Partition function of N composite bosons
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] D ec Partition function of N composite bosons Shiue-Yuan Shiau , Monique Combescot and Yia-Chung Chang , ∗ Department of Physics, National Cheng Kung University, Tainan, 701 Taiwan Institut des NanoSciences de Paris, Universit´e Pierre et Marie Curie, CNRS, 4 place Jussieu, 75005 Paris and Research Center for Applied Sciences, Academia Sinica, Taipei, 115 Taiwan (Dated: November 8, 2018)The partition function of composite bosons (“cobosons” for short) is calculated in the canonicalensemble, with the Pauli exclusion principle between their fermionic components included in anexact way through the finite temperature many-body formalism for composite quantum particleswe recently developed. To physically understand the very compact result we obtain, we first presenta diagrammatic approach to the partition function of N elementary bosons. We then show how toextend this approach to cobosons with Pauli blocking and interaction between their fermions. Thesediagrams provide deep insights on the structure of a coboson condensate, paving the way towardthe determination of the critical parameters for their quantum condensation. I. INTRODUCTION
A century ago, Albert Einstein suggested that as tem-perature decreases, non-interacting elementary bosonsmust undergo a phase transition with a macroscopicnumber of these bosons “condensed” into the systemground state. Such a condensation occurs below a crit-ical temperature which decreases with the boson num-ber N as N / . Interest in Bose-Einstein condensation(BEC) has been revived a decade ago by its first exper-imental realization thanks to advanced cooling and gastrapping techniques . These techniques now allow thestudy of condensation in geometrically different or low-dimensional potential wells in which a fixed number ofbosons are trapped. In addition, highly controllable Fes-hbach resonances opened the route to the study of theBEC-BCS crossover in atomic systems .As the effect of interaction between particles decreaseswith particle density, a condensation similar to thecondensation of non-interacting elementary bosons pre-dicted by Einstein should in principle occur in a dilutegas of bosonic particles, i.e., composite particles madeof an even number of fermions. And indeed, such aphase transition is now commonly produced in ultra-cold atomic vapors . Yet, Bose-Einstein condensationin the case of semiconductor excitons has been searchedfor decades − , even though these particles were for along time considered as the most promising candidateto evidence this remarkable macroscopic quantum effect:due to their very light effective mass, the exciton quan-tum degeneracy at density easy to experimentally achieveshould occur below a few kelvins while temperatures aslow as micro-kelvins are required for atoms. By contrast,evidence of condensation in exciton-polaritons has beendemonstrated in semiconductor quantum well embeddedinside a microcavity and more clearly in a trap .One reason for such a long time search could be that,due to their internal degrees of freedom, semiconductorexcitons exist in bright and dark states, i.e., excitons cou-pled or not coupled to light. This coupling goes alongwith an increase of the bright exciton energy, leaving darkexcitons in the lowest-energy state. So, the Bose-Einstein condensate of excitons must be dark, i.e., not coupled tolight . Another reason could be that, in addition toCoulomb interaction between carriers, excitons also in-teract in a non-standard way through carrier exchangesinduced by the Pauli exclusion principle between elec-trons and between holes. We may wonder if the Pauliexclusion principle at density necessary for condensationdoes not substantially affect the quantum condensationof a coboson gas. In relation to this question, we wishto mention that, although the BCS wave function ansatzwith all Cooper pairs condensed into the same state suc-cessfully explains the physical properties observed in con-ventional superconductors, this Pauli exclusion principlestill makes the exact wave function for N Cooper pairs,as deduced from the Richardson-Gaudin procedure, quitedifferent from the BCS wave function ansatz .Although quite successful in treating systems of inter-acting elementary particles, either bosonic and fermionic,conventional many-body formalism is inadequate whenit comes to cobosons like the excitons: first, conven-tional many-body theory such as the Green’s functionformalism is constructed in the grand canonical ensem-ble whereas a X -size excitons dissociate through a Motttransition when their number reaches L /a X , which isthe maximum number a sample volume L can accom-modate. Secondly, conventional many-body theory pre-sumes some kind of Hamiltonian which normally con-sists of a part for the particle kinetic energy and a partfor interaction between particles. But, attempts to con-struct energy-like effective scatterings between cobosonsthrough a “bosonization procedure” fail, by nature, toallow exchanges between the particle fermionic compo-nents because their fermions must be frozen into a fixedconfiguration: the problem comes from the fact thatfermion exchanges are dimensionless; so, they cannot leadto energy-like scatterings in order to possibly appear inthe Hamiltonian. These two reasons led us to seek fora new many-body formalism in which the number of co-bosons is fixed.A zero temperature formalism for composite quantumparticles which allows handling fermion exchanges in-duced by the Pauli exclusion principle in an exact waywas proposed by Combescot et al . We then extendedthis coboson formalism to finite temperature , pavingthe way to solving a large variety of coboson many-bodyeffects. The goal of this work is to derive the partitionfunction in the canonical ensemble based on this finitetemperature formalism. Through it, all statistical ther-modynamic properties, including the critical temperaturefor quantum condensation, should be possible to obtain.To start, we reconsider the partition function of non-interacting elementary bosons. The one commonlyknown is in the grand canonical ensemble. From it, wecan mathematically extract the partition function in thecanonical ensemble; in practice, however, its numericalimplementation is quite tricky. Here, we instead proposea direct derivation of this canonical partition functionbased on a recursion relation. Through this recursion re-lation we are directly led to the well-known compact formfor the canonical partition function of non-interacting el-ementary bosons given in Eq. (4). Its diagrammatic rep-resentation has the great advantage to allow easy iden-tification of the fully uncondensed, partially condensedand fully condensed contributions.To show the power of our diagrammatic approach, nextwe consider interacting elementary bosons. We show howto perform a many-body expansion of the canonical parti-tion function through a recursion relation similar to theone used for non-interacting bosons. Interestingly, wefind that the partition function for interacting elemen-tary bosons maintains the same recursion relation—andthe same compact form—as for ideal elementary bosonsprovided that we add interactions in each n -particle en-tangled configuration. While this is reminiscent of clusterexpansion for quantum systems , here we do not needto assume the property that the partition functions canbe divided into groups of “connected” particles. Theyautomatically show up.We then turn to the canonical partition function of N cobosons made of two fermions, like the excitons. Afterrecalling the key commutators of the coboson many-bodyformalism, we first calculate the recursion relation of thispartition function at first order in fermion exchange inthe absence of interaction scatterings between these co-bosons. Although this can be done through a brute-forceuse of commutators, we have here chosen to present aphysically intuitive way in getting this partition functionthrough the extension of the diagrammatic approach weused for non-interacting elementary bosons. Surprisingly,we find that the coboson partition function can be cast inthe same compact form as for non-interacting elementarybosons provided that we take into account the possibilitythat cobosons exchange their fermionic components dueto the indistinguishability in each n -particle entangledconfiguration. Since fermion exchange does not lead toa normal particle-particle potential, this canonical par-tition function is fundamentally different from the oneof interacting elementary bosons previously considered.These diagrams allow us to understand how an elemen-tary boson condensate is affected by fermion exchanges induced by the Pauli exclusion principle.Then, taking into account interaction between thefermionic components of the cobosons becomes ratherstraightforward due to similarities between interactingelementary bosons and interacting cobosons, differencescoming from additional Pauli exchange processes.The key result of this work is the recursion relationgiven in Eq. (85) for the canonical partition functions of N cobosons. This recursion relation leads to the parti-tion function in the same compact form as the one ofnon-interacting elementary bosons. Our result evidencesthat cobosons do not all condense into the same state,as non-interacting elementary bosons do in a BEC con-densate. The similar structure of the elementary bosonand coboson partition functions may help us build possi-ble links between condensate wave functions and criticalparameters for the BEC’s of elementary bosons and ex-citons. Moreover, the statistical entropy derived fromthe partition function enables us to study the relationbetween quantum entanglement in quantum informationlanguage and the composite particle bosonic nature .The present paper is organized as follows: In Sec. II,we briefly introduce the compact form for the canonicalpartition function of non-interacting elementary bosons.Next we present the diagrammatic approach to derive therecursion relation between canonical partition functions.Then we extend this diagrammatic approach to interact-ing elementary bosons. In Sec. III, we first briefly discusscomplexities intrinsic in the coboson systems. We thenintroduce the interaction expansion which allows us tosplit the coboson partition function into a non-interactingpart and an interacting part. Finally, we use a diagram-matic approach to calculate the partition function at ze-roth order and also at first order in interaction scatteringwith Pauli exchange treated at first order. Consequencesand significances of our results are discussed in the end. II. ELEMENTARY BOSONSA. Ideal(non-interacting) Bose gas
We consider a gas of non-interacting elementary bosonswith kinetic energy ε k = ~ k / m . Since these bosonsdo not interact, the energy of each k state occupied by N k bosons simply is N k ε k ; so, the partition function forthis ideal Bose gas in the canonical ensemble reads, for β = 1 /k B T , as ¯ Z (0) N = X { N k } N e − β P k N k ε k , (1)the sum being taken over all possible boson numbers sub-ject to P k N k = N .
