Permutation Phase and Gentile Statistics
Qiang Zhang1, ∗ and Bin Yan2, 3, † 1Beijing National Laboratory for Condensed Matter Physics, and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China 2Center for Nonlinear Studies, Los Alamos National Laboratory, New Mexico, 87544 3Theoretical Division, Los Alamos National Laboratory, New Mexico, 87544 (Dated: January 10, 2020) This paper presents a new way to construct single-valued many-body wavefunctions of identical particles with intermediate exchange phases between Fermi and Bose statistics. It is demonstrated that the exchange phase is not a representation character but the word metric of the permutation group, beyond the anyon phase from the braiding group in two dimensions. By constructing this type of wavefunction from the direct product of single-particle states, it is shown that a finite capacity q – the maximally allowed particle occupation of each quantum state, naturally arises. The relation between the permutation phase and capacity is given, interpolating between fermions and bosons in the sense of both exchange phase and occupation number. This offers a quantum mechanics foundation for Gentile statistics and new directions to explore intermediate statistics and anyons. PACS numbers: 05.30.Pr, 05.30.-d, 05.70.Ce, 03.65.Vf
I. INTRODUCTION anyonic system and has explicit distribution [6, 13, 14]. Specifically, anyons are quasi-particle (or quasi-hole) Quantum statistics distinguishes its classical counter- excitations in two dimensional systems, e.g., the excita- part by virtue of the intrinsic properties of the under- tion of a factional quantum Hall ground state with frac- ling quantum particles, i.e., occupations of single parti- tional charges. The nontrivial exchange phases of anyons cle states are restricted; the many-body wavefunctions are generally understood as the geometric phase accu- acquire phases upon particle exchanges [1]. The latter mulated via adiabatic exchange between two particles could refer to the variable exchanges in a many-body [1,4,5]. However, constructions of the many-anyon wave- wavefunction, or the actual adiabatic exchange of par- functions and discussions of the exchange properties are ticle positions in real spaces. For fermions and bosons, model related [15, 16], and the statistics is transmuted these two types of exchanges are essentially the same, from the underlying Hamiltonian [1, 17, 18]. There were giving raise to geometric phases π or 0. Consequently, not generic many-anyon states, nor anyonic distributions the allowed occupation numbers in a given single parti- were obtained. Alternatively for FES [6, 13], it is ar- cle state are restricted to one or arbitrary, for fermions gued that in condensed matter system the one-particle and bosons respectively. Hilbert space dimension dα in general depends on the P It has been realized for a long time that there exist occupation of states Nα, ∆dα = − β gαβ∆Nβ, as the intermediate regimes between the Fermi and Bose statis- generalized exclusion principle. The statistical parame- tics. Considerations of the possibility for the interme- ter gαβ might relate to the interchange phase in some diate statistics from the view points of occupation and system, e.g. gαβ = θαβ/(2π) in fractional quantum Hall exchange phases are both of long history and great inter- effect (FQHE) [6] and gαβ = δαβθ/π in spin chain [13, 14] est [2–8]. However, the relation between allowed particle with possible verification [19]. There have also been a lot occupations and the two types of exchanges phases (vari- of discussions on their relations with Gentile statistics able exchange or adiabatic position exchange), as well as [20–27]. the their implications to the statistical distributions, are In this work, we tackle the problem of relating the far from well-understood. many-body wavefunction exchange phases to the statis- For instances, Gentile statistics [2], starting from al- tical distributions from a different angle. Namely, we lowing any no more than q particles occupying the same show that there is a nature way to construct a single- quantum state, appeared early in 1940 but is still lack valued many-body wavefunction with non-trivial phases of quantum mechanics foundation; Fractional statistics upon variable exchanges; and intriguingly, this leads to arXiv:2001.02830v1 [cond-mat.stat-mech] 9 Jan 2020 (originated from studies of anyons), defined through any the well-known Gentile statistics. exchange