Complex-valued fractional statistics for D-dimensional harmonic oscillators

Physics Letters A 378 (2014) 100–108

Contents lists available at ScienceDirect

Physics Letters A www.elsevier.com/locate/pla

Complex-valued fractional statistics for D-dimensional harmonic oscillators Andrij Rovenchak ∗ Department for Theoretical Physics, Ivan Franko National University of Lviv, Ukraine

a r t i c l e

i n f o

Article history: Received 5 September 2013 Received in revised form 19 October 2013 Accepted 3 November 2013 Available online 15 November 2013 Communicated by C.R. Doering Keywords: Phase transition Quantum gas Finite system Fractional statistics

a b s t r a c t A system of isotropic harmonic oscillators obeying the Polychronakos fractional statistics with a complex parameter in a space having the dimension D > 1 is studied. Temperature dependences of the thermodynamic functions are analyzed in different temperature domains. The nature of the observed phase transitions is clarified. Both numerical and analytical estimates for the critical temperature are made depending on the number of particles, space dimensionality, and statistics parameter. Approximate correspondence with some other fractional statistics types is established. © 2013 Elsevier B.V. All rights reserved.

1. Introduction The notion of fractional statistics for quantum systems can be approached in a number of ways [1,2]. A vast spectrum of problems, which can be analyzed within different types of statistics fractionalization, include high-temperature superconductivity [3], fractional quantum Hall effect [4], low-dimensional interacting systems [5], Bose systems with a finite number of particles [6], etc. This work contains the generalization of two previous studies [7,8] devoted to the analysis of the fractional statistics of Polychronakos [9] with the parameter being a complex number on the unit circle: α = e i π ν . Calculations are made for the “bosonic” side 0  ν  1/2 with a special attention to the bosonic limit ν → 0. Complex physical quantities are usually linked to dissipative processes, which is probably best known from the complex refractive index in classical physics. With its roots in quantum chromodynamics [10], complex chemical potential was successfully applied also in other domains of physics [11,12], in particular in studies of the Bose condensate decay [13]. The notion of complex temperature was utilized in [14,15]. Some aspects of complex energy can be exemplified by papers [16,17]; influence of the laser field on atoms moving in crystals can be modeled via a complex external potential [18]. Imaginary term in the discrete nonlinear Schrödinger equation was used in [19] to describe Bose–Einstein condensates in leaking optical lattices. In the study of a deformed gas of p .q-bosons [20] the authors suggested that complex defor-

*

Tel.: +380 32 2614443. E-mail address: [email protected].

0375-9601/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physleta.2013.11.011

mation parameters can effectively account for interparticle interactions. The above applications are not intended as a comprehensive list but only to show a variety of possibilities for complex physical characteristics to appear in realistic problems. With respect to various types of fractional statistics, complex parameters are mostly applied when studying q-deformed algebras [21–23]. Some approaches, cf. [24], lead to the Gentile statistics [25] with occupation numbers limited by some finite s > 1. In [7], the extension of the statistics parameter to a complex plane was proposed and some preliminary results were obtained. In particular, it was shown that in the limits of ν → 0 and ν → 1 the respective systems effectively correspond to Bose or Fermi systems with a small dissipative part in the elementary excitation spectrum. Also, heat capacities of a two-dimensional oscillator system with non-conserving number of particles (zero chemical potential) were calculated for different values of the statistics parameter ν = 0 ÷ 1. A detailed study of a one-dimensional harmonic oscillator system was given in [8], where analytical and numerical results regarding the peculiarities of thermodynamic functions were obtained and estimations about possible experimental observations of the predicted critical behavior were made. In the present work, properties of a system of isotropic harmonic oscillators in a space having the dimension D > 1 are studied. The oscillators obey the Polychronakos fractional statistics with a complex parameter as defined above. Heat capacity of such a system is calculated and its behavior is studied in a wide temperature domain for different values of space dimension and the statistics parameter with a special attention paid to the bosonic limit. Letting the statistics parameter to step out from the real domain significantly enriches the variety of phenomena in such

A. Rovenchak / Physics Letters A 378 (2014) 100–108

systems. Such a problem is not only of purely academic or mathematical interest. With quite simple underlying techniques it can serve for description of complex physical processes. For instance, the observed several phase transitions can be thought as a possible mathematical model for cascade phase transitions [26–28] or a recently suggested two phase transitions in liquid helium [29]. The obtained results can be easily transferred from oscillators to systems of free particles in dimensions 2  D  4 due to similarities of the density-of-state functions. Indeed, for the linear spectrum of harmonic oscillators the density of states is g (ε ) ∝ ε D −1 while for free particles g (ε ) ∝ ε D /2−1 . Therefore, the obtained approach can be used to study the properties of a continuous transition between planar and bulk geometries, including processes in helium films of different thickness [30,31]. The paper is organized as follows. Section 2 contains a brief introduction of the statistics and derivation of expressions to be used for calculations of thermodynamic functions. Upon an analysis of the high-temperature limit in Section 3, the phase transition domain is studied in Section 4. It contains in particular analytical approximations for the critical temperatures, which are based on expressions given in Appendix A, as well as some observations regarding the issue of the so-called critical statistics parameter. A brief analysis of the low-temperature behavior of the energy and heat capacity is presented in Section 5 together with summary of the results. Finally, a discussion in Section 6 concludes the paper. 2. Thermodynamic functions In the Polychronakos statistics [9] the occupation numbers of the jth level with energy ε j at temperature T are given by

nj =

1 z −1 e ε j / T − α

(1)

