Permutation invariant matrix quantum thermodynamics and negative specific heat capacities in large N systems

TL;DR

The authors study the thermodynamics of a gauged permutation-invariant matrix quantum mechanics (GPIMQM) and show that, at large $N$, the canonical ensemble exhibits a cross-over with a vanishing Hagedorn temperature and a sharp SHC peak, while the microcanonical ensemble develops negative specific heat due to super-exponential degeneracy growth. They develop exact finite-$N$ partition functions via Molien-Weyl formulas, analyze high-temperature limits as free oscillator behavior, and reveal a breakdown scale $x_c=\frac{\log N}{N}$ that governs the transition and ensemble inequivalence. The work extends these phenomena to systems with continuous symmetries such as $U(N)$, including tensor models, where a similar vanishing $T_H$, negative SHC, and high-temperature oscillator limits appear; path-integral formulations and explicit counting via Kronecker/Littlewood-Richardson coefficients support these findings. The results bear on gravitational thermodynamics, offering a controlled toy-model perspective on negative specific heat in AdS/CFT and the microcanonical-canonical correspondence for complex gauge-invariant sectors.

Abstract

We study the thermodynamic properties of the simplest gauged permutation invariant matrix quantum mechanical system of oscillators, for general matrix size $N$. In the canonical ensemble, the model has a transition at a temperature $T$ given by $x = e^{ -1/ T } \sim x_c=e^{-1/T_c}=\frac{\log N}{N}$, characterised by a sharp peak in the specific heat capacity (SHC), which separates a high temperature from a low temperature region. The peak grows and the low-temperature region shrinks to zero with increasing $N$. In the micro-canonical ensemble, for finite $N$, there is a low energy phase with negative SHC and a high energy phase with positive SHC. The low-energy phase is dominated by a super-exponential growth of degeneracies as a function of energy which is directly related to the rapid growth in the number of directed graphs, with any number of vertices, as a function of the number of edges. The two ensembles have matching behaviour above the transition temperature. We further provide evidence that these thermodynamic properties hold in systems with $U(N)$ symmetry such as the zero charge sector of the 2-matrix model and in certain tensor models. We discuss the implications of these observations for the negative specific heat capacities in gravity using the AdS/CFT correspondence.

Figures (20)

• Figure 1: Energy versus temperature, parameterised by $x = e^{ - \beta } = e^{ - 1\over T }$ for $N=20$ : showing a cross-over
• Figure 2: Energy versus temperature : Cross-over sharpens and approaches zero temperature as $N$ increases. Blue, Green and Red curves are for $N =10,15,20$
• Figure 3: Specific heat capacity versus temperature : Sharp peak approaches zero temperature as $N$ increases. Blue, Green and Red curves are for $N =10,15,20$
• Figure 4: Energy curve showing location of $x_{ {\rm{max}}}$ (in yellow) at the centre of the energy transition region and $x_{ {\rm{min}}}$ (in green) near the high temperature end of the transition. for $N =20$.
• Figure 5: Entropy versus temperature : Sharp peak approaches zero temperature as $N$ increases. Blue, Green and Red curves are for $N =10,15,20$