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arXiv 1006.4792hep-thcond-mat.str-elgr-qchep-lat
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Fuzzy 2-球面上的反常热力学

Unusual Thermodynamics on the Fuzzy 2-Sphere

Sanatan Digal, Pramod Padmanabhan

备注：
19 pages, 7 figures. The paper has been significantly modified. Main results are unchanged
发表信息：
JHEP 1010:091,2010
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摘要：

连续球（$S^2$）和它的模糊类似物（$S^2_F$）上的高自旋狄拉克算子都与反对易手征算子配对。这样的结果是看到这些运营商的类费米子谱，这是特别真实的情况下，甚至整数自旋狄拉克运营商。受$S_F^2$上自旋为1的狄拉克算符谱的这一特征的启发，我们假设自旋为1的粒子服从费米-狄拉克统计。尽管在诸如$S^2$的紧凑表面上缺乏良好定义的自旋-统计关系，但这种选择仍然存在。比热计算的情况下，自旋$\frac{1}{2}$和自旋1狄拉克运营商。值得注意的是，自旋为$\frac{1}{2}$的粒子系统的比热大于自旋为1的粒子系统的比热，尽管自旋为1的粒子系统的自由度更多。其原因是推断通过研究的狄拉克运营商在这两种情况下的频谱。研究了自旋为1的狄拉克算符的零模随截止角动量$L$的变化，发现零模遵循简单的幂律.该数目使得自旋1和自旋$\frac{1}{2}$系统的具有正能量的状态的数目变得相当。通过研究高自旋狄拉克算符的零模及其谱随简并度的变化，对高自旋狄拉克算符的谱也作了一些讨论。在自旋$\frac{1}{2}$和自旋1的情况下，作为温度的函数的平均能量进行了研究。在2+1维空间中，它们偏离了标准的理想气体定律.

英文摘要：

Higher spin Dirac operators on both the continuum sphere($S^2$) and its fuzzy analog($S^2_F$) come paired with anticommuting chirality operators. A consequence of this is seen in the fermion-like spectrum of these operators which is especially true even for the case of integer-spin Dirac operators. Motivated by this feature of the spectrum of a spin 1 Dirac operator on $S_F^2$, we assume the spin 1 particles obey Fermi-Dirac statistics. This choice is inspite of the lack of a well defined spin-statistics relation on a compact surface such as $S^2$. The specific heats are computed in the cases of the spin $\frac{1}{2}$ and spin 1 Dirac operators. Remarkably the specific heat for a system of spin $\frac{1}{2}$ particles is more than that of the spin 1 case, though the number of degrees of freedom is more in the case of spin 1 particles. The reason for this is inferred through a study of the spectrums of the Dirac operators in both the cases. The zero modes of the spin 1 Dirac operator is studied as a function of the cut-off angular momentum $L$ and is found to follow a simple power law. This number is such that the number of states with positive energy for the spin 1 and spin $\frac{1}{2}$ system become comparable. Remarks are made about the spectrums of higher spin Dirac operators as well through a study of their zero-modes and the variation of their spectrum with degeneracy. The mean energy as a function of temperature is studied in both the spin $\frac{1}{2}$ and spin 1 cases. They are found to deviate from the standard ideal gas law in 2+1 dimensions.

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