白色热串位的颜色字符
Color Characters for White Hot String Bits
摘要：
通用串位模型的状态空间由作用于真空状态的$N\times N$矩阵创建算子所跨越。这样的创建操作符在颜色组$U(N)$（或$SU(N)$，如果矩阵是无迹的）的伴随表示中进行变换。我们考虑一个由$b$类玻色子比特和$f$类费米子比特组成的系统。出现在$N\to\infty$极限中的字符串标识$P^+=mM\sqrt{2}$，其中$M$是位数运算符，$P^-=H\sqrt{2}$是系统哈密尔顿算子。我们研究了这种弦位系统的热性质的情况下$H=0$，这可以被认为是无张力弦极限：唯一的动力学是限制物理状态的颜色单态。然后可以识别热配分函数${\rm Tr} e^{-βmM}$，将$x=e^{-βm}$与生成函数$χ_0^{bf}(x)$一起放置，其中$x^n$在其关于$x=0$的展开中的系数是具有位数$M=n$的颜色单态的数目。这个函数是一个纯粹的群论对象，在文献中得到了很好的研究。我们发现，在$N=\infty$这个系统显示了Hagedorn发散在$x=1/(b+f)$与最终温度$T_H=m/\ln(b+f)$。有限$N$的相应函数对于$0<x<1$是完全有限的，因此$N=\infty$系统在温度$T_H$处表现出相变，这对于任何有限$N$都是不存在的。我们证明，低温相是不稳定的$T_H$以上。最低阶$1/N$渐近校正，在高温阶段的$x\to1$，计算大$N$。值得注意的是，这与带有$N$节点的标记欧拉有向图的数量有关。系统的方法来扩展我们的结果，更高的订单$1/N$。
英文摘要：
The state space of a generic string bit model is spanned by $N\times N$ matrix creation operators acting on a vacuum state. Such creation operators transform in the adjoint representation of the color group $U(N)$ (or $SU(N)$ if the matrices are traceless). We consider a system of $b$ species of bosonic bits and $f$ speciesof fermionic bits. The string, emerging in the $N\to\infty$ limit, identifies $P^+=mM\sqrt{2}$ with $M$ the bit number operator and $P^-=H\sqrt{2}$ with $H$ the system Hamiltonian. We study the thermal properties of this string bit system in the case $H=0$, which can be considered the tensionless string limit: the only dynamics is restricting physical states to color singlets. Then the thermal partition function ${\rm Tr} e^{-βmM}$ can be identified, putting $x=e^{-βm}$, with a generating function $χ_0^{bf}(x)$, for which the coefficient of $x^n$ in its expansion about $x=0$ is the number of color singlets with bit number $M=n$. This function is a purely group theoretic object, which is well-studied in the literature. We show that at $N=\infty$ this system displays a Hagedorn divergence at $x=1/(b+f)$ with ultimate temperature $T_H=m/\ln(b+f)$. The corresponding function for finite $N$ is perfectly finite for $0<x<1$, so the $N=\infty$ system exhibits a phase transition at temperature $T_H$ which is absent for any finite $N$. We demonstrate that the low temperature phase is unstable above $T_H$. The lowest-order $1/N$ asymptotic correction, for $x\to1$ in the high temperature phase, is computed for large $N$. Remarkably, this is related to the number of labeled Eulerian digraphs with $N$ nodes. Systematic methods to extend our results to higher orders in $1/N$ are described.
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