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Green Function of the Sutherland Model with SU(2) internal symmetry 7 9 Yusuke Kato∗ 9 1 Department of Physics, Tohoku University, Sendai 980-77, Japan n a (Nov. 11, 1996) J 1 3 ] l Abstract e - r t s . We obtain the hole propagator of the Sutherland model with SU(2) internal t a m symmetryfor couplingparameter β = 1, whichis thesimplestnontrivialcase. - d One created hole with spin down breaks into two quasiholes with spin down n o c and one quasihole with spin up. While these elementary excitations are ener- [ getically free, the form factor reflects their anyonic character. The expression 1 v 3 for arbitrary integer β is conjectured. 0 0 2 0 75.10.Jm, 05.30.-d 7 9 / t a m - d n o c : v i X r a Typeset using REVTEX 1 There has been a remarkable development in the study of one dimensional models of particles with the inverse-square interactions [1]. Stimulated by the pioneering work by Simons, Lee, and Al’tshuler [2], the dynamical correlation functions of the Sutherland model for spinless particles were obtained explicitly with the help of various techniques, such as the supermatrix method [3], 2D Yang-Mills theory [4], and the theory of the Jack symmetric polynomial [5,6]. The amazing success in the dynamics is, however, limited to the spinless case and much lessisknownforthatofthemulticomponent system. Themulticomponent Sutherlandmodel (MSM) [7,8] realizes the multicomponent Tomonaga-Luttinger liquid in the simplest man- ner. The deeper understanding of the MSM would be helpful to understand other strongly correlated electron systems, especially t-J model [9] and the singlet factional quantum Hall effect [10]. Thus the study on the dynamics of the MSM is highly desirable. In thisLetter, we present theresults ontheone-particle Greenfunctionof theSutherland model for particles with SU(2) spin (σ = 1). This model is the simplest example of i ± the MSM. This is the first calculation of the dynamical correlation function of integrable continuum models with internal symmetry. We consider the following Hamiltonian: [7] N ∂2 2π2 β(β +P ) ˆ ij = + (1) H − ∂x2 L2 sin2[π(x x )/L] Xi=1 i Xi<j i − j for N-particle system. Here P is the operator that exchanges the spin of particles i and ij j [11]. The dimensionless coupling parameter β takes non-negative integer values. L is the linear dimension of the system. The statistics of particles are chosen as boson (fermion) for odd (even) β, so that we can set P = ( 1)β+1M with the coordinate exchange operator ij ij − M [11]. ij For the model (1), we calculate the hole propagator: Gβ(x,t) = g,N ψˆ†(x,t)ψˆ (0,0) g,N / g,N g,N (2) h | ↑ ↑ | i h | i for β = 1, which is the simplest nontrivial case. Here ψˆ (x,t) represents eiHˆtψˆ (x)e−iHˆt. ↑ ↑ 2 g,N is the ground state for N-particle system. The main result of this paper is the | i following explicit expression in the thermodynamic limit with ρ = N/L: 0 1 1 1 v v 4/3 v v −2/3 v v −2/3 Gβ=1(x,t) = a dv dv dv | 1 − 2| | ↑ − 1| | ↑ − 2| 1Z−1 1Z−1 2Z−1 ↑ [ǫ1(v1)ǫ1(v2)ǫ1(v↑)]1/3 exp[ it ǫ (v )+ǫ (v )+ǫ (v ) ζ +i(v +v +v )πxρ /2], (3) 1 1 1 2 1 ↑ 1 1 2 ↑ 0 × − { − } with a = ρ ζ Γ[2/3]/(72Γ2[1/3]), ζ = (3πρ /2)2, and ǫ (v) = 3(πρ /2)2(1 v2). 