1. Canonical partition function starting from grandcanonical ensemble
To lift the constraint in the sum of Eq. (1), one com-monly turns to the grand partition function with µ fixedinstead of N , defined as¯ Z ( GC ) = ∞ X N =0 e βµN ¯ Z (0) N . (2)A compact form for ¯ Z ( GC ) is easy to obtain by notingthat it also reads¯ Z ( GC ) = ∞ X N =0 X { N k } N e − β P k N k ( ε k − µ ) (3)= Y k ∞ X N k =0 e − βN k ( ε k − µ ) = Y k − e − β ( ε k − µ ) . The chemical potential µ is ultimately adjusted for themean value of the particle number in the grand canonicalensemble to equal the number of bosons at hand.Equation (2) shows that the partition function in thecanonical ensemble, ¯ Z (0) N , is just the prefactor of e βµN in¯ Z ( GC ) . This prefactor can be obtained from the N th derivative of ¯ Z ( GC ) with respect to e βµ . It has beenshown that this yields a compact form to the canonicalpartition function which reads as ¯ Z (0) N = X { p i } p ! (cid:18) z ( β )1 (cid:19) p p ! (cid:18) z (2 β )2 (cid:19) p · · · p N ! (cid:18) z ( N β ) N (cid:19) p N . (4)The p i ’s are a set of non-negative integers such that N = 1 p + 2 p + · · · + N p N , (5)while z ( nβ ) is defined as z ( nβ ) = X k e − nβε k . (6)
2. Direct approach to the canonical partition function
The above derivation of the canonical partition func-tion, based on derivatives of the partition function inthe grand canonical ensemble, is smart but completelyformal. It moreover presupposes the knowledge of thepartition function in the grand canonical ensemble. Wehere present a direct derivation of the canonical parti-tion function for a boson number N . This derivation isnot only useful for possible extension to cobosons, but,through its diagrammatic representation, it provides aphysical understanding of the various terms as comingfrom the fully uncondensed, partially condensed and fullycondensed configurations. ! ! ! ! " ! " ! " FIG. 1: Scalar product of N elementary bosons appearing inthe canonical partition function given in Eq. (12). Let | ¯ ψ { N k } N i be normalized N -particle eigenstate ofthe system Hamiltonian ¯ H with N k bosons having an en-ergy ε k . The canonical partition function given in Eq. (1)can be rewritten as¯ Z (0) N = X { N k } N h ¯ ψ { N k } N | e − β ¯ H | ¯ ψ { N k } N i . (7)We can circumvent the difficulty coming from the re-striction, P k N k = N , in the sum over all possible con-figurations { N k } N by using the closure relation in the N -elementary boson subspace written in terms of sin-gle boson operators ¯ B † k . These operators are such that( ¯ H − ε k ) ¯ B † k | v i = 0 where | v i denotes the vacuum state,with a commutation relation given by h ¯ B k ′ , ¯ B † k i − = δ k ′ k . (8)This closure relation reads as¯I N = 1 N ! X { k } ¯ B † k ¯ B † k · · · ¯ B † k N | v ih v | ¯ B k N · · · ¯ B k ¯ B k , (9)as can be checked from ¯I = ¯I and to generalize to ¯I N .Since the | ¯ ψ { N k } N i ’s are eigenstates of ¯ H , a closure re-lation also exists for normalized | ψ { N k } N i ’s, reading as¯I N = X | ¯ ψ { N k } N ih ¯ ψ { N k } N | . (10)By injecting Eq. (9) in front of | ¯ ψ { N k } N i in Eq. (7) andby getting rid of the | ¯ ψ { N k } N i states through Eq. (10),we can rewrite ¯ Z (0) N as¯ Z (0) N = 1 N ! X { k } h v | ¯ B k · · · ¯ B k N e − β ¯ H ¯ B † k N · · · ¯ B † k | v i . (11)The Hamiltonian ¯ H for non-interacting elementarybosons reads as ¯ H = P k ε k ¯ B † k ¯ B k ; so, the above canon-ical partition function readily reduces to¯ Z (0) N = 1 N ! X { k } e − β ( ε k + ··· + ε k N ) h v | ¯ B k · · · ¯ B k N ¯ B † k N · · · ¯ B † k | v i . (12)Note that (i) the k ’s in the sum now take all possiblevalues without restriction. (ii) a given { N k } N configu-ration appears once only in Eq. (10), while it appearsmany times in Eq. (12), which explains the presence ofthe 1 /N ! prefactor.
3. Recursion relation for ¯ Z (0) N The scalar product in the above equation can be cal-culated using the commutator (8). It allows us to replace¯ B k N ¯ B † k N by δ k N k N + ¯ B † k N ¯ B k N . The δ k N k N term, wheninserted into Eq. (12), readily gives1 N ! z ( β ) h ( N − Z (0) N − i . (13)To evaluate the ¯ B † k N ¯ B k N term, we push the operator ¯ B k N to the right according to the commutator (8). This yields( N −
1) terms like δ k N k N − h v | ¯ B k · · · ¯ B k N − ¯ B † k N ¯ B † k N − · · · ¯ B † k | v i (14)which are equivalent when inserted into Eq. (12) througha relabeling of the dummy indices k n ’s. Repeatingthe same procedure as above, we replace ¯ B k N − ¯ B † k N by δ k N − k N + ¯ B † k N ¯ B k N − . The term in δ k N − k N , when in-serted into Eq. (12), readily gives1 N ! ( N − z (2 β ) h ( N − Z (0) N − i . (15)The term in ¯ B † k N ¯ B k N − , calculated by pushing ¯ B k N − tothe right, yields ( N −
2) equivalent terms; and so on...So, we end with a nicely compact recursion relationwhich simply reads as¯ Z (0) N = 1 N h z ( β ) ¯ Z (0) N − + z (2 β ) ¯ Z (0) N − + · · · + z ( N β ) i = 1 N N X p =1 z ( pβ ) ¯ Z (0) N − p , (16)with ¯ Z (0)0 taken as 1. Using this recursion relation, it iseasy to recover the expression of the canonical partitionfunction obtained from the grand canonical ensemble ,as given in Eq. (4). As illustration, we give the lowestfew ¯ Z (0) N ’s in Appendix I.
4. Diagrammatic procedure
It is possible to recover the recursion relation (16) be-tween the canonical partition functions using diagrams.The diagram of Fig. 1 represents the scalar product of N elementary bosons ( k , · · · , k N ). We can set up reduc-tion rules to relate this scalar product to those of lowernumber of bosons. As depicted in Fig. 2, this is done byconnecting k N on the left to one of the k ’s on the right;this k can be either k N as in Fig. 2(a) (leaving behinda scalar product of N − k n ’s like k N − as in Fig. 2(b), which leads to ( N −
1) similar termsonce summation over dummy k indices is performed. Inthe diagram of Fig. 2(b), we can connect k N − on theleft either to k N as in Fig. 2(c) (leaving behind a scalar ! ! " (cid:16) ! " ! " ! ! " (cid:16) ! " " ! " (cid:16) " ! (cid:16) ! " " ! " (cid:16) ! " ! " ! ! " (cid:16) ! ! " (cid:16) ! " " ! " (cid:16) " ! (cid:16) ! " " ! " (cid:16) ! " ! " ! " ! ! " (cid:16) ! ! (cid:16) ! " ! ! " (cid:16) ! " ! ! " (cid:16) ! " ! " ! ! (cid:16) ! " ! ! " (cid:16) $ % & FIG. 2: Diagrams leading to the recursion relation (16) be-tween the canonical partition functions of non-interacting el-ementary bosons. product of N − k n ’s like k N − as in Fig. 2(d), which leads to ( N −
2) similar terms oncesummation over dummy k ’s is performed; and so on...We then readily find that the process of Fig. 2(a) givesto ¯ Z (0) N a contribution equal to (1 /N !) z ( β )[( N − Z (0) N − ].The process of Fig. 2(c), which imposes k N = k N − ,gives a contribution equal to ( N − /N !) z (2 β )[( N − Z (0) N − ]; and so on... So, we do recover the recursionrelation between the ¯ Z (0) N ’s as given in Eq. (16), z ( pβ )being the partition function for a condensate made of p elementary bosons, all in the same state.We are going to show that the partition function for N cobosons obeys a similar recursion relation, providedthat we take into account fermion exchanges and inter-action scatterings between the composite particles en-tangled in a condensate. However, before turning to co-bosons, let us go one step further by considering inter-acting elementary bosons. We are going to show that arecursion relation exists provided that we replace z ( nβ )for a non-interacting n -boson condensate by a modifiedˆ z ( nβ ) which contains interaction between bosons. B. Interacting Bose gas
