Sum Rule for the Eight-Vertex Model on Its Combinatorial Line

Abstract

We investigate the conjectured ground state eigenvector of the 8-vertex model inhomogeneous transfer matrix on its combinatorial line, i.e., at η=π/3, where it acquires a particularly simple form. We compute the partition function of the model on an infinite cylinder with certain restrictions on the inhomogeneities, and taking the homogeneous limit, we obtain an expression for the squared norm of the ground state of the XYZ spin chain as a solution of a differential recurrence relation.

Appendices

Appendix A: The ζ→0 Trigonometric Limit

The trigonometric limit is obtained by sending ζ to 0. The Boltzmann weights (1) of the eight-vertex model turn into those of the six-vertex (the weight d go to zero). In this limit the results of this paper should be closely related to the computations of [14]. Note that the “quadratic” sum rule considered here was actually not computed in [14]—instead the quantity ∑ _α Ψ _α (z ₁,…,z _L )² was used there. However, the same argument of degeneracy of the scalar product allows to conclude that

(53)

where s _λ is the Schur function associated to partition λ, and Y _L =(⌊(L−i)/2⌋) _i=1,…,L . In the homogeneous limit,

$$s_{Y_n}(1,\ldots,1)=3^{n(n-1)/2} \prod _{j=1}^n \frac{(3j)!(j-1)!}{(2j)!(2j-1)!} $$

and together we have Z _L =A _HT (L), where A _HT (L)=1,3,25,588… is the number of Half-Turn Symmetric Alternating Sign Matrices [23, 27].

The half-specialization of Sect. 3.5 produces the following factorization:

$$ s_{Y_L}\bigl(1,z_1,z_1^{-1}, \ldots,z_n,z_n^{-1}\bigr)=\prod _{i=1}^n \bigl(1+z_i+z_i^{-1} \bigr) \chi_{Y_n}(z_1,\ldots,z_n) \chi_{Y_{n+1}}(z_1,\ldots,z_n,\omega) $$

(54)

where χ _λ is the symplectic character, defined by:

$$\chi_\lambda(z_1,\ldots,z_n)= \frac{\det(z_i^{\lambda_j+n-j+1}-z_i^{-\lambda_j-n+j-1})}{\det(z_i^{n-j+1}-z_i^{-n+j-1})}, $$

and ω=e ^iπ/3; this formula can be proved by induction, or can be seen as a byproduct of this paper, as we now show.

In the limit ζ→0, the parameterization w is related to the multiplicative spectral parameter z by w=(z−1)²/(1+z+z ²); this way we find

$$h\bigl(z,z'\bigr)=\frac{9(z^2+zz'+z'^2)(1+zz'+z^2z'^2)}{(1+z+z^2)^2(1+z'+z'^2)^2} $$

The denominator factors out of Pfaffians and determinants.

1.1 A.1 Pfaffians

We now recognize the Pfaffian A _n (Eq. (42)) in even size:

which up to some prefactors is exactly the Pfaffian given in [11] (Eq. (3.27)) for the square of the partition function Z _UASM of U-turn symmetric ASMs of [23]. The latter is known to coincide with $\chi_{Y_{2m}}(z_{1},\ldots,z_{2m})$ [25] and so we reproduce the first factor of the l.h.s. of Eq. (54). More precisely, we find $A_{2m}(w_{1},\ldots,w_{2m})= 3^{2m^{2}}\prod_{i=1}^{2m} (1+z_{i}+z_{i}^{-1})^{-2m+1}\chi_{Y_{2m}}(z_{1},\ldots,z_{2m})^{2}$. The odd case can be reduced to the even case by sending one of the z _i to zero (something which did not make sense in the elliptic setting), so that for both parities we have

$$A_n(w_1,\ldots,w_n)=3^{2\lfloor n/2 \rfloor \lfloor(n+1)/2\rfloor} \prod _{i=1}^{n} \bigl(1+z_i+z_i^{-1} \bigr)^{-n+1} \chi_{Y_{n}}(z_1,\ldots,z_{n})^2 $$

or in terms of the original quantities,

$$\mathrm{A}_n(x_1,\ldots,x_n)=3^{-2\lfloor n/2\rfloor \lfloor (n-1)/2\rfloor} \chi_{Y_n}(z_1,\ldots,z_n)^2 $$

The second factor is simply obtained by noting that w=J ₂=−1/2 corresponds to z=ω=e ^iπ/3, so

$$B_n(w_1,\ldots,w_n)=2^{-n}3^{2\lfloor n/2+1 \rfloor \lfloor(n+1)/2\rfloor} \prod_{i=1}^{n} \bigl(1+z_i+z_i^{-1} \bigr)^{-n} \chi_{Y_{n+1}}(z_1,\ldots,z_n, \omega)^2 $$

or $\mathrm{B}_{n}(x_{1},\ldots,x_{n})=3^{-2\lfloor n/2 \rfloor \lfloor(n+1)/2\rfloor} \chi_{Y_{n+1}}(z_{1},\ldots,z_{n},\omega)^{2}$. Finally,

which is consistent with Eqs. (53) and (54).

