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

Denjoe O’Connora,∗, Sanjaye Ramgoolama,b,c,†

aSchool of Theoretical Physics

Dublin Institute of Theoretical Physics, 10 Burlington Road, Dublin 4, Ireland

bSchool of Physics and Astronomy , Centre for Research in String Theory
Queen Mary University of London, London E1 4NS, United Kingdom
dSchool of Physics and Mandelstam Institute for Theoretical Physics,

University of Witwatersrand, Wits, 2050, South Africa

E-mails<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### 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.

###### Contents

  1. 1  Introduction
  2. 2  Bosonic GPIMQM Partition functions : Review
  3. 3  GPIMQ-thermodynamics in the canonical ensemble : finite $N$ cross-over transition and vanishing Hagedorn temperature
    1. 3.1  Super-exponential degeneracies and vanishing Hagedorn temperature
    2. 3.2  Thermodynamic quantities in the canonical ensemble
      1. 3.2.1  Energy versus temperature
      2. 3.2.2  Specific heat capacity versus temperature
      3. 3.2.3  Entropy versus temperature and Entropy as a function of Energy
    3. 3.3  High temperature limit and $N^{2}$ simple harmonic oscillator thermodynamics
    4. 3.4  Zeroes in the complex plane
  4. 4  PIMQ-Thermodynamics in the micro-canonical ensemble : Negative specific heat and (in)equivalence of ensembles
    1. 4.1  Micro-canonical ensemble and negative specific heat capacities
    2. 4.2 Thermodynamic quantities in the micro-canonical ensemble
      1. 4.2.1 Energy versus micro-canonical temperature
      2. 4.2.2  Specific heat capacity versus micro-canonical temperature
    3. 4.3  Equivalence and in-equivalence of ensembles
  5. 5  High temperature expansion of PIMQ-Thermo and small cycle dominance
    1. 5.1  Cycle structures and the high temperature expansion
    2. 5.2  Breakdown of the high $T$ expansion and the characteristic scale $x_{c}={\log N\over N}$.
  6. 6 Path Integral for Tensor Models
    1. 6.1 Complex-Vectors
    2. 6.2 Two-Tensors
    3. 6.3 Three-Tensors
    4. 6.4 Hermitian Matrices with adjoint $U(N)$ action.
    5. 6.5 The Adjoint Matrix system with $d=2$ and the $U(1)$-Charge-0 Sector
  7. 7 High Temperature Limits from path integrals
    1. 7.1 Matrix Models under Adjoint Action of $U(N)$
    2. 7.2 Tensor Models
      1. 7.2.1 Counting Complex Vector Invariants
  8. 8  Hagedorn transitions in $U(N)$ invariant models and negative SHC in tensor models
    1. 8.1  3-index complex tensor model : Phase structure and computational evidence.
  9. 9  Negative SHCs in matrix quantum mechanical models and AdS/CFT
    1. 9.1  Super-exponential and sub-exponential growths of combinatorial sequences and negative specific heat capacities
    2. 9.2  Negative SHCs from zero-charge sector of complex matrix quantum mechanics and AdS/CFT
    3. 9.3  Negative SHC with super-exponential growth of degeneracies in multi-matrix models
  10. 10 Summary and Outlook
  11. A Multi-harmonic oscillator and ensemble equivalence
  12. B Proving that the first sub-leading term in the high temperature expansion is from $p=[2,1^{N-2}]$
  13. C Integer sequences, Hagedorn and heat capacity
  14. D  SAGE computation for zero charge sector of 2-matrix models
  15. E  SAGE computation for negative SHC in multi-matrix model with $d=n$ scaling

##  1  Introduction

Matrix quantum mechanics (MQM) models with $U(N)$ gauge symmetry groups
provide avenues to inform the solution of questions in quantum gravity through
gauge-gravity dualities [1, 2, 3, 4, 5]. In this paper we demonstrate that
simple non-interacting matrix models, where $S_{N}$ replaces $U(N)$ gauge
symmetry, provide a setting where negative specific heat capacities and
associated in-equivalence of thermodynamic ensembles emerge naturally. These
features are of interest in connection with gravitational thermodynamics in
holographic quantum mechanical systems. This work relies on analytic and
computational development of the results in [6, 7].

An important strand of research in gauge-gravity duality investigates the
relation between the gravitational thermodynamics in AdS and the CFT
thermodynamics. The transition in semi-classical gravity between the vacuum
AdS solution and the AdS black hole geometry was shown to imply a phase
transition at large ’t Hooft coupling in the large $N$ limit of $N=4$ SYM
theory [4]. Motivated by a study of phase structures in weakly coupled gauge
theories, it was shown in [8] that the $U(N)$ invariant sector of the 2-matrix
harmonic oscillator has an exponential growth of the degeneracies at low
energy levels and a finite Hagedorn temperature in the large $N$ limit. This
mirrors properties of weakly coupled gauge theories [9] in the large $N$
limit. Finite $N$ partition functions for 2-matrix quantum mechanics were
analysed for $N$ up to $7$ [10] and the Hagedorn transition related to a
convergence of zeroes to a point on the real axis $x=e^{-1/T}=e^{-\beta}$.
Recent work has shown that super-symmetric matrix models with known gravity
duals can have non-supersymmetric analogs which capture similar large $N$
thermodynamics [11, 12].

MQM models also play a role in earlier examples of gauge-string duality such
as large $N$ two-dimensional Yang-Mills theories [14] where the unitary matrix
models [15] were understood to capture the state space and amplitudes.
Particular sectors of the AdS/CFT correspondence are also controlled by MQM
models, e.g. the half-BPS sector by one-matrix models [16, 17, 18] and less
super-symmetric sectors by multi-matrix models [19, 20]. Permutation groups
arise as hidden symmetries, i.e. groups which are not manifest symmetries of
the action, but which control the combinatorics of many problems in gauge-
string duality through the mathematics of Schur-Weyl duality, as explained in
the reviews [21, 22]. Permutation groups also arise as more manifest
symmetries in the AdS/CFT correspondence, notably in the case of AdS3/CFT2 [3]
where the CFT is an orbifold by $S_{N}$.

In addition to their role as time-dependent variables in quantum mechanical
systems, matrices also arise as integration variables in zero-dimensional
matrix models. An example, related to the BFSS model, is the zero-dimensional
IKKT model [23]. Simple matrix models also have a long history of applications
in modelling statistical characteristics of physical data ranging from nuclear
energy levels to financial correlation matrices [24, 25, 26]. The standard
matrix models widely studied in the context of holography or of matrix
statistics have continuous manifest symmetries such as $U(N)$.

The study of permutation groups as manifest, as opposed to hidden, symmetries
in matrix models has been initiated in the context of applications of matrix
models to ensembles of matrices in computational linguistics [27]. The general
13-parameter Gaussian model [28] has been shown to describe approximate
gaussianity in matrix representations of words [29, 30] and a reduction to a
4-parameter model appropriate for symmetric traceless matrices has been used
to demonstrate approximate gaussianity for statistical correlation matrices
[31].

While there is not yet a gauge-string duality conjecture relating these matrix
models to appropriate gravitational or stringy duals, it is important to
investigate whether the large $N$ characteristics of these matrix models bear
significant similarities to those of matrix models which do have such known
duals. With these motivations, a large $N$ factorisation property for matrix
model correlators, and for matrix quantum mechanics inner products, was
established using properties of partition algebras which arise as hidden
symmetries through the mechanisms of Schur-Weyl duality [32, 33]. An important
aspect of the permutation invariant matrix observables is their relation to
the counting of directed graphs, which was observed in [27, 28], then studied
in detail and generalised to multi-matrix permutation invariants in [34] . The
relevant number sequence for the case of 1-matrix invariants is [35].