We now consider interacting elementary bosons. TheirHamiltonian reads¯ H = ¯ H + ¯ V (17)= X k ε k ¯ B † k ¯ B k + 12 X kk ′ q V q ¯ B † k + q ¯ B † k ′ − q ¯ B k ′ ¯ B k , the operators ¯ B † k still obeying the commutation rela-tion (8). The canonical partition function reads interms of the N -boson eigenstates of the system, ( ¯ H − ¯ E N,ξ ) | ¯ ψ N,ξ i = 0, as¯ Z N = X ξ e − β ¯ E N,ξ = X ξ h ¯ ψ N,ξ | e − β ¯ H | ¯ ψ N,ξ i . (18)To get rid of these unknown eigenstates, we follow thesame procedure as in Sec. II A 2: we insert the closurerelation (9) for N elementary bosons in front of | ¯ ψ N,ξ i inEq. (18) and use the fact that ¯I N = P | ¯ ψ N,ξ ih ¯ ψ N,ξ | . Thecanonical partition function then reads as¯ Z N = 1 N ! X { k } h v | ¯ B k · · · ¯ B k N e − β ¯ H ¯ B † k N · · · ¯ B † k | v i . (19)Next, we perform a many-body expansion of ¯ Z N . Wefirst rewrite e − β ¯ H using the Cauchy integral formula as e − β ¯ H = Z C dz πi e − βz z − ¯ H , (20)where the integration path C is a circle of finite radiuscentered at the ¯ H value on the complex plane. ( Forsimplicity, we omit this subscript C in the following.)The operator 1 / ( z − ¯ H ) is expanded for ¯ H = ¯ H + ¯ V through 1 z − ¯ H = 1 z − ¯ H + 1 z − ¯ H ¯ V z − ¯ H . (21)This leads us to split the partition function as¯ Z N = ¯ Z (0) N + ¯ Z (1) N + · · · (22)The zeroth-order ¯ Z (0) N in interaction reads as in Eq. (11)while the first order is given by¯ Z (1) N = 1 N ! X { k } Z dz πi e − βz (23) ×h v | ¯ B k · · · ¯ B k N z − ¯ H ¯ V z − ¯ H ¯ B † k N · · · ¯ B † k | v i . ¯ H acting on ¯ B † k N · · · ¯ B † k | v i gives ε k + · · · + ε k N while Z dz πi e − βz ( z − ¯ ε ) = − βe − βε ; (24)so, ¯ Z (1) N appears as¯ Z (1) N = − β N ! X { k } e − β ( ε k + ··· + ε k N ) ×h v | ¯ B k · · · ¯ B k N ¯ V ¯ B † k N · · · ¯ B † k | v i . (25)A convenient way to calculate the above matrix elementis to introduce commutators h ¯ V , ¯ B † p i − = X kq V q ¯ B † p + q ¯ B † k − q ¯ B k = ¯ V † p , (26) h ¯ V † p , ¯ B † p ′ i − = X q V q ¯ B † p + q ¯ B † p ′ − q . (27)By pushing ¯ V in Eq. (25) to the right using these com-mutators, we get ( N −
1) + · · · + 1 = N ( N − / ! ! " (cid:16) ! " ! " ! " ! " ! ! ! " (cid:16) (cid:16) ! " (cid:14) " " ! " " " ! " ! " " " $ % FIG. 3: (a) Diagrammatic representation of ¯ Z (1) N . (b) Dia-gram contributing to ¯ Z (1)2 . which contribute equally to ¯ Z (1) N through a relabeling ofthe dummy indices k n ’s. By symmetrizing the process,i.e., by also pushing ¯ V to the left, we end with the first-order term in interaction reading as¯ Z (1) N = − β N ! C N X { k } e − β ( ε k + ··· + ε k N ) X q V q (28) × h h v | ¯ B k · · · ¯ B k N ¯ B † k N + q ¯ B † k N − − q ¯ B † k N − · · · ¯ B † k | v i + c.c. i . This matrix element is shown in the diagram of Fig. 3(a).For N = 2, we readily get, since ¯ V ¯ B † k ¯ B † k | v i = P q V q ¯ B † k + q ¯ B † k − q | v i ,¯ Z (1)2 = − β V ( β, β ) (29)with V ( β, β ) defined through V ( n β, n β ) = X k k e − β ( n ε k + n ε k ) ( V + V k − k ) . (30) V ( β, β ) corresponds to the two processes shown inFig. 3(b), indicated by two columns of k vectors ( k , k )and ( k , k ) separated by a dashed line on the leftof the diagram. To understand how the result devel-ops for large N , we have explicitly derived ¯ Z (1) N for N = 4 in Appendix II. For arbitrary N , we isolate¯ Z (0) N − , ¯ Z (0) N − , · · · from the diagram of Fig. 3(a) in thesame way as for ideal elementary bosons. The prefactorof ¯ Z (0) N − is made of the processes involving ( k N , k N − )shown in Fig. 4(a). Their contribution to ¯ Z (1) N reads as − β N ! C N V ( β, β ) h ( N − Z (0) N − i = − β V ( β, β ) ¯ Z (0) N − . (31)The prefactor of ¯ Z (0) N − is made of processes involving( k N , k N − ) and one of the ( k , · · · , k N − ), let say k N − .As shown in Fig. 4(b) there are four such entangled pro-cesses indicated by the four columns of k vectors sep-arated by dashed lines on the left of the diagram. InFig. 4(b), k N − “condenses” either with k N − or with k N . Since there are C N − ways to choose this k N − boson among ( k , · · · , k N − ), the contribution of suchprocesses to ¯ Z (1) N reads as − β N ! C N C N − (cid:16) V ( β, β ) + V (2 β, β ) (cid:17)h ( N − Z (0) N − i = − β (cid:16) V ( β, β ) + V (2 β, β ) (cid:17) ¯ Z (0) N − . (32) ! ! " (cid:16) ! ! " (cid:16) ! " " ! " (cid:16) ! " " ! " (cid:16) ! " " ! " (cid:16) ! " " ! " (cid:16) $ ! " " ! " (cid:16) ! " " ! " (cid:16) ! " ! ! " (cid:16) " ! " (cid:16) " ! " (cid:16) ! " % ! " (cid:16) ! ! " (cid:16) ! " " ! " (cid:16) % ! ! " (cid:16) " ! " (cid:16) ! ! " (cid:16) % ! " (cid:16) " ! " (cid:16) ! ! " (cid:16) ! ! " (cid:16) ! ! " (cid:16) " ! " (cid:16) % ! " (cid:16) ! " % ! " (cid:16) % ! " (cid:16) ! " " ! " (cid:16) ! " ! ! " (cid:16) % ! " (cid:16) ! ! " (cid:16) ! " % ! " (cid:16) ! ! " (cid:16) % ! " (cid:16) " ! " (cid:16) ! " ! ! " (cid:16) ! " % ! " (cid:16) % ! " (cid:16) " ! " (cid:16) ! " % ! " (cid:16) " ! " (cid:16) "" " " ! ! ! " (cid:16) "" " ! ! " (cid:16) "" """ "" """ " % """ ! "" ! "" " ! " " ! """" % """ " " ! " (cid:16) ! " ! " % ! " (cid:16) ! " % ! " (cid:16) " ! " (cid:16) " ! " (cid:16) ! " " ! " (cid:16) " ! " (cid:16) ! ! " (cid:16) ! ! " (cid:16) ! " ! ! " (cid:16) FIG. 4: Three diagrams (a,b,c) corresponding to the prefac-tors of ¯ Z (0) N − , ¯ Z (0) N − and ¯ Z (0) N − . To get the prefactor of ¯ Z (0) N − , we isolate two k ’sout of ( k , · · · , k N − ), let say ( k N − , k N − ). Sincethere are C N − ways to choose these two k ’s, the con-tribution to ¯ Z (1) N of the entangled processes between( k N , k N − , k N − , k N − ) reads as − β N ! C N C N − (cid:16) V ( β, β ) + 2 V (2 β, β ) + 2 V (3 β, β ) (cid:17) × h ( N − Z (0) N − i = − β (cid:16) V ( β, β ) + V (2 β, β ) + V (3 β, β ) (cid:17) ¯ Z (0) N − . (33)The three terms in the parentheses originate from the 12processes shown in Fig. 4(c). They correspond to all pos-sible permutations of ( k N , k N − , k N − , k N − ) on the leftwhich make the same four k ’s on the right entangled, i.e.,the ( k N − , k N − ) must not “condense” with themselves;and so on...So, we finally get¯ Z (1) N = − β N X n =1 ˆ V ( nβ ) ¯ Z (0) N − n (34)with ˆ V ( nβ ) = n − X p =1 V ( pβ, ( n − p ) β ) . (35)By using the recursion relation between the ¯ Z (0) N ’s givenin Eq. (16), we get the partition function of N interactingelementary bosons at first order in interaction as¯ Z N ≃ N N X n =1 h z ( nβ ) − βN V ( nβ ) i ¯ Z (0) N − n . (36)Note that the second term in the brackets depends ondensity N/L since the V q scattering depends on samplevolume as 1 /L , which is physically reasonable for many-body effects.It actually is possible to write ¯ Z N in a compact formlike Eq. (4). For that, we must transform Eq. (36) into a recursion relation between the ¯ Z N ’s similar to Eq. (16).To do it, we rewrite ¯ Z (0) N − n on the right-hand side ofEq. (36) in terms of ¯ Z N − n using Eq. (22). Equation(36) then becomes¯ Z N ≃ N N X n =1 z ( nβ ) h ¯ Z N − n − ¯ Z (1) N − n (cid:3) + ¯ Z (1) N . (37)Next, we note that, due to Eq. (34), − N N X n =1 z ( nβ ) ¯ Z (1) N − n = 1 N β N X n =1 z ( nβ ) N − m X m =1 ˆ V ( mβ ) ¯ Z (0) N − n − m . (38)As the right-hand side also reads1 N β N X n =1 z ( nβ ) N − m X m =1 ˆ V ( mβ ) ¯ Z (0) N − n − m = 1 N β N X n =1 ˆ V ( nβ )( N − n ) ¯ Z (0) N − n = − ¯ Z (1) N − N β N X n =1 n ˆ V ( nβ ) ¯ Z (0) N − n , (39)we end with ¯ Z N correct up to first order in interactionreading as ¯ Z N ≃ N N X n =1 ˆ z ( nβ ) ¯ Z N − n (40)with, for n ≥ z ( nβ ) = z ( nβ ) − βn V ( nβ ) . (41)It is then straightforward to transform Eq. (40) into acompact form like Eq. (4) with z ( nβ ) replaced by ˆ z ( nβ ).We have demonstrated that, up to first order in inter-action, the canonical partition function for interactingelementary bosons takes the same compact form as fornon-interacting elementary bosons provided that we re-place z ( nβ ) by ˆ z ( nβ ) of Eq. (41). For the perturbativeregime to be valid, N βV must be smaller than 1. Since V scales as 1 /L , this imposes N ˜ V /L ≪ k B T . Higherorders in interaction are obtained in the same way usingEq. (21). We then rewrite ¯ Z (0) N − n ’s in terms of ¯ Z N − n ’s toobtain a recursion relation similar to Eq. (40). III. COMPOSITE BOSONSA. Intrinsic difficulties with cobosons
We now consider cobosons made of two fermions likethe excitons. Some difficulties immediately arise whencompared to the ideal Bose gas we previously considered. !" $ !" $ !" $ (cid:14) ! " % ! " & FIG. 5: (a) Pauli scattering Λ (cid:0) n jm i (cid:1) associated with the ex-change of fermion α or β in the absence of fermion-fermioninteraction. (b) Interaction scattering ξ (cid:0) n jm i (cid:1) between thefermions of the cobosons i and j , in the absence of fermionexchange. It is clear that, in order for cobosons to be formed, an at-tractive interaction between their fermionic components( α, β ) has to exist. Except for the very peculiar reducedBCS potential in which an up-spin electron with momen-tum k interacts with a down-spin electron with momen-tum − k only, such fermion-fermion interaction automat-ically brings an interaction between cobosons.In addition to this interaction, cobosons also feel eachother through the Pauli exclusion principle between theirfermionic components. This “Pauli interaction” in factdominates most coboson many-body effects. As a result,it is impossible to avoid considering interaction betweenbosons once we have decided to take into account theircomposite nature. To properly handle many-body effectsbetween cobosons with creation operators B † i = X k α k β a † k α b † k β h k β , k α | i i , (42)where a † k α and b † k β are creation operators of theirfermionic components, we adopt the commutation for-malism introduced in Ref. 23:(i) Fermion exchanges in the absence of fermion-fermioninteraction follow from h B m , B † i i − = δ mi − D mi , (43) h D mi , B † j i − = X n Λ (cid:0) n jm i (cid:1) B † n , (44) D mi being such that D mi | v i = 0. The Pauli scatteringΛ (cid:0) n jm i (cid:1) associated with fermion exchange is shown inFig. 