1, odd ν ν= (5) 2 |j| . even ν For example, N = 6 correspond to n = 5. When N = 6, N-annulene is aromatic, and we have j = 0, ±1, ±2, 3. There are six states, √1 6
|ψ0 i = |ψ+1 i = |ψ−1 i = |ψ+2 i = |ψ−2 i = |ψ3 i =
√1 6 √1 6 √1 6 √1 6
|si
= |0i5 ,
s=1
6 P
√1 6
p b |νin = p hνin |ν − 1in , hνs+ 1in |ν + 1in a† |νin = , ν−1 p P i 2πk n+1 hνin = e .
According to equations (3), (4) and (5), the annihilation operator b annihilates one particle each time, and we have ν+1 ν+1 j = (−1) Int . (10) 2 The occupation number ν decreases one each time, the wave function of N-annulenes changes a phase ν exp [−i (−1) 2πνs/N ]. So we use |ji denotes equation (3). We have b |ji =
e
s=1 6 P
s=1 6 P
= |2i5 , (6)
e−i
s=1 6 P
|si
ei
2π 3 s
2π 3 s
2πk N
(−1)ν 2πνs E −i N (j) , e odd ν
(11)
even ν
For instance, when b |ν = 4i5 =
e−iπs |si = |5i5 .
The states are the same for antiaromatic and nonaromatic N-annulenes. We construct a consistent one-toone match between wave functions of N-annulenes and Gentile oscillators. The corresponding matches of energy spectrum and partition functions have been discussed in our paper [27].
ei
k=0
|si = |3i5 ,
|si = |4i5 ,
ν−1 P
sk=0 ν−1 P i 2πk e N |j − νi , k=0 s = ν−1 P ei 2πk N |j + νi .
e−i 3 s |si = |1i5 , iπ 3s
(9)
k=0
π
s=1 6 P
s=1
is the commutation relation of fractional statistics named n bracket. And [u, υ]n→∞ = [u, υ], [u, υ]n=1 = {u, υ}. According to [3], the effects of the creation and annihilation operators are creating or annihilating a particle of the states respectively.
s
6 P
(8)
q h4i5 |3i5 ,
ν = 2 |j| is even, and N = 6. We rewrite it in the site picture of N-annulenes as v u 3 E uX πk 4πs b |j = −2i = t ei 3 e−i 3 (j = −2) ,
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P −iπs 0 † hs | a |si = 0, e s P N −1 ei N πs hs0 | a† |si = 0,
which means P i 2πs √b e 3 |si , 6
b |ψ−2 i = =
3 P k=0
s
P 4πs 2πs πk ei 3 √16 e−i 3 ei 3 |si ,
=
s 2j−2 P i 2πk −i (1−2j)2πs N e N e |si , k=0 s b |si = 2|j|−1 P i 2πk −i |j|4πs N e N e |si ,
s
3 P k=0
πk ei 3 √16
s
3 P
=
ei
P −i 2πs e 3 |si , s
πk 3
even ν
(15)
s
s
s
odd ν
odd ν (16) even ν
k=0
|ψ+2 i .
k=0 †
In like manner, the creation operator a creates one particle each time, we also have equation (10). At this time, the occupation number νincreases one each time, the wave function of N-annulenes changes a phase ν exp [−i (−1) 2π(ν + 1)s/N ]. s (−1)ν 2π(ν+1)s E ν P 2πk N a† |ji = ei N e−i (j) , k=0 s ν P 2πk ei N |j − ν − 1i , odd ν (12) k=0 s = ν P 2πk ei N |j + ν + 1i . even ν k=0
And s a |ji =
ν−1 P
e−i
2πk N
k=0
s b† |ji =
ν P
(−1)ν 2πνs E −i N (j) , e
s 2j−1 i 4jπs P ei 2πk N N e |si , k=0 † s a |si = 2|j| P i 2πk −i (2|j|+1)2πs N e N e |si .
e−i
2πk N
k=0
In this case, The basic operator relation (7) reads a† b |ψj i = ba† |ψj i =
2j−2 P k=0 2j−1 P
ei
2πk N
IV.
