Phases Of Melonic Quantum Mechanics

We explore in detail the properties of two melonic quantum mechanical theories which can be formulated either as fermionic matrix quantum mechanics in the new large $D$ limit, or as disordered models. Both models have a mass parameter $m$ and the transition from the perturbative large $m$ region to the strongly coupled "black-hole" small $m$ region is associated with several interesting phenomena. One model, with ${\rm U}(n)^2$ symmetry and equivalent to complex SYK, has a line of first-order phase transitions terminating, for a strictly positive temperature, at a critical point having non-trivial, non-mean-field critical exponents for standard thermodynamical quantities. Quasi-normal frequencies, as well as Lyapunov exponents associated with out-of-time-ordered four-point functions, are also singular at the critical point, leading to interesting new critical exponents. The other model, with reduced ${\rm U}(n)$ symmetry, has a quantum critical point at strictly zero temperature and positive critical mass $m_*$. For $0<m<m_*$, it flows to a new gapless IR fixed point, for which the standard scale invariance is spontaneously broken by the appearance of distinct scaling dimensions $\Delta_+$ and $\Delta_-$ for the Euclidean two-point function when $t\rightarrow +\infty$ and $t\rightarrow -\infty$ respectively. We provide several detailed and pedagogical derivations, including rigorous proofs or simplified arguments for some results that were already known in the literature.