5(a). It corresponds to an exchange of fermion α or β between cobosons in states ( i, j ), which then end instates ( m, n ). Note that Λ (cid:0) n jm i (cid:1) and Λ (cid:0) m jn i (cid:1) correspondto the same exchange processes. For simplicity, in thefollowing, we shall use the first diagram with crossingdashed-lines to represent the Pauli scattering Λ (cid:0) n jm i (cid:1) .(ii) Interaction in the absence of fermion exchange follows from h H, B † i i − = E i B † i + V † i , (45) h V † i , B † j i − = X mn ξ (cid:0) n jm i (cid:1) B † m B † n , (46) V † i being such that V † i | v i = 0. The associated interactionscattering ξ (cid:0) n jm i (cid:1) is shown in Fig. 5(b).These four commutators allow us to calculate anymany-body effect between cobosons made of fermions( α, β ), in terms of Λ (cid:0) n jm i (cid:1) and ξ (cid:0) n jm i (cid:1) , with the Pauliexclusion principle between the fermionic components ofthese cobosons included in an exact way. The dimension-less parameter which rules many-body effects between N Wannier excitons with Bohr radius a X in a 3D samplewith size L , reads as η = N (cid:16) a X L (cid:17) , (47)this parameter appearing as η n − in processes in which n excitons are involved. B. Formal expression of the canonical partitionfunction for cobosons
The canonical partition function of N cobosons isdefined in terms of N -pair eigenstate energies, ( H −E N,ξ ) | ψ N,ξ i = 0, as Z N = X ξ e − β E N,ξ = X ξ h ψ N,ξ | e − βH | ψ N,ξ i . (48)We can get rid of these unknown eigenstates by insertingthe closure relation for N cobosons made of two fermions.Instead of Eq. (9), this closure relation has been shownto read as I N = (cid:18) N ! (cid:19) X { i } B † i B † i · · · B † i N | v ih v | B i N · · · B i B i . (49)The fact that these cobosons are made of two fermionsappears through the prefactor (1 /N !) instead of 1 /N !.By inserting Eq. (49) in front of | ψ N,ξ i in Eq. (48) andby using the closure relation I N = P ξ | ψ N,ξ ih ψ N,ξ | forthe N -pair eigenstates, we can rewrite Eq. (48) as Z N = (cid:18) N ! (cid:19) X { i } h v | B i · · · B i N e − βH B † i N · · · B † i | v i . (50)We wish to stress that difference with the canonical par-tition function for elementary bosons given in Eq. (11) isnot so much the prefactor change from 1 /N ! to (1 /N !) as the fact that the coboson operators B † i ’s now commutein a different way from the elementary boson operators.In addition, since these cobosons interact, the Hamilto-nian H in e − βH cannot be simply replaced by the sumof individual boson energies as in Eq. (12).To calculate the scalar product of Eq. (50), we use thecommutators for coboson operators given in Eqs. (43-46). As for interacting elementary bosons, we first usethe Cauchy integral formula (20) to rewrite e − βH in or-der to possibly perform an interaction expansion. Thisinteraction expansion follows from1 z − H B † i = B † i z − H − E i + 1 z − H V † i z − H − E i , (51)as easy to check using Eq. (45). So, e − βH B † i = B † i e − β ( H + E i ) + Z dz πi e − βz z − H V † i z − H − E i . (52)By symmetrizing the expansion procedure, as necessarysince we are going to truncate the interaction expansion,as usual in many-body problems, we are led to split Z N as Z N = X i N e − βE iN (cid:2) Γ N ( i N ) + I N ( i N ) (cid:3) . (53)The I N ( i N ) part, which comes from the second term ofEq. (52), is given by I N ( i N ) = 12 (cid:18) N ! (cid:19) X i ··· i N − Z dz πi e − β ( z − E iN ) (54) × h h v | B i · · · B i N z − H V † i N z − H − E i N B † i N − · · · B † i | v i + c.c. i To obtain I N ( i N ) at first order in V † i , we can push theoperator 1 / ( z − H − E i N ) to the right by only keeping thefirst term in Eq. (51). This leads to replacing H on theright of the above matrix element by E i N − + · · · + E i and H on the left by E i N + · · · + E i . Since Z dz πi e − βz ( z − E i − · · · − E i N ) = − βe − β ( E i + ··· + E iN ) , (55) I N ( i N ) appears as I N ( i N ) ≃ − β (cid:18) N ! (cid:19) X i ··· i N − e − β ( E i + ··· + E iN − ) (56) × h h v | B i · · · B i N V † i N B † i N − · · · B † i | v i + c.c. i . To go further, we use Eq. (46) to push V † i N to the right.By noting that V † i N B † i N − · · · B † i | v i gives ( N −
1) terms like X mn ξ (cid:16) n i N m i N − (cid:17) B † m B † n B † i N − · · · B † i | v i (57) ! ! " (cid:16) ! " ! " ! " ! " ! $ FIG. 6: Scalar product appearing in ˜ Z (1) N given in Eq. (67). which give equal contribution to I N ( i N ) when relabelingthe dummy indices i n ’s, we end with I N ( i N ) ≃ − β (cid:18) N ! (cid:19) ( N − X i ··· i N − e − β ( E i + ··· + E iN − ) (58) × h X mn h v | B i · · · B i N B † m B † n B † i N − · · · B † i | v i ξ (cid:16) n i N m i N − (cid:17) + c.c. i . This term physically corresponds to the diagram of Fig. 6in which two out of the N cobosons interact before pos-sibly exchanging their fermions with the other cobosons.We now consider the Γ( i N ) term of Z N which comesfrom the first term of Eq. (52). It readsΓ N ( i N ) = (cid:18) N ! (cid:19) X i ··· i N − h v | B i · · · B i N B † i N e − βH B † i N − · · · B † i | v i . (59)The same equation (52) leads us to split Γ N ( i N ) asΓ N ( i N ) = X i N − e − βE iN − (cid:2) Γ N ( i N , i N − ) + I N ( i N , i N − ) (cid:3) , (60)in which in the second term appears h v | B i · · · B i N B † i N z − H V † i N − z − H − E i N − B † i N − · · · B † i | v i , (61)which is similar to the scalar product appearing inEq. (54) except that we now have B † i N on the left. Itslowest order in V † is obtained by replacing the right H operator by E i N − + · · · + E i and the left H operator by − E i N + E i N + · · · + E i . Integration over z in Eq. (52)again gives − βe − β ( E i + ··· E iN − ) . So, by symmetrizingthe above process, we get I N ( i N , i N − ) ≃ − β (cid:18) N ! (cid:19) X i ··· i N − e − β ( E i + ··· + E iN − ) × h h v | B i · · · B i N B † i N V † i N − B † i N − · · · B † i | v i + c.c. i . (62)To go further, we again use Eq. (46). V † i N − B † i N − · · · B † i | v i then leads to ( N −
2) terms similar to X mn ξ (cid:16) n i N − m i N − (cid:17) B † m B † n B † i N − · · · B † i | v i (63)which ultimately gives I N ( i N , i N − ) as I N ( i N , i N − ) ≃ − β N − N !) X i ··· i N − e − β ( E i + ··· + E iN − ) (64) × hX mn h v | B i · · · B i N B † i N B † m B † n B † i N − · · · B † i | v i ξ (cid:16) n i N − m i N − (cid:17) + c.c. i . To calculate Γ N ( i N , i N − ), we proceed in the same way,namely we push e − βH in the scalar product to the rightusing Eq. (52); and so on... After summing over i N and i N − , the I N ( i N ) and I N ( i N , i N − ) terms actually giveequal contribution through a relabeling of the i ’s. So, byconsidering all equivalent terms, namely ( N −
1) + ( N −
2) + · · · + 1 = N ( N − /
2, we end with Z N ≃ Z (0) N + Z (1) N ≡ N ! (cid:2) ˜ Z (0) N + ˜ Z (1) N (cid:3) , (65)where the zeroth-order term in interaction scattering is˜ Z (0) N = 1 N ! X { i } e − β ( E i + ··· E iN ) h v | B i · · · B i N B † i N · · · B † i | v i , (66)while the first-order term in ξ reads as ˜ Z (1) N = − β N ! C N X { i } e − β ( E i + ··· + E iN ) (67) × hX mn h v | B i · · · B i N B † m B † n B † i N − · · · B † i | v i ξ (cid:16) n i N m i N − (cid:17) + c.c. i . Note that in Eq. (65), we have turned from Z N to ˜ Z N inorder to better see the consequences of the boson com-posite nature, ˜ Z (0) N in Eq. (66) and ¯ Z (0) N in Eq. (12) thenbeing formally identical: their unique but major differ-ence lies in the commuting relations these ¯ B † k and B † i operators have.The canonical partition function Z N in Eq. (65) ap-pears as an expansion in interaction scattering ξ . In thecase of electrons and holes bound into excitons throughCoulomb processes, ξ scales as the exciton Rydberg R X multiplied by the exciton volume a X and divided by thesample volume L . So, for N excitons, ˜ Z (1) N / ˜ Z (0) N scalesas N βξ ≃ βR X η where η is the dimensionless many-bodyparameter defined in Eq. (47). The many-body interac-tion expansion we perform is thus valid for βR X η ≪ η ≪ k B T /R X . This ratio is small compared to 1if the lowest relative motion exciton state only is popu-lated. Note that η actively controls the exciton physicsbecause, for η >
1, excitons dissociate into an electron-hole plasma through a Mott transition.
