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Complete state counting for Gentile's generalization of the Pauli exclusion principle ARTICLE in PHYSICA A: STATISTICAL MECHANICS AND ITS APPLICATIONS · OCTOBER 2007 Impact Factor: 1.73 · DOI: 10.1016/j.physa.2007.05.036

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Physica A 384 (2007) 297–304 www.elsevier.com/locate/physa

Complete state counting for Gentile’s generalization of the Pauli exclusion principle R. Herna´ndez-Pe´rez, Dionisio Tun1 Sate´lites Mexicanos, S.A. de C.V., Centro de Control Satelital Iztapalapa, Av. de las Telecomunicaciones, S/N CONTEL Edif. SGA-II, Me´xico, D.F. 09310, Mexico Received 11 January 2007; received in revised form 7 May 2007 Available online 18 May 2007

Abstract In the present work, we perform the complete state counting for Gentile’s approach to the generalized Pauli exclusion principle (GPEP), which has been lacking in the literature. We count the total number of ways to allocate n identical particles occupying a group of g states with up to q particles in each state, in order to derive an exact expression for the statistical weight. Our obtained expression for the statistical weight gives the fermionic one for q ¼ 1; and for q41, it tends fast to a bosonic weight. Moreover, we perform a numerical comparison between our state counting and Wu’s (corresponding to the Haldane–Wu’s formulation of the GPEP), which implies that Gentile’s formulation gives rise to more boson-like behavior while Haldane–Wu’s approach to more fermion-like behavior; this difference lies on the fact that each formulation has its own state-occupation rules on which correlation plays a key role. r 2007 Published by Elsevier B.V. PACS: 05.30.d Keywords: Quantum statistics; Gentile statistics; Intermediate statistics; Fractional exclusion statistics

1. Introduction The study of certain low-dimensional quantum systems has revealed a very interesting property: their elementary excitations obey exotic quantum statistics not restricted to the well known either Bose–Einstein (BE) or Fermi–Dirac (FD) statistics, in which conventional quantum statistics is classified. The most known example of these systems is the quantum Hall effect, on which it has been reported that the Laughlin quasiparticles have many-body wave function that has a general phase eiy (not restricted to be 1 or 1) under particle exchange. These quasi-particles are known as anyons [1–3], i.e., particles whose quantum statistics

Corresponding author.

E-mail address: [email protected] (R. Herna´ndez-Pe´rez). In absence from: Departamento de Fı´ sica, Escuela Superior de Fı´ sica y Matema´ticas, Instituto Polite´cnico Nacional, Edif. 9 U.P. Zacatenco, Me´xico, D.F. 07738, Mexico. 1

0378-4371/$ - see front matter r 2007 Published by Elsevier B.V. doi:10.1016/j.physa.2007.05.036

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interpolates between that of bosons and fermions. This statistics interpolation has suggested a generalization of the Pauli exclusion principle, on which conventional quantum statistics is based on. There are two approaches to formulate the generalized Pauli exclusion principle (GPEP): the one given by Gentile’s [4] and that proposed by Haldane–Wu [9,12]. Even though in both formulations a single quantum state can accommodate up to q particles, the occupation rules are different in each case [5]. The difference is based on whether the number of available quantum states depends or not on which states have been already occupied. Gentile proposed his generalization as an intermediate statistics [4], on which the maximum number of particles in any quantum state is neither 1 (FD) nor 1 (BE), but equal to a certain finite number d; also, he provided the most probable distribution of the occupation numbers ni . Moreover, the state-occupation rules in Gentile’s approach are such that when a new particle is added to the system, it can occupy any state that is not full, regardless which states are already occupied, i.e., no correlation at all on the way the particles occupy the states. Even though different theoretical studies about Gentile’s statistics have continued to be developed [5–7], a complete combinatorial state counting for Gentile’s formulation is lacking, and actually this complete counting is the aim of the present work. Both FD and BE statistics are recovered from Gentile’s generalized statistics; although, as reported recently by Dai and Xie [8], BE statistics is recovered from Gentile’s statistics only when the fugacity zo1. On the other hand, Haldane proposed another approach to generalized quantum statistics called fractional exclusion statistics (FES) [9]. Rather than modifying the many-body exchange phases, Haldane generalized the Pauli principle by introducing new rules for occupying single-particle quantum states, establishing exclusion rules that appear while single-particle states are being occupied. The basic idea of FES is that adding to a system a number of particles DN, they block Dd of the states available for the next particle according to the linear relation Dd ¼ aDN, where the parameter a is called statistical interaction. This corresponds to a correlation between the particles in the sense that the available states now depend on which states have been already occupied. This also could be seen as a statistical repulsion between the particles in the phase space, and only very special types of interactions give rise to this type of exclusion of single-particle states [10]. One of Haldane’s original examples was a one-dimensional spin chain; moreover, he showed that the Laughlin quasiparticles (or generally anyons in a strong magnetic field, i.e., confined to the lowest Landau level) also obey exclusion statistics [9,11]. Also, fractional statistics is known to be applicable to realistic correlated systems, such as 1D spin chains and 2D quantum Hall systems with anyonic quasi-particles [9,12–16]. Following Haldane’s approach, Wu proposed the following interpolation for the statistical weight of ni identical particles occupying a group of gi states [12]: Wi ¼