In [6] we gave, following earlier work in the context of the BFSS model [13],
the path integral formulation for the partition function of permutation
invariant matrix models, by discretising the Euclidean time direction so that
parallel transport is implemented by group elements as in lattice gauge
theory, and taking a continuum limit. This allowed us to recover the Molien-
Weyl formula in the Gaussian case. In [7] we derived explicit formulae for the
partition functions of the general 11-parameter Gaussian permutation invariant
harmonic oscillator quantum mechanical systems.

The different thermodynamic characteristics of single-matrix, multi-matrix and
tensor models at large $N$ are largely due to properties of associated
combinatorial integer sequences. An important feature of the counting of
permutation invariants of a single matrix is the very rapid growth of the
dimension of the space of invariants as a function of degree $k$ for $k\leq
N/2$. We will refer to the counting in this regime as the stable large $N$
regime. This growth is related to a counting of directed graphs with $k$ edges
and any number of vertices. It is super-exponential as we will discuss in more
detail in section 3 of this paper.

A similar very rapid growth of invariants occurs in tensor models with
invariance under $U(N)$ or other continuous symmetry groups. The 3-index
complex tensor model with $U(N)\times U(N)\times U(N)$ gauge symmetry, which
has been studied with motivations from quantum gravity [36, 37] is a case
where the large $N$ stable counting is related to the counting of bi-partite
ribbon graphs [38] [40]. Super-exponential growth in degeneracies in the
stable regime have been recognised to lead to a vanishing Hagedorn temperature
in the large $N$ limit [41][42]. An open question has been to clarify the
physics of these zero-temperature Hagedorn transitions in the context of
matrix and tensor models at large $N$. This paper addresses the question.

We summarise the principal results of this paper, starting with results
pertaining to the simplest gauged permutation invariant quantum matrix (GPIQM)
model

  * •

Using the exact finite $N$ formula (2.17) for the canonical partition function
from [7], we derive the all-orders high temperature expansion organised in
terms of the degrees of singularities of the partition function at $x=1$. The
formula for the degrees (5.7) is a function of partitions $p$ of $N$.

  * •

We prove, using the equation (5.7), that the leading term in the high
temperature expansion comes from $p=[1^{N}]$. The high-temperature limit is
also derived from a path integral point of view.

  * •

We prove, using equation (5.7), that the second term in the high temperature
expansion comes from $p=[2,1^{N-1}]$. By comparing the first and second terms,
we obtain an all orders large $N$ formula for the breakdown scale
$x_{\rm{bkdn}}$ of the high temperature expansion. The first term in this
large $N$ formula identifies a characteristic scale $x_{c}={\log N\over N}$.

  * •

In the canonical ensemble, there is a Hagedorn-like cross-over transition at a
temperature which approaches zero in the large $N$ limit. The transition is
associated with a rapid increase in the expectation value of the energy as the
temperature is increased through the transition. This is signalled by a peak
in the specific heat capacity located at $x_{{\rm{max}}}\sim
x_{c}=\frac{\log{N}}{N}$. The zero temperature limit is evidenced in Figure
17.

  * •

The microcanonical and canonical ensembles are only equivalent above the
transition temperature. This is illustrated in Figure 15.

  * •

In the microcanonical ensemble the GPIQM exhibits negative heat capacity at
energies below the transition and for asymptotically large $N$ the specific
heat capacity diverges to positive infinity above and negative infinity below
the transition with the micro-canonical transition temperature approaching
zero.

The basic mechanism responsible for the region of negative specific heat
capacity, is super-exponential, or as we will find, near-exponential growth of
degeneracies at low energies $k\lesssim N\log N$. At higher energies, finite
$N$ effects tame the rapid growth of degeneracies and the micro-canonical
specific heat is positive.

It is natural to ask if the same mechanism exists in matrix or tensor systems
of size $N$ in the large $N$ limit for cases with continuous symmetries such
as $U(N)$. In the second part of the paper, we demonstrate that this is indeed
the case. Our main results here are:

  * •

We write down the path-integrals for complex tensors with indices transforming
in a product of $U(N)$ groups. This allows us to derive the asymptotic
divergence of the partition function near $x=1$ and hence to count the number
of physical degrees of freedom. For a system of $d$ complex $s$ tensors
transforming under the group $U(N)^{\times s}$ we find for $d>1$ and $s>1$
that $N_{\rm{phys}}=2dN^{s}-s(N^{s}-1)+1$ (see (7.8)). We note that this
number is known in mathematics as the Krull dimension of the commutative
algebra of invariants, see [43].

  * •

We find indications of similar thermodynamic characteristics to the gauged
permutation invariant matrix model in a system with gauged $U(N)^{\times 3}$
acting on complex $3$-index tensors. This includes a Hagedorn-like cross-over
transition which is expected to vanish as $N$ goes to infinity, in-equivalence
between the canonical and micro-canonical ensembles in the low temperature
region, and negative specific heat capacity at low micro-canonical
temperature. The evidence involves a combination of the high temperature
behaviour we have derived from the path integral formulation, and
computational evidence based on known group-theoretic formulae involving
Kronecker coefficients (see eqn (8.7)) for the finite $N$ counting of tensor
model invariants. The SAGE code needed to do the computations is included in
the discussion.

  * •

We find indications of the features of in-equivalence of ensembles and
negative specific heat capacity in the zero-charge sector of a complex one-
matrix model (equivalently 2-Hermitian matrix model) having $U(N)$ gauge
symmetry. This model has a finite temperature Hagedorn phase transition as $N$
goes to infinity. The negative specific heat capacity, which is the heat
capacity rescaled by $1/N^{2}$, at generic fixed temperatures below the
Hagedorn temperature vanishes in the large $N$ limit, but remains finite when
the temperature is infinitesimally close to the Hagedorn temperature in the
large $N$ limit. The evidence again consists of the high temperature limit
which is obtained from the path integral combined with computation with group-
theoretic formulae for the counting in terms of Littlewood-Richardson
coefficients (see eqn (9.29)). The SAGE code is again provided.

  * •

Examples of small $N$ canonical partition functions are presented for complex
vectors (section 6.1), complex two tensors (section 6.2), the $3$-tensor case
(section 6.3), Hermitian matrices (section 6.4 )and the charge zero sector of
two matrix models (section 6.5). These are obtained by performing the contour
integrals in the Molien-Weyl formula for the generating function of invariants
to obtain explicit rational functions of $x=e^{-\beta}$.

The paper is organised as follows: We begin with a review of gauged
permutation invariant matrix quantum mechanics (GPIMQM) in section 2. We
review analytic expressions for the micro-canonical and the canonical
partition functions for the simplest GPIMQM, where the quadratic potential is
${\rm{tr}}(MM^{T})$. This is a point of enhanced symmetry in the
$11$-dimensional parameter space of harmonic oscillator potentials considered
in [33, 6, 7], where the Hamiltonian commutes with $U(N)$, but the state space
includes $S_{N}$ invariant polynomials in the oscillators.