,

where z is the fugacity and α is the statistics parameter. Note that such an expression was also used to model anyons [32]. In [7,8] the generalization of this statistics was suggested, where this parameter was a complex number of the unit circle, α = e i π ν = α  + i α  . Further in this work, primes denote the real part of the respective quantity and its imaginary part is marked by a double prime. While a more general approach would consist in considering the complex statistics parameter in the form α = a + ib (cf. [22, 23]), restriction to the unit circle is a convenient way to represent physically interesting limits α → ±1 corresponding to the Bose and Fermi distributions. Moreover, this reduces the number of parameters: one ν instead of the pair a, b. Physically interesting problems include the limits of ν → 0 and ν → 1 reproducing the Bose or Fermi statistics, respectively. If the elementary excitation spectrum ε j of a Bose/Fermi system has a small dissipative correction γ j , so that ε j =  j + i γ j , an approximate equivalence with a system obeying the defined statistics can be established in the form: γ j = πν T  j , where T is a temperature [7]. In the standard calculation scheme, fugacity z is defined by the number of particles N:

N=



g jn j =

j



gj

j

z −1 e ε j / T − α

,

(2)

In this work, the system of isotropic harmonic oscillators in a D-dimensional space is considered. The jth level energy and degeneracy are given by [33,34]:

ε j = h¯ ω j ,

 j

g j ε jn j =



g jε j

j

z −1 e ε j / T − α

and other thermodynamic quantities.

(3)

Γ ( j + D) , Γ ( j + 1)Γ ( D )

gj =

j = 0, 1 , 2 , 3 , . . . ,

(4)

where the space dimension D is not limited to integer values for the sake of generality. In particular, fractional dimensions can model a porous medium [35–37] or effectively account for the presence of an external potential [38]. To simplify the analysis it is convenient to substitute the summation in Eqs. (2) and (3) with integration introducing the density of state function g (ε ):

∞ N=

g (ε ) dε z −1 e ε / T − α

0

∞ E=

ε g (ε) dε z −1 e ε / T − α

0

,

(5)

.

(6)

Numerical solutions of both Eqs. (2)–(3) and Eqs. (5)–(6) demonstrate only some temperature shifts of peculiar points in thermodynamic functions, while the qualitative behavior remains the same irrespectively of discrete or continuous approach used. When passing from summation to integration, it is essential to write the contribution of the ground state ε = 0 explicitly, similarly to the standard analysis of the Bose–Einstein condensation in a three-dimensional ideal gas. Indeed, for space dimensions D > 1 the density of state function

g (ε ) =

1

N0 =

1

Γ ( D ) (h¯ ω) D

ε D −1

(7)

ε = 0 while the respective term in the sum is finite:

vanishes at

z 1 − zα

(8)

.

Note that such a procedure is not required for D = 1, cf. [8]. Integrals in (5)–(6) can be expressed using the polylogarithm function:

N=

z

D

1

+

1 − zα

E = h¯ ω





D

T

α h¯ ω

Li D ( zα ),

(9)

 D +1

T

Li D +1 ( zα ),

α h¯ ω

(10)

where

Lis (x) =

∞ k  x k =1

(11)

.

ks

Differentiating energy with respect to temperature we obtain heat capacity in the following form:

C=

dE dT

=

D



D

1

α h¯ ω 

×

where g j is the degeneracy of the jth level. This equation gives z = z( T , N ), which can be used to calculate an energy:

E (T , N ) =

101

∂ z/∂ T z



T D +1 Li D ( zα ) + ( D + 1) T D Li D +1 ( zα ) . (12)

The above equation can be rewritten via N and E as follows:

C=

∂z DT E ( N − N 0 ) + ( D + 1) ∂T z T

(13)

102

A. Rovenchak / Physics Letters A 378 (2014) 100–108

(πν )2

suggesting possible critical behavior to have origins both in energy and fugacity dependences on the temperature. Further in this work special notations are used for the real and imaginary parts of both energy and heat capacity:

α  = cos πν 1 −

E = E + iΓ ;

Similar expressions are available for some other types of fractional statistics, including several q-oscillator models [20,40–44].

C = C + i Θ.

(14)

It is worth noting to facilitate further analysis that for D-dimensional oscillator systems the thermodynamic limit is formulated by the following condition:

ω D N = const,

(15)

which can be obtained from several considerations [6,39]. By virtue of the above expression it is immediately clear that Eqs. (10), (12) represent extensive quantities E , C ∝ N, as one would expect. 3. High-temperature limits

1



D

T

Li D ( zα ) =

α h¯ ω

1



T

α h¯ ω

D  zα +

( zα )2 2D

 + · · · . (16)

Taking the first term in the series only one can easily obtain the high-temperature behavior of the fugacity as follows:

 z=N

h¯ ω

D

α 

4.1. Estimation of the critical temperature In the model under consideration, phase transitions caused by the complex nature of the statistics parameter occur [8]. Mathematically this is explained by a singularity in the denominator of the integrals in (5), (6) when the imaginary part of zα is zero:

 

2D

N

h¯ ω

 D 2 (18)

.