C. Partition function at zeroth order in ξ To grasp how the Pauli exclusion principle affects thecanonical partition function of N cobosons, let us concen-trate on its zeroth-order term in interaction scatteringgiven in Eq. (66). The calculation of the scalar prod-uct appearing in ˜ Z (0) N can be done through a brute-force ! ! " (cid:16) " ! " ! " ! " $ ! " (cid:16) $ ! " (cid:16) $ ! (cid:16) ! ! " (cid:16) ! " ! ! " (cid:16) ! " ! ! " (cid:16) ! " ! ! " (cid:16) ! " ! ! " (cid:16) ! " ! " ! " $ ! " (cid:16) $ ! " (cid:16) " $ $ ! (cid:16) $ ! " (cid:16) " % ! " ! " ! " % ! " (cid:16) % ! " (cid:16) % ! (cid:16) $ ! " (cid:16) ! ! " (cid:16) ! ! " (cid:16) ! ! " (cid:16) ! " ! ! " (cid:16) $ ! " (cid:16) ! ! " (cid:16) ! " " & ! ! " (cid:16) ! " ! " $ ! " (cid:16) ! " $ ! " (cid:16) $ ! " (cid:16) ! ! " (cid:16) ! ! " (cid:16) ! " $ ! " (cid:16) ! " $ ! " (cid:16) FIG. 7: Diagrams (a,b) correspond to the prefactor of ˜ Z (0) N − while diagrams (c,d) correspond to the prefactor of ˜ Z (0) N − . use of Eqs. (43) and (44). However, as for elementarybosons, calculating this scalar product diagrammaticallygreatly helps the understanding of the physical processesthis part of the partition function contains. This is whyhere we present a diagrammatic derivation of the recur-sion relation existing between the ˜ Z (0) N ’s, which is similarto the one we gave for elementary bosons. For readersnot at ease with diagrams, we also give in Appendix IIIthe brute-force calculation of ˜ Z (0) N for low N ’s.
1. Diagrammatic derivation of recursion relation for ˜ Z (0) N The scalar product appearing in ˜ Z (0) N looks very muchlike the scalar product of N elementary bosons shown inFig. 1, except that the k n lines are now replaced by i n double-lines representing the fermions α n and β n of thecoboson i n . As for elementary bosons, we can connectthe i N double-line on the left to the i N double-line onthe right, leaving the ( N −
1) other cobosons unaffected,in the same way as in Fig. 2(a). This process readilyleads to a contribution to ˜ Z (0) N given by1 N ! z ( β ) h ( N − Z (0) N − i . (68)We can also connect the i N double-line on the left to oneof the ( N −
1) other double-lines on the right, let say i N − .The i N − double-line on the left then has to be connectedto one of the i n ’s on the right; this can be either to i N or to one of the ( N −
2) double-lines like i N − . Thefirst process leads to the diagrams (a,b) of Fig. 7: sincethe cobosons i N and i N − can exchange their fermions,these two cobosons appear either as in Fig. 7(a) or as inFig. 7(b). The physical processes corresponding to thesetwo diagrams bring a contribution to ˜ Z (0) N given by( N −
1) 1 N ! ˜ z (2 β ) h ( N − Z (0) N − i , (69)0 ! ! " (cid:16) " ! " (cid:16) ! ! " (cid:16) ! " ! ! " (cid:16) " ! " (cid:16) ! ! " (cid:16) ! " ! ! " (cid:16) " ! " (cid:16) " ! " (cid:16) ! " ! " (cid:16) ! " ! " (cid:16) " ! " (cid:16) ! " (cid:16) ! " ! " (cid:16) " ! " (cid:16) ! " ! " (cid:16) ! ! " (cid:16) ! " (cid:16) ! " (cid:32) " ! " (cid:16) (cid:32) " ! ! " " (cid:16) (cid:16) (cid:32) (cid:32) ! ! " (cid:16) " ! " (cid:16) " ! " (cid:16) ! " ! " ! " (cid:16) ! ! " (cid:16) ! " (cid:16) ! ! " (cid:16) " ! " (cid:16) ! ! " (cid:16) ! " ! " (cid:16) ! " ! " (cid:16) " ! " (cid:16) ! ! " " (cid:16) (cid:32) (cid:32) ! ! ! " " (cid:16) (cid:32) (cid:32) ! ! " (cid:16) " ! " (cid:16) ! ! " (cid:16) ! " ! " (cid:16) ! " ! " (cid:16) " ! " (cid:16) ! " ! ! " " (cid:16) (cid:16) (cid:32) (cid:32) ! ! " (cid:16) " ! " (cid:16) " ! " (cid:16) ! " ! " ! " (cid:16) ! ! " (cid:16) ! " (cid:16) ! ! " (cid:16) (cid:32) $ % $ % $ $ % % $ % & $ % ’ $ % ( $ % ) ! " (cid:16) (cid:32) FIG. 8: Diagrams corresponding to the prefactor of ˜ Z (0) N − . where ˜ z (2 β ) = z (2 β ) − L ( β, β ), the fermion exchangepart L ( β, β ) being defined through L ( n β, n β ) = X i i e − β ( n E i + n E i ) Λ (cid:0) i i i i (cid:1) . (70)We now consider processes in which i N − on the left isconnected to i N − on the right (in the same way as inFig. 2(d)). We can then connect i N − on the left to i N or to any of the other ( N −
3) cobosons like i N − on theright. The first possibility leads to the diagrams shown inFigs. 7(c,d), in which the three cobosons ( i N , i N − , i N − )possibly exchange their fermions. If we restrict to onefermion exchange only, we get the three processes shownin Fig. 7(d) in which two cobosons are in the same state,while in the process of Fig. 7(c) the three bosons arecondensed into the same state. So, the processes ofFigs. 7(c,d) bring a contribution to ˜ Z (0) N given by( N − N −
2) 1 N ! ˜ z (3 β ) h ( N − Z (0) N − i , (71)where ˜ z (3 β ) at first order in fermion exchange is equalto z (3 β ) − L (2 β, β ). To go one step further, we iso-late the cobosons ( i , · · · , i N − ), while the cobosons( i N , i N − , i N − , i N − ) form a condensate in which theypossibly exchange their fermions as shown in Fig. 8. Ifwe restrict to one fermion exchange only, we must con-nect any two double-lines by exchange, leaving unaffectedthe other two double-lines, these lines imposing their co-bosons to be in the same state. This brings a contributionto ˜ Z (0) N given by( N − N − N −
3) 1 N ! ˜ z (4 β ) h ( N − Z (0) N − i , (72)where ˜ z (4 β ) at first order in fermion exchange is equal to z (4 β ) − L (3 β, β ) − L (2 β, β ). The z (4 β ) term comesfrom diagram (a), the four L (3 β, β ) term come from dia-grams (b,d,e,g) while the two L (2 β, β ) term come fromdiagrams (c,f). ! ! " (cid:16) " ! " (cid:16) ! ! " (cid:16) ! " ! " ! " ! " " " ! $ % " (cid:14) (cid:16) " $ % " (cid:14) (cid:16) ! $ " (cid:14) $ % " (cid:14) ! $ " (cid:16) $ " ! $ % " (cid:14) (cid:16) $ " " $ " (cid:16) ! " ! " ! $ " (cid:16) ! ! " (cid:16) " ! " (cid:16) ! ! " (cid:16) ! " $ % " (cid:14) ! $ % " (cid:14) (cid:14) & %%% %%% %%% %%% FIG. 9: Diagrams contributing to ˜ z ( nβ ). Using the same procedure, we end with the followingrecursion relation between the ˜ Z (0) N ’s˜ Z (0) N = 1 N N X n =1 ˜ z ( nβ ) ˜ Z (0) N − n . (73)This is just the one for elementary bosons (16) but with z ( nβ ) replaced by ˜ z ( nβ ): ˜ z ( β ) = z ( β ) while, for n ≥ z ( nβ ) reads, at lowest order in fermion exchange,˜ z ( nβ ) ≃ z ( nβ ) − n n − X m =1 L ( mβ, ( n − m ) β ) (74)with L ( n β, n β ) = L ( n β, n β ), as seen from Eq. (70).The recursion relation (73) allows us to write ˜ Z (0) N inthe same form as ¯ Z N in Eq. (4) with z ( nβ ) simply re-placed by ˜ z ( nβ ). We must however note that, in order toget ˜ Z (0) N at first order only in fermion exchange, we haveto keep one ˜ z ( nβ ) only, while taking the other p -bosoncondensates as z ( pβ ).