1 0 1 1 0 1 0 − In the following paragraphs, we present the derivation of (3). In the last part of this paper, we conjecture the expression for arbitrary non-negative integer β on the basis of the result for β = 1. We consider only the case where N is even and M = N/2 is odd. The ground state is singlet and its wavefunction ΦN ( X exp[i2πx /L],σ ) is given g { i ≡ i i} by [7] ΦN ( X ,σ ) = N (X X )β+δ[σi,σj](i)(σi−σj)/2 g { i i} i<j i − j n o Q N X−{β(N−1)+M−1}/2 PN [ σ ], (4) × i=1 i S { i} S Q X withPN [ σ ] = δ[σ ,1] δ[σ , 1]. ThesumwithrespecttoS runsoverthesubset S { i} i∈S i j6∈S j − of 1,2,3 ,N Qwith M elemQents. The norm of (4) is given by C D(M ,M ;β)LN, with N M 1 2 { ··· } C = N!/[M!(N M)!] and N M − M1 dz M2 dw M1 M2 M1 M2 D(M ,M ;β) = i j z z 2β+2 w w 2β+2 z w 2β. 1 2 i j k l i j i2πz i2πw | − | | − | | − | i=1ZC i j=1ZC j i<j k<l i=1j=1 Y Y Y Y Y Y (5) Here C is the contour counterclockwise on the unit circle. The value of D(M ,M ;β) was 1 2 derived in ref. [12] as M1 A(k 1,k;β) A(M ,M ;β) − , (6) 1 2 A(k,k 1;β) kY=1 − with J2−1 Γ[(β +1)(j +1)+J ] 1 A(J ,J ;β) = (J !) for J J 1. (7) 1 2 2 2 1 Γ[(β +1)j +J +1] ≥ − j=0 1 Y When J = 1, the product is taken as unity. 2 3 We consider the action of ψ on the ground state. The wavefunction of the state ψ (0,0) g,N is given by ↑ | i N ( 1)(β+1)kΦN ( X ,σ ,X = 1,σ = 1). (8) − g { i i} k k k=1 X Since particles are identical, we pick out the term k = N in Eq. (8): N−1 ΦN ( X ,σ ;X = 1,σ = 1) = ( i)M X−β/2(X 1)β+δ[σi,1]ΦN−1, (9) g { i i} N N − i i − g i=1 Y where ΦN−1 represents the ground state wavefunction with momentum π(M 1)/L. We g − − calculate the propagator for Eq. (9) and then multiply it by N. The factor N comes from the number of ways to choose which particle is removed and added. For taking the trace over spin variables in Eq. (2), we replace the Hamiltonian (1) by ˆ = N ∂2 + 2π2 β β −(−1)βMij . (10) Hβ − ∂x2 L2 sin(cid:16)2[π(x x )/L(cid:17)] Xi=1 i Xi<j i − j Spin variable for each particle is then a good quantum number since Hamiltonian (10) does not contain spin variables. Hence we can fix spin for each particle and multiply the result by C as an equivalent procedure to trace over spin variables. It is from this advantage N−1 M−1 that we consider (10) instead of (1). In the following, we set σ = 1 for 1 i M 1 and i ≤ ≤ − σ = 1 for M i 2M 1. i − ≤ ≤ − After tracing over σ , we arrive at the following expression: i { } N−1 dX M−1 M Gβ(x,t) = c i ΦÑ−1 (z¯ 1)β+1 (w¯ 1)β 0 i=1 I i2πXi! g i=1 i − j=1 j − Y Y Y M−1 M e−i (Hˆβ−EgN)t−Pˆx (z 1)β+1 (w 1)βΦÑ−1, (11) × { } i − j − g i=1 j=1 Y Y with c = C ρ /[ C D(M ,M ,β)]. The overlines represent complex conjugate. 