Elliptic supersymmetric integrable model and multivariable elliptic functions Kohei Motegi
∗
arXiv:1704.07035v1 [math-ph] 24 Apr 2017
Faculty of Marine Technology, Tokyo University of Marine Science and Technology, Etchujima 2-1-6, Koto-Ku, Tokyo, 135-8533, Japan
April 25, 2017
Abstract We investigate the elliptic supersymmetric gl(1|1) integrable model introduced by Deguchi and Martin, which is an elliptic extension of the Perk-Schultz model. We introduce and study the wavefunctions of the elliptic model. We first make a face-type version of the Izergin-Korepin analysis to give characterizations of the wavefunctions. We then show that the wavefunctions are expressed as a multivariable elliptic functions which is a product of elliptic factors and elliptic Schur-type symmetric functions. The correspondence obtained resembles the recent works by number theorists which the correspondence between the wavefunctions and the product of the deformed Vandermonde determinant and Schur functions was established.
1
Introduction
In the field of statistical physics and field theory, integrable models [1, 2, 3, 4] play special roles not only because many physical quantities can be computed exactly but also because of its deep connections with mathematics. One of the highlights is the discovery of quantum groups [5, 6] through the investigations of the R-matrix which is the most local object in integrable models. From the point of view of integrable models, quantum groups are related with the trigonometric models whose matrix elements are given by trigonometric functions. There is a class of elliptic integrable models whose matrix elements are generalized to elliptic functions. The most famous one is the Baxter’s eight-vertex model [7]. Investigating the underlying structures, several versions of the elliptic quantum groups have been formulated and studied [8, 9, 10, 11, 12]. There are also developments on partition functions of elliptic integrable models, which are global objects constructed from R-matrices. For the case of elliptic models, the models are often analyzed by transforming to the face models which are connected with the vertex models via the vertex-face transformation. There are extensive studies on the ∗
E-mail:
[email protected]
1
partition functions of this type of elliptic models [13, 14, 15, 16, 17, 18, 19, 20, 21] , ranging from the domain wall boundary partition functions to wavefunctions, by developing various methods to study integrable models. The models we described above are about models related with the Baxter’s eight-vertex model. In this paper, we investigate another type of elliptic integrable models, which was introduced by Deguchi and Martin [22]. The Deguchi-Martin model was introduced to construct an elliptic extension of the Perk-Schultz model [23], which is a trigonometric model which the quantum superalgebra structure is underlying [24], hence is called as supersymmetric integrable models. The elliptic versions of supersymmetric models seem to be interesting since the trigonometric version of the model were found to have connections with the field of algebraic combinatorics in recent years. Number theorists Bump-Brubaker-Friedberg [25] found that the wavefunctions of a free-fermion model in an external field gives rise to an integrable model realization of the Tokuyama formula for the Schur functions [26] (see also [27, 28, 29] for pioneering works on the variations of the Tokuyama formula). Tokuyama formula is a one-parameter deformation of the Weyl character formula for the Schur functions. This fundamental result lead people to find generalizations and variations of the Tokuyamatype formula for various types of symmetric functions [30, 31, 32, 33, 34, 35, 36, 37, 38]. The latest topic is the introduction of the notion of the metaplectic ice which is explicitly constructed in [36] by twisting the higher rank Perk-Schultz models. As for the domain wall boundary partition function, the celebrated Izergin-Korepin analysis [39, 40] was performed and observations of the factorization phenomena on the domain wall boundary partition functions was found for the Perk-Schultz model and the cloesely related Felderhof free-fermion model [41] in [42, 43] (see also [44] for an application to correlation functions). This factorization formula for the domain wall boundary partition functions cannot be observed in general for the case of the Uq (sl2 ) six-vertex