In section 3 we discuss the thermodynamics of the simplest harmonic oscillator
GPIMQM as a function of the temperature, conveniently parameterised by
$x=e^{-\beta}=e^{-1/T}$, and matrix size $N$. Using the formula (2.17) we are
able to compute the canonical partition functions for values of $N$ up to
around $N=40$, with the help of Mathematica with computation times less than a
minute to two hours. Some of our calculations take longer and extend to
$N=70$. We examine the expectation values of the energy, heat capacity and
entropy. We establish that there is a Hagedorn crossover transition at finite
$N$. This occurs at decreasing temperatures as $N$ is increased and goes to
zero approximately as $x\sim x_{c}=\frac{\log N}{N}$. The specific heat
capacity has a sharp peak in the transition region and remains positive for
all $x$, as is required in the thermodynamics of a Hamiltonian system where
the partition function ${\rm{tr}}(e^{-\beta H})$ is well-defined (see (3.17)).
The specific heat capacity has a maximum at $x=x_{{\rm{max}}}\sim x_{c}$ and
there is a narrow critical region around it. It is easily seen that
$x_{{\rm{max}}}$ decreases as $N$ increases, see Figure 2. A good fit to the
data for large $N$ is given by $x_{{\rm{max}}}=a{\log N\over
N}+\frac{b}{N}+c\frac{\log N}{N^{2}}$, with $N$-independent constants which we
determine numerically with the help of Mathematica. The form of the ansatz
follows the derivation of a formula for the scale of breakdown of the high
temperature expansion in section 5. The fitted result is presented in Figure
17.

In section 4 we switch to a discussion of the GPIMQM system in the
microcanonical ensemble. We find that the specific heat capacity in the low
temperature region is negative. The negative sign is accompanied by an
inequivalence between the canonical and micro-canonical ensembles in this low
temperature region, as illustrated by Figure 15. The possibility of this type
of inequivalence associated with a non-concave entropy function $S(U)$ in the
micro-canonical ensemble has been discussed in the statistical physics
literature e.g. [44] and the example at hand is aligned with these
discussions. The region of inequivalence between ensembles, $x\lesssim
x_{c}={\log N\over N}$, goes to zero in the large $N$ limit. The two ensembles
agree in the high temperature region above the cross-over transition.

In section 5 we derive the high temperature expansion and obtain a formula for
the breakdown scale of the high temperature expansion, denoted $x_{\rm{bkdn}}$
by comparing the leading two terms. The leading term is associated with the
partition $p=[1^{N}]$ ( corresponding to the identity permutation) in the
expansion for ${\cal Z}(N,x)$ in terms of partitions $p$ (2.17). The next-to-
leading term comes from $p=[2,1^{N-2}]$. We refer to the preponderance of
small parts, corresponding to small cycles in the cycle decomposition of these
permutations, as small cycle dominance. The formula for the $x_{\rm{bkdn}}$ is
a series involving powers of ${\log N\over N}$ and ${1\over N}$. The leading
term ${1\over 2}{\log N\over N}$ leads to the definition of $x_{c}={\log
N\over N}$ which is found to also play an important role as a characteristic
scale of the cross-over transition in the numerical study of the partition
functions in sections 3 and 4.

Section 6 discusses the path integral formulation of vector and tensor models
and we exhibit sample small $N$ partition functions. Subsection 6.5 deals with
the charge-zero sector of two matrix models. We derive a Molien-Weyl formula
for the partition function and evaluate explicit expressions for partition
functions for $N=2,3$ and $4$ and the large $N$ low temperature limit
observing that in the micro-canonical ensemble this charge-zero system also
exhibits negative specific heat (see section 9.2). Section 7 deals with the
high temperature scaling in the path integral formulation.

In section 8 we combine information from the high temperature limits with
representation theoretic formulae in two cases of interest, to deduce that
there is a negative specific heat capacity at low temperature in the micro-
canonical ensemble followed by a transition and a high temperature phase of
free oscillators. The two cases of primary interest involve a 3-index tensor
system and a complex-matrix system with zero charge.

Section 9 presents a discussion of general forms of degeneracies leading to
negative specific heat capacities along with connections with the physics of
small black holes via the AdS/CFT correspondence. This discussion includes
instances of multi-matrix thermodynamics which show negative specific heats
when the number of matrices scales with the energy. We conclude with a summary
of our results and a brief discussion of future research directions in section
10. The Appendices develop technical points arising in the main discussion.

##  2  Bosonic GPIMQM Partition functions : Review

In [7] we derived formulae for the partition functions of matrix harmonic
oscillators with a general $11$-parameter family of potentials invariant under
the symmetric group $S_{N}$ and with gauged $S_{N}$ symmetry. The partition
functions were expressed as sums over partitions of $N$. Each partition
corresponds to a cycle structure of permutations in $S_{N}$. The summand
associated to a given partition was given as a product involving the least
common multiples (LCM) and greatest common divisors (GCD) of pairs of cycle
lengths. In this paper, we focus on the case where the quadratic potential for
the matrix variables $M_{ij}$ with $i,j\in\\{1,\cdots,N\\}$ is simply
$tr(MM^{T})$. We review the formula for the partition function here.

For permutations $\sigma\in S_{N}$, let $U_{\sigma}$ be the linear operator
acting in the natural representation $V_{N}$ of $S_{N}$. Matrix bosonic
oscillators $A^{\dagger}_{ij}$ with $i,j\in\\{1,2,\cdots,N\\}$ admit an action
of $U_{\sigma}$ :

$\displaystyle\displaystyle
A^{\dagger}_{ij}\rightarrow(U_{\sigma})_{ik}A^{\dagger}_{kl}(U_{\sigma}^{T})_{lj}$
(2.1)

or in matrix notation

$\displaystyle\displaystyle A^{\dagger}\rightarrow UA^{\dagger}U^{T}\,.$ (2.2)

The action can also be written as :

$\displaystyle\displaystyle A^{\dagger}_{ij}\rightarrow
A^{\dagger}_{\sigma(i)\sigma(j)}$ (2.3)

The dimension of the subspace of the Fock space of these oscillators, at
degree $k$, which is invariant under the $S_{N}$ action has been computed in
eqn (B.9) of [27]. The discussion in the paper [27] is in the context of
polynomial functions of a classical matrix variable $M_{ij}$ invariant under
the action

$\displaystyle\displaystyle M_{ij}\rightarrow M_{\sigma(i)\sigma(j)}$ (2.4)

and the mathematics, of this invariant theory question, evidently applies
equally well to the same action on bosonic oscillators.