0 N−1 M−1 0 N M 1 2 Here ΦÑ−1 is given by g M−1 M M−1 M ΦÑ−1 = (z z )β+1 (w w )β+1 (z w )β g i − j k − l i − j i<j k<l i=1 j=1 Y Y Y Y M−1 M −[β(N−1)+M−1]/2 −[β(N−1)+M−1]/2 z w , (12) × i j i=1 j=1 Y Y 4 where z (1 i M 1) = X and w (1 j M) = X represent the complex i i j j+M−1 ≤ ≤ − ≤ ≤ coordinates of particles with spin up and down, respectively. Instead of calculating Gβ=1(x,t) directly, we first calculate the following correlation function for later convenience: N−1 dX M−1 M G˜β(x,t) c i ΦÑ−1 (z¯ 1)2 (w¯ 1) ≡ 0 i=1 I i2πXi! g i=1 i − j=1 j − Y Y Y M−1 M e−i (Hˆβ−EgN)t−Pˆx (z 1)2 (w 1)ΦÑ−1. (13) × { } i − j − g i=1 j=1 Y Y From (11) to (13), we just replace (z 1)β+1(w 1)β by (z 1)2(w 1) but i j i − j − i j i − j − Q Q Q Q leave the rest of β as an arbitrary integer. The first step in calculating G˜β(x,t) is to expand M−1(z 1)2 M (w 1)ΦÑ−1 in i=1 i − j=1 j − g ˆ Q Q terms of eigenfunctions of the Hamiltonian . Relevant eigenstates have the following β H form: K( z ,w )ΦÑ−1. (14) { i j} g Here K( z ,w ) represents a polynomial satisfying the following conditions: i j { } symmetric with respect to exchange between z ’s and exchange between w ’s i j • polynomial of degree not more than 2 with respect to z and of degree not more than i • 1 with respect to w . j Now we assign two types ofindices to eigenstates described by (14). The polynomial part K( z ,w ) is written by the linear combination of m˜ ( z ,w ) = m ( z )m ( w ). i j µ i j µ↑ i µ↓ j { } { } { } { } Here m ( z ) and m ( w ) are monomial symmetric functions of z M−1 and w M , µ↑ { i} µ↓ { j} { i}i=1 { j}j=1 respectively. These are given by m ( z ) = M−1zµ↑i , m ( w ) = M wµ↓j , (15) µ↑ { i} P(i) µ↓ { j} P(j) P i P j X Y XY respectively. ThesumwithrespecttoP runsoverdistinctpermutations. Hereµisdefinedby µ = µ↑;µ↓ , where µ↑ = µ↑,µ↑,µ↑ µ↑ and µ↓ = µ↓,µ↓,µ↓ ,µ↓ are partitions 1 2 3··· M−1 1 2 3··· M n o n o n o 5 arranged in decreasing order (µ↑ µ↑ µ↑ , µ↓ µ↓ µ↓ ). The weight M−1µ↑ 1 ≥ 2 ≥ 3··· 1 ≥ 2 ≥ 3··· i=1 i ( M µ↓) is denoted by µ↑ ( µ↓ ). P j=1 j | | | | PNext we define µ˜↑ and µ˜↓ by µ˜↑ = µ↑ + M 2,M 3, ,0 and µ˜↓ = µ↓ + { − − ··· } M 1,M 2, ,0 , respectively. Furthermore we introduce µ+ as the rearranged se- { − − ··· } ries of µ˜↑ µ˜↓ in decreasing order. Here we define the order of µ+ so that µ+ > ν+ if the ⊕ first nonvanishing difference µ+ ν+ is positive. Next we define the order of µ = µ↑;µ↓ . i − i n o If µ+ > ν+, then µ > ν. If µ+ = ν+ and µ↓ > ν↓ , then µ > ν. | | | | We define the polynomial K ( z ,w ) as µ i j { } K ( z ,w ) = m˜ ( z ,w )+ v m˜ ( z ,w ) • µ { i j} µ { i j} ν(<µ) µν ν { i j} P K ( z ,w )ΦÑ−1 is an eigenfunction of ˆ . • µ { i j} g Hβ K ( z ,w )ΦÑ−1’s are orthogonal with one another. • µ { i j} g Eigenfunctions of ˆ are obtained for distinguishable particles by the same method as used β H for the single component bosonic case [13,14]. Although the index µ seems natural to describe the polynomial K , here we introduce µ another index λ = λ ,λ ,λ , which is related directly with rapidities or velocities of 1 2 ↑ { } elementary excitations. λ (λ ) is the smaller (larger ) non-negative integer among the set 1 2 M,M 1, ,0 ν˜↑ and λ is among M,M 1, ,0 ν˜↓. In Fig. 1, we illustrate ↑ { − ··· }− { − ··· }− the case of µ = µ↑;µ↓ = 2211;1111 for example. We see that there are three vacant { } n o rapidities λ = M 5,M 1,M 4 . Now we identify λ ,λ and λ as rapidities of 1 2 ↑ { − − − } quasiholes