Clifford statistics and the temperature limit in the theory of fractional quantum Hall effect 2 Andrei A. Galiautdinov 0 0 School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332-0430 2 February 1, 2008 n a J 1 Abstract 1 UsingtherecentlydiscoveredCliffordstatisticsweproposeasimplemodelforthegrandcanonicalensemble of the carriers in the theory of fractional quantum Hall effect. The model leads to a temperature limit 3 associated with the permutational degrees of freedom of such an ensemble. We also relate Schur’s theory v of projective representations of the permutation groups to physics, and remark on possible extensions of the 2 second quantization procedure. 5 0 1 0 PACS numbers: 73.43.-f, 05.30.-d 2 0 / In a series of papers, building on the work on nonabelions of Read and Moore [1, 2], Nayak and Wilczek h [3, 4, 5] (see also [6] on how spinors can describe aggregates) proposed a startling new spinorial statistics t - for the fractional quantum Hall effect (FQHE) carriers. The prototypical example is furnished by a so- p called Pfaffian mode (occuring at filling fraction ν =1/2), in which 2n quasiholes form an 2n−1-dimensional e irreduciblemultipletofthecorrespondingbraidgroup. Thenewstatisticsisclearly non-abelian: itrepresents h the permutation group S on the N individuals by a non-abelian group of operators in the N-body Hilbert : N v space, a projective representation of S . N Xi We have undertaken a systematic study of this statistics elsewhere, aiming primarily at a theory of ele- mentary processes in quantum theory of space-time. Wehavecalled the new statistics Clifford, to emphasize r a itsintimaterelationtoCliffordalgebrasandprojectiverepresentationsofthepermutationgroups. Thereader is referred to [7, 8, 9] for details. Since the subject is new, many unexpected effects in the systems of particles obeying Clifford statistics may arise in future experiments. One simple effect, which seems especially relevant to the FQHE, might be observed in a grand canonical ensemble of Clifford quasiparticles. In this paper we give its direct derivation first. Following Read and Moore [2] we postulate that only two quasiparticles at a time can be added to (or removedfrom)theFQHEensemble. Thus,westartwithanN =2n-quasiparticleeffectiveHamiltonianwhose only relevant to our problem energy level E is 2n−1-fold degenerate. The degeneracy of the ground mode 2n with no quasiparticles present is taken to be g(E )=1. 0 Assumingthataddingapairofquasiparticlestothecompositeincreasesthetotalenergybyε,andignoring all the externaldegrees of freedom, we can tabulate theresulting many-body energy spectrum as follows: NumberofQuasiparticles, N =2n 0 2 4 6 8 10 12 ··· Degeneracy, g(E )=2n−1 0 1 2 4 8 16 32 (1) 2n ··· CompositeEnergy,E 0ε 1ε 2ε 3ε 4ε 5ε 6ε 2n ··· 1 Notice that the energy levels so defined furnish irreducible multiplets for projective representations of permutation groups in Schur’stheory [11], as was first pointed out byWilczek [4, 5]. Wenow consider a grand canonical ensemble of Clifford quasiparticles. The probability that thecomposite contains n pairs of quasiparticles, is g(E )e(nµ−E2n)/kBT P(n,T)= 2n 1+ ∞n=1g(E2n)e(nµ−E2n)/kBT 2n−1e−(nµ−E2n)/kBT P , (2) ≡ 1+ ∞ 2n−1e−n(ε−µ)/kBT n=1 where µ is the quasiparticle chemical potential.PThe denominator of this expression is the grand partition function of thecomposite, ∞ Z(T)=1+ g(E )e(nµ−E2n)/kBT 2n n=1 X ∞ 1+ 2n−1e−n(ε−µ)/kBT (3) ≡ Xn=1 at temperature T. Now, ∞ 2n−1e−nx = e−x[20e−0x+21e−1x+22e−2x+ ] ··· n=1 X = e−x ∞ en(ln2−x). (4) n=0 The partition function can therefore bewritten as P ∞ Z(T)=1+e−(ε−µ)/kBT en(ln2−(ε−µ)/kBT). (5) n=0 X This leads totwo interesting possibilities (assuming ε>µ): 1) Regime 0<T <T , where c ε µ Tc = k −ln2. (6) B Herethe geometric series converges and 1 e−(ε−µ)/kBT 2e−(ε−µ)/2kBT ε µ Z(T)= − = sinh − . (7) 1 2e−(ε−µ)/kBT 1 2e−(ε−µ)/kBT 2kBT − − (cid:16) (cid:17) The probability distribution is given by 2n−1e−n(ε−µ)/kBT(1 2e−(ε−µ)/kBT) P(n,T)= − . (8) 1 e−(ε−µ)/kBT − 2) Regime T T . c ≥ Underthis condition the partition function diverges: Z(T)=+ , (9) ∞ and theprobability distribution vanishes: P(n,T)=0. (10) 2 This result indicates that the temperatureT of (6) is theupperboundof theintrinsic temperatures that c thequasiparticleensemblecanhave. Raisingthetemperaturebringsthesystemtohigherenergylevelswhich are more and more degenerate, resulting in a heat capacity that diverges at thetemperature T . c To experimentally observe this effect, a FQHE system should be subjected to a condition where quasi- particles move freely between the specimen and a reservoir, without exciting other degrees of freedom of the system. A similar limiting temperature phenomenon seems to occur in nature as the Hagedorn limit in particle physics[10]. Knowing the partition function allows us to find various thermodynamic quantities of the quasiparticle system for sub-critical temperatures 0 < T < T . We are particularly interested in the average number of c pairs in the grand ensemble: ∂lnZ n =λ , (11) h iCliff ∂λ where λ=eµ/kBT, or after some algebra, e−(ε−µ)/kBT n(T) = . (12) h iCliff (1 e−(ε−µ)/kBT)(1 2e−(ε−µ)/kBT) − − Wecan compare this with thefamiliar Bose-Einstein, 1 e−(ε−µ)/kBT n(T) = , (13) h iBE e(ε−µ)/kBT 1 ≡ 1 e−(ε−µ)/kBT − − and Fermi-Dirac, 1 e−(ε−µ)/kBT n(T) = , (14) h iFD e(ε−µ)/kBT +1 ≡ 1+e−(ε−µ)/kBT distributions. For theClifford oscillator, n(T) + as T T ,as had to beexpected. Cliff c h i → ∞ → − Let usnow turntoprojective representations of thesymmetric (permutation) groupsthat havelong been knownto mathematicians, butreceived little attention from physicists. Suchrepresentations were overlooked in physics much like projective representations of the rotation groups were overlooked in the early days of quantummechanics. For convenience, following [11, 12, 13] (cf. also [4, 5]), we briefly recapitulate the main results of Schur’s theory. Oneespecially useful presentation of thesymmetric group S on N elements is given by N SN = ht1,...,tN−1 : t2i =1, (tjtj+1)3=1, tktl=tltk i, 1 i N 1, 1 j N 2, k l 2. (15) ≤ ≤ − ≤ ≤ − ≤ − Heret are transpositions, i t1 =(12),t2=(23),...,tN−1 =(N 1N). (16) − Closely related to S is thegroup S˜ , N N S˜N = hz,t′1,...,t′N−1 : z2=1, zt′i =t′iz, t′i2 =z, (t′jt′j+1)3=z, t′kt′l =zt′lt′k i, 1 i N 1, 1 j N 2, k l 2. (17) ≤ ≤ − ≤ ≤ − ≤ − A celebrated theorem of Schur(Schur,1911 [11]) states thefollowing: (i) The group S˜ has order 2(n!). N (ii) Thesubgroup 1,z is central, and is contained in thecommutator subgroup of S˜ , provided n 4. N (iii) S˜ / 1,z S{ . } ≥ N N { }≃ (iv) If N <4, then every projective representation of S is projectively equivalent to a linear representation. N 3 (v)If N 4, then every projective representation of S is projectively equivalent to a representation ρ, N ≥ ρ(SN) = ρ(t1),...,ρ(tN−1):ρ(ti)2 =z,(ρ(tj)ρ(tj+1))3 =z, h ρ(t )ρ(t)=zρ(t)ρ(t ) , k l l k i 1 i N 1, 1 j N 2, k l 2, (18) ≤ ≤ − ≤ ≤ − ≤ − where z= 1. In thecase z =+1, ρ is a linear representation of S . N The gro±up S˜ (17) is called therepresentation group for S . N N Themostelegantwaytoconstructaprojectiverepresentationρ(S )ofS isbyusingthecomplexClifford N N R algebra CliffC(V,g) N associated with thereal vector space V =N , ≡C γ ,γ = 2g(γ ,γ ). (19) i j i j { } − Here γ N is an orthonormal basis of V with respect to thesymmetric bilinear form { i}i=1 g(γ ,γ )=+δ . (20) i j ij Clearly, any subspace V¯ of V = NR generates a subalgebra CliffC(V¯,g¯), where g¯ is the restriction of g to V¯ V¯. A particularly interesting case is realized when V¯ is × N N V¯ := αkγ : α =0 (21) k k ( ) k=1 k=1 X X of codimension one, with thecorresponding subalgebra denoted by ¯N−1 [13]. If we consider a special basis t′ N−1 V¯ (which is not orthonoCrmal) defined by { k}k=1 ⊂ 1 t′ := (γ γ ), k=1,...,N 1, (22) k √2 k− k+1 − then the group generated by this basis is isomorphic to S˜ . This can be seen by mapping t to t′ and z to N i i -1,and bynoticing that 1) For k=1,...,N 1: − t′2 = 1; (23) k − 2) For N 2 j: − ≥ (t′t′ )3= 1; (24) j j+1 − 3) For N 1 m>k+1: − ≥ t′t′ = t′ t′, (25) k m − m k as can bechecked bydirect calculation. Onechoice for thematrices is provided by thefollowing construction (Brauer and Weyl,1935 [14]): γ =σ σ (iσ ) 1 1, 2k−1 3 3 1 ⊗···⊗ ⊗ ⊗ ⊗···⊗ γ =σ σ (iσ ) 1 1, 2k 3 3 2 ⊗···⊗ ⊗ ⊗ ⊗···⊗ i=1, 2, 3,..., M, (26) for N =2M. Here σ , σ occur in the k-th position, the product involves M factors, and σ , σ , σ are the 1 2 1 2 3 Pauli matrices. If N =2M +1, we first add one more matrix, γ =iσ σ (M factors), (27) 2M+1 3 3 ⊗···⊗ 4 and then define: Γ := γ γ , 2k−1 2k−1 2k−1 ⊕ Γ := γ γ , 2k 2k 2k ⊕ Γ := γ ( γ ). (28) 2M+1 2M+1 2M+1 ⊕ − Therepresentation ρ(SN) so constructedis reducible. Anirreducible moduleof ¯N−1 restricts that repre- sentation to theirreducible representation of S˜N, since {t′k}Nk=−11 generates C¯N−1 as aCn algebra [13]. TorelateSchur’stheorytophysicswemaytrytodefineanew,purelypermutationalvariableoftheClifford composite, whose spectrum would reproduce thedegeneracy of Read and Moore’s theory. A convenient way to define such a variable is by the process of quantification (often called second quan- tization), which is used in all the usual quantum statistics — by mapping the one-body Hilbert space into a many-bodyoperator algebra. This procedurewas described in detail in [8]. Let us thus assume that if there is just one quasiparticle in the system, then there is a limit on its localization, so that the quasiparticle can occupy only a finite number of sites in the medium, say N = 2n. We further assume that the Hilbert space of the quasiparticle is real and N = 2n-dimensional, and that a one-body variable (which upon quantification corresponds to the permutational variable of the ensemble) is an antisymmetric generator of an orthogonal transformation of theform 0 1 G:=A n n , (29) 1 0 n n (cid:20)− (cid:21) where A is a constant coefficient. Note that in the complex case this operator would be proportional to the imaginary unit i, and the corresponding unitary transformation would be a simple multiplication by a phase factor with no observable effect. Since the quantified operator algebra for N >1 quasiparticles will be complex,theeffectofjustonesuch“real”quasiparticleshouldberegardedasnegligibleinthegrandcanonical ensemble. In the non-interacting case the process of quantification converts G into a many-body operator Gˆ by the rule N Gˆ := eˆGl eˆj, (30) l j l,j X whereusually eˆ and eˆj are creators and annihilators, butin more general situations are thegenerators (that i appear in the commutation or anticommutation relations) of the many-body operator algebra. If M is the ij metric (not necessarily positive-definite) on theone-body Hilbert space then eˆ† =M eˆj. (31) i ij In thepositive-definitecase, M =δ with eˆ†=eˆi,as usual. ij ij i InClifford statistics [8]thegenerators ofthealgebraareClifford unitsγ =2eˆ = γ †,soit isnaturalto i i i − assume that quantification of G proceeds as follows: n Gˆ = A (eˆ eˆk eˆ eˆk+n) k+n k − − k=1 Xn = +A (eˆ eˆ eˆ eˆ ) k+n k k k+n − Xkn=1 = 2A eˆ eˆ k+n k k=1 X 5 n 1 A γ γ . (32) ≡ 2 k+n k k=1 X By Stone’s theorem, the generator Gˆ acting on the spinor space of the complex Clifford composite of N = 2n individuals can be factored