The dimension of the space of $S_{N}$ invariants at degree $k$ is given as a
sum of partitions of $N$ and $k$

$\displaystyle\displaystyle{\cal Z}(N,k)$ $\displaystyle=$
$\displaystyle\sum_{p\vdash N}{1\over{\rm{Sym}}~{}p}{\cal Z}(N,p,k)$ (2.5)

where

$\displaystyle\displaystyle{\cal Z}(N,p,k)=\sum_{q\vdash
k}{1\over{\rm{Sym}}~{}q}\prod_{i}\left(\sum_{l|i}lp_{l}\right)^{2q_{i}}\,.$
(2.6)

Define the generating function

$\displaystyle\displaystyle{\cal Z}(N,x)$
$\displaystyle=\sum_{k=0}^{\infty}x^{k}{\cal Z}(N,k)$ (2.9)
$\displaystyle=\sum_{k=0}^{\infty}\sum_{p\vdash
N}{x^{k}\over{\rm{Sym}}~{}p}{\cal Z}(N,p,k)$ $\displaystyle=\sum_{p\vdash
N}{1\over{\rm{Sym}}~{}p}\sum_{k}x^{k}{\cal Z}(N,p,k)$

It is also useful to define a generating function for fixed $N$ and fixed
partition $p$ of $N$ by summing over $k$

$\displaystyle\displaystyle{\cal Z}(N,p,x)=\sum_{k}x^{k}{\cal Z}(N,p,k)$
(2.10)

${\cal Z}(N,x)$ can therefore be written as a sum

$\displaystyle\displaystyle{\cal Z}(N,x)=\sum_{p\vdash
N}{1\over{\rm{Sym}}~{}p}{\cal Z}(N,p,x)\ $ (2.11)

For partitions of $N$ the form
$p=[a_{1}^{p_{1}},a_{2}^{p_{2}},\cdots,a_{s}^{p_{s}}]$, where $a_{i}$ are
distinct non-zero parts with $1\leq a_{i}\leq N$ and $p_{i}$ are positive
integers. It is often useful to think of the numbers to be ordered as
$a_{1}<a_{2}<\cdots<a_{s}$.

$\displaystyle\displaystyle N=\sum_{i=1}^{s}a_{i}p_{i}$ (2.12)

In [7] we derived the following formula for ${\cal Z}(N,p,x)$ which we refer
to as the LCM formula:

$\displaystyle\displaystyle{\cal
Z}(N,p,x)=\prod_{i}{1\over(1-x^{a_{i}})^{a_{i}p_{i}^{2}}}\prod_{i<j}{1\over(1-x^{L(a_{i},a_{j})})^{2G(a_{i},a_{j})p_{i}p_{j}}}$
(2.13)

$L(a_{i},a_{j})$ is the LCM of $a_{i}$ and $a_{j}$ and $G(a_{i},a_{j})$ is the
GCD of $a_{i},a_{j}$. A useful fact is that

$\displaystyle\displaystyle a_{i}a_{j}=L(a_{i},a_{j})G(a_{i},a_{j})$ (2.14)

The expression $2G(a_{i},a_{j})p_{i}p_{j}$ in (2.13) can also be written as :

$\displaystyle\displaystyle
2G(a_{i},a_{j})p_{i}p_{j}={2a_{i}a_{j}p_{i}p_{j}\over L(a_{i},a_{j})}$ (2.15)

We can also present $p$ as $[i^{p_{i}}]$ where $i$ are all integers in the set
$\\{1,\cdots,N\\}$ and $p_{i}$ are non-negative integers (possibly zero). In
this case we can write

$\displaystyle\displaystyle{\cal
Z}(N,p,x)=\prod_{i}{1\over(1-x^{i})^{ip_{i}^{2}}}\prod_{i<j}{1\over(1-x^{L(i,j)})^{2G(i,j)p_{i}p_{j}}}$
(2.16)

The terms with $p_{i}=0$ all give factors of $1$ so this immediately reduces
to the previous formula. In summary, a very useful formula for the canonical
partition function is

$\displaystyle\displaystyle{\cal Z}(N,x)=\sum_{p\vdash
N}{1\over{\rm{Sym}}~{}p}\prod_{i}{1\over(1-x^{i})^{ip_{i}^{2}}}\prod_{i<j}{1\over(1-x^{L(i,j)})^{2G(i,j)p_{i}p_{j}}}$
(2.17)

##  3  GPIMQ-thermodynamics in the canonical ensemble : finite $N$ cross-over
transition and vanishing Hagedorn temperature

The formula (2.17) for the generating function of permutation invariants of
matrices reviewed above, allows a detailed study of the thermodynamics of a
gauged permutation invariant quantum mechanical $N\times N$ matrix system of
oscillators. In this section, we will focus on thermodynamics in the canonical
ensemble. The variable $x$ in the generating function is interpreted
physically as $e^{-\beta}$ where $\beta={1\over T}$ is the inverse
temperature. In the path integral formulation of the gauged permutation
invariant matrix oscillator [6], $\beta$ is the periodicity in the Euclidean
time direction. The fixed energy degeneracies ${\cal Z}(N,k)$ have a very
rapid super-exponential growth as a function of the energy $k$, when $N\geq
2k$. This is explained in section 3.1 below. Whereas the exponential
degeneracies arising from multi-matrix models lead to a finite Hagedorn
temperature (see e.g. [8][10]), super-exponential degeneracies lead to a
vanishing Hagedorn temperature, as has been discussed recently in the context
of tensor models [41][42].

In section 3.2 we give a brief review of the key formulae for the thermodyamic
quantities in the canonical ensemble - energy, entropy and heat capacity - and
describe the behaviour of these quantities as a function of temperature for a
range of fixed values of $N$ up to $N=40$. Since the system under
consideration is one of $N^{2}$ particles, subject to a gauge symmetry
constraint, it is natural to consider the energy, specific heat capacity and
entropy divided by $N^{2}$, which are obtained from ${\log{\cal Z}(N,x)\over
N^{2}}$ and its derivatives. We will refer to the heat capacity divided by
$N^{2}$ as the specific heat capacity. These quantities are finite as
$N\rightarrow\infty$ for generic $x$. We present numerical evidence that at
$x\sim x_{c}={\log N\over N}$ there is a rapid cross-over in the energy and
entropy curves, signalled by a specific heat capacity which scales like $N\log
N$ (or heat capacity of $N^{3}\log N$). An analytic derivation of the ${\log
N\over N}$ scale as the scale of breakdown of a high temperature expansion
developed in 5, is given in section 5.2.

It should be noted that obtaining analogous closed form generating functions
for continuous gauge symmetries such as $U(N)$ in the case of multi-matrix and
tensor degrees of freedom is a challenging problem. For a nice review of
matrix invariant theory see [43]. For example the two variable generating
functions, for $U(N)$ invariants of 2-matrix systems, are given in [46] up to
$N=6$ and the single variable $N=7$ case was evaluated in [10]. No larger $N$
examples are available in print.

The high temperature limit of the permutation invariant thermodynamics is
simple, since it is essentially that of $N^{2}$ free harmonic oscillators -
the analytic formulae for this high temperature limit is reviewed in Appendix
A. We conclude this section with a description of the zeroes of the partition
function ${\cal Z}(N,x)$ in the complex plane : a set of closely spaced zeroes
approaching the origin in the large $N$ limit gives an additional perspective
on the vanishing Hagedorn temperature. The role of such zeroes in connection
with the finite temperature Hagedorn transition in 2-matrix systems has been
given in [10].

###  3.1  Super-exponential degeneracies and vanishing Hagedorn temperature

The sequence of numbers ${\cal Z}(N,k)$, the dimensions of the vector space of
$S_{N}$ invariant polynomials in a matrix $X$ of degree $k$, for fixed $N$ and
varying $k$ has a universal behaviour for $k$ up to ${N\over 2}$. This means
that

$\displaystyle\displaystyle\hbox{ If }~{}~{}~{}~{}~{}k\leq{N\over
2}~{}~{},~{}~{}{\cal Z}(N,k)={\cal Z}(M,k)~{}~{}\hbox{ for all }~{}~{}M\geq
N\,.$ (3.1)

We may also write

$\displaystyle\displaystyle\hbox{ If }~{}~{}~{}~{}~{}k\leq{N\over
2}~{}~{},~{}~{}{\cal Z}(N,k)={\cal
Z}(\infty,k)\equiv\lim_{M\rightarrow\infty}Z(M,k)\,.$ (3.2)

We refer to $k\leq{N\over 2}$ as the stable region in the counting of $S_{N}$
matrix invariants. This is analogous to the behaviour of the counting of
$U(N)$ invariant polynomials in a matrix $X$ of degree $k$ which has a stable
form for $k\leq N$, but for $k>N$ has corrections due to finite $N$ relations
(Caley-Hamilton relations). The existence of a stable region is a typical
property of the counting of multi-matrix and tensor invariants, of interest in
multi-matrix models and tensor models.

