INTEGRAL REPRESENTATION FOR THE GRAND PARTITION FUNCTION IN QUANTUM STATISTICAL MECHANICS OF EXCLUSION STATISTICS

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1 International Journal of Modern Physics B, Vol. 4, No. 5 (000) c World Scientific Publishing Company INTEGRAL REPRESENTATION FOR THE GRAND PARTITION FUNCTION IN QUANTUM STATISTICAL MECHANICS OF EXCLUSION STATISTICS KAZUMOTO IGUCHI 70-3 Shinhari, Hari, Anan, Tokushima , Japan KAZUHIKO AOMOTO Department of Poly Mathematics, University of Nagoya, Chikusa, Nagoya, Aichi , Japan Received 30 September 999 We derive an exact integral representation for the grand partition function for an ideal gas with exclusion statistics. Using this we show how the Wu s equation for the exclusion statistics appears in the problem. This can be an alternative proof for the Wu s equation. We also discuss that singularities are related to the existence of a phase transition of the system. PACS number(s): d, 7.0.-w. Introduction In quantum statistical mechanics (QSM) the grand partition function (GPF) Q plays a fundamental role. In fact, all physical quantities such as the pressure P, the energy E, the total number N of the system are given by the GPF and its factorization has been very important for the purpose. Furthermore, it has been known that the existence of a singularity in the GPF is related to the presence of a phase transition. For example, for an identical particle gas we can write the GPF as Q Q N z N, () N0 where Q N is the microcanonical partition function (MPF) and z is the fugacity of the system given by z e βµ with β /k B T and the chemical potential µ. The kazumoto@stannet.ne.jp aomoto@math.nagoya-u.ac.jp 485

2 486 K. Iguchi & K. Aomoto MPF is given by Q N W N e β j Njεj, () N i0,n j Nj where N j is the number of particles occupied in the quantum number j and W N the number of states of the system corresponding to the set of occupation numbers N j, and hence it gives the entropy S of the system such that S k B ln W N. The equation of state of the system is given by P k B T V ln Q b l z l F (z), (3) l and N V z z F(z) lb l z l. (4) l These are called cluster expansions. The energy of the system is also given as E ln Q. (5) β Then, Lee and Yang s theorem tells us that if there exists a singularity of F (z) on the positive real axis of a complex z-plane, then there is a phase transition in the system. Therefore, the convergence of the cluster expansion and the existence of a singularity in the GPF are exactly related to the presence of a phase transition of the system. To find such a singularity the factorization of an infinite series expansion of the GPF with respect to the fugacity has played a very crucial role in the theory of a phase transition., To do so, we have used a variational method taking an extremum condition in order to find a most probable contribution in the GPF as we regard the GPF as an asymptotic series. This has been only a known method which may factorize the GPF. Therefore, Wu 3 also followed this approach when he considered to obtain the distribution function for an ideal gas with fractional exclusion statistics. However, it has not yet been known an exact method that factorizes the GPF and enables one to find the singularities in the GPF for the exclusion statistics. In this paper we present an integral representation of the GPF for an ideal gas with exclusion statistics, 4 using an exact analytical method. And we show that the factorization can be done analytically by using the integral representation, from which we can find the singularities in the GPF. The organization of the paper is the following: In Sec., we will review the standard argument in QSM for the boson, fermion, and g-on gases with identical particles. In Sec. 3, we will give an integral representation for the GPF for the single species gas case with exclusion statistics. In Sec. 4, we will give an integral representation for the GPF for the multi-species gas case with mutual exclusion statistics. In Sec. 5, we will give an integral representation for the GPF in the energy and momentum representations. In Sec. 6, we give a conclusion.

3 Integral Representation for the Grand Partition Function 487. Grand Partition Function and the Equation of State Let us first consider the GPF for identical particle gases with a quantum statistics using the standard argument for later purposes. For the Bose gas case, W N is the number of permutations of the N j particles plus the G j partitions that give rise to distinct arrangement such that WN b (N j + G j )!, (6) N j j!(g j )! and for the Fermi gas case, W N is the number of ways in which N j things can be chosen from G j things such that W f N G j! N j j!(g j N j )!, (7) where G j is the number of the available single particle states of the system. For the g-on gas case Wu 3 has generalized the above two cases to the following: W g N (N j + D j )!, (8) N j j!(d j )! using the Haldane s definition of the exclusion statistics: 4 D j G j g(n j ). (9) To derive the distribution function n j of the system we usually consider most probable contribution of the asymptotic series of the GPF by taking the extremum δ condition: δn j [ln W N + β j (ε j µ)n j ] 0 and using the Stirling formula: N! N ln N N wherewealwaysassumeg j N j. We then obtain the pressure, the density and the distribution function of the system of Bose and Fermi gases as PV k B T lnq j G j ln( x j ) j P j V k B T, (0) ( ) P, () k B T ρ N V n j z z j n j z ( ) Pj G jx j, () z k B T x j where (+) sign stands for the Bose(Fermi) statistics and x j ze βεj. Similarly, for the g-on gas, we obtain PV k B T ) j G j ln (+ wj, (3) n j G j w j + g, (4) w g j ( + w j) g x j, (5)

