Matrix product operator representations for the local conserved quantities of the spin-$1/2$ XYZ chain

Abstract

We present explicit matrix product operator (MPO) representations for the local conserved quantities of the spin-$1/2$ XYZ chain. Through these MPO representations, we simplify the coefficients appearing in the local conserved quantities originally derived by one of the authors, and reveal their combinatorial meaning: the coefficients prove to be a polynomial generalization of the Catalan numbers, defined via weighted monotonic lattice paths. Furthermore, we obtain a new simple $3 \times 3$ Lax operator for the XYZ chain that, unlike Baxter's R-matrix, does not involve elliptic functions.

Figures (8)

• Figure 1: Combinatorial representation of $S_{n}^{\left(N_x, N_y, N_z\right)}$ as a weighted sum over monotonic lattice paths. (a) General case: The grid has $n+1$ columns and $N = N_x + N_y + N_z$ rows. Each row $j$ is assigned a flavor $\alpha_j \in \{x, y, z\}$, with exactly $N_\alpha$ rows of flavor $\alpha$ for each $\alpha \in \{x, y, z\}$. The value of $S_{n}^{\left(N_x, N_y, N_z\right)}$ depends only on $(N_x, N_y, N_z)$, not on the arrangement of flavors. Paths run from $(0,1)$ to $(n, N)$, with each East step along row $j$ contributing a factor $\alpha_j$ to the path weight. The red path illustrates one trajectory with weight $\alpha_1 \alpha_3^2 \cdots \alpha_j \cdots \alpha_{N-1} \alpha_N$. (b) Example: $n=2$, $(N_x, N_y, N_z) = (2,1,0)$. The red path represents one of the paths with weight $xy$. The table enumerates all path weights and their multiplicities. Summing all path contributions gives $S_{2}^{\left(2, 1, 0\right)} = 3x^2 + 2xy + y^2$.
• Figure 2: Combinatorial representation of $R_{n}^{\left(N_x, N_y, N_z\right)}$ as a weighted sum over good paths. The lower region is a rectangular grid identical to Fig. \ref{['fig:Sfunc-combinatoric']}, with $N = N_x + N_y + N_z$ flavored rows. As before, the ordering of flavors in the lower region does not affect the final result. The upper region contains $n$ bundles, each bundle consisting of three rows $\{x, y, z\}$ shown as double lines. Traversing a bundle follows the same rule as the lower region: the path crosses exactly one row, contributing that flavor to the weight. The diagonal line connects $(0, N)$ and $(n, N+3n)$ with slope $3$; good paths must stay weakly below this line, and the forbidden region above it is shaded. The red path illustrates an example good path. The magnified view shows the internal structure of one bundle.
• Figure 3: Example of $R_{2}^{\left(2, 1, 0\right)}$ with $n=2$, $N_x = 2$, $N_y = 1$, and $N_z = 0$. The diagonal line connects $(0, N) = (0, 3)$ and $(n, N+3n) = (2, 9)$. Good paths must not cross the diagonal, and the forbidden region is shaded. The table enumerates all path weights and their multiplicities. Summing all path contributions gives $R_{2}^{\left(2, 1, 0\right)} = 5x^2 + 5xy + 2y^2 + 2xz + yz$. The red path illustrates an example good path with weight $xy$. The diagonal constraint prevents paths from traversing any row in bundle 2.
• Figure 4: Combinatorial proof of the identity \ref{['eq:sfunc-R1-identity']}. The lattice represents $R_{n}^{\left(N_x+m, N_y+m, N_z+m\right)}$. We arrange the $N + 3m$ flavored rows of the lower region as $N = N_x + N_y + N_z$ rows followed by $m$ bundles, and place $n$ bundles in the upper region. Good paths must stay below the diagonal line (which begins after the $N$ rows and $m$ bundles). Any path can be decomposed at column $\widetilde{n}$ where it crosses the horizontal line $y = N$: the lower red region contributes $S_{\widetilde{n}}^{\left(N_x, N_y, N_z\right)}$, while the upper blue region contributes $R_{n-\widetilde{n}}^{\left(m+\widetilde{n}\right)}$. Summing over $\widetilde{n}$ establishes the identity.
• Figure 5: Decomposition of paths for proving Eqs. \ref{['eq:sfunc-decomp-1']} and \ref{['eq:sfunc-decomp-2']}. The lattice consists of $N = N_x + N_y + N_z$ flavored rows in the lower region in the same way as in Fig. \ref{['fig:Sfunc-combinatoric']}, and $n$ bundles in the upper region, where each bundle contains one row of each flavor $x$, $y$, $z$ in the same way as explained in Fig. \ref{['fig:Rfunc-combinatoric']}. The diagonal line connects $(0, N)$ and $(n, N+3n)$ with slope $3$. Any path from $(0,1)$ to $(n, N+3n)$ can be uniquely decomposed at the column $\tilde{n}$ where it first crosses the diagonal. The portion before the crossing (red region) is a good path that does not cross the diagonal, contributing to $R_{\tilde{n}}^{\left(N_x, N_y, N_z\right)}$, while the portion after crossing (blue region) contributes to $S_{n-\tilde{n}}^{\left(n-\tilde{n}\right)}$. Summing over $\tilde{n} = 0, 1, \ldots, n$ yields Eq. \ref{['eq:sfunc-decomp-1']}. For Eq. \ref{['eq:sfunc-decomp-2']}, one adds an $x$-flavored row on top, so that paths end at $(n, N+3n+1)$; the diagonal remains unchanged, and the remaining portion contributes to $S_{n-\tilde{n}}^{\left(n-\tilde{n}+1, n-\tilde{n}, n-\tilde{n}\right)}$.

Theorems & Definitions (5)

• Theorem 1
• Theorem 2
• Conjecture 1
• Lemma 1
• proof