The large $N$ sequence for $S_{N}$ invariants is given by directed multi-
graphs with $k$ edges (any number of nodes) [35]. The stable sequence ${\cal
Z}(k,\infty)$ grows approximately as $k!$ at large $k$, by using the graph
theoretic interpretation. The counting of directed multi-graphs is certainly
as large as the counting of undirected graphs. The set of all undirected
graphs with fixed number of edges includes the counting with fixed valency
types. It is known [45] that the leading large $k$ counting of graphs with $k$
quartic vertices grows as $k\log k$, a fact familiar to quantum field
theorists from the factorial growth of Feynman graph counting. Given such a
growth of ${\cal Z}(k,\infty)$ as a function of $k$ it is evident that

$\displaystyle\displaystyle\sum_{k=0}^{\infty}{\cal Z}(k,\infty)e^{-\beta k}$
(3.3)

diverges for any finite $x$ or $\beta$, or equivalently any finite
temperature. There is a similar growth of degeneracies in the stable region in
the context of tensor models, and this has been interpreted as a zero-
temperature Hagedorn transition [41] [42]. We will return to the tensor model
case in more detail in sections 6 and 7.

The numerical studies in this section show that if we keep $N$ finite, the
partition function converges for finite temperatures, and has a sharp
transition localised within a region of size $x\lesssim x_{c}={\log N\over
N}$, which goes to zero as $N\rightarrow\infty$. We will explain this scaling
of the transition region from the point of view of the high temperature
expansion in section 3.3. In the next section 4, we will find that the region
$x\lesssim{\log N\over N}$ also displays the phenomenon of negative specific
heat capacity and ensemble in-equivalence.

###  3.2  Thermodynamic quantities in the canonical ensemble

The canonical partition function is defined as a trace of the Hilbert space
${\cal H}$

$\displaystyle\displaystyle{\rm Tr}_{{\cal H}}e^{-\beta
H}=\sum_{k=0}^{\infty}{\cal Z}(N,k)e^{-\beta k}={\cal Z}(N,x)$ (3.4)

where $x=e^{-\beta}$ and $\beta={1\over T}$ is the inverse temperature. The
energy levels in this matrix harmonic oscillator system are non-negative
integers and the degeneracies of the energy levels are dimensions of spaces of
permutation invariant polynomials of degree $k$ in matrix variables of size
$N$.

We will define the dimensionless version ${\cal W}$ of the Helmholtz free
energy ${\cal F}$ as

$\displaystyle\displaystyle{\cal W}=-{\beta{\cal F}}=\log{\cal Z}$ (3.5)

Dividing by $N^{2}$ we define the dimensionless free energy per particle

$\displaystyle\displaystyle W={{\cal W}\over N^{2}}={1\over N^{2}}\log{\cal
Z}\equiv\log Z$ (3.6)

$W$ is finite in the large $N$ limit for generic $x$, i.e. ${\cal W}$ scales
as $N^{2}$.

The internal energy is the expectation value of the energy

$\displaystyle\displaystyle{\cal U}=-{1\over{\cal Z}}{\partial{\cal
Z}\over\partial\beta}$ (3.7)

while the internal energy per particle is

$\displaystyle\displaystyle U={{\cal U}\over N^{2}}=-{1\over Z}{\partial
Z\over\partial\beta}=-{\partial W\over\partial\beta}={x\partial W\over\partial
x}$ (3.8)

The heat capacity is

$\displaystyle\displaystyle{\cal C}_{{\rm{sh}}}={\partial{\cal U}\over\partial
T}$ (3.9)

while the heat capacity per particle, or specific heat capacity

$\displaystyle\displaystyle C_{{\rm{sh}}}={\partial U\over\partial T}={1\over
N^{2}}{\cal C}_{{\rm{sh}}}$ (3.10)

is finite as $N\rightarrow\infty$ for generic $T$. The specific heat capacity
is

$\displaystyle C_{{\rm{sh}}}={\partial U\over\partial
T}={\partial\beta\over\partial T}{\partial U\over\partial\beta}$ (3.11)
$\displaystyle=(\log x)^{2}\left(x^{2}{\partial^{2}\over\partial
x^{2}}W+x{\partial\over\partial x}W\right)$ (3.12) $\displaystyle=(\log
x)^{2}\left({x^{2}\over Z}{\partial^{2}Z\over\partial x^{2}}-{x^{2}\over
Z^{2}}\left({\partial Z\over\partial x}\right)^{2}+{x\over Z}{\partial
Z\over\partial x}\right)$ (3.13)

It is also useful to write

$\displaystyle C_{{\rm{sh}}}=-\beta{\partial
U\over\partial\beta}=\beta^{2}{\partial^{2}W\over\partial\beta^{2}}$ (3.14)
$\displaystyle=\beta^{2}{\partial\over\partial\beta}\left({1\over Z}{\partial
Z\over\partial\beta}\right)=\beta^{2}\left(-{1\over Z^{2}}{\partial
Z\over\partial\beta}{\partial Z\over\partial\beta}+{1\over
Z}{\partial^{2}Z\over\partial\beta^{2}}\right)$ (3.15)
$\displaystyle=\beta^{2}\left(\langle H^{2}\rangle-\langle
H\rangle^{2}\right)$ (3.16)

The dispersion

$\displaystyle\displaystyle\langle H^{2}\rangle-\langle
H\rangle^{2}=\langle(H-\langle H\rangle)^{2}\rangle$ (3.17)

is positive. Thus we conclude that

$\displaystyle\displaystyle C_{{\rm{sh}}}\geq 0$ (3.18)

Note that this positivity of the specific heat capacity does not hold for the
micro-canonical ensemble, as discussed in the literature on statistical
thermodynamics, see [44] and references therein.

The probability of energy levels $E$ at temperature $T$ in canonical ensemble
is given by

$\displaystyle\displaystyle P(E,T)={1\over{\cal Z}}e^{-E\over T}=e^{(-E+{\cal
F})\over T}$ (3.19)

where the Helmholtz Free energy ${\cal F}$ is defined in terms of the
partition function ${\cal Z}$ by

$\displaystyle\displaystyle{\cal F}=-T\log{\cal Z}$ (3.20)

The entropy in the canonical ensemble is defined as

$\displaystyle\displaystyle{\cal S}(T)$ $\displaystyle=\langle\log
P(E)\rangle={(E-{\cal F})\over T}=\log{\cal Z}+{{\cal U}\over T}$ (3.22)
$\displaystyle=\log{\cal Z}-\beta{\partial\log{\cal
Z}\over\partial\beta}=\log{\cal Z}-\log x~{}\left(x{\partial\over\partial
x}\right)\log{\cal Z}$

In the case at hand we will be interested in

$\displaystyle\displaystyle S={{\cal S}\over N^{2}}$ (3.23)

which will be finite in the large $N$ limit for generic $x$ or generic
$\beta$. In the canonical ensemble, approaching from $x\sim 1$ (corresponding
to $\beta\rightarrow 0,T\rightarrow\infty$) towards $x\sim 0$ (corresponding
to $\beta\rightarrow\infty,T\rightarrow 0$), we find a small region near
$x\sim{\log N\over N}$ where the high temperature expansion breaks down. In
this region, vanishingly small as $N\rightarrow\infty$, the specific heat
capacity rises to a very large value and then drops to zero.