4 488 K. Iguchi & K. Aomoto where w j w(x j ). 3 This interpolates between Bose (g 0)andFermi(g) statistics continuously. Thus, from using Eq. (3) we inversely obtain the factorization of the GPF with respect to the distribution function n j : 5,6 Q j Q j j ( + w j ) Gj. (6) In this way, we are able to obtain the equation of state by Eqs. () and (3) as well as the energy E of the system. 3. Integral Representation for the GPF: The Single Species Case In spite of the above very elegant formulation of the theory, it has not been known whether or not there is an exact method without taking such a most probable contribution of the GPF, so far. We present such a method here. Let us first consider the Bose gas case. In this case the GPF is given by Q j Using the binomial expansion: N j0 Γ(α) ( u) α (N j + G j )! x Nj j. (7) N j!(g j )! n0 where Γ(α) is the Gamma function, 7 we obtain Γ(α + n) u n, (8) n! ( ) Gj. (9) Q j x j For the Fermi gas case, we have Q j G j N j0 G j! N j!(g j N j )! xnj j. (0) This is nothing but the binomial expansion: Hence, we obtain ( + x) G G n0 G! n!(g n)! xn. () Q j Q j j ( + x j ) Gj. () These coincide with the results of Eq. (0), and only the difference here is that we do not use a most probable distribution for the system nor any approximation.

5 Integral Representation for the Grand Partition Function 489 Let us next consider the g-on case. We now have Q j N j0 [G j + g +( g)n j )! x Nj j. (3) N j![g j + g gn j ]! To evaluate this we use the following theorem: Theorem. F n0 Γ(α)Γ(α + β n) n!γ(α )Γ(α + βn) xn B(α α,α ) πi B(α α,α ) πi t C0 α ( t + xt β ) γ dt t C0 α ( + t) α [ + xt β ( + t) β dt, (4) ] γ where γ +α α, β β +,B(p, q) B(q, p) Γ(p)Γ(q)/Γ(p + q), called the β-function, 7 and C 0 means a closed loop path moving from 0 to itself, encircling t 0 anticlockwise, where t 0 (t ) is the smaller (larger) solution of t + xt β 0or +xt β ( + t) β 0. Proof. Using the Euler integral of the first kind 7 : B(p, q) we first find the following relation: 0 t p ( t) q dt 0 t p dt, (5) ( + t) p+q Γ(α + β n) Γ(α + βn) Γ(α α t α +β n ( t) α α n dt n) 0 t α +β n Γ(α α dt. (6) n) 0 ( + t) α+βn Substituting Eq. (6) into Eq. (4), we obtain F Γ(α) Γ(α ) 0 t α ( t) α α Γ(α) t α Γ(α ) 0 ( + t) α n0 n0 Γ(α α n)n! Using the relation for the Gamma function 7 : Γ(λ)Γ( λ) Γ(α α n)n! ( ( xt β ( + t) β ) n xt β dt t ) n dt. (7) π sin πλ, (8)

6 490 K. Iguchi & K. Aomoto we find Γ(α α n) sin(π(α α n))γ(γ + n) π ( )n sin(π(α α )) Γ(γ + n) π Substituting Eq. (9) into Eq. (7), we find F Γ(α) Γ(α α )Γ(α )Γ(γ) n0 Γ(γ + n) n! 0 ( ) n Γ(α α Γ(γ + n). (9) )Γ(γ) t α ( t) α α ) n ( xt β dt (30) t Γ(α) t α Γ(α α )Γ(α )Γ(γ) 0 ( + t) α Using Eq. (8) for Eq. (30) we obtain n0 Γ(γ + n) n! ( ) n xt β ( + t) β dt. (3) Γ(α) t α F Γ(α α )Γ(α dt (3) ) 0 ( t + xt β ) γ Γ(α) t α Γ(α α )Γ(α ) 0 ( + t) α [ + xt β ( + t) β dt. (33) ] γ However, the integrations of Eqs. (3) and (33) may diverge at some value of t within the interval of integration, since at some value of x the denominator vanishes. Also the series expansion of Eq. (7) may diverge at t (ort ) in the first (second) expression of the right hand side of Eq. (7). To escape from such divergences, we analytically continue the integration for the complex variable t containing the integral interval. Since in general t + xt β 0or+ xt β ( + t) β 0 consists of two real solutions t 0 and t such that <t 0 <t,we need to make a contour integral along a loop C 0, which moves from the origin to itself, encircling t 0 anticlockwise such that xt β /( t) <, xt β /(+t) β < for t C 0, respectively. This provides the expression of Eq. (4). Hence, the theorem is proved. Recently we have discussed this kind of functions and called them the quasihypergeometric functions. 8 The detail of the proof of the above theorem has been discussed in the second paper of Ref. 8 and we have used them in order to represent many physical quantities in the systems with exclusion statistics. 9

