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arXiv 2002.02066hep-thcond-mat.str-el
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大$N$ Majorana量子力学中的Hagedorn温度

Hagedorn Temperature in Large $N$ Majorana Quantum Mechanics

Gabriel Gaitan, Igor R. Klebanov, Kiryl Pakrouski, Preethi N. Pallegar, Fedor K. Popov

备注：
27 pages, 8 figures, v3: minor improvements, references added
发表信息：
Phys. Rev. D 101, 126002 (2020)
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摘要：

我们讨论两种类型的量子力学模型，耦合大量的马约拉纳费米子，并具有正交对称群。在向量类型的模型中，只有一个对称群具有大秩。采用大的$N$极限，保持$gN=λ$固定，其中$g$乘以四次哈密顿量。我们引入了一个简单的模型与$O(N)\times SO(4)$对称性，其能量表示的对称群的平方Casimir。该模型可以变形，使得对称性为$O(N)\times O(2)^2$，并且哈密顿量简化为arXiv：1802.10263中研究的哈密顿量。我们发现大$N$态密度和自由能的解析表达式。在这两种矢量模型中，对于很宽的能量范围，大的$N$态密度大致随$e^{-|E|/λ}$变化。当温度接近哈格多恩温度$T_{\rm H} = λ$时，这会产生临界行为。在正式的大$N$极限中，比热膨胀为$(T_H- T)^{-2}$，这意味着$T_H$是极限温度。然而，在任何有限的$N$，有可能达到任意大的温度。因此，有限的$N$效应平滑了哈格多恩转变。我们还研究了矩阵型模型，它有两个大秩的$O(N)$对称群。具有$O(N)^2\times O(2)$对称性的马约拉纳矩阵模型提供了一个示例，该模型在arXiv：1802.10263中进行了研究。与矢量模型相比，态密度是平滑的，并且在谱的中间附近接近高斯分布。

英文摘要：

We discuss two types of quantum mechanical models that couple large numbers of Majorana fermions and have orthogonal symmetry groups. In models of vector type, only one of the symmetry groups has a large rank. The large $N$ limit is taken keeping $gN=λ$ fixed, where $g$ multiplies the quartic Hamiltonian. We introduce a simple model with $O(N)\times SO(4)$ symmetry, whose energies are expressed in terms of the quadratic Casimirs of the symmetry groups. This model may be deformed so that the symmetry is $O(N)\times O(2)^2$, and the Hamiltonian reduces to that studied in arXiv:1802.10263. We find analytic expressions for the large $N$ density of states and free energy. In both vector models, the large $N$ density of states varies approximately as $e^{-|E|/λ}$ for a wide range of energies. This gives rise to critical behavior as the temperature approaches the Hagedorn temperature $T_{\rm H} = λ$. In the formal large $N$ limit, the specific heat blows up as $(T_H- T)^{-2}$, which implies that $T_H$ is the limiting temperature. However, at any finite $N$, it is possible to reach arbitrarily large temperatures. Thus, the finite $N$ effects smooth out the Hagedorn transition. We also study models of matrix type, which have two $O(N)$ symmetry groups with large rank. An example is provided by the Majorana matrix model with $O(N)^2\times O(2)$ symmetry, which was studied in arXiv:1802.10263. In contrast with the vector models, the density of states is smooth and nearly Gaussian near the middle of the spectrum.

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