models. Instead, a determinant formula called as the Izergin-Korepin determinant was already obtained in 1980s [39, 40]. The factorization phenomena for the Perk-Schultz and Felderhof models was extended to the elliptic models [45, 46]. For example, an extensive study on the domain wall boundary partition functions of the Deguchi-Martin model was performed in the PhD thesis of Zuparic [46]. In this paper, focusing on the gl(1|1) Deguchi-Martin model, we introduce and investigate the explicit forms of the wavefunctions, which is a more general class of partition functions and include the domain wall boundary partition functions as a special case. We apply the Izergin-Korepin technique to study wavefunctions which was first applied to the Uq (sl2 ) sixvertex model in [47]. We view the wavefunctions as an elliptic polynomial of the parameters in the quantum spaces, characterize its properties by using the Izergin-Korepin analysis, and connect wavefunctions between different sizes. We then prove that the wavefunctions are expressed as a certain multivariable elliptic functions, which is a product of elliptic factors and symmetric functions which can be regarded as an elliptic version of the Schur functions. The result resembles the one for the trigonometric model whose wavefunctions are given as the product of a one-parameter deformation of the Vandermonde determinant and the (factorial) Schur functions [25, 33], hence can be viewed as an elliptic analogue of the result for the trigonometric model. This paper is organized as follows. In the next section, we first review the properties of elliptic functions, and introduce the gl(1|1) Deguchi-Martin model and the wavefunctions of the model. We make the Izergin-Korepin analysis in section 3 and list the properties needed 2
to uniquely determine the explicit form of the wavefunctions. We prove in section 4 that a certain product of elliptic factors and symmetric functions satisfies all the required properties which are extracted from the Izergin-Korepin analysis. Section 5 is devoted to the conclusion of this paper.
2
Elliptic functions and Deguchi-Martin model
In this section, we first review the properties of elliptic functions, and introduce the DeguchiMartin gl(1|1) model, following Deguchi-Martin [22] and the PhD thesis of Zuparic [46].
2.1
Elliptic functions
We first introduce elliptic functions and list the properties needed in this paper. The half period magnitudes K1 and K2 are defined for elliptic nome q (0 < q < 1) as ∞
1 Y K1 = π 2
n=1
(
1 + q 2n−1 1 − q 2n−1
)(
1 − q 2n 1 + q 2n
)!2
,
1 K2 = − K1 log(q), π
(2.1) (2.2)
The theta functions H(u) is defined using K1 , K2 and q as ) ! ∞ ( ! 1 πu πu Y 2n 4n H(u) = 2q 4 sin {1 − q 2n }. 1 − 2q cos +q 2K1 K1
(2.3)
n=1
The important properties of the theta functions are quasi-periodicities H(u + 2mK1 ) = (−1)m H(u), 2
H(u + 2inK2 ) = (−1)n q −n exp
−
!
inπu H(u), K1
(2.4) (2.5)
for integers m and n. Also note that H(u) is an odd function H(−u) = −H(u). In the analysis of face model domain wall boundary partition functions by PakuliakRubtsov-Silantyev [14] (see also Felder-Schorr [48]), the notions and properties of elliptic polynomials were used. Let us recall here. A character is a group homomorphism χ from multiplicative groups Γ = Z + τ Z to C× . An N -dimensional space ΘN (χ) is defined for each character χ and positive integer N , which consists of holomorphic functions φ(y) on C satisfying the quasi-periodicities φ(y + 1) = χ(1)φ(y), −2πiN y−πiN τ
φ(y + τ ) = χ(τ )e
(2.6) φ(y).
(2.7)
The elements of the space ΘN (χ) are called elliptic polynomials. The fact that the space ΘN (χ) is N -dimensional and the following proposition
3
Proposition 2.1. [14, 48] Suppose there are two elliptic polynomials P (y) and Q(y) in ΘN (χ), where χ(1) = (−1)N , χ(τ ) = (−1)N eα . If those two polynomials are equal P (yj ) = P Q(yj ) at N points yj , j = 1, . . . , N satisfying yj − yk 6∈ Γ, N k=1 yk − α 6∈ Γ, then the two polynomials are exactly the same P (y) = Q(y). was used for the analysis on the domain wall boundary partition functions of the eightvertex solid-on-solid model [49]. This is an elliptic version of the fact that if P (y) and Q(y) are polynomials of degree N − 1 in y, and if these polynomials match at N distinct points, then the two polynomials are exactly the same. We use these properties for the Izergin-Korepin anlaysis of the wavefunctions of the elliptic Deguchi-Martin model.