phase θ in two dimensions [3–5] attracts much attention [8–12] but has no explicitly deduced statisti- The paper is structured as the following. We begin cal distribution; Fractional exclusion statistics (FES) [6] with the general requirement for phase shift of the from generalized Pauli exclusion principle can be used in identical particles wavefunctions in Sec.II and then construct a wavefunction with the aid of word metric in Sec.III. We prove that any interchange or permutation of the arbitrary particles yield a total phase shift on the ∗ [email protected] many-body wave function. We shall note that, instead † [email protected] of the adiabatic phase constrained in two dimensions 2 for anyon, the phase shift of the permutation on the corresponding exchange phases Θ(P2P1), i.e., many-body wavefunction can exist in any dimensions. iΘ(P2P1) Applying it to the wavefunction with some particles P2P1Ψ = e Ψ. (2) occupying the same state in Sec.IV, we find that the interchange phase limits the maximal occupation num- On the other hand, interpreting P2P1 as a sequential ap- ber named capacity q of each quantum state and that plication of P1 and P2 gives an alternative expression the mutual phase gives no constraint on the occupation, iP2Θ(P1) iP2Θ(P1) iΘ(P2) as a new q-exclusion principle. Thus, the identical P2P1Ψ = e P2Ψ = e e Ψ. (3) particles respecting permutation phase shift shall obey Gentile statistics, different from FES. Some additional This leads to a consistency condition discussion on the possible concern is also given. P2Θ(P1) + Θ(P2) = Θ(P2P1). (4)
Once equation (4) is satisfied, the single-valueness of the wavefunction would not be violated. This is the per- II. EXCHANGE PHASE OF IDENTICLE mutation constraint on many-body wavefunction for iden- PARTICLES tical particles we proposed. For the wavefunction as the basis of the representation of the permutation group with Quantum mechanically, the indistinguishability of eiθ as the character, there are only two kinds of statistics identical particles is expressed by the permutation phase (Fermi and Bose) since the one-dimensional representa- shift of the many-body wavefunction: tion of the permutation group can ony have character ±1. In the case of P2Θ(P1) 6= Θ(P1), this is not a representa- iΘ Ψ(P (q1, q2, ··· , qN )) = e Ψ(q1, q2, ··· , qN ), (1) tion of the permutation group, beyond the fermionic and bosonic statistics. where the qi denotes the quantum number of particle-i It seems redundant [3] but still straightforward and and P denotes any permutation among those quantum non-confusing to define the wavefunction on labeled local states. Given that the wavefunction is single-valued, in- variables. As an analogy to Slater determinant, one can terchanging the coordinates of two particles twice must build the N-body wavefunction (unnormalized) as a sum leave the wavefunction unchanged, resulting in the only over functions of permutations on variables: possibilities for a phase shift of 0 or π, corresponding to bosons and fermions, respectively. X iΘ(˜ P ) Ψ(q1, q2 ··· qN ) ≡ e Φ(P (q1, q2 ··· qN )), (5) The possibility for having arbitrary phase shift opens P up once we lift the restriction of single-valueness. Multi- valued many-body functions with well-defined phase shift where Φ is a N-variable function with qi being local vari- upon particle exchanges can exist in two dimensions [3, ables such as coordinate numbers and Θ(˜ P ) is a function 5]. Indeed, this gives raise to a new type of particles, of the permutation P . Applying P1 on the wavefunction, P iP1Θ(˜ P ) iΘ(P1) known as anyons. The existence of anyons is not merely we have P1Ψ = P e Φ(P1P ) = e Ψ, if a mathematical marvel, but has physical correspondence to quasi-particle excitations in two-dimensional systems P1Θ(˜ P ) = Θ(˜ P1P ) + Θ(P1) (6) [4]. It is worth to emphasize that, rather than the phase for any P . Identity (4) can be obtained by applying shift from coordinate permutation in the wavefunction, P2 on both sides of equation (6). Without lose of the exchange phase of these anyon-like quasi-particles is generality, it is natural to let the phase of identity be typically referred to the geometric phase accumulated zero: Θ(˜ PId) = 0, and then one can find Θ(˜ P ) = −Θ(P ). during physical exchange in the sense of adiabatic trans- The price paid is that the permutation phase is label port [1], without dealing with the multi-valued wavefunc- dependent. It is acceptable since this