T

With the high-temperature expression for the fugacity (17) one can immediately obtain the energy and heat capacity in this limit as:

E = E = DNT ,

C = C = D N.

(19)

These are classical expression for the energy and heat capacity of D-dimensional harmonic oscillators. As expected, no influence of statistics is observed in the classical limit. The first correction to the classical limit can be found by inverting the polylogarithm series, see Eq. (A.2) in Appendix A. This gives

 zα = N α

h¯ ω T

D

   D 2 h¯ ω − 2− D N α + ···, T

(20)

 Nα =

D

T

Li D ( zα ).

h¯ ω



E = D N T 1 − 2− D −1 α

N (h¯ ω) D



TD

(21)

The above equation is an analog of the virial expansion. Comparing it to the results obtained within some other fractional statistics one can establish approximate correspondence between different types of statistics [2]. For instance, considering only the real part of energy E of a two-dimensional system and comparing it to the anyonic gas with the second virial coefficient b2 = −(1 − 4η + 2η2 )/4, where η ∈ [0; 1] is the statistics parameter, or to the Haldane–Wu gas with b2 = (2g − 1)/4, where g is the statistics parameter, we can see the relations:



αN

h¯ ω

D  (25)

.

T

(k)

The condition defining critical temperatures T c (generally speaking, this can be a set, not just a single value) is as follows:

 Im invLi D



αN

h¯ ω

D  = 0.

(k)

Tc

(26)

Unfortunately, there is no closed-form expression for the inverse polylogarithm via the elementary or known special functions, so the only reliable treatment of this problem is the numerical solution. Some estimations with several approximate expressions for invLi D ( w ) from Appendix A are presented below. Using the series expansion given by Eq. (A.2) up to w 2 only, after simple transformations one obtains the following expression for T c :

T c = h¯ ω

(2N cos πν )1/ D 2

(27)

.

With series up to w 3 a rather cumbersome but still affordable analytical form is obtained:

N

 + ··· .

(24)

Its solution can be formally written using the inverse polylogarithm function invLi D ( w ), see Appendix A. For the fugacity we immediately obtain:



which, being inserted in Eq. (10), after simple transformations yields:

(23)

For temperatures far enough from zero, the contribution of the ground state is negligible and Eq. (9) reduces to the form:

zα = invLi D

which is a real number. To obtain the first non-vanishing contribution from the imaginary part of the fugacity, two terms of the polylogarithm series in (16) must be considered giving:

z = −

for anyons, (22) for Haldane–Wu statistics.

4. Phase transition domain

(17)

,

T

∝

Im( zα ) = 0.

As T → ∞ the value of N 0 is negligible and the argument of the polylogarithm in (9) satisfies | zα |  1:

N=

2 1 − 4η + 2η 2 1 − 2g



=

h¯ ω

D

Tc 2− D sin 2πν −



2−2D sin2 2πν − 4(21−2D − 3− D ) sin πν sin 3πν 2(21−2D − 3− D ) sin 3πν

.

(28) Note, however, that for some values of the statistics parameter ν it leads to a complex-valued T c , cf. Tables 2–3. This is solely an effect of the approximation and complex temperatures are not applicable in the problems under consideration. From Eq. (A.9) the critical temperature is defined by:

 N

h¯ ω Tc

2 =

π2 sin2 πν



5 + cos 2πν 6

 − cos πν .

(29)

A. Rovenchak / Physics Letters A 378 (2014) 100–108

Table 1 Critical temperature estimation from different expressions for the inverse polylogarithm function, D = 2. The values are given for N = 1000 in h¯ ω units.

103

Table 5 Critical values of the statistics parameter space dimension D.

νc for different number of particles N and

ν

Exact

Eq. (27)

Eq. (28)

Eq. (30)

Eq. (29)

Eq. (31)

Eq. (50)

D

N = 102

N = 103

N = 104

N = 105

N = 106

0.05 0.10 0.20 0.25 0.30 0.40 0.50

22.5 20.9 18.2 17.0 15.8 13.5 11.2

22.2 21.8 20.1 18.8 17.1 12.4 0

19.8 19.5 18.4 17.6 16.6 14.1 10.9

17.4 17.1 16.3 15.7 15.0 13.2 11.1

23.0 22.0 20.1 19.1 18.0 15.4 12.2

22.4 20.8 17.8 16.4 15.0 11.9 8 .9

24.5 24.0 22.2 20.7 18.9 13.7 0

3.0 2.0 1.5 1.4 1.3 1.2 1.1

0.07355 0.1013 0.1508 0.1687 0.1912 0.2199 0.2566

0.02340 0.03592 0.06674 0.07994 0.09781 0.1223 0.1560

0.007241 0.01256 0.03026 0.03936 0.05279 0.07280 0.1028

0.001510 0.004326 0.01389 0.01982 0.02947 0.04538 0.07171

0.0002819 0.001471 0.006413 0.01012 0.01680 0.02915 0.05198

Table 2 Critical temperature estimation from different expressions for the inverse polylogarithm function, D = 1.1. The values are given for N = 1000 in h¯ ω units.