½gi þ ðni  1Þð1  aÞ! , ni !½gi  ani  ð1  aÞ!

(1)

where a ¼ 0 corresponds to bosons ðW bi ¼ ðgi þnnii 1ÞÞ and a ¼ 1 to fermions ðW fi ¼ ðgnii ÞÞ. Wu showed that this statistics respects Haldane’s generalization of Pauli exclusion principle. Also, he provided the most probable distribution of the occupation numbers ni ; however, no detailed combinatorial state counting was provided [12]. Later, Polychronakos performed a microscopic formulation of Haldane’s exclusion statistics and provided a concrete model to realize Wu state counting [17]; from which he obtained the following expression for the statistical weight: W i ¼ gi

½gi þ ð1  aÞni  1! , ni !½gi  ani !

(2)

where, as in Eq. (1), the bosonic and fermionic weights are recover when a ¼ 0 and 1, respectively. Even though this equation differs from Eq. (1), it leads to the same statistics than Wu’s weight does [17]—which also will be showed by our numerical results. Few years ago, Aringazin and Mazhitov [18] reported a combinatorial interpretation of Haldane–Wu FES, counting the configurations having a maximal possible number of totally occupied states (exactly q particles in state), corresponding to a kind of FES. However, this approach does not include other configurations where there are more than one partially occupied state. In the present work we perform the complete combinatorial state counting for Gentile’s generalization of the Pauli exclusion principle, which has been not performed in the literature before. Following two different