phase shift is not tion. from any measurable physical process, but a constraint Here, we would like to attract attention to a new on the wavefunction. This constraint affects the many- possibility of having arbitrary exchange phases for a body spectrum and statistics. many-body wavefunction. Our approach does not as- sume multi-valueness of the wavefunction, nor the path dependent adiabatic exchange of particles. It is demon- strated that for single-valued wavefunctions there is in- III. CONSTRUCTING MANY-BODY deed a consistent way to have arbitrary exchanges phases WAVEFUNCTION rather than 0 and π; Namely, the permutation operator acts on the phases as well, and consequently makes the It will be more convenient to represent each permuta- exchange phase permutation-dependent. tion P with a (labeled particles) sequence, a1a2 ··· aN . Consider the composed permutation P2P1. By defini- The identity operator is the natural ordering sequence: tion, when applied to the wavefunction Ψ, it generates a PId = 12 ··· N. After applying P , the sequence becomes 3
a1a2 ··· aN . Define the pair parameter gmn = Sgn(n−m) (a) 213231 (b) 132123 (c) 231 213 (g ) (g ) (g ) and (g12+g13) 12 (g31+g32) 31 31
X 321 123 213 312 321 123 l(P ) = gmn, (7) (g12+g13+g23) (g31+g32+g12) (g23) (g31+g21) (mn) (g +g ) (g +g ) (g +g )(g +g +g ) 23 13 (g23) 12 32 (g12) 23 21 23 21 31 312 132 231 321 312 132 here (mn) means inversion pair: m appears on the right of n for m < n in sequence P . From sequence P to Id FIG. 1. Graphic illustration of word metric (Cayley Graph) P , particle-m and -n must be interchanged, so the num- of permutation group S3. Each vertex is associated with one ber of inversion pairs equals to the number of least adja- sequence and each edge connects two sequences different by cent interchange [28] (least braiding in two dimension) , once adjacent interchange. In (a) we enumerate all the se- called the length of P in word metric [29]. Any permu- quences and phase factors starting from sequence 123 (red). tation can be decomposed as least adjacent interchange, (b) is a cyclic permutation 1 → 3 → 2 of (a). In (c) we so the phase of P could be interpreted as Θ(P ) = θl(P ) restart from sequence 231 (red). This is equivalent to a gauge with θ the characteristic interchange phase of this kind transformation of the wavefunction. of identical particles, alike to the anyon in the sense of any interchange phase. Thus the wavefunction is
X −iθl(P ) −iθl(P ) Ψ ≡ e Φ(P ). (8) Typically, l(P1P2) 6= l(P1) + l(P2) and e is not P a representation character of the permutation group, un- −iθ less e = ±1. Moreover, l(P1P2) 6= l(P2P1) in gen- When θ = 0/π, it recovers the bosonic/fermionic wave- eral, so the phase shift is indeed non-Abelian. We shall function. Explicitly, the wavefunction for two-particle is clarify that this is due to the property of word met- ric of non-Abelian permutation group, rather than the Ψ = Φ(1, 2) + e−iθg12 Φ(2, 1). (9) path-dependent braiding like non-Abelian anyons [30] or Interchange particle -1 and -2 denoting as T 12, charges [31]. In the coordinate space X, coordinates can be used 12 −iθg21 −iθg21 T Ψ = Φ(2, 1) + e Φ(1, 2) = e Ψ, (10) as labels: g(x, y) = Sgn(y − x). The sign function could be generalized in high dimensions, since the Carte- and certainly repeated interchange sian product of totally ordered set is still totally or- 12 12 −iθg12 −iθg21 dered. For example, the arg(z) function in two dimen- T (T Ψ) = e e Ψ = Ψ (11) P sion [15] or Sgn( i(Sgn(yi − xi))) in any dimension. leaves the wavefunction unchanged, respecting the single- The “q-symmetrize” [32, 33] wavefunction for two par- valueness of the wavefunction. As mentioned before, the ticles φ(x, y) + q(x, y)φ(y, x) is the same to ours with ij iθg(x,y) essence is that T also changes gnm. A three-particle q(x, y) = e . Yet, without the help of word metric, example can be visualized in Fig.1. Diagram (a) and the many-body permutation properties and statistics can (b) enumerates all the sequences and the corresponding not be explored. phase terms for Ψ and T 132Ψ, with cyclic permutation T 132 meaning 1 → 3 → 2. Comparing the phase differ- ence of the same sequences before and after the permu- tation, the wavefunction feels a phase shift:
132 iθ(g +g ) iθl(T 132) T Ψ = e 13 23 Ψ = e . (12) IV. GENTILE STATISTICS