ν

Exact

Eq. (27)

Eq. (28)

Eq. (30)

Eq. (50)

0.16 0.20 0.30 0.40 0.50

142 152 175 186 183

444 413 309 172 0

∈C ∈C

136 150 174 181 174

55.4 51.5 38.5 21.5 0

246 268 216

4.2. Critical value of the statistics parameter

Table 3 Critical temperature estimation from different expressions for the inverse polylogarithm function, D = 1.5. The values are given for N = 1000 in h¯ ω units.

ν

Exact

Eq. (27)

Eq. (28)

Eq. (30) or (50)

0.10 0.20 0.30 0.40 0.50

52.7 50.2 46.8 42.3 36.8

76.8 68.9 55.7 36.3 0

∈C ∈C

51.0 45.8 37.0 24.1 0

50.0 47.5 38.6

Table 4 Critical temperature estimation from different expressions for the inverse polylogarithm function, D = 3. The values are given for N = 1000 in h¯ ω units.

ν

Exact

Eq. (27)

Eq. (28)

Eq. (50)

0.05 0.10 0.20 0.30 0.40 0.50

7.63 6.83 5.65 4.73 3.91 3.19

6.27 6.20 5.87 5.28 4.26 0

6.71 6.62 6.22 5.44

9.37 9.25 8.76 7.88 6.36 0

∈C ∈C





αN

Im invLi D

h¯ ω

D  =0

Tc

(30)

exists only for D  2. If the space dimension is expressed as D = 1 + 1/n, where n is an integer (n > 1), the above equation becomes an nth order algebraic equation with respect to 1/ T cD . In particular, for D = 3/2 critical temperature T c coincides with (50). And finally, yet another T c can be obtained numerically from (A.13): {4}

Im invLi2





αN

h¯ ω Tc

2  = 0.

It can be shown that in the limit of small ν the imaginary part Im( zα ) remains positive for all the temperatures hence no critical point T c is observed. Solutions of Eq. (9) demonstrate existence of some critical value of the statistics parameter νc so that for ν < νc there is no T c unlike the domain ν > νc where Im( zα ) reaches zero at some T c . Below νc , energy and heat capacity remain continuous for finite N, and there exist discontinuities corresponding to phase transitions above νc . At fixed space dimensionality D, the value of νc tends to zero as the number of particles N increases. On the other hand, at fixed N the value of νc increases as D decreases, see Table 5. The values of this critical parameter were obtained from the numerical calculations. The simplest formula to describe these data is the power dependence model:

νc = aN b .

(31)

Comparison of the critical temperature values in different approximations with exact values calculated numerically are presented in Tables 1–4. As one can see, for D = 2 several approximations yield quite satisfactory results for the critical temperature at different values of the statistics parameter. Even the simplest form given by Eq. (27) can be used.

(32)

It was tested for different values of space dimensions in the domain of N = 102 ÷ 106 giving the following fitting results:

D = 1.1:

νc = (0.60 ± 0.04) N −0.19±0.01 ,

D = 1.5:

νc = (0.76 ± 0.01) N

−0.35±0.003

,

(34)

νc = (0.81 ± 0.01) N

−0.45±0.002

,

(35)

νc = (0.75 ± 0.03) N

−0.50±0.01

D = 2.0:

Other approximations of the inverse polylogarithm function given in Appendix A cannot be reduced to a closed expression for the critical temperature. From Eqs. (A.6), (A.14) the solution of {2}

For 1 < D < 2, Eq. (30) seems the most appropriate for estimations of critical temperature. For D > 2, Eqs. (27), (28) can be utilized.

D = 3.0:

(33)

.

(36)

Such values of the b exponent may suggest a more complicated dependence of νc on the number of particles than a simple model (32), including N ln N or some other combinations. While analytical results for νc are not known yet, it is possible to analyze small-ν (bosonic) limit of the statistics in order to verify whether thermodynamic functions are continuous. Eq. (9) linking the fugacity, temperature, and the number of particles yields in the leading order

Nα =



zα 1 − zα

+

T h¯ ω

D ζ ( D ).

(37)

In the ν → 0 limit the statistics parameter low-temperature fugacity can be written as

z = 1 +  z = 1 +  z + i  z , where  z and

 z  < 0,

α 1 + i πν and the (38)

 z are small corrections to the real and imaginary

parts of z, respectively. Therefore,

zα = 1 +  z + i  z + i πν , zα 1 1 − zα

=−

 z + i ( z + πν )

(39)

.

(40)

104

A. Rovenchak / Physics Letters A 378 (2014) 100–108

Fig. 1. (Color online.) Re z (left) and Im z (right) of the two-dimensional system compared to approximations (43) and (44), respectively. The data are calculated for N = 1000, ν = 0.01 (green circles) and ν = 0.02 (red boxes). Approximating functions are shown by green and red lines, respectively.