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combinatorial procedures, we count all the possible combinations to allocate n identical particles occupying a group of g states, with up to qpn particles in each state. Our state counting is valid for arbitrary dimensions, since no a priori assumption on the number of dimensions has been made to perform the state counting. Moreover, we perform a numerical comparison between the obtained statistical weight and both Wu’s and Polychronakos’, from which it is shown that our statistical weight grows faster than both Wu’s and Polychronakos’ for increasing q, tending faster to the bosonic weight, which suggests that for q41, Gentile’s formulation predicts bosonic-like statistics, while Haldane–Wu’s predicts fermionic-like statistics. 2. Calculation of the statistical weight We are interested in obtaining the statistical weight for the Gentile generalization of the Pauli exclusion principle, on which the available states are all those that are not full, without considering any correlation with the currently occupied states. Then, we have to count all the combinations to allocate n identical particles on a group of g single-particle states with up to qpn particles in a certain state. Let us define the statistical weight as Rq ðg; nÞ. For fermions, q ¼ 1 and R1 ðg; nÞ ¼ ðgnÞ; for bosons, q ¼ n and Rn ðg; nÞ ¼ ðgþn1 n Þ. To obtain the weight Rq ðg; nÞ for other values of q, for instance q ¼ n  1, we could start from the bosonic weight (Rn ðg; nÞ) and then subtract the number of combinations having a state occupied by the n particles (in this case there are g possible states). Once we have obtained Rn1 (to simplify we will omit the arguments ðg; nÞ in subsequent discussion), we could subtract the number of combinations with n  1 particles in one state to obtain Rn2 . Repeating this procedure, we could obtain the weight for a certain q ¼ n  m, for 1pmpn  1. A second procedure is starting from the fermionic weight (R1 ) and then add up the number of combinations with at least one state occupied by two particles to obtain R2 . Then, by adding up the combinations with at least one state with three particles, we could obtain R3 . Repeating this procedure, we could obtain Rq , for 1oqpn. In the following subsections we obtain a binomial formula for the statistical weight when proceeding in the first fashion described above, while the second strategy leads to a statistical weight expressed in a recursive formula. 2.1. First approach: starting from the bosonic weight For q ¼ n  1, we start with Rn and then subtract the number of combinations having a state occupied by the n particles, i.e. ðn; 0; . . . ; 0Þ. In this case, there are g possible combinations. Therefore, Rn1 ¼ Rn  g ¼ Rn  gðg1 0 Þ. Now, to calculate Rn2 we must subtract to Rn1 the number of arrangements of the form ðn  1; 1; 0; . . . ; 0Þ, which are a total of gðg  1Þ combinations, thus Rn2 ¼ Rn1  gðg  1Þ ¼ Rn1  gðg1 1 Þ. Similarly, for q ¼ n  3, we subtract to Rn2 the number of possible combinations with the following configurations: ðn  2; 2; 0; . . . ; 0Þ and ðn  2; 1; 1; . . . ; 0Þ. There are gðg  1Þ of the first kind, and there are g1 g1 g1 gðg1 2 Þ of the second kind, therefore Rn3 ¼ Rn2  g½ðg  1Þ þ ð 2 Þ ¼ Rn2  g½ð 1 Þ þ ð 2 Þ. For Rn4 , we must subtract to Rn3 the number of combinations with the configurations: ðn  3; 3; 0; . . . ; 0Þ, ðn  3; 2; 1; . . . ; 0Þ and ðn  3; 1; 1; 1; . . . ; 0Þ, i.e., gðg  1Þ, gðg  1Þðg  2Þ and gðg1 3 Þ, respectively. In this way, g1 g1 g1 Rn4 ¼ Rn3  g½ðg  1Þ þ ðg  1Þðg  2Þ þ ðg1 Þ ¼ R  g½ð Þ þ 2ð Þ þ ð n3 3 1 2 3 Þ. Before continuing, let us point out that the algorithm described above provides the exact statistical weight for qX½n=2 (where ½x is the integer part of x), but for qo½n=2 it subtracts some combinations more than once; and therefore, the obtained statistical weight will be lower than the actual amount of combinations allowed. For instance, for n even and for q ¼ n=2  1, the algorithm subtracts from Rn=2 all the combinations with at least n=2 particles in a certain state, including the configuration ðn=2; n=2; 0; . . . ; 0Þ. In this case, the algorithm counts gðg  1Þ combinations for that configuration, instead of gðg  1Þ=2, therefore in this case the algorithm will subtract twice the correct number of combinations. For q ¼ n=2  2, the algorithm subtracts from Rn=21 all the combinations with at least n=2  1 particles in a certain state, for example ðn=2  1; n=2 þ 1; 0; . . . ; 0Þ, but this configuration is not present in Rn=21 as it was subtracted already from Rn=2þ1 while obtaining Rn=2 , thus the algorithm will be subtracting more combinations that needed. This is not the case for qX½n=2, when the described algorithm counts exactly the number of possible states.

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In summary, the weights obtained up to this point are ! g1 Rn1 ¼ Rn  g , 0 ! g1 , Rn2 ¼ Rn1  g 1 " ! !# g1 g1 þ , Rn3 ¼ Rn2  g 1 2 ! ! " !# g1 g1 g1 þ2 þ Rn4 ¼ Rn3  g . 3 1 2 By adding up these equations member to member, we obtain the following expression for Rn4 :  3   X 3 g1 . Rn4 ¼ Rn  g k k k¼0 Generalizing for the case q ¼ n  m, with 1pmpn  1, we obtain:   m1 X m1 g1 , Rnm ¼ Rn  g m1k k k¼0 x Þ. The sum in the previous equation can be written as a binomial where we have used the identity: ðxyÞ ¼ ðxy coefficient by means of the identity [19]: ! ! p   X s rþs r ¼ . pk p k k¼0