7 Integral Representation for the Grand Partition Function 49 Let us now consider Eq. (3). Identifying α α G j + g, β g,and β g,weobtain Q j Q j j j πi C 0 G t j+g j t j + xt g j dt j (34) G t j+g j πi C 0 ( + t j ) Gj+g +xt g j ( + t j ) dt g j. (35) If we change the variable in the last expression as t j t j t j without changing g, then we obtain Q j πi C 0 G t j+g j t j + xt g j dt j (36) ( + t j ) πi C0 Gj+g t Gj+g xt g j j ( + t j) g dt j. (37) The expressions of Eqs. (36) and (37) are very important since the expressions are exact without any approximation such as taking a most probable distribution and the singularity of the integration occurs when the denominator in the integrand diverges at ψ(t j ) t j + xt g j 0, (38) ϕ(t j ) xt g j ( + t j) g 0, (39) which are exactly equivalent to the Sutherland s equation 0 and the Wu s equation 3 of Eq. (5), respectively. Let us use Eq. (39), for example. Since this provides a simple root t j w j,we can use the expansion around the root as ϕ(t j )ϕ (w j )(t j w j ). Substituting this into Eq. (35) and applying the Cauchy theorem: πi C 0 f(t) dt f(w), (40) t w where C 0 means a loop encircling w anticlockwise, we obtain Q j ( + t j ) πi C0 Gj+g t Gj+g ϕ (w j )(t j w j ) dt j j ( + w j) Gj+g. (4) Gj +g wj ϕ (w j ) From Eq. (39) we find ϕ (w j ) w j+g w j ( + w j ). (4)

8 49 K. Iguchi & K. Aomoto Substituting Eq. (4) into Eq. (4), finally we derive the exact expression: ( ) Gj+g +wj w j Q j w j w j +g. (43) But if we adopt a usual condition G j g, then it turns out to be ( ) Gj +wj Q j. (44) w j This coincides perfectly with Eq. (6). From Eqs. (36) and (37), it becomes clear how the singularity in the power series of the GPF appears. The singularity is obtained as follows: Let us consider the solution of the Wu s equation, Eq. (5). The singularity is obtained as follows: 5,0, Taking log of Eq. (5), we find This yields x c g ln w j +( g)ln(+w j ) ln x j. (45) x j dx j g dw j w j ( g) g ( g) g +( g) dw j +w j (w j +g) w j ( + w j ) dw j, (46) which provides a nontrivial singularity at w j g. Substituiting this into Eq. (5), we find the singularity of ζ occurs at x x c : e ±iπg g g ( g) g for 0 <g< (g )g g g for <g. (47) Let us regard w j as a function of x. We find the following: () For 0 <g<, Eq. (5) has one real solution for w j for <x<,wherew j ± as x ±0. () For <g, there are two real solutions w j, and w j, for x c <xsuch that w j, ± and w j, ± 0asx ±0, there is one real solution, w j g at x x c, and there is no solution for x<x c. Since a physical solution of Eq. (5) must satisfy w j at x 0, we choose the upper branch of w j for x>x c when g>. Thus, the convergence of the power series of x is dominated by the presence of a singularity in the series. Hence, the series expansions can converge for x x c and the expansion is given by analytic continuation of the power series for x > x c. The existence of a singularity in the GPF is crucially important since it is related to the presence of a phase transition. Using the above result of Eq. (47) together with the Lee Yang theorem, it was proved that a true phase transition occurs only when the pure boson case of g 0, since in this case the singularity of the GPF lies at z in the positive real axis of the fugacity. 5, In this way, the explicit determination of singularities of the GPF is very important for the theory of phase transition.