####  3.2.1  Energy versus temperature

Figure 1 is a plot of the energy $U$ as a function of $x=e^{-\beta}$ over the
range $x\in[0,1]$ which corresponds to $T\in[0,\infty]$, for $N=20$. There is
a sharp transition at $x\sim 0.1$ from low $U$ to high $U$. In Figure 2 we
have plotted the same curve for $N=10,15,20$, demonstrating that the
transition moves to lower $x$ as $N$ increases.

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 $N=10,15,20$

####  3.2.2  Specific heat capacity versus temperature

The Figure 3 shows a plot of the specific heat capacity $C_{{\rm{sh}}}$ as a
function of $x$.

Figure 3: Specific heat capacity versus temperature : Sharp peak approaches
zero temperature as $N$ increases. Blue, Green and Red curves are for
$N=10,15,20$

There is a sharp peak at a critical $x\sim x_{{\rm{max}}}$ which approaches
zero temperature, corresponding to $x=0$, as $N$ increases. There is also a
shallow minimum at $x>x_{c}$ followed by a monotonic rise towards
$x\rightarrow 1$.

In section 5 we will derive an estimate of $x\sim x_{c}$ for the breakdown of
the high temperature expansion

$\displaystyle x_{c}={\log(N)\over N}$ (3.24)

Numerical plots show that, as expected, the high temperature expansion starts
to deviate visibly from the exact partition function near the sharp transition
region. This leads us to expect that the minimum and maximum of
$C_{{\rm{sh}}}$ occur at locations $x_{{\rm{max}}},x_{{\rm{min}}}$ which obey

$\displaystyle\lim_{N\rightarrow\infty}{x_{{\rm{max}}}\over
x_{c}}=\hbox{finite }$ (3.25)
$\displaystyle\lim_{N\rightarrow\infty}{x_{{\rm{min}}}\over
x_{c}}=\hbox{finite }$ (3.26)

It is thus useful to define $a_{{\rm{max}}}$ and $a_{{\rm{min}}}$ by

$\displaystyle x_{{\rm{max}}}={x_{c}\over a_{{\rm{max}}}}$ (3.27)
$\displaystyle x_{{\rm{min}}}={x_{c}\over a_{{\rm{min}}}}$ (3.28)

The data for $x_{{\rm{max}}},a_{{\rm{max}}}$ for a number of sample values of
$N$ are as follows :

$N$ | $x_{{\rm{max}}}$ | $a_{{\rm{max}}}$
---|---|---
10 | 0.14855 | 1.55004
15 | 0.118935 | 1.51794
20 | 0.101350 | 1.47791
25 | 0.089096 | 1.44513
30 | 0.079880 | 1.41929
40 | 0.06654 | 1.38596

The data for $x_{{\rm{min}}},a_{{\rm{min}}}$ are as follows :

$N$ | $x_{{\rm{min}}}$ | $a_{{\rm{min}}}$
---|---|---
10 | 0.22875 | 1.00659
15 | 0.159517 | 1.13177
20 | 0.124674 | 1.20143
25 | 0.104143 | 1.23633
30 | 0.0905245 | 1.2524

The value of the specific heat capacity at the maximum, although this is the
heat capacity divided by $N^{2}$ (see equation (3.9)), has a peak which grows
as $N$ tends to infinity. We find that this maximum value which we denote as
$C_{{\rm{sh}};{\rm{max}}}$ is of order $1$ in units of $N\log N$ in the range
of $N$ up to $40$, and conjecture that this will be the case as
$N\rightarrow\infty$.
Conjecture 1:

$\displaystyle\displaystyle\lim_{N\rightarrow\infty}{C_{{\rm{sh}};{\rm{max}}}\over
N\log N}=\hbox{ finite }$ (3.29)

Sample data illustrating the evidence which suggests this conjecture is

N | ${C_{{\rm{sh}};{\rm{max}}}\over N\log N}$
---|---
10 | 0.104449
15 | 0.105513
20 | 0.108528
25 | 0.108454
30 | 0.106822
40 | 0.102552

On the other hand, the value of the specific heat capacity at the minimum,
i.e. at $x=x_{{\rm{min}}}$, denoted $C_{{\rm{sh}};{\rm{min}}}$ scales like
$x_{c}={\log N\over N}$. The suggests the conjecture.
Conjecture 2:

$\displaystyle\displaystyle\lim_{N\rightarrow\infty}{C_{{\rm{sh}};{\rm{min}}}\over
x_{c}}=\lim_{N\rightarrow\infty}{NC_{{\rm{sh}};{\rm{min}}}\over\log N}=\hbox{
finite }$ (3.30)

A sample of the data suggesting this conjecture is

$N$ | ${C_{{\rm{sh}};{\rm{min}}}\over x_{c}}$
---|---
10 | 3.72009
15 | 4.32296
20 | 4.82531
25 | 5.26598
30 | 5.66654
40 | 6.38326

It is also useful to tabulate the energy $U$, evaluated at $x_{min}$ and
$x_{max}$. In contrast to $C_{{\rm{sh}};{\rm{max}}}$, and like
$C_{{\rm{sh}},{\rm{min}}}$, the natural scale for $U$ in the transition region
is, based on evidence from the data, ${\log N\over N}=x_{c}$ In the following
table we give the value $U_{{\rm{min}}}\equiv U(x_{{\rm{min}}})$:

$N$ | ${U_{{\rm{min}}}\over x_{c}}$
---|---
10 | 1.28359
15 | 1.04787
20 | 0.948274
25 | 0.900938
30 | 0.876539

Likewise the energy at the maximum $U_{{\rm{max}}}\equiv U(x_{{\rm{max}}})$ is
given as:

$N$ | ${U_{{\rm{max}}}\over x_{c}}$
---|---
10 | 0.458389
15 | 0.412405
20 | 0.399507
25 | 0.395996
30 | 0.395297

The figure 4 illustrates the location of $x_{{\rm{min}}}$ and $x_{{\rm{max}}}$
relative to the sharp transition in the energy curve.

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 blue) near
the high temperature end of the transition. for $N=20$.

An interesting problem is to develop a model for the these large $N$
characteristics of the specific heat capacity and energy in the transition
region, e.g. along the lines of [47].

Figure 5: Energy per particle and specific heat capacity per particle divided
by $N\log N$, plotted together versus temperature $T$ at $N=20$ : illustrating
macroscopic scale of the specific heat capacity in the transition region

####  3.2.3  Entropy versus temperature and Entropy as a function of Energy

Figure 6: Entropy versus temperature : Sharp peak approaches zero temperature
as $N$ increases. Blue, Green and Red curves are for $N=10,15,20$

As discussed at the start of this section, we calculate the entropy

$\displaystyle\displaystyle S={1\over N^{2}}\left(\log{\cal Z}-\log
x~{}\left(x{\partial\over\partial x}\right)\log{\cal Z}\right)$ (3.31)

See the plot in Figure 6.