Therefore, after replacing m by n  q, we obtain the following expression for the statistical weight (named binomial formula): !   gþnq2 gþn1 Rq ¼ g . (3) nq1 n The advantage of this equation is that it is very short and it can be evaluated straightforwardly, and even when it does not provided the exact value for the counting for qo½n=2, it is a good approximation in this range (as will be discussed later); while for qX½n=2 it indeed provides the exact value. 2.2. Second approach: starting from the fermionic weight The other approach to the exclusion statistical weight is starting from the fermionic weight ðR1 Þ and then add up the number of combinations with up to two particles per energy state. Thus, we need to add up to R1 the number of all the combinations of the following forms: ð2; 1; . . . ; 1; 0; . . . ; 0Þ, ð2; 2; 1; . . . ; 1; 0; . . . ; 0Þ, g g1 g g2 ð2; 2; 2; 1; . . . ; 1; 0; . . . ; 0Þ, and so on. There are gðg1 n2Þ ¼ ð1Þðn2Þ combinations of the first kind, ð2Þðn4Þ of the g g3 second, ð3Þðn6Þ of the third, and so on. Therefore, the statistical weight R2 (for q ¼ 2) is given by ! ! ! ! ! ! g g  ½n=2 g1 g2 g g R2 ðg; nÞ ¼ R1 ðg; nÞ þ þ þ  þ ½n=2 n  2½n=2 n2 n4 1 2 ! ½n=2 ! ! gj X g g ¼ þ j n  2j n j¼1 ! ½n=2 g X ¼ R1 ðg  j; n  2jÞ, j j¼0 where we have used that R1 ðk; lÞ  ðklÞ.

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The above expression is exact as it includes all the possible combinations of states with up to two particles per state, following Gentile’s state-occupation rules. In general, the exact statistical weight correspondent to the number of quantum states of n identical particles occupying a group of g states with up to qpn in a certain state, is given by the expression (named recursive formula) ! ½n=q X g Rq ðg; nÞ ¼ (4) Rq1 ðg  j; n  qjÞ. j j¼0 This expression states in compact form the complete combinatorial state counting for the Gentile’s generalization of the Pauli exclusion principle.

3. Comparison between the Gentile and Haldane–Wu statistical weights With the obtained expressions of the statistical weight for Gentile’s formalism, we are in position to perform a numerical comparison between Gentile statistical weight (Eqs. (3) and (4)), and those correspondent to the Haldane–Wu formalism (Eqs. (1) and (2)); in order to study the differences between them. To perform such comparison, we picked up different values of g, n and q. Fig. 1 depicts the comparison for g ¼ 500 and n ¼ 50, for instance. As can be seen both Gentile and Haldane–Wu weights start at the pure fermionic one.2 However, there is a remarkable difference between them: for the same values of g and n, Haldane–Wu statistical weights predict a smaller number of possible combinations than Gentile statistical weight. Moreover, Gentile weights tend faster to the pure bosonic one than those of Haldane–Wu. These two signatures suggest that Haldane–Wu statistical weight predicts a more fermion-like statistics for particles whose a is not 0 or 1 [12]; while Gentile weight predicts a more bosonic-like behavior. This difference in the predicted behavior can be understood considering the distinct state-occupation rules established by each formalism. Namely, in the Haldane–Wu formalism every particle added to the system blocks a certain number of states which not only depends on the number of particles already present, but also on which states are already occupied [9,12], therefore, strong correlation in the state occupation is present. This is not the case in the Gentile formulation on which every particle added to the system can be allocated in any state that is not full already, i.e., no correlation to which states are already occupied. Thus, the correlation affects significantly the availability of states as it imposes a more stringent condition for the different possibilities to occupy the states, and therefore, the state counting on each formalism will take different values. In Gentile formulation, for instance, the number of available states decreases slowly with the number of added particles (as not every added particle decreases the number of available states), while in the Haldane–Wu formulation every added particle will block states. Then, the larger the number of available states to occupy, the larger is the number of combinations to occupy them; and, therefore, the larger is the statistical weight. In addition, Fig. 2 depicts the difference between the statistical weights given by the recursive formula and Wu interpolation. As can be seen, this deviation is greater for q ¼ 2 and it decreases for both increasing g and q. Moreover, Fig. 1 shows that both Wu and Polychronakos weights almost match, what is expected since both are compliant with the Haldane–Wu formulation [17]. On the other hand, for the Gentile statistical weight we obtained, we can see that the binomial formula gives a smaller value of the statistical weight for fermions (q ¼ 1) than expected. This is due to the above-mentioned fact that for qo½n=2, while subtracting combinations to previous configurations, some of them were already subtracted. Nevertheless, it is a good approximation in this range; while it indeed provides the complete statistical weight for qX½n=2. This can be seen from Fig. 3, which shows the deviation of the binomial formula (Eq. (3)) from the recursive one (Eq. (4)) for different values of g. The deviation is maximum for q ¼ 1 (fermionic weight), while for 1oqo½n=2 the binomial formula approximates very well to the recursive one.