9 Integral Representation for the Grand Partition Function Integral Representation for the GPF: The Multi-Species Case Let us next consider the ideal gas of multi-species quasiparticles with mutual exclusion statistics. The Haldane s exclusion statistics is generalized to D p G p q g pq (N q δ pq ). (48) Here p, q include both the labels of species a and quantum numbers j such as p {a, j} where a {,...,S}. 4 Applying this to Eq. (), Wu generalized W N to W (S) N ({g pq}) p [G p +N p q g pq(n q δ pq )]! N p![g p q g. (49) pq(n q δ pq )]! The GPF is then represented as Q W N ({g pq }) N0 p NpN p N 0 N S0 x Np p W N N S ({g pq })x N x NS S, (50) where x p e β(εp µa) e β(εa j µa). Let us find the integral representation for the GPF of Eq. (50). Let us first consider the case of two species of S. And for the sake of simplicity, we assume that the two species of particles belong to the same quantum state such that G p G a δ j,0 and g pq g ab δ jk for a, ; but the generalization to taking into account both the species and the quantum numbers are straightforward. In this case, Eq. (49) becomes W () N [G + N g (N ) g N ]! N![G g (N ) g N ]! [G + N g N g (N )]! N![G g N g (N )]! [G + g +( g )N g N ]! N![G + g g N g N ]! [G + g g N +( g )N ]!. (5) N![G + g g N a g N ]! Substituting this into Eq. (50), we obtain Q N 0 N 0 Γ(G + g +( g )N g N ) N!Γ(G + g g N g N ) Γ(G + g g N +( g )N ) N!Γ(G + g g N g N ) xn xn. (5)

10 494 K. Iguchi & K. Aomoto Let us replace as α a α a G a + g aa, a,, (53) ( ) ( ) β ˆB β g g, (54) β β g g ( ) ( ) β β g g ˆB. (55) β β g g Then, Eq. (5) is simply represented as Γ(α Q + β N + β N ) Γ(α + β N + β N ) N 0 N 0 Γ(α + β N + β N ) Γ(α + β N + β N ) x N N! x N N!. (56) Following the same procedure as for the proof of Theorem, we use Eq. (6) in order to obtain the integral representation of Eq. (56). After some algebra, we prove the generalization of Theorem : Theorem. F N 0 N 0 Γ(α )Γ(α + β N + β N ) Γ(α )Γ(α + β N + β N ) Γ(α )Γ(α + β N + β N ) Γ(α )Γ(α + β N + β N ) x N N! x N N! ( ) α t t α B πi C C [ψ (t)] γ [ψ (t)] dt dt γ ( ) B πi C C where we have defined as α t t α ( + t ) α ( + t ) α dt dt, (57) [v (t)] γ [v (t)] γ γ +α α, (58) γ +α α, (59) B B(α α,α )B(α α,α ), (60) ψ (t) t +x t β t β, (6) ψ (t) t +x t β t β, (6) v (t) +x t β ( + t ) β t β ( + t ) β, (63)

11 Integral Representation for the Grand Partition Function 495 v (t) +x t β ( + t ) β β t ( + t ) β, (64) and C j (j,...,s) means a loop encircling the root of ψ j (t) 0or v j (t) 0 anticlockwise. Applying this theorem to Eq. (56), we obtain ( ) G +g t G+g Q dt dt (65) πi ψ (t)ψ (t) ( ) πi C C C t C G t +g t G+g ( + t ) G+g ( + t ) G+g v (t)v (t) dt dt. (66) This is exactly a generalized integral representation for the single species case of Eqs. (34) and (35) to the two species case. If we change the variables such as t t t andt t t in Eq. (66), then we find Q ( πi ) ( ) πi C C C wherewehavedefinedas G t +g t G+g ψ (t)ψ (t) C ( + t ) G+g ( + t ) G+g t G+g t G+g ϕ (t)x t β ( + t ) β dt dt (67) ( t +t ϕ (t)ϕ (t) dt dt, (68) ) β, (69) ( ) β ϕ (t) x t β t ( + t ) β. (70) +t The singularities in the GPF of Q appear when the denominator vanishes, which is provided by ψ (t) ψ (t)0, (7) ϕ (t) ϕ (t)0. (7) These are nothing but the Sutherland s equations : ζ +x ζ g ζ g, and the Wu s equations 3 : ζ +x ζ g ζ g, (73) w g ( + w ) g ( w ) g, +w x ( ) g w g ( + w w ) g, (74) +w x