with spin down and up, respectively. In order to calculate G˜β, we have to know the expansion coefficient C which appears in λ M−1 M (z 1)2 (w 1) = C K (16) i j λ λ − − iY=1 jY=1 0≤Xλ1<λ21≤Xλ2≤M0≤Xλ↑≤M and the norm N for each wavefunction: λ N = N−1 dXi K ΦÑ−1 2. (17) λ λ i2πX iY=1 I i (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) 6 In terms of C and N , the hole propagator is written as λ λ G˜β(x,t) = c C2N exp i EN−1 EN t+iP x . (18) 0 λ λ − λ − g λ λ1X,λ2,λ↑ h (cid:16) (cid:17) i Now we present the explicit forms of EN−1, P , C , and N . Momentum of the state λ is λ λ λ λ given by P = π(5M 1)/L (2π/L)(λ +λ +λ )+πβ(1 2M)/L. (19) λ 1 2 ↑ − − − In terms of λ = λ ,λ ,λ , we write the expression EN−1 EN as { 1 2 ↑} λ − g EN−1 EN = λ − g (2π/L)2 (M 1/2)Mβ2 M2 10M +1 /4 β 2M2 +7M 1 /2+(2M 1)β2/4 − − − − − − − h (cid:16) (cid:17) (cid:16) (cid:17) (1+2β) λ2 +λ2 +λ2 +(M +2βM +β 1)(λ +λ +λ ) 2β λ˜ λ˜ . (20) − 1 2 ↑ − 1 2 ↑ − 1 − 3 (cid:16) (cid:17) (cid:16) (cid:17)i ˜ ˜ ˜ Here λ , λ , and λ are the smallest, the second smallest and the largest of λ ,λ ,λ , 1 2 3 1 2 ↑ { } respectively. General expressions of C and N for the Jack polynomial have been derived in math- λ λ ematical literature [15]. In the case of K , however, these expressions have not yet been λ derived. Hence we calculate K for system with finite number of particles (M 11) up λ ≤ to degree µ = 20. Based on these data, we obtain the general form of C and N . The λ λ | | coefficient C is given by λ C = Γ2[(1+β)η]/(Γ[(2+β)η]Γ[η]) λ Γ[λ2−λ1+η] Γ[λ↑−λ1+(1−β)η] Γ[λ↑−λ2+η]  Γ[λ2−λ1−βη] Γ[λ↑−λ1+η] Γ[λ↑−λ2+(1+β)η] ×  Γff[ooΓλ[rr2λ−2λλ−λ11λ+1<(−12−λλββ2η)]η≤<]Γλλ[λΓ↑↑[−λλ↑−1+λ(11++ηβ])η]ΓΓ[[λλ22−−λλ↑↑−−2ββηη]] (21) 1 ↑ 2 ≤ with η = 1/(1+2β). The nΓΓfo[oλ[rλr2m2−−λλλ↑i11s−+<βgηηi]λ]vΓ1eΓ[n[λ<λ11b−−λyλλ2↑↑,−−2ββηη]]ΓΓ[[λλ22−−λλ↑↑−−23ββηη]] 7 D(M 1,M;β)Γ[M]Γ[M +1 βη] , λ λ N = − − A B , (22) λ Γ[M βη]Γ[M +1 3βη]Γ3[(1+β)η] − − with ˜ ˜ 3 Γ λ +1 kβη Γ M λ +(1+(k 2)β)η k k = − − − (23) Aλ Γ λ˜ h+1+(1 k)iβηhΓ M λ˜ +(1+(k 1)βi)η k=1 k k Y − − − h i h i and Γ[λ2−λ1−βη]Γ[λ2−λ1+(1+β)η]Γ2[λ↑−λ1+1−2βη]Γ2[λ↑−λ2+1−βη] for λ < λ λ  Γ[λ2−λ1]Γ[λ2−λ1+η]Γ[λ↑−λ1+1−3βη]Γ[λ↑−λ1+1−βη]Γ[λ↑−λ2+1−2βη]Γ[λ↑−λ2+1] 1 2 ≤ ↑ Bλ =  Γ[λ2−λ1+Γ(1[λ−2β−)λη]1Γ+[λη2]Γ−[λλ21−−λβ1η−]Γ2[βλη↑]−Γλ2[1λ+↑1−]Γλ[1λ+↑(−1+λ1β+)ηη]]ΓΓ2[[λλ22−−λλ↑↑−−2ββηη]]Γ[λ2−λ↑] for λ1 ≤ λ↑ < λ2  Γ[λ2−λΓ1[λ]Γ2[−λλ2−1−λβ1+η]ηΓ][Γλ[2λ−1λ−1λ+↑(]1Γ+[λβ1)−η]λΓ↑2−[λ21β−ηλ]Γ↑[−λβ2η−]λΓ↑2−[λβ2η−]Γλ[↑λ−22−βλη↑]−3βη] for λ↑ < λ1 < λ2. (24) Now we consider the thermodynamic limit M , L with ρ = 2M/L. For 0 → ∞ → ∞ λ , λ , λ , λ λ , λ λ , λ λ (M), we obtain the asymptotic forms of C , 1 2 ↑ 1 2 1 ↑ 2 ↑ λ | − | | − | | − | ∼ O N , EN−1 EN, and P . In the case of C and N , we can use the Stirling formula: λ λ − g λ λ λ Γ[N +1] √2πNN+1/2e−N. The sums with respect to λ ,λ , and λ turn into the multiple 1 2 ↑ → integral: 2 3λ2−1 M M 1 1 1 1 dv dv dv , (25) 1 2 ↑ M → 2 (cid:18) (cid:19) λX1=0λX2=1λX↑=0 Z−1 Z−1 Z−1 with v = 1 (2λ /M) (i = 1,2, ). From the above results, G˜β(x,t) reduces to i i − ↑ 1 1 1 v v 2(1+β)η v v −2βη v v −2βη G˜β(x,t) = a dv dv dv | 1 − 2| | ↑ − 1| | ↑ − 2| βZ−1 1Z−1 2Z−1 ↑ [ǫβ(v1)ǫβ(v2)ǫβ(v↑)]βη exp[ it ǫ (v )+ǫ (v )+ǫ (v ) ζ +i(2 2β +v +v +v )πxρ /2], (26) β 1 β 2 β ↑ β 1 2 ↑ 0 × − { − } − with the prefactor Γ[(1+β)η]ρ (ηζ )3βηMγ 0 β a = , (27) β 21+6η(2+β)βΓ2[(2+β)η]Γ2[η] 8 the chemical potential ζ = [π(1 + 2β)ρ /2]2, the quasihole energy ǫ (v) = β 0 β (πρ /2)2(1+2β)(1 v2), and the exponent γ = (5 + β)η (1 + β). By setting β = 1 0 − − in Eq. (26), we obtain the expression (3). Now we discuss the physical implication of (26). We can see that the three quasiholes are energetically free while the form factor has nontrivial structure. From the study of thermodynamics [16], on the other hand, we have learned that quasiholes are free but obey nontrivial (exclusion) statistics. Here we attributethe nontrivialformfactor to theexclusion statistics. Namely we read the form factor in (26) as v v 2g↓↓ v v 2g↑↓ v v 2g↑↓ 1 2 ↑ 1 ↑ 2 | − | | − | | − | , (28) [ǫ (v )ǫ (v )]1−g↓↓ [ǫ (v )]1−g↑↑ β 1 β 2 β ↑ where the matrix g↑↑ g↑↓ 1 β +1 β gh =   =  −  (29) 1+2β g g β β +1  ↓↑ ↓↓   −          represents the statistical interaction among quasiholes of the Sutherland model with SU(2) internal symmetry. Actually we meet the matrix (29) in the thermodynamics of the present model. Lastly we consider the hole propagator Gβ(x,t) with arbitrary integer coupling β. In this case, we know that one created hole with spin down breaks into β +1 quasiholes with spin down and β quasiholes with spin up. Energies and momenta of intermediate states are obtained easily. In the thermodynamic limit, we obtain EN−1 EN β+1ǫ (v ) + λ − g → i=1 β i β ǫ (u ) ζ and PN−1 (πρ /2) β+1v + β u , where v (1 iP β +1) and j=1 β j − β λ → 0 i=1 i j=1 j i ≤ ≤ (cid:16) (cid:17) P P P u (1 j β) are normalized velocities of quasiholes with spin down and up, respectively. j ≤ ≤ Although we do not have the expression for the form factor, we make a conjecture relying on the interpretation (28). The resultant expression for Gβ(x,t) is given by β+1 1 β 1 β+1 v v 2g↓↓ β u u 2g↑↑ β+1 β v u 2g↓↑ Gβ(x,t) dv du i<j | i − j| k<l| k − l| i=1 j=1| i − j| ∼ i=1 Z−1 ij=1Z−1 jQ βi=+11[ǫQβ(vi)]1−g↓↓ βj=1[ǫQβ(uj)Q]1−g↑↑ Y Y β+1 β Q β+1 Q β exp it ǫ (v )+ ǫ (u ) ζ +i v + u πxρ /2 , (30) × −  β i β j − β  i j 0  Xi=1 jX=1  Xi=1 jX=1       9 up to a constant factor. The expression (30) would be plausible in view of the known form factor in the single component model (with integer coupling β), which is given by [4–6] β v v 2g i<j| i − j| . (31) Q βi [ǫ(vi)]1−g Q Here v is the velocity and ǫ(v) is the energy of quasihole. The statistical parameter g i was derived as 1/β from the study of the thermodynamics [17]. We can see that the form factor in (30) is a natural generalization of (31). The conjecture (30) should be checked microscopically, e.g. with the use of the theory of the nonsymmetric Jack polynomial [18]. This is left as a future work. The author thanks Y. Kuramoto and T. Yamamoto for useful comments. This work was supported by a Grant-in-Aid for Encouragement of Young Scientists (08740306) from the Ministry of Education, Science, Sports and Culture of Japan. 10