2. Partition function of a n -coboson condensate at zerothorder in ξ ˜ z ( nβ ) appears as the partition function of a n -coboson condensate with fermion exchange between theirfermionic components. The diagrammatic representa-tion of the partition function for a n -elementary bo-son condensate is shown in Fig. 9(a) with the double-lines replaced by single lines. This diagram indeed im-poses i n = i n − = · · · = i . As these n bosons havethe same energy, their partition function is given by P i e − nβE i = z ( nβ ). To get the partition function ofa n -coboson condensate, we must add fermion exchangeto this diagram. At first order, this corresponds to pro-cesses like the one of Fig. 9(b) with one fermion exchangebetween any two double-lines. The cobosons unaffectedby this exchange imposes i p + r − = i p + r − = · · · = i p and1 i p − = · · · = i = i n = i n − = · · · = i p + r . So, the dia-gram (b) brings an exchange term equal to L ( rβ, ( n − r ) β )to the partition function of the n -coboson condensate.Due to the various ways p can be chosen and the factthat L ( rβ, ( n − r ) β ) = L (( n − r ) β, rβ ), such an exchangeleads to a contribution to the partition function of a n -coboson condensate given by ( n/ L ( rβ, ( n − r ) β ) + L (( n − r ) β, rβ )]. Note that, as scatterings involving n cobosons bring a factor ( a X /L ) ( n − , keeping fermionexchange between two cobosons corresponds to perform-ing a many-body expansion at lowest order in density. D. Partition function at first order in ξ We now turn to the contribution at first order in in-teraction scattering to the canonical partition function of N cobosons, as given in Eq. (67) . It is fundamentallysimilar to the canonical partition function of N interact-ing elementary bosons given in Eq. (28). One just has toinclude fermion exchanges in the processes considered inour previous calculations.Let us first consider it for N = 2. It reads˜ Z (1)2 = − β X e − β ( E i + E i ) h h v | B i B i B † m B † n | v i ξ (cid:0) m i n i (cid:1) + c.c. i . (75)Using the commutators (43,44), we find that the scalarproduct in the above relation reads as δ i m δ i n + δ i n δ i m − Λ (cid:0) i i i i (cid:1) ; so, ˜ Z (1)2 is equal to˜ Z (1)2 = − β ξ ( β, β ) , (76)where ˆ ξ ( β, β ) follows fromˆ ξ ( n β, n β ) = X i i e − β ( n E i + n E i ) ˆ ξ ( i , i ) . (77)The scattering ˆ ξ ( i , i ) corresponds to all possible directand exchange interaction processes between incoming co-bosons ( i , i ) ending in states ( i , i ). It precisely readsˆ ξ ( i , i ) = ξ (cid:0) i i i i (cid:1) + ξ (cid:0) i i i i (cid:1) − ξ in (cid:0) i i i i (cid:1) − ξ in (cid:0) i i i i (cid:1) . (78)Precise definition of the exchange scattering ξ in can befound in Ref. 23.˜ Z (1) N for arbitrary N is calculated by writing it as a sumof terms proportional to ˜ Z (0) N − p . This can be done througha brute-force calculation using the key commutators ofthe coboson many-body formalism. In Appendix IV, weshow the calculation for N = 3. Instead, we here givea more enlightening derivation based on diagrams. Thescalar product appearing in ˜ Z (1) N is shown in Fig. 6. Theprefactor of ˜ Z (0) N − is made of ( i N , i N − ) cobosons only(see Fig. 10(a)). It just corresponds to the four direct and ! ! ! ! " " ! (cid:16) " " ! (cid:16) " " (cid:16) ! " ! (cid:16) ! " ! (cid:16) " ! " ! $ " ! " ! (cid:16) ! " ! (cid:16) " ! " " ! (cid:16) " " " ! (cid:16) " ! ! " ! (cid:16) " ! " " ! (cid:16) " " ! (cid:16) ! " ! (cid:16) " ! % ! FIG. 10: Interaction processes involving two cobosons (a) andthree cobosons (b,c). exchange interaction processes appearing in ˜ Z (1)2 . Wereadily get their contribution to ˜ Z (1) N as − β N ! C N ˆ ξ ( β, β ) h ( N − Z (0) N − i = − β ξ ( β, β ) ˜ Z (0) N − . (79)To get the prefactor of ˜ Z (0) N − , we isolate one more co-bosons out of ( N − i N − , and we draw allentangled processes. This imposes i N − not to be con-nected with itself, as in the diagram of Fig. 10(b). Bynoting that ˆ ξ ( β, β ) = ˆ ξ (2 β, β ), these two processes leadto − β N ! C N C N − h ˆ ξ ( β, β ) + ˆ ξ (2 β, β ) ih ( N − Z (0) N − i . (80)Note that we can also have exchange processes like theones of Fig. 10(c) which connect three cobosons. The as-sociated scatterings, however, are ( a X /L ) smaller thandiagram (b). So, the dominant prefactor of ˜ Z (0) N − is theone given in Eq. (80).As for interacting elementary bosons (see Eq. (33)),the prefactor of ˜ Z (0) N − in ˜ Z (1) N − is obtained by iso-lating two cobosons out of ( i , · · · , i N − ), let say( i N − , i N − ), and by drawing all entangled processesbetween ( i N , i N − , i N − , i N − ), like in the diagrams ofFig. 4(c). This brings a contribution to ˜ Z (1) N given by − β N ! C N C N − h ξ ( β, β ) + 2 ˆ ξ (2 β, β ) + 2 ˆ ξ (3 β, β ) i × h ( N − Z (0) N − i . (81)So, we end with an expansion of ˜ Z (1) N similar to the onefor interacting elementary bosons, ¯ Z (1) N , namely˜ Z (1) N = − β N X n =2 ˆ ξ ( nβ ) ˜ Z (0) N − n (82)2with ˆ ξ ( nβ ) = n − X p =1 ˆ ξ ( pβ, ( n − p ) β ) . (83)By adding the Pauli part ˜ Z (0) N of the N -coboson partitionfunction given in Eq. (73), we find that the canonicalpartition function of these composite quantum particlesis given at lowest order in ( a X /L ) by˜ Z N ≃ N N X n =1 h ˜ z ( nβ ) − βN ξ ( nβ ) i ˜ Z (0) N − n . (84)We can go further and transform the above equationinto a recursion relation between the ˜ Z N ’s by followingthe procedure we have used for interacting elementarybosons. We then end with ˜ Z N correct up to first orderin both, Pauli exchange and interaction scattering, as˜ Z N ≃ N N X n =1 ˜˜ z ( nβ ) ˜ Z N − n . (85)where the partition function for a n -coboson condensateis given, for n ≥
2, by˜˜ z ( nβ ) = ˜ z ( nβ ) − βn ξ ( nβ ) . (86)It is then straightforward to show that Eq. (85) leads toa compact form for ˜ Z N similar to Eq. (4) with z ( nβ ) re-placed by ˜˜ z ( nβ ). A similar compact form for the canoni-cal partition function of cobosons to all orders in interac-tion and fermion exchange appears to us as conceptuallyobvious, although beyond the scope of the present work. IV. CONCLUSIONS
We propose a diagrammatic approach to the canon-ical partition function of N cobosons. In addition tothe usual diagrams representing the condensation pro-cesses existing for elementary bosons, the Pauli exclu-sion principle generates new diagrams for fermion ex-changes between the fermionic components of cobosons.The partition function we obtain provides grounds forthe study of coboson quantum condensation. Here, wecalculate in details the canonical partition functions ofnon-interacting elementary bosons as well as interactingelementary bosons and interacting composite bosons atfirst order in interaction and fermion exchange. In allcases, the partition function takes the same compact formas the one of non-interacting elementary bosons providedthat we include interaction and fermion exchange in thepartition function z ( nβ ) of the n -particle condensate. Acknowledgments
This work is supported by National Cheng-Kung Uni-versity, National Science Council of Taiwan under Con- ! ! " ! ! $ % " & ! ! ! " ! ! & ! $ % ! ! " ! ! & ! $ % $ ! ! " ! ! & ! ! ! & ! ! ! " ! ! & ! ! ! " ! ! & ! " ! ! FIG. 11: Diagrams appearing in the scalar product for 4 ele-mentary bosons at first order in interaction (wavy lines). tract No. NSC 101-2112-M-001-024-MY3, and AcademiaSinica, Taiwan. M.C. wishes to thank the NationalCheng Kung University and the National Center for The-oretical Sciences (South) for invitations.
