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arXiv 1407.4262cond-mat.stat-mechmath-phmath.MPnlin.SIquant-ph
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关于键维数为$D=4$的可积矩阵乘积算子

On integrable matrix product operators with bond dimension $D=4$

Hosho Katsura

备注：
13 pages, 3 figures, submitted to a special issue of JSTAT on "Quantum Entanglement in Condensed Matter Physics"; Conjectures presented in version 1 have been proved in version 2; typos corrected
发表信息：
J. Stat. Mech., P01006 (2015)
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摘要：

构造并研究了键维$D=4$的双参数矩阵乘积算子族。运算符$M(x,y)$作用于$({\mathbb C}_2)^{\otimes N}$，即，一个长度为$N$的自旋$1/2$链的状态空间。对于$x=1/3$和$y=1/\sqrt{3}$这两个参数的特定值，算符与六角梯上价键固态的约化密度矩阵的平方根成正比。我们发现，$M(x,y)$有几个有趣的属性时，$(x,y)$在于单位圆中心在原点：$x^2 + y^2=1$。在这种情况下，我们发现$M(x,y)$与各向同性自旋$1/2$海森堡链的哈密顿量和所有守恒电荷对易。此外，如果$x^2_i + y^2_i=1$用于$i=1$和$2$，则$M(x_1,y_1)$和$M(x_2,y_2)$相互交换。$M(x,y)$的这些显著性质被证明是杨-巴克斯特方程的结果。

英文摘要：

We construct and study a two-parameter family of matrix product operators of bond dimension $D=4$. The operators $M(x,y)$ act on $({\mathbb C}_2)^{\otimes N}$, i.e., the space of states of a spin-$1/2$ chain of length $N$. For the particular values of the parameters: $x=1/3$ and $y=1/\sqrt{3}$, the operator turns out to be proportional to the square root of the reduced density matrix of the valence-bond-solid state on a hexagonal ladder. We show that $M(x,y)$ has several interesting properties when $(x,y)$ lies on the unit circle centered at the origin: $x^2 + y^2=1$. In this case, we find that $M(x,y)$ commutes with the Hamiltonian and all the conserved charges of the isotropic spin-$1/2$ Heisenberg chain. Moreover, $M(x_1,y_1)$ and $M(x_2,y_2)$ are mutually commuting if $x^2_i + y^2_i=1$ for both $i=1$ and $2$. These remarkable properties of $M(x,y)$ are proved as a consequence of the Yang-Baxter equation.

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