into a Hermitian operator O˜ and an imaginary unit i that commutes strongly with O˜: Gˆ =iO˜. (33) We suppose that O˜ corresponds to the permutational many-body variable mentioned above, and seek its spectrum. WenotethatGˆ isasumofncommutinganti-Hermitianalgebraically independentoperatorsγ γ , k= k+n k 1,2,...,n, (γ γ )† = γ γ , (γ γ )2 = 1. If we now use 2n 2n complex matrix representation of k+n k k+n k k+n k − − × BrauerandWeyl(26)fortheγ-matrices,wecansimultaneouslydiagonalizethe2n 2n matricesrepresenting thecommuting operators γ γ , and usetheir eigenvalues, i, to findthe spectru×m of Gˆ, and consequently k+n k of O˜. The final result is obvious: there are 22n eigenkets of±O˜, corresponding to the dimensionality of the spinor space of CliffC(2n). In the irreducible representation of SN this number reduces to 2n−1, as required byRead and Moore’s theory. Notethatinthisapproachthepossiblenumberofthequasiholesintheensembleisfixedbythenumberof theavailablesites,N =2n. Achangeinthatnumbermustbeaccompaniedbyachangeinthedimensionality of the one-quasiparticle Hilbert space. It is natural to assume that variations in the physical volume of the entiresystem would prividesuch a mechanism. With regard to the quantification procedure (31, 32) mentioned above, we point out that a priori there is no compelling reason for using only the formalism of creation and annihilation operators in setting up a many-bodytheory. Forexample,ifwechoosetoworkexclusivelywithanN-bodysystem,thenalltheinitialandfinalselective actions (projections, or yes-no experiments) on that system can be taken as simultaneous sharp production and registration of all the N particles in the composite with no need for one-body creators and annihilators. Thetheorywouldresemblethatofjustoneparticle. Theelementarynon-relativisticquantumtheoryofatom providessuch an example. Of course in real experiments much more complicated processes occur. The number of particles in the composite may vary, and if a special vacuum mode is introduced, then those processes can conveniently be described by postulating elementary operations of one-body creation and annihilation. Using just the notion of the vacuum mode and a simple rule by which the creation operators act on the many-body modes, it is possible to show (Weinberg [15]) that any operator of such a many-body theory may be expressed as a sum of productsof creation and annihilation operators. Inphysicsshifts in description are veryfrequent,especially in thetheory of solids. Thestandard example is the phonon description of collective excitations in crystal lattice. There the fundamental system is an ensemble of a fixed number of ions without any special vacuum mode. An equivalent description is in terms of a variable numberof phonons, their creation and annihilation operators, and thevacuum. It is thuspossible that a deepertheory underlyingthe usual physics might be based on a completely new kindofdescription. Finkelsteinsometimeago[16]suggestedthattheroleofatomicprocessesinsuchatheory might be played by swaps (or permutations) of quantum space-time events. Elementary particles then would be the excitations of a more fundamental system. The most natural choice for the swaps is provided by the differences of Clifford units(22) defined above. All that prompted us to generalize from the common statistics to more general statistics, as was done in [8]. There, the quantification rule (31) is a consequence of the so-called representation principle. 6 Acknowledgements This work was aided by discussions with James Baugh and David Finkelstein. It was partially supported by theM. and H. Ferst Foundation. References [1] G. Moore, and N. Read,Nucl. Phys. 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