Appendix I. ¯ Z (0) N FOR LOW N ’S For N = 1, the canonical partition function reduces to¯ Z (0)1 = z ( β ) . (A.1)For N = 2, the recursion relation (16) gives¯ Z (0)2 = 12! (cid:2) z ( β ) + z (2 β ) (cid:3) (A.2)in agreement with Eq. (4) taken for ( p = 2) or ( p = 1).This ¯ Z (0)2 taken in the recursion relation for ¯ Z (0)3 gives¯ Z (0)3 = 13! (cid:2) z ( β ) + 3 z ( β ) z (2 β ) + 2 z (3 β ) (cid:3) , (A.3)which agrees with Eq. (4) taken for ( p = 3), ( p =1 , p = 1) or ( p = 1).These ¯ Z (0)1 , ¯ Z (0)2 and ¯ Z (0)3 taken in the recursion rela-tion (16) for N = 4 give¯ Z (0)4 = 14! (cid:2) z ( β ) + 6 z ( β ) z (2 β ) + 8 z ( β ) z (3 β )+6 z (4 β ) + 3 z (2 β ) (cid:3) (A.4)in agreement with Eq. (4) taken for ( p = 4), ( p =2 , p = 1), ( p = p = 1), ( p = 1) or ( p = 2). We notethat the sum of prefactors in these partition functions,e.g., (1 + 6 + 8 + 6 + 3) /
4! in the case of 4 bosons, is equalto 1. So, these prefactors physically correspond to theprobability of the condensation process at hand.3 ! ! " ! ! $ ! ! ! " ! ! ! $ ! ! ! $ ! ! ! " ! ! $ ! % & " " ! ! ! ! $ ! ! ! $ ! ! ! ! " ! $ ! ! ! $ ! " ! ! ! ! $ ! ! ! $ ! " ! ! ! $ ! ! ! ! $ ! ! " ! ! ! " ! ! $ ! ! % & " ! ! ! ! $ ! ! % & $ ! ! " ! ! $ ! ! % & % ! ! " ! ! $ ! ! % & & " ! ! ! ! $ ! ! % & ’ ! !! ! ! ! ! !! ! ! " ! ! ! " ! " ! ! ! ! $ ! " ! ! ! " ! FIG. 12: Diagrams following from the diagrams of Fig. 11after we have chosen to connect k − q to one of the three k ’son the left. ! ! " ! ! $ ! $ ! " ! ! ! $ ! % & " ! " ! " ! ! % & ! ! " ! ! $ ! ! $ ! % & $ ! ! " ! " ! ! ! ! ! " ! ! ! " ! ! $ ! ! ! " ! ! $ ! % & % ! ! $ ! $ ! ! ! ! ! $ ! $ ! ! !!!! !!!!! ! ! ! ! $ ! " ! ! ! " ! ! $ ! FIG. 13: Diagrams (a,b) follow from the diagrams ofFigs. 12(b,d), while diagrams (c,d) follow from the diagramof Fig. 12(f).
Appendix II. CALCULATION OF ¯ Z (1)4 The interaction part of the partition function for 4 in-teracting elementary bosons appears as¯ Z (1)4 = − β C X e − β ( ε k + ··· + ε k ) X q V q (B.1) × h h v | ¯ B k ¯ B k ¯ B k ¯ B k ¯ B † k + q ¯ B † k − q ¯ B † k ¯ B † k | v i + c.c. i . The above scalar product is shown in Fig. 3(a) takenfor N = 4. To get it, we can connect k + q to anyof the ( k , k , k , k ) on the left, as shown in Fig. 11.Since connecting k + q to k or to k is equivalent, theprocesses of diagram 11(c) are going to appear twice.(i) To start, we can connect k − q to k in diagram 11(a),and we can connect k − q to k in diagram 11(b). Thesetwo processes lead to the diagrams shown in Fig. 12(a).Their contribution to ¯ Z (1)4 reads as − β C V ( β, β ) h
2! ¯ Z (0)2 i = − β V ( β, β ) ¯ Z (0)2 . (B.2)In diagrams 11(a) or (b), we can also connect k − q to k or to k , which gives equivalent contribution; so, theseprocesses, shown in Figs. 12(b,c), will appear twice.Finally, from diagram 11(c), we can connect k − q to k , k or k , as shown in Figs. 12(d,e,f).(ii) To go further, we consider diagrams 12(b,d), and weconnect k to k or to k , as shown in Figs. 13(a,b). ! ! ! ! " ! " ! " ! ! ! ! " ! " ! ! " ! ! " ! ! ! " ! ! $ % ! ! " ! ! ! " ! (cid:32) ! ! ! ! " ! " ! ! & " ! ! ! " ! (cid:32) ! ! FIG. 14: Diagram (a) represents the scalar product of twocobosons. Diagrams (b,c,d) show the three distinct configu-rations for these two cobosons.
Diagram 13(a) gives a contribution to ¯ Z (1)4 equal to − β C V ( β, β ) ¯ Z (0)1 = − β V ( β, β ) ¯ Z (0)1 , (B.3)while diagram 13(b) gives a contribution to ¯ Z (1)4 equal to − β C V ( β, β ) = − β V ( β, β ) . (B.4)If we now consider diagram 12(e), we note that it fol-lows from diagram 12(b) by interchanging k and k .This interchange also transforms diagram 12(c) into di-agram 12(d). So, diagrams 12(c) and (e) give the samecontribution as diagrams 12(d) and (b).Finally, in diagram 12(f) we can connect k to k or to k as shown in Figs. 13(c,d). This brings a contributionto ¯ Z (1)4 given by − β C V (2 β, β ) = − β V (2 β, β ) . (B.5)Collecting all the terms and noting that V ( n β, n β ) = V ( n β, n β ), we end with¯ Z (1)4 = − β n V ( β, β ) ¯ Z (0)2 + (cid:2) V ( β, β ) + V (2 β, β ) (cid:3) ¯ Z (0)1 + V ( β, β ) + V (2 β, β ) + V (3 β, β ) o . (B.6) Appendix III. DIRECT CALCULATION OF ˜ Z (0) N We here show how to calculate the canonical partitionfunction of N cobosons at zeroth order in interactionscattering by using the key commutators (43) and (44)of the many-body formalism. This part of the partitionfunction reads as Z (0) N = ˜ Z (0) N /N ! with˜ Z (0) N = 1 N ! X { i } e − β ( E i + ··· E iN ) h v | B i · · · B i N B † i N · · · B † i | v i . (C.1)4 ! ! ! ! " ! " ! " ! ! ! ! ! ! ! ! ! " " ! % ! " ! % ! " $ % " ! % ! " ! % ! & % ! % ! ! ! ! ! ! " ! % ! " ! % ! FIG. 15: Diagrams (a,b) representing the first two terms ofthe three-coboson scalar product in Eq. (C.6). Diagrams (c,d)follow from diagram (a).
To understand how the recursion relation for the ˜ Z (0) N ’sgiven in Eq. (73) develops, let us explicitly calculate ˜ Z (0) N for N = 2 and N = 3. A. Two cobosons
Equation (43) allows us to write the scalar product oftwo cobosons shown in Fig. 14(a) as h v | B i B i B † i B † i | v i = h v | B i ( δ i i − D i i + B † i B i ) B † i | v i (C.2)By inserting the term in δ i i into ˜ Z (0)2 , we readily get itscontribution to ˜ Z (0)2 as12! z ( β ) ˜ Z (0)1 . (C.3)The corresponding diagram is shown in Fig. 14(b).Using Eq. (44) for the term in D i i , we get − X m h v | B i B † m | v i Λ (cid:0) m i i i (cid:1) = − Λ (cid:0) i i i i (cid:1) . (C.4)The corresponding diagram is shown in Fig. 14(c). Wheninserted into ˜ Z (0)2 , this term leads to − L ( β, β ).Finally, the term in B † i B i gives h v | B i B † i | v i δ i i asshown in Fig. 14(d). This imposes i = i and yields z (2 β ). So, we end with˜ Z (0)2 = 12! h z ( β ) ˜ Z (0)1 + ( z (2 β ) − L ( β, β )) i (C.5)with ˜ Z (0)1 = z ( β ). We can rewrite this expression asEq. (73), with ˜ z (2 β ) = z (2 β ) − L ( β, β ) in agreement withEq. (74) taken for N = 2. B. Three cobosons
Equation (43) gives the scalar product of three co-bosons as h v | B i B i B i B † i B † i B † i | v i (C.6)= h v | B i B i ( δ i i − D i i + B † i B i ) B † i B † i | v i . The term in δ i i , when inserted into Eq. (C.1) taken for N = 3, readily yields a contribution to ˜ Z (0)3 given by13! z ( β ) h
2! ˜ Z (0)2 i , (C.7)which corresponds to the diagram of Fig. 15(a).For the term in D i i of Eq. (C.6), we use Eq. (44) toreplace D i i B † i by P m B † m Λ (cid:0) m i i i (cid:1) + B † i D i i and weuse again Eq. (44) for D i i B † i . This leads to − X m h v | B i B i B † m h Λ (cid:0) m i i i (cid:1) B † i +Λ (cid:0) m i i i (cid:1) B † i i | v i . (C.8)When inserted into Eq. (C.1), these two terms contributeequally through a relabeling of ( i , i ). So, the term in D i i gives a contribution to ˜ Z (0)3 given by − X e − β ( E i + E i + E i ) Λ (cid:0) m i i i (cid:1) h v | B i B i B † m B † i | v i . (C.9)This term is shown in Fig. 15(b). The scalar product inthe above equation gives two delta terms, namely δ mi and δ mi , plus one exchange term in Λ (cid:0) i mi i (cid:1) that we canneglect if we only want the first-order correction to ˜ Z (0)3 .The two delta terms shown in Figs. 15(c) and (d) give − / L ( β, β ) ˜ Z (0)1 and − / L (2 β, β ) respectively.Finally, the term in B † i B i of Eq. (C.6) is calculatedby pushing B i to the right according to Eq. (43), h v | B i B i B † i ( δ i i − D i i + B † i B i ) B † i | v i . (C.10) B † i B i B † i | v i = δ i i B † i | v i leads to a contribution similarto the term in δ i i through a relabeling of the ( i , i )indices, while D i i B † i | v i is calculated using Eq. (44).So, the term in B † i B i yields two terms given by13! X e − β ( E i + E i + E i ) h δ i i h v | B i B i B † i B † i | v i− X m h v | B i B i B † m B † i | v i Λ (cid:0) m i i i (cid:1) i . (C.11)The scalar product in the first term of the above equa-tion, shown in Fig. 16(a), is calculated by replacing B i B † i with δ i i − D i i + B † i B i according to Eq. (43).Since D i i B † i | v i gives P m B † m | v i Λ (cid:0) m i i i (cid:1) , these threeterms shown in Fig. 16(c) ultimately yield a contributionto ˜ Z (0)3 given by13! 2 h z (2 β ) ˜ Z (0)1 − L (2 β, β ) + z (3 β ) i . (C.12)5 ! ! ! ! " ! " ! " ! ! ! ! " " $ " ! % ! " ! % ! % ! % ! ! $ ! ! % ! ! ! " ! ! ! ! ! " ! " ! % ! % ! ! " ! % ! ! ! ! ! " ! " ! % ! % ! ! % ! ! ! ! " ! % ! " ! % ! & ! ! ! ! " ! % ! " ! % ! FIG. 16: Diagrams (a,b) representing the two terms inEq. (C.11) for the three-coboson scalar product. Diagramsin (c) follow from (a), while diagrams in (d) follow from (b).