Fig. 2. (Color online.) Imaginary (left) and real (right) parts of zα for two-dimensional systems with N = 1000 at values of statistics parameter ν close to the critical one νc = 0.035917 . . . . Temperature T is in the h¯ ω units. Dashed lines correspond to discrete summations. Vertical lines mark the Bose–Einstein temperatures without finite-size effects (T 24.7) and with finite-size effects taken into account (T 23.6). Color correspondence is as follows: green — ν = 0.03; red — ν → νc − 0; blue — ν → νc + 0; magenta — ν = 0.037; black — ν = 0.04. Here and further, vertical lines connecting graph branches at critical temperatures are shown for better visualization.

For the real and imaginary parts of Eq. (37) we obtain:

 N−

D

T

ζ (D) = −

h¯ ω

πν N =

( z )2

 z , + ( z + πν )2

(41)

 z + πν . ( z )2 + ( z + πν )2

(42)

z = − 

zα

1

[N

− ( h¯Tω ) D ζ ( D )] + (

 z + πν =

πν πν N

zα = 1 + x + iy ,

N )2 /[ N

(45)

so that

The solutions to these equations are: 

instead of the continuous approach lead only to a shift in the temperature dependences leaving the qualitative picture unchanged. This shift is proportional to N −1/ D , cf. the analysis for Bose systems in [45]. In the vicinity of T c , the following approximation can be used:

− ( h¯Tω ) D ζ ( D )]

[ N − ( h¯Tω ) D ζ ( D )]2 + (πν N )2

= Im( zα ).

,

(43) (44)

Comparison of these solutions with exact calculations is shown in Fig. 1 demonstrating a very good accuracy in the low-temperature domain. As one can see, in this approximation the imaginary part Im( zα ) remains positive, thus no critical temperature is observed in the bosonic limit as is confirmed by numerical calculations yielding smooth temperature dependences of the thermodynamic functions for ν < νc . 4.3. Behavior of thermodynamic functions Fig. 2 shows the temperature dependence of the imaginary and real parts of zα in the vicinity of the transition point for values of the statistics parameter below and above the critical one νc for D = 2. Results for some other space dimensions are shown in Fig. 3. Note that calculations involving discrete summations (2), (3)

1 − zα

= −1 −

x x2

+

y2

+i

y x2

+ y2

(46)

,

where x, y are small. For the polylogarithm one has in a similar way:

Li D ( zα ) = Li D (1 + x + iy ) = ζ ( D ) +  + i 

(47)

with  ,  standing for small corrections to the real and imaginary parts, respectively;  ,  ∝ x D −1 or  ∝ x ln x for D = 2. As T → T c , y → 0, so in this case from Eq. (9), dropping off the negligible unity in (46), we obtain: 

Nα +

1 x

 =

T h¯ ω

D ζ ( D ),

N α  =

y x2

.

(48)

Thus, in the leading order the real part Re( zα ) is

Re( zα ) = 1 + x = 1 +

1

( h¯Tω ) D ζ ( D ) − N α 

,

(49)

while Im( zα ) = y remains a small positive quantity. This solution correctly reproduces the observed sign change in the x correction, see Fig. 4 and suggests a rough approximation for the critical temperature as follows:

A. Rovenchak / Physics Letters A 378 (2014) 100–108

105

Fig. 5. (Color online.) Real part of the specific heat C / N for D = 2 at values of statistics parameter ν close to the critical one νc . Colors as in Fig. 2.

 T c = h¯ ω

N α

1/ D (50)

.

ζ (D)

Obviously, the solution given by Eq. (49) gives a good estimate only for temperatures satisfying

    T D    ζ ( D ) − N α  h¯ ω  > 1.

(51)

The discontinuity in z leads to a discontinuity in the temperature dependence of energy suggesting this critical point to correspond to a first-order phase transition. As seen in Fig. 2, at the critical value of the statistics parameter νc there is a cusp in the Im(zα ) dependence on the temperature at some T p > T c . It should correspond to another phase transition, which is of the second order, see Fig. 5. It must be also taken into consideration that in the reference Bose system (ν = 0) the only phase transition is the Bose–Einstein condensation (BEC) in the limit of N → ∞ at the temperature

 T BEC = h¯ ω

Fig. 3. (Color online.) Im( zα ) for D = 1.1 (top), D = 1.5 (middle), and D = 3 (bottom) at values of statistics parameter ν close to the critical one νc . Number of particles N = 1000. Color correspondence is as follows: green — ν < νc ; red — ν → νc − 0; blue — ν → νc + 0; magenta — ν > νc . Vertical lines not coinciding with the grid mark the BEC temperature.

N

1/ D (52)

,

ζ (D)

where D > 1. However, only for D > 3/2, which corresponds to the condition D > 3 for a homogeneous system, a discontinuity in C ( T ) (a cusp for D = 3) is observed, cf. Fig. 6. Thus the third transition point should exist for D > 1 also in the case of the fractional statistics as in the limit N → ∞ the critical parameter νc → 0. A detailed analysis of this issue is beyond the scope of the present work; it will be studied elsewhere. Note that an analog of the Bose–Einstein condensation in ordinary systems with the Polychronakos statistics (with a real parameter) was reported in [46]. 5. Low temperature side and summary of results In the low-temperature domain, the fugacity remains nearly constant with zα close to unity, and the polylogarithm can be expressed via the zeta-function:

Li D ( zα ) ζ ( D )

(53)

with corrections, both to the real and imaginary parts, having the order of 1/ N. The energy is thus to a good accuracy as follows:

E = α  − i α  h¯ ω D Fig. 4. (Color online.) Re( zα ) in the vicinity of critical statistics parameter νc for D = 2 compared to the analytical approximation given by Eq. (49) [black dotted lines]. Other colors as in Fig. 2.