2.2
Deguchi-Martin model
We introduce the Deguchi-Martin gl(1|1) (elliptic supersymmetric integrable) model. DeguchiMartin model is an elliptic generalization of the trigonometric Perk-Schultz vertex model, and is a face model, which is a convenient description for constructing elliptic integrable models. The face models have additional state variables coming from state vectors besides the spectral parameters and inhomogeneous parameters which the vertex models also have. The state variables are essentially the state vectors of the face model, and for the case of the Deguchi-Martin gl(1|1) model, The state variables are elements of Z2 . Keeping this is mind, we introduce two unit vectors eˆ1 = {1, 0} and eˆ2 = {0, 1} of Z2 . We also introduce the complex variables ω12 and ω21 connected by ω12 = −ω21 , and ω11 = ω22 = 0, ǫ1 = 1, ǫ2 = −1, which are introduced for the purpose of describing the model. Each face is labeled by the four state vectors associated with the four vertices around the face, and the spectral parameters and inhomogeneous parameters. One can think that the spectral and inhomogeneous parameters are carried by lines penetrating the vertical and → → − − → → horizaontal edges. We denote the weights associated with the state vectors − a , b ,− c, d, spectral parameter u and inhomogeneous parameter v by (Figure 1) → ! − → − a b (2.8) W − → u v . − → c d To define the weights, one introduces the following notation for convenience [u] = H(λu), for a fixed complex variable λ. In this notation, the quasi-periodicities are # " 2K1 = −[u], u+ λ # ! " iπλu K2 −1 = −q exp − [u]. u + 2i λ K1
4
(2.9)
(2.10) (2.11)
Figure 1: A graphical description of a face of the face model. The state vectors are associated at four vertices, and a spectral parameter u and an inhomogenous parameter v are also associated with the integrable lattice models. A weight is associated for each fixed state vectors, which is denoted by (2.8). The weights for the gl(1|1) Deguchi-Martin model is defined as (Figure 2) ! → − → − [1 + ǫj (u − v)] a a + eˆj , j = 1, 2, W − u v = → → − a + eˆj a + 2ˆ ej [1] ! → − → − [u − v][ajk − 1] a a + eˆk W − , (j, k) = (1, 2), (2, 1), u v = → → − a + eˆj a + eˆj + eˆk [1][ajk ] ! → − → − [ajk − (u − v)] a a + eˆj , (j, k) = (1, 2), (2, 1), W − u v = → → a + eˆj − a + eˆj + eˆk [ajk ]
(2.12) (2.13) (2.14)
→ when the state vector on the top left corner vertex is − a ∈ Z2 . Here, ajk is defined as ajk = ǫj aj − ǫk ak + ωjk . Since we are dealing with the gl(1|1) model, the ajk s appearing in this paper are a12 = a1 + a2 + ω12 and a21 = −a1 − a2 + ω21 . Moreover, since ω12 = −ω21 , we have a12 = −a21 and we essentially need only a12 . → ! − → − a b All the other weights of W − → u v which cannot be written in the form of (2.12), − → c d (2.13), (2.14) are defined as zero. The weights (2.12), (2.13), (2.14) satisfy the face-type Yang-Baxter relation (Figure 3) X
W
− → g ∈Z2
=
X
− → g ∈Z2
W
! − → → f − g u1 u3 W → − → − a b → ! − → − e d u1 u3 W → − → g − c
− ! → − → e d u2 u3 W → − − f → g ! → − → g − c → u2 u3 W − → − a b 5
! − → → d −c u2 u1 → − → − g b ! → − → e − g → − − u2 u1 . f → a
(2.15)
Figure 2: The weights for the gl(1|1) Deguchi-Martin model (2.12), (2.13), (2.14) associated with the configurations. One next introduces the notation for the product of wavefunctions ! →j − − →j −→ j a a · · · a j 0 1 M u v , . . . , v TM → − → − −→ 1 M j j j b0 b1 · · · bM →j − − →j ! →j − − →j ! a0 a1 a1 a2 =W − u v1 W − → → − → → u v2 · · · W − j j b b bj bj 0
1
1
2
weights (Figure 4) to describe the
−− −→ ajM −1 −− −→ bjM −1
−→ ajM −→ bjM
! u vM .