1.2 A.2 Determinants

Similarly, the determinants simplify as ζ→0. Noting that w=J ₃ and w=J ₄ both correspond to z=0, we conclude that there are only two distinct determinants for each parity; Tsuchiya’s determinant [23, 34] is known to be equal at a cubic root of unity to the symplectic character introduced above [25]

and then we have:

1.3 A.3 More Determinants

The expression (39) of S_2m as a Slater determinant reduces to the numerator of our definition of the symplectic character $\chi_{Y_{2m}}$ (since k _j =Y _2m;m+1−j +2m−j+1, j=1,…,2m)

$$\mathrm{S}_{2m}(z_1,\ldots,z_{2m})= \det_{i,j=1,\ldots,2m} \bigl(z_i^{k_{j}}-z_i^{-k_{j}} \bigr) $$

The differential equation (49) reduces to

$$\sum_{i=1}^{2m} \biggl(z_i \frac{\partial}{\partial z_i} \biggr)^2 \mathrm{S}_{2m}(z_1, \ldots,z_{2m}) =m\bigl(6m^2-1\bigr)\mathrm{S}_{2m}(z_1, \ldots,z_{2m}) $$

1.4 A.4 Homogeneous Limit

Finally, $A_{2m}^{1/2}=H_{2m}=3^{-m(m-1)} \chi_{Y_{2m}}(1,\ldots,1)=1,1,3,26,646\ldots$ is the number of Vertically Symmetric Alternating Sign Matrices of size 2m+1 (also, the number of Off-diagonally Symmetric Alternating Sign Matrices of size 2m, and the number of Descending Plane Partitions of size m which are symmetric w.r.t. all reflections, i.e., Cyclically Symmetric Transpose Complement Plane Partitions of a hexagon of size (m+1)×(m−1) with a triangular hole cut out), while $A_{2m-1}^{1/2}=H_{2m}(J_{3/4})=3^{-(m-1)^{2}} \chi_{Y_{2m-1}}(1,\ldots,1)=1,2,11,170\ldots$ is the number of Cyclically Symmetric Transpose Complement Plane Partitions of size m (also, the number of VSASMs of size (2m−1)×(2m+1) with a defect on the m ^th row, the symmetry line). Note that the square of the number of VSASMs also appears in the observations of [28].

The sequence of numbers

$$2^{m}B_{2m}^{1/2}=2^{m}H_{2m}(J_2,J_{3/4})=3^{-m(m-1)} \chi_{Y_{2m+1}}(1,\ldots,1,\omega)=1,5,66,2431\ldots $$

appears as one of the factors of the enumeration of UUASMs in [23]. The last sequence,

$$2^m (B_{2m-1}/3)^{1/2}=H_{2m}(J_2)=3^{-(m-1)^2} \chi_{Y_{2m}}(1,\ldots,1,\omega)=1,7,143,8398,\ldots $$

is the number of ASMs of order 2m+1 divided by the number of VSASMs of size 2m+1.

As mentioned before, the last two cases, namely H _2m (J ₃,J ₄), and H _2m (J ₂,J ₃,J ₄), are related to H _2m and H _2m (J ₂) by multiplication by powers of 3 and 2.

Appendix B: The ζ→1 Limit

Besides the ζ→0 limit, there is another trigonometric limit, namely ζ→1 or α→0. It is expected to be somewhat trivial since the corresponding Hamiltonian is the Ising Hamiltonian with interaction σ ^x σ ^x. Indeed, we find that the building block H _2m of the partition function becomes:

This formula is valid as long as the w _i stay finite as ζ→1. One special case is if one w _i is equal to J ₄=1/(1−ζ). Then we find instead

$$H_{2m}(w_1,\ldots,w_{2m-1},J_4)|_{\zeta=1}=2^{(m-1)^2} $$

so that

$$X_n(\ldots)=2^{n(n+1)+1} $$

This is compatible with a constant value of Ψ _n,α =2^n(n−1)/2 since $X_{2m}=2^{2n+1} \varPsi_{2m,\alpha}^{2}$.

