Spectra of eigenstates in fermionic tensor quantum mechanics
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 group invariant states for any set of $${N}_{i}$$. It is non-vanishing only when each $${N}_{i}$$ is even. For equal ranks the number of singlets exhibits rapid growth with $$N$$: it jumps from 36 in the $$O(4{)}^{3}$$ model to 595 354 780 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.
- Research Organization:
- Harvard Univ., Cambridge, MA (United States); Princeton Univ., NJ (United States)
- Sponsoring Organization:
- USDOE Office of Science (SC), High Energy Physics (HEP); National Science Foundation (NSF); US Army Research Office (ARO)
- Grant/Contract Number:
- SC0007870; PHY-1620059; W911NF-14-1-0003
- OSTI ID:
- 1439739
- Alternate ID(s):
- OSTI ID: 1498896
- Journal Information:
- Physical Review. D., Journal Name: Physical Review. D. Vol. 97 Journal Issue: 10; ISSN 2470-0010
- Publisher:
- American Physical Society (APS)Copyright Statement
- Country of Publication:
- United States
- Language:
- English
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