12 496 K. Iguchi & K. Aomoto respectively, where we have the relation: ζ a +/w a,fora,.theabove results are all exact without adopting any approximation such as taking a most probable distribution and the reason why we must encounter the Wu s equation when we consider the exclusion statistics. Let us now consider an integral: ( ) f(t,t ) πi C C ϕ (t)ϕ (t) dt dt ( ) f(t,t ) πi C C ϕ (t)ϕ (t) (ϕ,ϕ ) dϕ dϕ, (75) (w,w ) where (ϕ,ϕ) (w,w ) means the Jacobian: ϕ ϕ (ϕ,ϕ ) (w,w ) w w ϕ ϕ. (76) w w Using to Eq. (75) the generalized Cauchy s theorem for two-variable complex functions 3 : ( ) f(ϕ,ϕ ) πi C C ϕ m dϕ dϕ f(0, 0)δ m, δ n,, (77) ϕn where C and C mean closed loops encircling 0 anticlockwise, we are led to the following theorem: ( ) f(t,t ) πi C C ϕ (t)ϕ (t) dt dt Using this theorem, we finally obtain f(t (ϕ),t (ϕ)) (ϕ,ϕ ) (w,w ) ϕ ϕ 0. (78) Q ( + w ) G+g ( + w ) G+g w G+g w G+g (ϕ,ϕ ). (79) (w,w ) This is the generalized result of Eq. (4) to the two species case. Thus, the factorization has been performed for the case of the two species gas with mutual exclusion statistics. Let us now consider the location of singularities in the GPF. The appearance of a singularity is now very clear. It comes from when the denominator of Eq. (78)

13 Integral Representation for the Grand Partition Function 497 vanishes such that the Jacobian vanishes. Hence, the singularities are located on a curve given by ϕ ϕ w w ϕ ϕ 0. (80) w w Let us find the explicit expression of Eq. (80). To do so, let us take total derivative of ϕ and ϕ. We then obtain dϕ dx + w + g x w ( + w ) dw g + w ( + w ) dw, (8) dϕ dx + w + g x w ( + w ) dw g + w ( + w ) dw. (8) From this we find w + g g w ( + w ) w ( + w ) w + g g g w + g w ( + w ) w ( + w ) g w + g 0. w ( + w ) w ( + w ) (83) Substituting Eq. (83) into Eq. (79), we obtain ( ) G+g ( ) G+g +w +w w w Q w w w + g g, (84) g w + g which is the generalization of Eq. (43) to the two species case. Hence, we finally find the singularity curve as w + g g g w + g (w +g )(w + g ) g g 0. (85) This is also related to the singularities in the distribution function n a.these are given by n a x a ln Q, (86) x a V for a,.assumingg a g aa for a,,weobtain n G V w +g g (w + g )(w + g ) g g, (87) n G w + g g. (88) V (w + g )(w + g ) g g This particular aspect is interesting. The reason is as follows: The Jacobian in Eq. (79) once disappears when we take log of the Q, compared to the leading

14 498 K. Iguchi & K. Aomoto terms of order G a since G a. Nevertheless, once we consider the distribution functions, it appears once again. In this way, the divergence of the GPF is related to the divergence of the distribution functions. The above argument can be straightforwardly generalized to the case of an ideal S species gas with mutual exclusion statistics. In this case, the GPF of Eq. (50) can be transformed into an integral form: where ( ) S G t +g t GS+gSS S Q dt dt S (89) πi C C S ψ (t) ψ S (t) ( ) S ( + t ) πi C CS G+g ( + t S ) GS+gSS t G+g t GS+gSS S ϕ (t) ϕ S (t) dt dt S, (90) ψ a (t) t a + x a t gaa a ϕ a (t) x a t gaa a ( + t a) gaa b( a) b( a) ( tb t g ba b, (9) +t b ) gba, (9) for a, b,...,s and C j means a loop encircling the zero of ϕ j 0. Following the same procedure as when we obtain Eq. (79), we use for Eq. (89) the generalized Cauchy theorem for multi complex variable functions: 3 ( ) S f(ϕ,...,ϕ S ) dϕ πi C C S ϕ n dϕ S f(0,...,0)δ n, δ ns,, ϕns S (93) where C j means a loop encircling 0. This provides the following theorem: ( ) S f(t,...,t S ) πi C C S ψ (t) ψ S (t) dt dt S f(t (ϕ),...,t S (ϕ)) (ϕ,...,ϕ S ). (94) (w,...,w S ) ϕ ϕ S0 Applying the above theorem to Eq. (90), we obtain the factorization Q ( + w ) G+g ( + w S ) GS+gSS w G+g w GS+gSS (ϕ,...,ϕ S ). (95) S (w,...,w S )

