量子Grassmann矩阵的多项式解
Polynomial solution of quantum Grassmann matrices
摘要：
我们研究了Anninos和Silva最近提出的具有矩阵状指数结构（指数为$N$和$L$）和四次相互作用的量子力学费米子模型。对于不同的$L$值和任意的$N$，我们用$q$变形的正交多项式（Stieltjes-Wigert多项式）精确地计算配分函数.从显式的热配分函数的评估，能级和简并确定。对于一个给定的$L$，不同能量的状态数在$N$中是二次的，这意味着能级的指数简并。我们还表明，在高温下，我们有一个高斯矩阵模型，这意味着交换$N$和$L$的对称性，连同一个威克旋转的光谱参数。在这个极限中，我们还将一般$L$和$N,$的配分函数写成一个广义Hermite多项式。
英文摘要：
We study a model of quantum mechanical fermions with matrix-like index structure (with indices $N$ and $L$) and quartic interactions, recently introduced by Anninos and Silva. We compute the partition function exactly with $q$-deformed orthogonal polynomials (Stieltjes-Wigert polynomials), for different values of $L$ and arbitrary $N$. From the explicit evaluation of the thermal partition function, the energy levels and degeneracies are determined. For a given $L$, the number of states of different energy is quadratic in $N$, which implies an exponential degeneracy of the energy levels. We also show that at high-temperature we have a Gaussian matrix model, which implies a symmetry that swaps $N$ and $L$, together with a Wick rotation of the spectral parameter. In this limit, we also write the partition function, for generic $L$ and $N,$ in terms of a single generalized Hermite polynomial.
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