# Phases Of Melonic Quantum Mechanics

## TL;DR

This paper investigates two melonic quantum mechanical models, revealing complex phase transitions, critical phenomena, and IR fixed points, with detailed analysis of their thermodynamics, critical exponents, and symmetry breaking.

## Contribution

It provides a detailed analysis of phase transitions and critical behavior in two melonic quantum models, including new critical exponents and IR fixed points, with pedagogical derivations.

## Key findings

- First-order phase transition line with a critical point and non-mean-field exponents
- Singular behavior of quasi-normal frequencies and Lyapunov exponents at criticality
- Existence of a zero-temperature quantum critical point with spontaneous scale invariance breaking

## Abstract

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.