Appendix C: Proof of Symmetry of H _2m

The symmetry of H_2m , defined by (33) can be seen as a particular case of a general result, which can be formulated as follows: (see also Thm. 4.2 in [24])

Proposition

Let ϕ ₁,ϕ ₂ be two functions (with values in ℂ) such that

• (i) ϕ(x,y)=−ϕ(y,x),

• (ii) ϕ(x ₁,x ₂)ϕ(x ₃,x ₄)−ϕ(x ₁,x ₃)ϕ(x ₂,x ₄)+ϕ(x ₁,x ₄)ϕ(x ₂,x ₃)=0 for ϕ=ϕ ₁,ϕ ₂.

Then, in the domain of the (x _i )_1≤i≤2m such that ϕ ₂(x _i ,x _j )≠0 for all 1≤i,j≤2m,

$$\varDelta _{2m}(x_1,\ldots,x_{2m})= \frac{\det_{\substack{i=1,\ldots,m\\j=m+1,\ldots,2m}} (\frac{\phi_1(x_i,x_j)}{\phi_2(x_i,x_j)} )}{\prod_{\substack{1\le i<j\le m\\\mathrm{or}\\m+1\le i<j\le 2m}} \phi_2(x_i,x_j)} $$

is symmetric in all arguments {x ₁,…,x _2m }.

Actually it is well-known that functions that satisfy (i) and (ii) are 2×2 determinants , so that, removing symmetric factors, one may without loss of generality write ϕ _i (x ₁,x ₂)=ϕ _i (x ₁)−ϕ _i (x ₂), i=1,2. The proposition then follows from the following representation (characteristic of Toda chain tau functions): starting from $\frac{\phi_{1}(x_{i})-\phi_{1}(x_{j})}{\phi_{2}(x_{1})-\phi_{2}(x_{j})} = \frac{1}{2\pi i}\oint_{C} \frac{dy}{(y-\phi_{2}(x_{i}))(y-\phi_{2}(x_{j}))}\phi_{1}(y)$ where C is any contour that surrounds once counterclockwise the ϕ ₂(x _j ), j=1,…,2m, and expanding the determinant in Δ _2m we get

which is explicitly symmetric in the x _i .

The application to H_2m consists in writing ϕ ₂(x,y)=h(x,y)ϑ(x−y)ϑ(x+y), ϕ ₁(x,y)=ϑ(x−y)ϑ(x+y) and checking that they satisfy (i) and (ii), so that H_2m (x ₁,…,x _2m )=∏_1≤i<j≤2m h(x _i ,x _j )Δ _2m (x ₁,…,x _2m ). It is slightly easier to apply it to H _2m , i.e., after the change of variables from x to w, since we then have the more explicit expressions ϕ ₁(w)=w, ϕ ₂(w)=w/(1+(3+ζ ²)w ²−(1−ζ ²)w ³).

Note that other identities following from integrability of the Toda chain, for example the Hankel determinant form

They also provide an alternative derivation of Eq. (39) (“first quantized” form of the tau function).

Appendix D: Differential Equations

We provide here analogues of Eqs. (51) and (52) when H _2m (that is, the function H _2m with all arguments set to zero) is replaced with H _2m (S), S⊂{J ₂,J ₃,J ₄} (again, with all other arguments set to zero). Because of the permutation symmetry w.r.t. {J ₂,J ₃,J ₄}, we only need to provide one formula for each possible cardinality of S. When taking derivatives w.r.t. u or v, the convention is that the arguments that are specialized to J ₂,J ₃,J ₄ are among the u’s.

After transposition (for display purposes), Eq. (51) is of the form

For H _2m itself, the matrix P is

For H _2m (J ₂):

For H _2m (J ₃,J ₄):

For H _2m (J ₂,J ₃,J ₄):

As to Eq. (52):

$$C_0 H_{2(m+1)}H_{2(m-1)}= C_1H_{2m} H''_{2m} -C_2 \bigl(H'_{2m}\bigr)^2 +C_3H_{2m} H'_{2m}+C_4 H_{2m}^2 $$

The coefficients for H _2m are:

For H _2m (J ₂):

For H _2m (J ₃,J ₄):

For H _2m (J ₂,J ₃,J ₄):