The general phase shift P Ψ = eiθl(P )Ψ is proven in Ap- pendix (A). To explore the occupation of each quantum state, the Not only the phase dependent on the artificial la- wavefunction should be constructed from the direct prod- 1 2 N bels, it is also sensitive to the selection of zero phase uct of single particle states: Φ(P ) = φaφb ··· φc where i sequence. Different selection means to re-define a new φa is the wavefunction of particle-i at state-a, which is origin rather than the identical operator in word metric impossible for the path-dependent braiding anyon ex- and l(P ) would change correspondingly. Yet, this does change phase [10, 16]. Consider two and three particles not hurt the metric, indistinguishability and statistics occupying the same state (ignoring the particle labels for since re-selection is equivalent to a permutation of parti- them are at the same state), the wavefunctions become cles and the wavefunction is shifted by a phase as seen in the example of three particles case in Fig.1(c). Starting from sequence 231 and defining l(P ) as the sum of the −iθ Ψ2a =(1 + e )φaφa, (13) inverted pair parameters referring to sequence 231, the −iθ −iθ −2iθ phase shift is Ψ0 = eiθl(231)Ψ. Ψ3a =(1 + e )(1 + e + e )φaφaφa. (14) 4
achieved by replacing θl(P ) in equation (8) with Θ(P ) = TABLE I. Some examples of phase-capacity relation from P θ g , where α is the specie particle-m belong- equation (16). The capacities are degenerate some fractions αmαn mn m due to the r ± 2k on the numerator and we take the smallest ing to. For Na particles at state-a of specie α and Nb positive fractions. particles at state-b of specie β,
θ/π 1 2/3 1/2 2/5 1/3 2/7 ··· 0 Na−1 n Nb−1 n Y X −i∗kθαα Y X −i∗kθββ q 1 2 3 4 5 6 · · · ∞ Ψ α β = ( e ) ( e ) Na Nb n=0 k=0 n=0 k=0
X −iΘ(P ) Na Nb × e P φα φβ . (17) By induction (see Appendix.B ) the wavefunction of Na particles at the same state-a is In the last sum of permutation terms, there are (Na + Nb)!/(Na!Nb!) distinct permutations and all the Na−1 n mutual exchange phases θαβ appear in these terms. Y X −i∗kθ ΨNa = ( e ) × φa...φa They are mutual orthogonal and the superposition will n=0 k=0 never cancel. Thus the wavefunction can only vanish
Na due to the front Q-factorial terms from the permutation Y 1 − e−i∗nθ = ( ) × φ ...φ . (15) phase of the same state in the first line. Therefore, 1 − e−iθ a a n=1 neither the occupation numbers of the same species different states (α = β, a 6= b) nor of the different Indeed, this is the Q-factorial form for the number op- species (α 6= β) mutually affect. Only the interchange −iθ erator in quantum group [34] with Q = e . The Q- phase of the same specie at the same state constrains Qn−1 Pi k factorial of n is defined as n!Q ≡ i=0 ( k=0 Q ). Anal- the capacity of each single state, quite different from the ogy to the exclusion of fermion, where the wavefunc- mutual statistics parameter gαβ from FES [6, 13]. tion of fermions occupying the same state vanishes, i.e., 1 + e−iπ = 0, the many-particle wavefunction (15) van- Finite capacity of each quantum state means Gentile −iθ 1+q ishes when Na > q + 1, providing (e ) = 1. As statistics [2]. The thermodynamics of Gentile statistics a consequence, the maximal occupation number named such as the heat capacity, condensation, equation of state capacity of state-a is q. and relation with FES are well studied [7, 21, 26, 42]. In the (real) Q-analogy quantum group approach [34– At zero temperature, the occupation number hnii = q 37](aa† + Qa†a = QN with −1 < Q < 1), it is noticed for those states below the Fermi-like surface, similar to that the number operator is positive definite unless Q is the FES without mutual statistics gαβ = αδαβ [13]. the root of unity (impossible in their case). If we were Plausibly, one may consider that Nj particles occupy at allowed to extend the real Q to the interchange phase least dNj/qe (ceiling function) states among Gj states. −iθ e , the capacity q of quantum state should be found. Thereafter the effective Fock space dimension is dB = In the case that θ/π is a simple fraction r/p, where the Gj + Nj − dNj/qe ≈ Gj + (Nj − 1)(1 − 1/q) and the thermodynamic limit can be achieved via a sequence of number of ways W 0 = CNj is the same to Wu’s count- j dB systems with different particle numbers [4,6], the phase- ing [13] for FES, providing q = 1/α. Here the Binomial capacity relation is explicit: k constant CG ≡ G!/(k!(G − k)!). Indeed, the exact statis- tical weight of assigning N particles into G states with 2p − 1, r is odd j j q = (16) capacity q (combination with limited repetition [43]): p − 1, r is even.