T h¯ ω

 D +1 ζ (D)

(54)

with any significant deviations possibly occurring in the immediate vicinity of the critical temperature only.

106

A. Rovenchak / Physics Letters A 378 (2014) 100–108

Fig. 6. (Color online.) Specific heat of the Bose system of oscillators in spaces with different dimensionality D for different number of particles N. Top row: D = 1.3 (left), D = 3/2 (right); bottom row: D = 2 (left), D = 3 (right). For the latter two cases, in the limit of N → ∞ there is a discontinuity with C | T → T c −0 > C | T → T c +0 . Black line — N = 102 ; red — N = 103 , green — N = 104 , blue — N = 105 , magenta — N = 106 .

Fig. 7. (Color online.) Real part of the specific heat C / N of a 2D system, N = 1000, for different values of the statistics parameter. Black line — ν = 0.0; red — ν = 0.1; light-green — ν = 0.2; blue — ν = 0.3; cyan — ν = 0.5. Dotted lines correspond to values close to νc : ν = 0.03591 < νc (dark-green) and ν = 0.03592 > νc (magenta).

The low-temperature behavior of the heat capacity reproduces that of a D-dimensional Bose gas of harmonic oscillators to the factor of α ∗ :

D 

 T  C = α − i α D ( D + 1) ζ ( D ). h¯ ω

(55)

In Fig. 7 the real part of the specific heat for D = 2, N = 1000, is shown in a wide temperature domain covering both high temperatures (above the phase transitions) and low temperatures. The imaginary part of the specific heat is shown in Fig. 8. Results for some other space dimensions are shown in Fig. 9. Certainly, only those systems are physically interesting, where the imaginary part of energy is small thus dissipations still allow to treat the system as nearly equilibrium. From (54) it is clear that in the low-temperature domain the condition |Γ / E |  1 holds if πν  1, while for high temperatures the imaginary part vanishes,

Fig. 8. (Color online.) Imaginary part of the specific heat C / N of a 2D system, N = 1000, for different values of the statistics parameter. Colors as in Fig. 7.

cf. Section 3. As demonstrated by Fig. 10, the ratio |Γ / E | is in fact controlled by the low-temperature behavior of the energy, cf. Eq. (54), and thus

 Γ  E

  α    α  = tan πν .

(56)

6. Discussion The presented paper contains a rather comprehensive analysis of thermo dynamic properties of a D-dimensional harmonic oscillator system obeying the Polychronakos fractional statistics with a complex parameter. In the high-temperature limit, the classical result was obtained showing no influence of the statistics parameter ν , as one would expect. As the temperature decreases, the first-order phase transitions occur in the systems if ν is larger than some critical νc ,

A. Rovenchak / Physics Letters A 378 (2014) 100–108

Fig. 9. (Color online.) Real part of the specific heat C / N of a D-dimensional system, N = 1000, for different values of the statistics parameter. Top: D = 1.1, middle: D = 1.5, bottom: D = 3. Black line — ν = 0.0; yellow — ν = 0.05; red — ν = 0.1; light-green — ν = 0.2; blue — ν = 0.3; cyan — ν = 0.5. Dotted lines correspond to values close to νc : ν < νc (dark-green) and ν > νc (magenta).

107

which is defined by the number of particles and space dimension. The values of critical temperature T c defined by the condition Im( zα ) = 0 decrease as the statistics parameter ν grows for most of the analyzed space dimensions. Only as D approaches unity (D = 1.1), this correlation changes. Relevant analytical estimations can be deduced from Eq. (30). For ν = νc , a non-analyticity in the temperature behavior of fugacity suggests existence of another phase transition (of the second order) at some T p > T c . In the low-temperature limit, a Bose-like behavior of thermodynamic functions was observed, to a factor of e −i π ν . In fact, from this limit a condition for the applicability of the considered statistics can be derived in a form πν  1, which ensures that the imaginary part of energy is small. Several tasks call for a detailed future study, namely the issue of the critical statistics parameter and the behavior of thermodynamic functions in the immediate vicinity of the phase transitions. The first one is closely linked to the critical point T p . Once analytical properties of νc are determined, the details of this phase transition become clearer. The specification of behavior of thermodynamic functions near T c requires more accurate expressions for the inverse polylogarithm. The phenomena described in this work are related to several real physical systems with small dissipation. First of all these are trapped bosons, for which the shape of the external potential determines the effective dimensionality D [38]. A Bose system in S space dimensions trapped to an external potential V (r ) ∝ r η is equivalent, by virtue of the density of state function, to a harmonic oscillator system with the effective dimension D = S (1 + 2/η)/2. Harmonic traps (η = 2) are typically used in experiments, thus is this case the values of D = 3 and especially D = 2 are of primary interest. Since fractional space dimensions 2 < S < 3 can be used to describe porous medium [36], bosonic systems with a quadratic excitation spectrum correspond to oscillators with 1 < D < 3/2. In turn, D = S in the case of a linear spectrum, for instance, the phonon branch of the excitation spectrum in liquid helium. Moreover, in helium films of different thickness the transition 2  D  3 smoothly interpolates between planar and bulk geometries. Preliminary analysis allows to establish approximate correspondence between the described statistics and some other types of fractional statistics, including anyonic, Haldane–Wu, and qdeformed bosons. As previous estimations for 1D systems of harmonic oscillators show, experimental tests of the predicted phase transitions are in principle within accessible accuracies [8]. A properly prepared system to be effectively described by the Polychronakos fractional statistics with a complex parameter remains so far an open issue. Acknowledgement This work was partly supported by Project ΦΦ -110Φ (registration number 0112U001275). Appendix A. Inversion of the polylogarithm function Define the inverse polylogarithm function invLi D ( w ) as follows:

Li D (x) = w

⇒

invLi D ( w ) = x.

(A.1)

In the vicinity of x = 0, the easiest way is to invert the power series [47] for the polylogarithm, which gives: Fig. 10. (Color online.) The ratio of imaginary and real parts of energy Γ / E at D = 1.8 for different values of the statistics parameter. Color correspondence is as follows (lines ordered top to bottom): red — ν = 0.05, green — ν = 0.1, blue — ν = 0.15, magenta — ν = 0.2, cyan — ν = 0.25.

{1}



invLi D ( w ) = w − 2− D w 2 + 21−2D − 3− D w 3 + · · · .

(A.2)

In the vicinity of x = 1, series expansions of the polylogarithm function depend on D:

108

A. Rovenchak / Physics Letters A 378 (2014) 100–108

Li D (x) = ζ ( D ) + Γ (1 − D )(1 − x) D −1 + · · · ,

1 < D < 2,

Li2 (x) = ζ (2) − (x − 1) ln(1 − x) + · · · , Li D (x) = ζ ( D ) + (x − 1)ζ ( D − 1) + · · · ,

(A.3) (A.4)

D > 2.

(A.5)

The solutions with the first leading term after ζ ( D ) taken into consideration read: {2}

invLi D ( w ) =

⎧  ( D ) 1/( D −1) ⎪ , 1 < D < 2, ⎨ 1 − Γw (−ζ 1− D ) ⎪ ⎩1+

w −ζ ( D ) ζ ( D −1 ) ,

(A.6)

D > 2.

The D = 2 case requires the so-called Lambert W function, which is considered below. For integer n there exists an expansion for |x| > 1 [48]:

Lin (x) = (−1)n−1

∞  1 j =1

+2

 n/2

jn x j

−

lnn (−x) n!

Li2 j (−1) lnn−2 j (−x)

(n − 2 j )!

j =1

,

(A.7)

where Li2 j (−1) = (21−2 j − 1)ζ (2 j ). It takes a simple form for n = 2

Li2 (x) = −

ln2 (−x) 2

− ζ (2) −

1 x

+ ···,

(A.8)

yielding with the first two terms taken into consideration √

{3} invLi2 ( w ) = −e −2[ w +ζ (2)] .

(A.9)

The expression in the n = 3 case can be written analytically as well but it is too cumbersome for analysis. It is also possible to express the inverse polylogarithm using the Lambert W function [49], which is a multivalued function solving the following equation:

W ( z)e W (z) = z,

z ∈ C.

(A.10)

For integer n series for Lin (x) in the vicinity of x = 1 can be expressed in the form [48], see also [50, p. 30],

Lin (x) = ζ (n) +

+

− ln(− ln x) + ψ(n) + γ n−1 ln x (n − 1)!

∞  ζ (n − j ) j ln x, j!

(A.11)

j =1 j =n−2

where ψ(n) is the polygamma function and γ = 0.57721 . . . is the Euler–Mascheroni constant. For n = 2 this expression simplifies to





Li2 (x) = ζ (2) − ln x ln − ln(x) + ln x + · · · ,

(A.12)

from which one obtains {4}



invLi2 ( w ) = exp −e W k (−

ζ (2)+ w e

)+1



(A.13)

with k to be chosen from additional considerations. In [49] authors suggest k = −1 for an ideal 4-dimensional Bose gas.

It is also possible to write the solution of (A.4) using the Lambert function: {2}







invLi2 ( w ) = 1 − exp W k w − ζ (2) .

(A.14)