(2.16)
We now introduce the wavefunctions of the elliptic supersymmetric Deguchi-Martin model using the product of weights (2.16) as the following X WM,N (u1 , . . . , uN |v1 , . . . , vM |x1 , . . . , xN |a12 ) = −−−→ − → − → { b0 },{ b1 },...,{bN−2 }
! − − a→ ˆ1 · · · − a→ e1 a→ ˆ1 12 + e 12 + (M − 1)ˆ 12 + M e −− −→ − → uN v1 , . . . , vM − ··· b0M −1 a→ ˆ1 + eˆ2 b01 12 + M e 12 1 ! −− → −− −→ N −2 j−1 − − Y b1j−1 · · · bM a→ ˆ1 + jˆ e2 a→ e1 12 + M e 12 + jˆ j −1 TM × uN −j v1 , . . . , vM →j − −− −→ j − → + M eˆ + (j + 1)ˆ − → + (j + 1)ˆ b · · · b a e a e j=1 12 1 2 12 1 1 M −1 ! −−−→ −− −→ −→ − N −2 a→ e1 b1N −2 · · · bM a + M e ˆ + (N − 1)ˆ e 12 + (N − 1)ˆ 12 1 2 N −1 ×TM u1 v1 , . . . , vM , −−−→ −−−→ −−−−1 → −−−→ cN −1 cN −1 · · · cN −1 cN −1 0 TM
− a→ 12 − → a + eˆ
0
1
M −1
M
(2.17)
6
Figure 3: The face-type Yang-Baxter relation (2.15). We take the sum over the inner state vectors. −−−→ −−−→ −− −→ −−−→ N −1 N −1 are fixed using the sequence of integers 1 ≤ x1 < x2 < where c0N −1 , c1N −1 , . . . , cM −1 , cM · · · < xN ≤ M as −− −→ −−−→ −− −→ −−−→ N −1 N −1 c0N −1 , c1N −1 , . . . , cM −1 , cM − → − → =a12 + N eˆ1 , . . . , a12 + (N + x1 − 1)ˆ e1 , − a→ e1 + eˆ2 , 12 + (N + x1 − 1)ˆ − → − → a12 + (N + x1 )ˆ e1 + eˆ2 , . . . , a12 + (N + x1 + x2 − 3)ˆ e1 + eˆ2 , − a→ e1 + 2ˆ e2 , 12 + (N + x1 + x2 − 3)ˆ − → − → a + (N + x + x − 2)ˆ e + 2ˆ e , . . . , a + M eˆ + N eˆ . (2.18) 12
1
2
1
2
12
1
2
The sequences of integers 1 ≤ x1 < x2 < · · · < xN ≤ M label the positions of bottom edges of the wavefunctions where the difference of the state vectors of adjacent vertices differ by eˆ2 . The integers 1 ≤ x1 < x2 < · · · < xN ≤ M is essentially labeling the state vectors of the bottom vertices. X The notation in (2.17) means that we take the sum over all inner states −−−→ → − − → { b0 },{ b1 },...,{bN−2 }
−−−→ −0 − → → bj , b1j , . . . , bjN −2 , j = 1, . . . , M − 1. (2.17) is a face-type analogue of the wavefunctions of the vertex models. We remark that from the definition of the weights (2.12),(2.13),(2.14), the dependence on the state vector − a→ 12 = (a1 , a2 ) in the top left corner vertex is reflected to the wavefunction (2.17) in the form of the scalar quantity a12 = a1 + a2 + ω12 , hence we write the wavefunction as WM,N (u1 , . . . , uN |v1 , . . . , vM |x1 , . . . , xN |a12 ). See Figures 5 and 6 for pictorial descriptions of (2.17). In the next section, we examine the properties of the wavefunctions.
7
Figure 4: The product of weights (2.16) associated with one row configurations.