k=bNj /(q+1)c This is the q-exclusion principle we obtained from the X k k Gj −1 Wj = (−1) CG ∗ C . (18) permutation constraint on the wavefunction. The capac- j Gj −1+Nj −k(q+1) k=0 ity is degenerate due to the r ± 2k freedom on the nu- merator. Alike to the accuracy of plateau resistance in The most probable occupation numbern ¯j is determi-
FQHE [38, 39], θ/π shall be exactly a fraction. In Table nant from identity [13] δnj log Wj|n¯j + log zj = 0 with −β(i−µ) I we list the fractions for several small capacity numbers zi ≡ e being the fugacity coefficient and β, i, µ and intriguingly they are the prominent filling fractions being the inverse temperature, energy level and chemical in FQHE [40, 41]. Taken the smallest positive fractions potential, respectively. It is well known [21, 26] that the for each capacity numbers, we also note that the phase- Gentile statistics and the FES do not produce the same capacity relation is monotone decreasing, smoothly inter- distribution. polating between fermion and boson in the sense of both Moreover, the relation between the statistics parame- exchange phase and occupation numbers. Besides, to ob- ters and the occupation number is different for the Gen- tain this exclusion in real space, namely the possibilities tile statistics and FES. In the generalized ideal gas of of particles at different states staying the same position, FES ,the statistics parameter is mapped to the phase by we shall let g(x, x) = 1. α = θ/π = r/p [13], so the occupationn ¯0 is p/r, <εF Many-particle wavefunction with mutual exchange not necessary an integer. This is different from the ca- phases θαβ among mixed species could be similarly pacity we found for the identical particles. For instance, 5 when θ = 2π/3, the state capacity is q = 2 while by Sciences, Materials Sciences and Engineering Division, FES,n ¯0 = 3/2 and for θ = π/2, we get q = 3 while Condensed Matter Theory Program, and in part by the <εF n¯0 = 2. Center for Nonlinear Studies at LANL. <εF
V. DISCUSSION Appendix A: Interchange properties
It appears confusing to talk about many particles at To explore the permutation phase of the wavefunc- the same state respecting P Ψ = eiθl(P )Ψ, since people tion, we decompose permutation into least adjacent in- might also expect P Ψ = Ψ for permutation of particles terchange σi’s called reduced word [29], where σi denotes at the same state seems to change nothing. Thereafter the interchange of the digits at position-i and i + 1 [5]. Ψ = eiθl(P )Ψ and the wavefunction seems to vanish for Although the decomposition is not unique, the inver- θ 6= 0, resulting in hard-core condition. In some proposed sion pairs are determinate. For example, P of sequence models, such as the magnon excitation in Heisenberg spin 2413 can be decomposed as P = σ2σ3σ1 = σ2σ1σ3, chain via Bethe ansatz [14, 44], the charge-flux model while the inversion pairs are (12), (14), (34) and l(2413) = [15] and the spin-anyon mapping from Jordan-Wigner g12 + g14 + g34. transformation [45, 46], the hard-core condition is always An example of permutation phase shift for three-body found or preinstalled, resulting in the conventional Fermi wavefunction is illustrated in Fig.1. To prove P Ψ = N iθl(P ) distribution [14]. Indeed in the coordinate space X /SN , e Ψ is to verify condition (6), or l(PP1) − P (P1) = the wavefunction is multi-valued [3, 15] in the sense of l(P ) for any P,P1. Since every permutation