2

In fact, only our recursive expression for Gentile starts at the fermionic weight, because the binomial one does not provide the exact value for q ¼ 1.

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x 1071

3

Statistical weight

2.5 2 Recursive Binomial Wu Polychronakos

1.5 1 0.5

0

10

20

30

40

50

q Fig. 1. Illustrative plot for the numerical comparison between the Gentile, Eqs. (3) and (4)), and Haldane–Wu statistical weights, Eqs. (1) and (2), as a function of the maximum allowable state occupation q and for g ¼ 500 and n ¼ 50, for instance. As can be seen, each formulation of the GPEP leads to different behaviors of the statistical weight.

Deviation: Wu from Recursive (%)

0

−20

−40

−60

g = 100 g = 200 g = 300 g = 400 g = 500 g = 600

−80

−100

0

10

20 q

30

40

Fig. 2. Deviation of Wu interpolation, Eq. (1), from the recursive formula for Gentile’s formulation, Eq. (4), as a function of the maximum allowable state occupation q. We used n ¼ 40 and different values of g.

4. Conclusions We performed the complete combinatorial state counting for Gentile’s generalization of the Pauli exclusion principle, which has been not performed in the literature before. We derived the statistical weight by counting the number of total ways to allocate n identical particles occupying a group of g states, with up to qpn particles in each state. We obtained two formulas for this weight depending on the approach followed: (i) starting from the pure-bosonic weight and then subtracting successively the combinations with states populated with more than q particles, resulting in Eq. (3) which provides an exact counting of states for qX½n=2, and a good approximation for 1oqo½n=2, while is a short expression easy to calculate and handle;

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x 104

Deviation: binomial from recursive (%)

0 −2 −4 −6 g = 200, n = 20 g = 300, n = 30 g = 400, n = 40 g = 500, n = 50 g = 600, n = 60

−8 −10 −12 −14 0

5

10

15

q Fig. 3. Deviation of the binomial formula, Eq. (3), from the recursive one, Eq. (4), as a function of the maximum allowable state occupation q and for different values of g and n. As can be seen, the deviation is maximum for lower values of q, but then both formulas match.

and (ii) starting from the pure-fermionic weight and then adding up successively the combinations with at least one state populated with 2; 3; . . . ; q  1 particles, resulting in Eq. (4) which provides an exact counting of states for 1pqpn, and it is suitable for numerical analysis. When comparing the Gentile’s statistical weight to Wu’s interpolation, we showed that they differ considerably since, as q increases, Wu’s interpolation tends to the pure-bosonic weight in a slower fashion than predicted by our statistical weights, which predict that the Gentile statistical weight tends rapidly to the purebosonic weight, even for small q. This implies that Gentile’s formulation gives rise to more boson-like behavior while Haldane–Wu’s approach to more fermion-like behavior. This difference in the predicted behaviors lies on the distinct state-occupation rules established by each formalism, mainly due to the presence of correlation: in the sense that the particles already in the system block available states for the upcoming particles in a different way. For Haldane–Wu formalism, the number of available states depend on which states are already occupied; whereas for the Gentile formalism, the available states are all those that are not full already. Thus, even when in both cases the occupation for each state is pq (as demanded by a generalization of the exclusion principle), the number of available states is significantly different between both formalisms; therefore, the larger the number of available states to occupy, the larger is the number of combinations to occupy them; and, therefore, the larger is the statistical weight for Gentile’s formulation. Acknowledgments We would like to thank the reviewer for his suggestions and criticisms, which helped to improve this manuscript. References [1] [2] [3] [4]

J.M. Leinaas, J. Myrheim, Nuovo Cimento 37B (1977). F. Wilczek (Ed.), Fractional Statistics and Anyon Superconductivity, World Scientific, Singapore, 1989. C. Nayak, F. Wilczek, Phys. Rev. Lett. 73 (1994) 2740. G. Gentile, Nuovo Cimento 17 (1940) 493; G. Gentile, Nuovo Cimento 19 (1942) 109. [5] A.K. Rajagopal, Phys. Lett. A 214 (1996) 127. [6] W.S. Dai, M. Xie, Physica A 331 (2004) 497.

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