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

A bstract 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 $$ x={e}^{-1/T}\sim {x}_c={e}^{-1/{T}_c}=\frac{\log N}{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>x</mml:mi> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>e</mml:mi> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mi>T</mml:mi> </mml:mrow> </mml:msup> <mml:mo>~</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>c</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>e</mml:mi> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>c</mml:mi> </mml:msub> </mml:mrow> </mml:msup> <mml:mo>=</mml:mo> <mml:mfrac> <mml:mrow> <mml:mo>log</mml:mo> <mml:mi>N</mml:mi> </mml:mrow> <mml:mi>N</mml:mi> </mml:mfrac> </mml:math> , 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.