15 Integral Representation for the Grand Partition Function 499 And explicitly calculating the Jacobian as in Eq. (83), we finally end up with ( ) G+g ( ) GS+g SS +w +ws w w S Q, (96) w w S D S wherewehavedefinedas w +g g g S g w + g g S D S (97). g S g S w S +g SS The above results are all exact without any approximation such as taking a most probable distribution in the GPF. Therefore, this is the main non-trivial result in this paper. Let us study the singularities in the GPF. A singularity appears when the denominator of Eq. (96) vanishes. Hence, it is determined by D S 0. (98) Therefore, this provides a singularity hypersurface in the space of R S (w,...,w S ). As was discussed in the case of two species, it also appears in the denominator of the distribution functions n a, which are given by n a ˆD S (a), (99) V D S for a,...,s, where we have defined as w +g g G g S g w + g G g S ˆD S (a)., (00) g S g S G S w S +g SS where elements in the ath column in the determinant D S are all replaced by G s. The proof of this theorem is rather straightforward using knowledge of linear algebra as follows: Starting from Eq. (86) assuming G a g aa,wehave ln Q S ( ) +wa G a ln. (0) a Differentiating this with respect to x j,weget x a ln Q x a b w a S G b w b ( + w b ) x w b a, (0) x a

16 500 K. Iguchi & K. Aomoto for a,...,s. On the other hand, we take log of Eq. (9), which yields g aa ln w a +( g aa )ln(+w a )+ S b( a) g ab [ln w b ln( + w b )] ln x a (03) for a,...,s. And by differentiation of Eq. (03), we obtain S (δ ab w a + g ab ) w b ( + w b ) x w b c δ ac, (04) x c b for a, c,...,s. Assuming matrices  (A ab) such that A ab δ ab w a + g ab and ˆX (X ab ) such that X ab ( /w b (+w b ))x a ( w b / x a ), Eq. (04) becomes  ˆX Ê, whereêis the S S unit matrix, and solving this for ˆX by using linear algebra, we obtain ˆX Â. Substituting them into Eq. (0) [i.e., n a b G bx ab ]and using the formula for an expansion of determinant, we obtain Eq. (99). In this way, the singularity hypersurface of Eq. (98) is very significant when we consider the location of singularities in the GPF as well as in the distribution functions. However, the treatment is not so simple as when we consider the g-on case, though. 5. Integral Representation for the GPF in the Energy and the Momentum Representations Let us consider when we have to take care of the quantum numbers as well as the species. In this case, we go back to Eq. (49). From generalizing Eq. (96) with taking care of quantum numbers j, we easily obtain Q j S D j S a ( +w a j w a j ) G a j +gjj aa wj a. (05) From this we find ln Q j,a (G a j + gaa jj )ln ( +w a j w a j ) + ln wj a j,a j ln D j S, (06) where w a j satisfies ( + wj a ) ( ) w b g ba kj k +w b b,k k x a. (07) j Following the argument in Ref. 5, we can convert the above into the energy representation: ( ) V ln Q +wa (ε) dεd a (ε)ln, (08) w a (ε) 0

17 Integral Representation for the Grand Partition Function 50 where D a (ε) is the density of states of the system and w a (ε) satisfies [ + w a (ε)] b,ε [ wb (ε ) +w b (ε ) ] gba (ε,ε) x a (ε). (09) We are able to convert the expression in the energy representation into that in the momentum representation, using an energy dispersion such as ε ε(k) where kis a D-dimensional wave vector. Then we have V ln Q a d D k (π) D ln ( +wa (k) w a (k) ). (0) This is also carried out in the same way as before. For the case of an ideal gas of multispecies quasiparticles with mutual exclusion statistics, the Haldane s definition in the momentum representation is given by n a (k)+ρ a (k) [g ab (k, k ) δ ab δ k,k ]n b (k ), () b,k where n a (k) is the particle distribution function for species a with momentum k and ρ a (k) the hole distribution function for species a with momentum k, respectively. Now, following the method of Yang and Yang, 4 the entropy term W N is given by W N [V (n a (k)+ρ a (k))d D k]! [Vn a (k)d D k]![vρ a (k)d D k]!, () a,k where Vn a (k)d D k means the total number of k s in d D k and Vρ a (k)d D k the total number of holes in d D k.thegpfqis now given by [V (n a (k)+ρ a (k))d D k]! Q [Vn a (k)d D k]![vρ a (k)d D k]! e β a,k [εa(k) µa]na(k). N0 Therefore, we get Q a,k na(k)n {n a(k)} a,k {n a(k)} a,k [V (n a (k)+ b,k g ab(k, k )n b (k ))d D k]! [Vn a (k)d D k]![v ( b,k g ab(k, k )n b (k ))d D k]! (3) [x a (k)] na(k), (4) where x a (k) e β[εa(k) µa]. To evaluate the above for the multispecies case, let us first consider the two species case assuming g ab (k, k) g ab δ k,k for a, b,, for the sake of simplicity. Now the W N becomes W N [V ( + ( g )n (k) g n (k))d D k]! [V ( g n (k) g n (k))d D k]! [V ( + ( g )n (k) g n (k))d D k]! [V ( g n (k) g n (k))d D. (5) k]!