could be permutation phase shift. The hard-core condition is a decomposed into independent (commutative) cyclic per- too strong constraint for identical particles [3] and the mutations, the general justification of any permutation coinciding points (some xi’s are equal) in our wavefunc- on wavefunction could be concluded from the following tion are still singular since only finite of them could be two cases. equal. Besides, phase shifted defined in our wavefunc- Firstly, consider the 2-circle permutation T ij. The fol- tion is a result of the Gedanken permutation of particles, lowing two sequences(i < j) rather than the phase evolution of any dynamic process P1 = a1a2...ak−1 i ak+1ak+2..al−1 j al+1al+2..aN , in low dimensions. Thus, this phase shift is not limited | {z } | {z } | {z } in spacial two dimensions, neither measurable during any s1 s2 s3 adiabatic process nor interference of different paths. The P2 = a1a2...ak−1 j ak+1ak+2..al−1 i al+1al+2..aN , influence of this phase buries into the q-exclusion princi- | {z } | {z } | {z } s1 s2 s3 ple and thus can be detected in any non-fermion/boson distribution related experiments [47–50]. with length In summary, we proposed a consistency condition (4) X X X X l(P ) = g + g + g + g for the single-valueness of the many-body wavefunction. 1 jm1 im1 jm2 m2i Constructed similar to the Slater determinant, with m1>j m1>i m2>j m2i m1>j m2>i m2
T ij itself is a 2-circle permutation as an element of the As to mutual statistics, replace θl(P ) with Θ(P ) = P permutation group and it could be represented as se- θαiαj and let θαiαj be antisymmetric, here αi is the quence species particle-i belonging to, the general permutation property (A10) becomes 12 ∗ ∗(i − 1)j(i + 1) ∗ ∗(j − 1)i(j + 1) ∗ ∗N. (A3) P Ψ = eiΘ(P )Ψ, (A11) with the starts parts “∗∗” being the natural increasing ordering. The length of T ij is in the exactly same spirit.
m
ijk mn lijk,mn ≡ T T l(S1) − l(S2). (A7) is indeed the generating function of IN (l) with argument −iθ e .Ψ1 = 1 and by recurrence relation (B2), Once gnm is antisymmetric, the phase shift is the same Na−1 n for all the ordered sequences in Ψ and Y X −i∗kθ ΨNa = ( e ) × φa...φa (B5) a
The first five terms sum over all the numbers except For the wavefunction of many particles involving dif- i, j, k, m, n. The first three of them are the inversion ferent states or different species with mutual interchange pair parameters for T ijk and the following two are for phases, the wavefunction can be checked out in the same way. Preset the order Φ = φ1 ··· φNa φNa+1 ··· φNa+Nb T mn. The last bracket term is the inversion pairs pa- 0 a a b b rameters among i, j, k, m, n themselves. Likewise, per- with zero phase and permute on the same state first, the mutation T ijkT mn is able to represented as sequence Q-factorial like phase terms for the same state appear. Then we permute on the different states to get the mixed 12 ∗ ∗j ∗ ∗i ∗ ∗n ∗ ∗k ∗ ∗m ∗ ∗N, (A9) distinct terms and the wavefunction with different states or species shall take the form: Then by definition (7), we can also find l(T ijkT mn) = Na−1 n Nb−1 n −lijk,mn. Longer circle and more complicated decompo- Y X −i∗kθαα Y X −i∗kθββ ΨN αN β = ( e ) ( e ) sition would not make things different. It thus is not hard a b n=0 k=0 n=0 k=0 to generalize to any permutation P , X −iΘ(P ) Na Nb × e P φα φβ . (B7) P Ψ = eiθl(P )Ψ. (A10) 7
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