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arXiv 1802.10263hep-thcond-mat.str-el
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费米子张量量子力学中的本征态谱

Spectra of Eigenstates in Fermionic Tensor Quantum Mechanics

Igor R. Klebanov, Alexey Milekhin, Fedor Popov, Grigory Tarnopolsky

备注：
42 pages, 1 figure. v2: minor improvements, references added. v3: minor corrections. v4: minor improvements
发表信息：
Phys. Rev. D 97, 106023 (2018)
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摘要：

研究了三指数Majorana费米子的$O(N_1)\times O(N_2)\times O(N_3)$对称量子力学.当秩$N_i$都相等时，该模型具有较大的$N$极限，该极限由Melonic Feynman图控制。我们推导出一个积分公式，计算的数目$SO(N_1)\times SO(N_2)\times SO(N_3)$不变状态的任何一组$N_i$。对于相同的秩，只有当$N$是偶数时，单线态的数量才是非零的，并且它表现出快速的增长：它从$O(4)^3$模型中的$36$跳到$O(6)^3$模型中的$595354780$。我们推导出的能量值的界限，这表明，它们的规模最多为$N^3$在大$N$限制，与预期一致。我们还表明，最低的单重态和非单重态之间的分裂是为了$1/N$。对于$N_3=1$的张量模型减少到$O(N_1)\times O(N_2)$费米子矩阵量子力学，我们发现一个简单的表达式的对称群的二次卡西米尔运营商的哈密顿量。对于具有$SU(N_1)\times SU(N_2)\times U(1)$对称性的复矩阵模型，也导出了类似的表达式。最后，我们研究了张量模型的$N_3=2$情形，它给出了一个更复杂的复矩阵模型，其对称性仅为$O(N_1)\times O(N_2)\times U(1)$。所有的能量都是以适当的单位表示的整数，我们导出了一个简明的谱公式。我们所研究的费米子矩阵模型具有标准的t Hooft大$N$极限，其中基态能量为$N^2$阶，而能隙为$1$阶。

英文摘要：

We study the $O(N_1)\times O(N_2)\times O(N_3)$ symmetric quantum mechanics of 3-index Majorana fermions. When the ranks $N_i$ are all equal, this model has a large $N$ limit which is dominated by the melonic Feynman diagrams. We derive an integral formula which computes the number of $SO(N_1)\times SO(N_2)\times SO(N_3)$ invariant states for any set of $N_i$. For equal ranks the number of singlets is non-vanishing only when $N$ is even, and it exhibits rapid growth: it jumps from $36$ in the $O(4)^3$ model to $595354780$ in the $O(6)^3$ model. We derive bounds on the values of energy, which show that they scale at most as $N^3$ in the large $N$ limit, in agreement with expectations. We also show that the splitting between the lowest singlet and non-singlet states is of order $1/N$. For $N_3=1$ the tensor model reduces to $O(N_1)\times O(N_2)$ fermionic matrix quantum mechanics, and we find a simple expression for the Hamiltonian in terms of the quadratic Casimir operators of the symmetry group. A similar expression is derived for the complex matrix model with $SU(N_1)\times SU(N_2)\times U(1)$ symmetry. Finally, we study the $N_3=2$ case of the tensor model, which gives a more intricate complex matrix model whose symmetry is only $O(N_1)\times O(N_2)\times U(1)$. All energies are again integers in appropriate units, and we derive a concise formula for the spectrum. The fermionic matrix models we studied possess standard 't Hooft large $N$ limits where the ground state energies are of order $N^2$, while the energy gaps are of order $1$.

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