It is also natural to consider the entropy as a function of the energy $E$. To
do this, we consider $E(T)$. This is a single-valued function, so we can
invert it to obtain $T(E)$. The entropy as a function of $E$ is then obtained
from $S(T)$ by replacing $T$ with $T(E)$ :

$\displaystyle\displaystyle S(E)=S(T(E))$ (3.32)

It will be instructive, in section 4, to compare this function of energy with
the entropy in the micro-canonical ensemble.

###  3.3  High temperature limit and $N^{2}$ simple harmonic oscillator
thermodynamics

As we show in section 5 , the high temperature limit of the PIMQ
thermodynamics is equivalent to that of $N^{2}$ particles in a harmonic
oscillator. We recall the thermodynamic properties of this system in Appendix
A. The energy $U$ per particle is linear as a function of temperature $T$ in
the high temperature regime. This behaviour of $U$ in the permutation
invariant matrix oscillator is illustrated in Figure 7.

Figure 7: Energy versus temperature $T$ at $N=15$ : Linear high temperature
behaviour sets in near the high temperature end of the transition region

###  3.4  Zeroes in the complex plane

We get an interesting insight into the nature of the transition at $x\sim
x_{c}$ by looking at the behaviour of ${\cal Z}(N,x)$ in the complex
$x$-plane. See Figures 8 and 9. In both figures, there are a number of zeroes
located close to a vertical line near $x=0$. As $N$ increases, the zeroes
become closer to each other and to the origin $x=0$. The table below gives the
values of the zeroes nearest to the origin, for a range of $N$, illustrating
how they approach the origin as $N$ increases. They are tabulated alongside
$x_{{\rm{max}}}$ the location on the real $x$-axis of the maximum of the
specific heat capacity $C_{{\rm{sh}}}$ in order to illustrate the proximity of
these complex zeroes to the real temperature transition point. This proximity
becomes sharper as $N$ increases.

$N$ |  nearest zeroes in complex $x$-plane | $x_{{\rm{max}}}$
---|---|---
5 | 0.20 $\pm$ 0.13 $i$ | 0.23
6 | 0.18 $\pm$ 0.09 $i$ | 0.20
7 | 0.17 $\pm$ 0.07 $i$ | 0.18
8 | 0.16 $\pm$ 0.05 $i$ | 0.17
9 | 0.15 $\pm$ 0.04 $i$ | 0.16

Note these zeroes are approaching the origin as $N\rightarrow\infty$. All the
poles of ${\cal Z}(N,x)$ are on the unit circle. This is evident from plots
and also from the expression 2.13.

Figure 8:  Zeroes of the canonical partition function in the complex $x$-plane
for $N=7$ : A line a zeroes approaches $x=0$ as $N\rightarrow\infty$ Figure 9:
Zeroes of the canonical partition function in the complex $x$-plane for $N=9$
: A line a zeroes approaches $x=0$ as $N\rightarrow\infty$

##  4  PIMQ-Thermodynamics in the micro-canonical ensemble : Negative specific
heat and (in)equivalence of ensembles

The canonical ensemble describes a physical system in equilibrium with thermal
baths of fixed temperatures, as a function of the temperature. As reviewed in
section 3.2 a notable feature of physics in the canonical ensemble is that the
specific heat capacity is necessarily positive. In the micro-canonical
ensemble, a system with discrete energy levels as is the system under study
here, is considered in isolation as a function of the discrete values of the
energy. In section 4.1, we review the key equations of thermodynamics in the
micro-canonical ensemble and some relevant discussions of in-equivalence
between the canonical and micro-canonical ensemble from the statistical
physics literature.

We will use the coefficients in the expansion of ${\cal Z}(N,x)$,
equation(2.17), to perform calculations in the micro-canonical ensemble. This
is equivalent to but more efficient than using the original micro-canonical
formula for ${\cal Z}(N,k)$ (equation (2.5)) in terms of a sum over partitions
of $N$ and $k$.

###  4.1  Micro-canonical ensemble and negative specific heat capacities

We will describe the translation between our generating functions ${\cal
Z}(N,x)$ and ${\cal Z}(N,k)$ into thermodynamic quantities. $k$ is the energy.
${\cal Z}(N,k)$ is the micro-canonical degeneracy. Following standard
treatments of thermodynamics, the entropy as a function of energy is the
logarithm of the degeneracy which is conventionally denoted $\Omega(N,k)$.
Here these degeneracies are obtained as the coefficients ${\cal Z}(N,k)$ in
the expansion of ${\cal Z}(N,x)$. Thus the micro-canonical entropy is

$\displaystyle\displaystyle{\cal S}(N,k)=\log{\cal Z}(N,k)=\log\Omega(N,k)$
(4.1)

We will occasionally emphasize that we are working with the micro-canonical
ensemble rather than the canonical and will use $\Omega(N,k)$ for the
degeneracies rather than ${\cal Z}(N,k)$. As in section 3, we define
thermodynamic quantities per particle to obtain quantities with finite limits
at generic $x$ in the large $N$ limit. The energy per particle is defined as
$E={k\over N^{2}}$. Thus the micro-canonical entropy per particle is

$\displaystyle\displaystyle S(N,E)={1\over N^{2}}\log{\cal Z}(N,k)={{\cal
S}(N,N^{2}E)\over N^{2}}$ (4.2)

When we discuss equivalence and in-equivalence of micro-canonical and
canonical ensemble, we we will compare $k$ in the micro-canonical ensemble to
the expectation value ${\cal U}$ (equation (3.7) ) in the canonical ensemble,
and equivalently in a per-particle comparison we compare $E={k\over N^{2}}$ in
the micro-canonical ensemble with $U={{\cal U}\over N^{2}}$ (equation (3.8))
in the canonical ensemble.

The temperature in the micro-canonical ensemble is defined as

$\displaystyle\displaystyle T_{\rm{micro}}=\left({\partial S\over\partial
U}\right)^{-1}=\left({\partial{\cal S}\over\partial{\cal U}}\right)^{-1}$
(4.3)

We use $\beta_{\rm{micro}}=T_{\rm{micro}}^{-1}$. Since the energy levels $k$
are discrete with unit spacing, the micro-canonical temperature is defined
using a discrete derivative $\Delta$

$\displaystyle\displaystyle T_{\rm{micro}}^{-1}(k)={\Delta(\log({\cal
Z}(N,k))\over\Delta k}$ (4.4)

For $\Delta$ we will use $D$ or $D_{\rm{sym}}$ where we define

$\displaystyle\displaystyle DF(k)=F(k)-F(k-1)$ (4.5)

or

$\displaystyle\displaystyle D_{\rm{sym}}F(k)={1\over 2}(F(k+1)-F(k-1))$ (4.6)

which give

$\displaystyle\displaystyle T_{\rm{micro}}^{-1}(k)=(\log{\cal
Z}(N,k)-\log{\cal Z}(N,k-1))$ (4.7)

or

$\displaystyle\displaystyle T_{\rm{micro};\rm{sym}}^{-1}(k)={1\over
2}(\log{\cal Z}(N,k+1)-\log{\cal Z}(N,k-1))$ (4.8)

Defining the temperature using the entropy and energy per particle

$\displaystyle\displaystyle T_{\rm{micro}}^{-1}(k)={\Delta S\over\Delta
E}={\Delta({\cal S}/N^{2})\over\Delta(k/N^{2})}={\Delta(\log({\cal
Z}(N,k)/N^{2})\over\Delta k/N^{2}}={\Delta(\log({\cal Z}(N,k))\over\Delta k}$
(4.9)

which is the same as (4.4). In this form, it is clear that at large $N$, we
are taking differences at points separated by vanishingly small separations,
thus reaching a continuum limit.