18 50 K. Iguchi & K. Aomoto Using Eq. (5), we obtain the following relation: Γ(V ( + ( g )n (k) g n (k))d D k +) Γ(V ( g n (k) g n (k))d D k +) t V(+( g)n(k) gn(k))dd k Γ( Vn (k)d D k) 0 ( + t ) dt V ( gn(k) gn(k))dd k+ ( )Vn(k)dDk Γ( + Vn (k)d D k) π [t g ( + t ) g ] Vn(k)dD k 0 ( +t Γ(V ( + ( g )n (k) g n (k))d D k +) Γ(V ( g n (k) g n (k))d D k +) t t VdD k ( + t ) VdD k+ ) gvn (k)d D k dt, (6) t V(+( g)n(k) gn(k))dd k Γ( Vn (k)d D k) 0 ( + t ) dt V ( gn(k) gn(k))dd k+ ( )Vn(k)dDk Γ( + Vn (k)d D k) π [t g ( + t ) g ] Vn(k)dD k 0 ( +t t t VdD k ( + t ) VdD k+ ) gvn (k)d D k dt, (7) where we have abbreviated as t a t a (k). Substituting Eqs. (6) and (7) into Eq. (4) for the two species case, we obtain Q(k) t VdD k t VdD k π dt dt ( + t ) VdD k+ ( + t ) VdD k+ 0 Vn (k)d D k Vn (k)d D k 0 [ ( ) g ] Vn(k)d D k +t x t g ( + t ) g t [ ( ) g ] Vn(k)d D k +t x t g ( + t ) g. (8) t Applying the binomial expansion of Eq. (8) to Eq. (8), we obtain Q(k) t VdD k t VdD k π dt dt ( + t ) VdD k+ ( + t ) VdD k x t g ( + t ) g [( + t )/t ] g. (9) +x t g ( + t ) g [( + t )/t ] g

19 Integral Representation for the Grand Partition Function 503 Extending first the integral intervals from [0, ) to(, ) dividing each integration by, secondly extending the real integration to the complex integration, and thirdly changing the variables as t a t a t a fora,, we get Q(k) (πi) C ( + t ) dt dt C Vd D k t VdD k+ ( + t ) VdD k t VdD k+ ϕ (t)ϕ (t), (0) where ϕ (t),ϕ (t) are defined by Eq. (74). By using the generalized Cauchy theorem, we end up with Q k Q(k) k ( + w ) VdD k ( + w ) VdD k w VdD k+ w VdD k+ (ϕ,ϕ ). () (w,w ) Hence, we have factorized the GPF for the two species case in the momentum representation. It is not difficult to generalize the above result to that for the case of multispecies case. Following the same procedure, we are able to obtain Q(k) (πi) S dt dt S C C S ( + t ) VdD k t VdD k+ ( + w ) VdD k w VdD k+ ( + t S) VdD k t VdD k+ S ( + w S) VdD k w VdD k+ S where ϕ ϕ S 0andϕ,...,ϕ S are defined by ϕ (t) ϕ S (t) ϕ a (t) x a (k)w a (k) gaa(k,k) [ + w a (k)] gaa(k,k) b,k ( a,k) () (ϕ,...,ϕ S ), (3) (w,...,w S ) [ wb (k ] gba (k ),k) +w b (k, (4) ) for a, b,...,s and x a (k) e β[εa(k) µa]. Equation (3) in the momentum representation is an equivalent expression to Eq. (95) in the state representation. Let us calculate the Jacobian in Eq. (3). Substituting Eqs. (4) into Jacobian, we then find (ϕ,...,ϕ S ) (w,...,w S ) D S (k), (5) S w a (k)[ + w a (k)] a