3
Izergin-Korepin analysis
In this section, we extract the properties of the wavefunctions of the elliptic supersymmetric face model WM,N (u1 , . . . , uN |v1 , . . . , vM |x1 , . . . , xN |a12 ). The method we use is the IzerginKorepin analysis [39, 40]. Proposition 3.1. The wavefunctions WM,N (u1 , . . . , uN |v1 , . . . , vM |x1 , . . . , xN |a12 ) satisfies the following properties. (1) When xN = M , the wavefunctions WM,N (u1 , . . . , uN |v1 , . . . , vM |x1 , . . . , xN |a12 ), regarded λ as a polynomial in yM := vM , is an elliptic polynomial in ΘN (χ). 2K1 (2) The wavefunctions WM,N (uσ(1) , . . . , uσ(N ) |v1 , . . . , vM |x1 , . . . , xN |a12 ) with the ordering of the spectral parameters permuted uσ(1) , . . . , uσ(N ) , σ ∈ SN are related with the unpermuted one WM,N (u1 , . . . , uN |v1 , . . . , vM |x1 , . . . , xN |a12 ) by Y [1 + uσ(k) − uσ(j) ]WM,N (u1 , . . . , uN |v1 , . . . , vM |x1 , . . . , xN |a12 ) 1≤jσ(k)
=
Y
[1 + uσ(j) − uσ(k) ]WM,N (uσ(1) , . . . , uσ(N ) |v1 , . . . , vM |x1 , . . . , xN |a12 ).
(3.1)
1≤jσ(k)
(3) The following recursive relations between the wavefunctions hold if xN = M (Figure 8): WM,N (u1 , . . . , uN |v1 , . . . , vM |x1 , . . . , xN |a12 )|vM =uN =
N −1 Y j=1
M −1 [1 − uj + uN ] Y [1 + uN − vj ] [1] [1] j=1
× WM −1,N −1 (u1 , . . . , uN −1 |v1 , . . . , vM −1 |x1 , . . . , xN −1 |a12 + 1). 8
(3.2)
Figure 5: The wavefunctions WM,N (u1 , . . . , uN |v1 , . . . , vM |x1 , . . . , xN |a12 ) (2.17). If xN 6= M , the following factorizations hold for the wavefunctions (Figure 9): WM,N (u1 , . . . , uN |v1 , . . . , vM |x1 , . . . , xN |a12 ) N [a21 − M − N + 1] Y [uj − vM ] = WM −1,N (u1 , . . . , uN |v1 , . . . , vM −1 |x1 , . . . , xN |a12 ). (3.3) [a21 − M + 1] [1] j=1
(4) The following holds for the case N = 1, x1 = M M −1 [−u + vM + a12 + M − 1] Y [1 + u − vk ] . WM,1 (u|v1 , . . . , vM |M |a12 ) = [a12 + M − 1] [1]
(3.4)
k=1
Proof. Property (1), (2) and (3) for the case xN = M can be proved essentially in the same way with the domain wall boundary partition functions, which is given in the PhD thesis of Zuparic [46] (note there is a small differnce in the setting. For example, the ordering of the spectral parameters is inverted). One should just have to be reminded that one has to treat the case xN 6= M separately, which is given as the factorization formula of the wavefunctions in Property (3). To show Property (1), we first decompose the wavefunctions satisfying xN = M in the following form as we do for the case of vertex models. We split the wavefunctions as a sum of the products of large partition functions and one-column partition functions (see Figure 7 for this decomposition) WM,N (u1 , . . . , uN |v1 , . . . , vM |x1 , . . . , xN |a12 ) =