References [1] G.S. Canright, M.D. Johnson, J. Phys. A, Math. Gen. 27 (1994) 3579. [2] A. Khare, Fractional Statistics and Quantum Theory, 2nd edition, World Scientific, Singapore, 2005. [3] R.B. Laughlin, Science 242 (1988) 525. [4] Gun Sang Jeon, J.K. Jain, Phys. Rev. B 81 (2010) 035319. [5] M.T. Batchelor, X.W. Guan, N. Oelkers, Phys. Rev. Lett. 96 (2006) 210402. [6] A. Rovenchak, Low Temp. Phys. 35 (2009) 400; A. Rovenchak, Fiz. Nizk. Temp. 35 (2009) 510. [7] A. Rovenchak, J. Phys. Conf. Ser. 400 (2012) 012064. [8] A. Rovenchak, Fiz. Nizk. Temp. 39 (2013) 1141; A. Rovenchak, Low Temp. Phys. 39 (2013), 888. [9] A.P. Polychronakos, Phys. Lett. B 365 (1996) 202. [10] I.M. Barbour, D.S. Henty, E.G. Klepfish, Nucl. Phys. B, Proc. Suppl. 34 (1994) 311. [11] E. Abraham, I.M. Barbour, P.H. Cullen, E.G. Klepfish, E.R. Pike, Sarben Sarkar, Phys. Rev. B 53 (1996) 7704. [12] P.K. Chakraborty, B. Nag, K.P. Ghatak, J. Phys. Chem. Solids 64 (2003) 2191. [13] G.E. Cragg, A.K. Kerman, Phys. Rev. Lett. 94 (2005) 190402. [14] Wim van Saarloos, Douglas A. Kurtze, J. Phys. A, Math. Gen. 17 (1984) 1301. [15] V. Matveev, R. Shrock, J. Phys. A, Math. Gen. 28 (1995) 4859. [16] T.K. Bailey, W.C. Schieve, Nuovo Cimento A 47 (1978) 231. [17] Carl M. Bender, Dorje C. Brody, Daniel W. Hook, J. Phys. A, Math. Theor. 41 (2008) 352003. [18] M.K. Oberthaler, R. Abfalterer, S. Bernet, J. Schmiedmayer, A. Zeilinger, Phys. Rev. Lett. 77 (1996) 4980. [19] G.S. Ng, H. Hennig, R. Fleischmann, T. Kottos, T. Geisel, New J. Phys. 11 (2009) 073045. [20] A.M. Gavrilik, A.P. Rebesh, Mod. Phys. Lett. B 25 (2012) 1150030. [21] M. Chaichian, R. Gonzalez Felipe, C. Montonen, J. Phys. A 26 (1993) 4017. [22] R.K. Gupta, C.T. Bach, H. Rosu, J. Phys. A, Math. Gen. 27 (1994) 1427. [23] A. Algin, Int. J. Theor. Phys. 48 (2009) 71. [24] Y. Yang, S. Xie, W. Feng, X. Wu, Mod. Phys. Lett. A 13 (1998) 879. [25] G. Gentile Jr., Nuovo Cim. Nuova Ser. 17 (1940) 493. [26] R.Z. Levitin, et al., JETP Lett. 79 (2004) 423. [27] M.M. Altarawneh, et al., Phys. Rev. Lett. 109 (2012) 037201. [28] Y. Kohama, et al., Phys. Rev. Lett. 109 (2012) 167204. [29] Yu.M. Poluektov, A.S. Rybalko, Fiz. Nizk. Temp. 39 (2013) 992; Yu.M. Poluektov, A.S. Rybalko, Low Temp. Phys. 39 (2013) 770. [30] D. Finotello, K.A. Gillis, A. Wong, M.H.W. Chan, Phys. Rev. Lett. 61 (1988) 1954. [31] V.E. Syvokon, Low Temp. Phys. 32 (2006) 48; V.E. Syvokon, Fiz. Nizk. Temp. 32 (2006) 65. [32] R. Acharya, P. Narayana Swamy, J. Phys. A 27 (1994) 7247. [33] V.P. Maslov, Theor. Math. Phys. 153 (2007) 1575. [34] A. Rovenchak, Acta Phys. Pol. A 118 (2010) 531. [35] E. Courtens, R. Vacher, Proc. R. Soc. Lond. Ser. A 423 (1989) 55. [36] S.-H. Kim, Ch.K. Kim, K. Nahm, J. Phys. Condens. Matter 11 (1999) 10269. [37] J. Kou, F. Wu, H. Lu, Y. Xu, F. Song, Phys. Lett. A 374 (2009) 62. [38] V. Bagnato, D. Kleppner, Phys. Rev. A 44 (1991) 7439. [39] F. Dalfovo, S. Giorgini, L.P. Pitaevskii, Rev. Mod. Phys. 71 (1999) 463. [40] M.A. R-Monteiro, I. Roditi, L.M.C.S. Rodrigues, Mod. Phys. Lett. B 9 (1995) 607. [41] M. Rego-Monteiro, L.M.C.S. Rodrigues, S. Wulck, Physica A 259 (1998) 245. [42] C. Ou, J. Chen, Phys. Rev. A 68 (2003) 026123. [43] G. Su, L. Chen, J. Chen, Open Syst. Inf. Dyn. 10 (2003) 135. [44] A. Algin, E. Ilik, Phys. Lett. A 377 (2013) 1797. [45] Mingzhe Li, Lixuan Chen, Jincan Chen, Zijun Yan, Chuanhong Chen, Phys. Rev. A 60 (1999) 4168. [46] B. Mirza, H. Mohammadzadeh, Phys. Rev. E 82 (2010) 031137. [47] M. Abramowitz, I.A. Stegun (Eds.), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, tenth printing, Dover, New York, 1972, p. 14. [48] http://functions.wolfram.com/ZetaFunctionsandPolylogarithms/PolyLog/. [49] J. Tanguay, M. Gil, D.J. Jeffrey, S.R. Valluri, J. Math. Phys. 51 (2010) 123303. [50] A. Erdélyi, et al., Higher Transcendental Functions, vol. 1, Krieger, New York, 1981.