COHERENT STATE OF N-ANNULENES
In this part, we discuss the coherent state of Nannulenes. The construction is not unique, we just offer the simplest one. The coherent state is the eign state of the annihilation operator b, b |Ψi = Ψ |Ψi ,
|Ψi = M
n X
δ(ν, n) |νin Ψν ,
ei
2πk N
|ψj i ,
where M is the normalization constant. In [3], we assumed
†
a b |ψj i = |ψj i .
In other words, ba† − e a† b = 1 (equation (7)). The basic operator relation of Gentile statistics is verified. In addition, when ν = 0, j = 0, we also get v u ν−1 Y uν−1 X 2πk † ν t (a ) |0i = ei N |ji , k=0
where ν = 2j − 1 or ν = 2 |j|. For the site states |si, we conclude X hs0 | b |si = 0, (14) s
(20)
and ν−1 Y
i 2π N
k=1
(19)
ν=0
Ψ |νin = λ(ν, n) |νin Ψ, ba |ψj i − e
(18)
where |Ψi is the coherent state, Ψ is the Grassmann number, and ΨN = 0. Grassmann number doesn’t commute with state, it is a kind of anticommuting c number. For Gentile statistics, the coherent state [3] is constructed as
and i 2π N
even ν
In summary, we have shown how creation and annihilation operator act on the states in the site picture of N-annulenes, and the basic operator relation-n-bracket of Gentile statistics has been verified.
|ψj i ,
k=0
†
(17)
k=0
(13) (−1)ν 2π(ν+1)s E −i N (j) . e
odd ν
δ(ν, n) =
p=0
λ(p, n) p . hp + 1in
(21)
The simplest method is using equation (3) and (5) to replace all |νin with |ψj i, and νwith j. We also give another construction here. According to equation (19) and (16), we reconstruct the coherent state of N-annulenes in the site picture as N N −1 (−1)m Int( m+1 ) M XX 2 2πs N |Ψi = √ δ(m, N − 1)ei |si Ψm , N s=1 m=0 (22)
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here, we assume 2π
Ψ |si = e−i N s |si Ψ.
(23)
In this case, bring equation (22) into equation (18), we can get the coefficient m Q ei [(−1)m−1 m−m−1]2πs √N , odd m hkin k=1 (24) δ(m, N ) = m−1 [(−1) m+m−1]2πs m Q ei √N . even m hkin
k=1
The normalization constant reads X ¯ m |δ(m, N )|2 ]− 12 . M = N [ (ΨΨ)
(25)
m
V.
annihilation of particles. Therefore it is very meaningful to simulate Gentile systems or even find real systems that obey Gentile statistics. Five years ago, we discovered that the cyclic hydrocarbon polyenes-N-annulenes in H¨ uckel model are possible Gentile systems in nature. The energy levels and partition functions of N-annulenes in H¨ uckel model and Gentile oscillators had been corresponded to each other’s. In this paper, through one to one correspondence of wave functions, the expressions of operators and the intermediate commutation relations of operators, we could confirm that N-annulenes in H¨ uckel model are real Gentile statistical systems. Furthermore, the coherent states of N-annulenes are discussed. It is a remarkable fact that we could also correspond Nannulenes in H¨ uckel model to anyon statistics using the method mentioned in our paper [9].
CONCLUSION
Fractional statistics, which is famous for its marvelous characteristics, has become a research focus in recent years [31–33]. Gentile statistics is a kind of fractional statistics, where the states are expressed with occupation numbers. Gentile picture is regarded as occupation number picture, which facilitates the research on creation and
The research was supported by The Opening Project of Shanghai Key Laboratory of Crime Scene Evidence No.2016XCWZK11.
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ACKNOWLEDGEMENT
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