N −1 X j=0
fj (u1 , . . . , uN |vM |a12 ), Pj (u1 , . . . , uN |v1 , . . . , vM −1 |x1 , . . . , xN −1 |a12 )W
9
(3.5)
Figure 6: An example W6,4 (u1 , . . . , u4 |v1 , . . . , v6 |1, 2, 4, 6|a12 ) of the wavefunctions (2.17). Note that x1 = 1, x2 = 2, x3 = 4, x4 = 6 labels the positions of the edges of the bottom part of the wavefunctions where the difference of the adjacent edges is eˆ2 . This essentially labels the state vectors in the bottom vertices. fj (u1 , . . . , uN |vM |a12 ) are the one-column partition functions corresponding to the where W shaded part in Figure 7 fj (u1 , . . . , uN |vM |a12 ) W
! W = uN −k vM 12 1 2 12 1 2 k=0 ! − − a→ e1 + jˆ e2 a→ ˆ1 + jˆ e2 12 + (M − 1)ˆ 12 + M e ×W uN −j vM − → − → a12 + M eˆ1 + (j + 1)ˆ e2 a12 + M eˆ1 + jˆ e2 ! N −2 − − Y a→ ˆ1 + kˆ e2 a→ ˆ1 + (k + 1)ˆ e2 12 + M e 12 + M e × W −→ . u v a12 + M eˆ1 + (k + 1)ˆ e2 − a→ ˆ1 + (k + 2)ˆ e2 N −k−1 M 12 + M e j−1 Y
− a→ e1 + kˆ e2 12 + (M − 1)ˆ − → a + (M − 1)ˆ e + (k + 1)ˆ e
− a→ ˆ1 + kˆ e2 12 + M e − → a + M eˆ + (k + 1)ˆ e
(3.6)
k=j
The large partition functions Pj (u1 , . . . , uN |v1 , . . . , vM −1 |x1 , . . . , xN −1 |a12 ) corresponding
10
Figure 7: A graphical description of a summand in the decompostion of the wavefunctions (3.5). The unshaded part and the shaded part corresponds to fj (u1 , . . . , uN |vM |a12 ) (3.6) , respecPj (u1 , . . . , uN |v1 , . . . , vM −1 |x1 , . . . , xN −1 |a12 ) (3.7) and W tively. to the unshaded part in Figure 7 is explicitly written as Pj (u1 , . . . , uN |v1 , . . . , vM −1 |x1 , . . . , xN −1 |a12 ) =
X
−−−→ − → − → { b0 },{ b1 },...,{bN−2 }
0 TM −1
×
j−1 Y
k TM −1
k=1 j ×TM −1
×
N −2 Y
k TM −1
k=j+1 N −1 ×TM −1
! − − a→ ˆ1 · · · − a→ e1 a→ e1 12 + e 12 + (M − 2)ˆ 12 + (M − 1)ˆ − − − → → − uN − → + (M − 1)ˆ 0 0 · · · b a e + e ˆ b 12 1 2 12 1 1 M −2 ! −−→ −−−→ − → + (M − 1)ˆ − k−1 a e + kˆ e b1k−1 · · · bM a→ e1 12 1 2 12 + kˆ uN −k → − −−−−2 → → − k b · · · bkM −2 − a12 + (M − 1)ˆ e1 + (k + 1)ˆ e2 a→ + (k + 1)ˆ e 12 1 1 ! −− → −− −→ j−1 − − → + (M − 1)ˆ a→ e1 b1j−1 · · · bM a e + jˆ e 12 + jˆ 12 1 2 uN −j → − −− −−2 → j j − → − a12 + (j + 1)ˆ e1 b1 · · · bM −2 a→ ˆ1 + jˆ e2 12 + M e ! −−→ −−−→ −→ − k−1 k−1 a + M e ˆ + (k − 1)ˆ e b · · · b a→ + kˆ e 12 1 2 12 1 1 M −2 uN −k → − −− −→ − → − → k k a12 + M eˆ1 + kˆ e2 a12 + (k + 1)ˆ e1 b1 · · · bM −2 ! −− −→ −− −→ −→ − N −2 N −2 a + M e ˆ + (N − 2)ˆ e b · · · b a→ + (N − 1)ˆ e 12 1 2 12 1 1 M −2 u1 , (3.7) −−−→ −−−→ −−−→ −−−→ dN −1 dN −1 dN −1 · · · dN −1 − a→ 12 − → a + eˆ
0
1
M −2
M −1
−→ −− −→ −− −−−→ −−−→ N −1 N −1 where d0N −1 , d1N −1 , . . . , dM −2 , dM −1 are fixed using the sequence of integers 1 ≤ x1 < x2