In the second term of Eq. (C.11), shown in Fig. 16(b),we just have to replace the scalar product by δ i m δ i i + δ i m δ i i if we want this term at first order in fermionexchange only. These two terms, shown in Fig. 16(d),yield − / L (2 β, β ).Collecting all the above terms, we end with˜ Z (0)3 = 13 h z ( β ) ˜ Z (0)2 + ( z (2 β ) − L ( β, β )) ˜ Z (0)1 +( z (3 β ) − L (2 β, β )) i , (C.13)in agreement with Eqs. (73,74). Appendix IV. CALCULATION OF ˜ Z (1)3 We here calculate the partition function at first orderin interaction scattering, ˜ Z (1) N , given in Eq. (67) for threecobosons, namely˜ Z (1)3 = − β X e − β ( E i + E i + E i ) (E.1) × h h v | B i B i B i B † m B † n B † i | v i ξ (cid:0) n i m i (cid:1) + c.c. i . The above scalar product is calculated by first replacing B i B † i with δ i i − D i i + B † i B i according to Eq. (43).(i) The δ i i term leads to a contribution to ˜ Z (1)3 givenby − β X e − β ( E i +2 E i ) h h v | B i B i B † m B † n | v i ξ (cid:0) n i m i (cid:1) + c.c. i . (E.2)As h v | B i B i B † m B † n | v i = δ i m δ i n + δ i n δ i m − Λ (cid:0) i mi n (cid:1) ,we ultimately get this contribution to ˜ Z (1)3 as − β ξ ( β, β ) (E.3) with ˆ ξ ( n β, n β ) defined in Eqs. (77,78), the factor of 2coming from the c.c. part.(ii) The term in D i i , inserted into Eq. (E.1), leads to β X e − β ( E i + E i + E i ) (E.4) × h h v | B i B i D i i B † m B † n | v i ξ (cid:0) n i m i (cid:1) + c.c. i . Using Eq. (44), we get D i i B † m B † n | v i as X p Λ (cid:0) p mi i (cid:1) B † p B † n | v i + ( m ←→ n ) . (E.5)So, Eq. (E.4) gives2 β X e − β ( E i + E i + E i ) (E.6) × h h v | B i B i B † p B † n | v i X m Λ (cid:0) p mi i (cid:1) ξ (cid:0) n i m i (cid:1) + c.c. i . The sum over m corresponds to a scattering representedby a diagram similar to the one of Fig. 10(c). As itinvolves three cobosons, this term leads to a contributionto ˜ Z (1)3 of the order of ( a X /L ) which can be neglectedin a first-order calculation.(iii) The term in B † i B i leads to a contribution to ˜ Z (1)3 given by − β X e − β ( E i + E i + E i ) (E.7) × h h v | B i B i B † i B i B † m B † n | v i ξ (cid:0) n i m i (cid:1) + c.c. i . To get it, we replace B i B † i by δ i i − D i i + B † i B i :The term in δ i i is equivalent to the one of Eq. (E.2) ifwe interchange i and i ; so, it gives a contribution equalto − β ξ (2 β, β ) . (E.8)The term in D i i leads to β X e − β ( E i + E i + E i ) (E.9) × h h v | B i D i i B i B † m B † n | v i ξ (cid:0) n i m i (cid:1) + c.c. i . Since the above scalar product already contains onefermion exchange associated with D i i , we can reduce B i B † m B † n | v i to δ i m B † n | v i + δ i n B † m | v i at lowest order in a X /L . When inserted into Eq. (E.9), we get2 β X e − β ( E i + E i + E i ) h h v | B i D i i B † n | v i ξ (cid:0) n i i i (cid:1) + c.c. i . As h v | B i D i i B † n | v i reduces to Λ (cid:0) i ni i (cid:1) , the term in D i i leads to a scattering involving three cobosons; so,it gives a contribution of the order ( a X /L ) which canbe neglected at lowest order. The term in B † i B i gives − β X e − β ( E i + E i + E i ) (E.10) × h h v | B i B † i B i B i B † m B † n | v i ξ (cid:0) n i m i (cid:1) + c.c. i . h v | B i B † i = h v | δ i i , the above contribution reducesto z ( β ) − β X e − β ( E i + E i ) (cid:2) h v | B i B i B † m B † n | v i ξ (cid:0) n i m i (cid:1) + c.c. (cid:3) = z ( β ) ˜ Z (1)2 = − β ξ ( β, β ) ˜ Z (0)1 . (E.11)All these terms combine to yield, with ˆ ξ ( nβ ) defined in Eq. (83),˜ Z (1)3 = − β h ˆ ξ (2 β ) ˜ Z (0)1 + ˆ ξ (3 β ) i , (E.12)in agreement with Eq. (82). ∗ Electronic address: [email protected] M. H. Anderson, J. R. Ensher, M. R. Mathhews, C. E.Wieman and E. A. Cornell, Science , 198 (1995). K. B. Davis, M.-O. Mewes, M. R. Andrews, N. J. vanDruten, D. S. Durfee, D. M. Kurn and W. Ketterle, Phys.Rev. Lett. , 3969 (1995). C. C. Bradley, C. A. Sackett, J. J. Tollett and R. G. Hulet,Phys. Rev. Lett. , 1687 (1995). S. Inouye, M. R. Andrews, J. Stenger, H.-J. Miesner, D.M. Stamper-Kurn and W. Ketterle, Nature , 151-154(1998). M. W. Zwierlein, J. R. Abo-Shaeer, A. Schirotzek, C. H.Schunck and W. Ketterle, Nature , 1047-1051 (2005). Bose-Einstein Condensation , Lev. P. Pitaevskii and San-dro Stringari (Oxford university Press, 2003). For a review, see D. W. Snoke and G. M. Kavoulakis,arXiv:1212.4705v1 and the references therein. S. Yang, A. T. Hammack, M. M. Fogler, L. V. Butov andA. C. Gossard, Phys. Rev. Lett. , 187402 (2006). A. A. High, J. R. Leonard, A. T. Hammack, M. M. Fogler,L. V. Butov, A. V. Kavokin, K. L. Campman and A. C.Gossard, Nature , 584 (2012). V. B. Timofeev and A. V. Gorbunova, J. App. Phys. ,081708 (2007). D. W. Snoke, Phys. Status, Solidi (b) , 389 (2003). Z. V¨or¨os and D. W. Snoke, Mod. Phys. Lett. , 701(2008). D. W. Snoke, arXiv:1208.1213v1. For a review on polariton condensates, see D. W. Snokeand P. Littlewood,
Physics Today , 42 (2010). H. Deng, G. Weihs, C. Santori, J. Bloch and Y. Yamamoto,Science , 199 (2002). H. Deng, G. Weihs, D. W. Snoke, J. Bloch and Y. Ya-mamoto, Proc. Natl. Acad. Sci. U.S.A. , 15318 (2003). H. Deng, D. Press, S. G¨otzinger, G. S. Solomon, R. Hey, K. H. Ploog and Y. Yamamoto, Phys. Rev. Lett. , 146402(2006). R. Balili, V. Hartwell, D. W. Snoke, L. Pfeiffer and K.West, Science , 1007 (2007). M. Combescot, O. Betbeder-Matibet and R. Combescot,Phys. Rev. Lett. , 176403 (2007). R. Combescot and M. Combescot, Phys. Rev. Lett. ,026401 (2012). M. Alloing, M. Beian, D. Fuster, Y. Gonzalez, L. Gon-zalez, R. Combescot, M. Combescot, and F. Dubin,arXiv:1304.4101. M. Crouzeix and M. Combescot, Phys. Rev. Lett. ,267001 (2011). M. Combescot, O. Betbeder-Matibet and F. Dubin,Physics Reports , 215 (2008). M. Combescot, S.-Y. Shiau and Y.-C. Chang, Phys. Rev.Lett. , 206403 (2011). B. Kahn and G. E. Uhlenbeck, Physica , 399 (1938); K.Huang, Statistical Mechanics (Wiley, New York, 1987). C. K. Law, Phys. Rev. A , 034306 (2005). C. Chudzicki, O. Oke and W. K. Wootters, Phys. Rev.Lett. , 070402 (2010). M. Combescot Europhys. Lett. , 60002 (2011). M. C. Tichy, P. A. Bouvrie and K. Mømer,arXiv:1310.8488v1. D. I. Ford, Am. J. Phys. , 215 (1971). Feynman elegantly counted the number of ways to asso-ciate bosons by using cyclic permutations; see R. P. Feyn-man,
Statistical Mechanics (Benjamin, 1972). Our way ofcounting through diagrams, which shares a similar spiritwith Feynman’s, can be extended to interacting systems. M. Combescot and M. A. Dupertuis, Phys. Rev. B78