20 504 K. Iguchi & K. Aomoto wherewehavedefinedas w (k)+g g g S g w (k)+g g S D S (k).. (6)..... g S g S w S (k)+g SS Substituting Eq. (5) into Eq. (3), we obtain Q ( ) Vd D k+ ( ) Vd D k+ +w +ws w w S, (7) w w S D S k which is exactly the generalization of Eq. (98) to that in the momentum representation. Thus, the singularities in the GPF are absolutely dominated by the singularity hypersurface: D S (k) 0. (8) Finally, taking log of Q and assuming Vd D k, we derive Eq. (0). Hence, we have completed the factorization of the GPF in the momentum representation. The singularity of the GPF is also related to that in the distribution functions n a (k). Defining the w a (k) by w a (k) ρ a(k) n a (k), (9) substituting this into Eq. (), we obtain [w a (k)δ ab δ k,k + g ab (k, k )]n b (k ). (30) b,k Solving for n a (k), we obtain n a (k) ˆD S (k) D S (k), (3) wherewehavedefinedas D S (k)det[w a (k)δ ab δ k,k + g ab (k, k )], (3) and ˆD S (k) is given by replacing all elements in the (a, k) column by. For example, when g ab (k, k )g ab δ k,k, we get the similar expressions to Eqs. (97) and (00), replacing w a by w a (k) such as Eq. (6) and w (k)+g g g S g w (k)+g g S ˆD S.. (33) g S g S w S (k)+g SS Thus, the same singularity hypersurface dominates the singularity in the distribution functions as well.

21 Integral Representation for the Grand Partition Function Conclusion In conclusion, we have derived an integral representation for the GPF for an ideal gas of multispecies quasiparticles with mutual exclusion statistics, for the first time. For the derivation, we have used only some knowledge of classical mathematics such as the Euler integral of the first kind, the Gamma function, the beta function, the binomial expansion, and the generalized Cauchy theorem for multi complex variable functions. From the integral representation, we have easily proved the factorization of the GPF without any approximation. There, indeed we have not used the assumption of a most probable distribution regarding the GPF as an asymptotic series, which is very crucial for the standard argument in QSM. Therefore, the results are all exact! This is a remarkable fact in QSM. We have shown that the factorized GPF consists of the Wu s equations 3 and equivalently, the Sutherland s equations 0 in its denominator when we consider the ideal multispecies gas with mutual exclusion statistics. This provides a deep reason why one must encounter the distribution functions that obey the Wu s equations. 3 Therefore, the factorized GPF tells us how a singularity appears in the GPF, which is given by Eq. (98). And this singularity coincides with that in the distribution functions [Eq. (99)]. Finally, we have generalized the scheme to obtain the factorized GPF in the energy and the momentum representations, respectively. In this way, we believe that the factorized GPF will prove to be very important when we answer whether or not a phase transition occurs in the system of an ideal gas of multispecies quasiparticles with mutual exclusion statistics. Acknowledgments One of us (K. I.) would like to thank the Mitsubishi Foundation for research fund and Kazuko Iguchi for her continuous financial support and encouragement. References. K. Huang, Statistical Mechanics, nd Edn. (John Wiley, New York, 987).. C. N. Yang and T. D. Lee, Phys. Rev. 87, 404 (95); T. D. Lee and C. N. Yang, ibid. 87, 40 (95). 3. Y.-S. Wu, Phys. Rev. Lett. 73, 9 (994). 4. F. D. M. Haldane, Phys. Rev. Lett. 67, 937 (99). 5. K. Iguchi, Phys. Rev. Lett. 78, 333 (997); W.-H. Huang, ibid. 8, 39 (998); K. Iguchi, ibid. 8, 393 (998). 6. C. Nayak and F. Wilczek, Phys. Rev. Lett. 73, 740 (994); S. B. Isakov, D. P. Arovas, J. Myrheim and A. P. Polychronakos, Phys. Lett. A, 99 (996). 7. E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, 4th edn., reprinted (Cambridge University, Cambridge, 97). 8. K. Aomoto and K. Iguchi, On quasi-hypergeometric functions, Methods and Applications of Analysis 6, 55 (998); Singularity and Monodromy of Quasi- Hypergeometric Functions, in Contemporary Mathematics (AMS, Providence, 999).

22 506 K. Iguchi & K. Aomoto K. Aomoto, Integral Representations of Quasi Hypergeometric Functions, to appear in Proc. Int. Workshop on Specikal Functions (City University of Hong Kong, 000). 9. K. Iguchi and K. Aomoto, Mod. Phys. Lett. B3, 039 (999). 0. B. Sutherland, J. Math. Phys., 50 (97); in Exactly Solvable Problems in Condensed Matter and Relativistic Theory, eds.b.s.shastry,s.s.jhaandv.singh (Springer-Verlag, New York, 985), p... K. Iguchi, Mod. Phys. Lett. B, 765 (997); Int. J. Mod. Phys. B, 355 (997).. K. Iguchi, Mod. Phys. Lett. B, (998); ibid. B, 63 (998); Phys. Rev. B58, 689 (998). 3. K. Kodaira, Theory of Complex Manifolds (Iwanami, Tokyo, 99) (in Japanese); C. L. Siegel, Topics in Complex Function Theory, Vol. III (Wiley, New York, 989). 4. C. N. Yang and C. P. Yang, J. Math. Phys. 0, 5 (969).