###  4.2 Thermodynamic quantities in the micro-canonical ensemble

It is instructive to display the dependence of the thermodynamic quantities,
notably the energy $k$ and the specific heat capacity
$C_{{\rm{sh}};\rm{micro}}(T_{\rm{micro}})$ in the micro-canonical ensemble.

####  4.2.1 Energy versus micro-canonical temperature

Figure 11 shows a plot of $T_{\rm{micro};\rm{sym}}(k)$ versus the energy $k$.
The point $(k=12,T=0.491)$ is at the minimum temperature. For lower
temperatures the energy decreases as the energy increaes. For $k\geq 12$, the
energy increases with temperature. We will refer to this value of energy where
the $T_{\rm{micro}}$ takes a minimum value as $k_{\rm{crit}}$. Using the
asymmetric discretisation of the derivative gives the same shape of curve.
There are a couple of points at very small $k$ where the variation of
$T_{\rm{micro}}$ with $k$ is different from that of $T_{\rm{micro};sym}$ with
$k$.

Figure 10:  Plot of micro-canonical temperature $k$ at $N=10$ for $k_{max}=40$
\- using symmetrised derivative – produces consistent negative SHC trend below
critical E

It is also useful to display the energy as a function of temperature, which
will be useful later for comparison of the canonical and micro-canonical
ensemble: see Figure 11. The micro-canonical specific heat capacity ${\partial
E\over\partial T_{\rm{micro}}}$ tends towards negative infinity as $k$
approaches $k_{\rm{crit}}$ from below and towards positive infinity as $k$
approaches $k_{\rm{crit}}$ from above.

Figure 11:  Plot of micro-canonical energy $E=k\over N^{2}$ versus micro-
canonical temperature at $N=15$ for $k_{min}=4,k_{max}=100$ \- using
descending derivative – produces consistent negative SHC trend below critical
E

Table 1 gives the values of $k_{\rm{crit}}$ for a range of values of $N$.

Table 1:  Table of values of $k_{\rm{crit}}$ as a function of $N$ $N$ | $k_{\rm{crit}}$ | ${k_{\rm{crit}}\over N\log N}$
---|---|---
10 | 12 | 0.5212
15 | 21 | 0.5170
20 | 30 | 0.5007
25 | 40 | 0.4971
30 | 51 | 0.4998
35 | 62 | 0.4982
40 | 73 | 0.4947
45 | 84 | 0.4904
50 | 97 | 0.4959
55 | 109 | 0.4945
50 | 122 | 0.4966
65 | 134 | 0.4939
70 | 147 | 0.4943
Figure 12: $k_{crit}$ versus $\frac{1}{N}$ for $N=10$ to $N=70$. The curve is
$\frac{1}{2}\frac{\log N}{N}$

All the integers in the second column are within $\pm 1$ of $0.5N\log N$.
Conjecture 3:  It is reasonable to conjecture that in the large $N$ limit, the
value of the un-normalised micro-canonical energy at the critical temperature
where the specific heat capacity diverges, i,e $k_{\rm{crit}}$, obeys

$\displaystyle\displaystyle\hbox{Limit}_{N\rightarrow\infty}k_{\rm{crit}}=0.5~{}~{}(N\log
N)$ (4.10)

The un-normalised energy is related to the number of edges in the graph in the
graph theory interpretation reviewed briefly in section 3.1. With this
conjecture, the critical energy per particle
$E_{\rm{crit}}={k_{\rm{crit}}\over N^{2}}$ obeys

$\displaystyle\displaystyle\hbox{Limit}_{N\rightarrow\infty}E_{\rm{crit}}=0.5~{}~{}{\log
N\over N}$ (4.11)

####  4.2.2  Specific heat capacity versus micro-canonical temperature

Figure 13:  Plot of specific heat capacity versus $E$ at $N=15$

From (4.3) we can calculate the derivative

$\displaystyle{\partial T_{micro}\over\partial E}=-{1\over({\partial
S\over\partial E})^{2}}{\partial^{2}S\over\partial E^{2}}$ (4.12)
$\displaystyle=-{1\over({1\over\Omega}{\partial\Omega\over\partial
E})^{2}}{\partial\over\partial E}({1\over\Omega}{\partial\Omega\over\partial
E})$ (4.13)

The specific heat capacity is then

$\displaystyle\displaystyle C_{{\rm{sh}};\rm{micro}}={\partial E\over\partial
T_{\rm{micro}}}=-({1\over\Omega}{\partial\Omega\over\partial
E})^{2}{1\over{\partial\over\partial
E}({1\over\Omega}{\partial\Omega\over\partial E})}$ (4.14)

This can be expressed in terms of finite differences and plotted. The finite
difference formula is

$\displaystyle\displaystyle C=-{(\Delta\log Z(k))^{2}\over\Delta(\Delta\log
Z(k))}$ (4.15)

The significant features are independent of the choice of discrete derivative.
Calculations done with the two definitions are very close to each other except
for very small $k$.

Figure 13 shows the negative specific heat separated from the region of
positive specific heat, with the transition occurring at $E\sim{\log N\over
N}$. From (4.15), in negative SHC branch, the second derivative of the micro-
canonical entropy as a function of energy is positive, in other words, the
entropy is convex. In the positive SHC branch the second derivative is
negative, in other words the entropy is a concave function of the energy.

It is worth noting that when $k$ exceeds $N/2$, we exit the stable regime of
degeneracies described in (3.1) (3.2). However, at these low energies these
departures from the stable regime do not cause a significant modification of
the thermodynamic behaviour compared to the stable regime. A significant
modification occurs at $k\sim N\log N$. This is analogous to a similar
phenomenon discussed in the context of the thermodynamics of two matrix
harmonic oscillators with $U(N)$ invariance. Here finite $N$ trace relations
lead to departures from stable behaviour at energies comparable to $N$,
however thermodynamically significant departures occur for energies comparable
to $N^{2}$ [48, 49].

###  4.3  Equivalence and in-equivalence of ensembles

The permutation invariant matrix quantum thermodynamics is well approximated
by $N^{2}$ decoupled harmonic oscillators at high temperature. In this regime,
we expect the micro-canonical ensemble and canonical ensemble to be
equivalent. The thermodynamics of the multi-harmonic oscillator system along
with an explicit account of ensemble equivalence in this context is discussed
in Appendix A. Based on the general discussion of specific heat capacities in
the canonical ensemble of statistical thermodynamics we have seen that these
are related to a dispersion of the energy distribution and therefore positive
(see (LABEL:SHCpos1)(3.17) (3.18)). As we observed in section 4.2.1 above, the
micro-canonical entropy fails to be concave in the negative SHC region. This
concavity plays a central role in the broad discussion of equivalence of
ensembles in [44]. We therefore expect that in the region of $T_{\rm{micro}}$
where the SHC is negative, we have a failure of the equivalence between the
canonical and micro-canonical ensemble.

In Figure 14 we have superposed for $N=15$, a plot of the expectation value of
the energy per particle $U$ in the canonical ensemble versus canonical
temperature $T$ with a plot of the rescaled per-particle energy level
$E={k\over N^{2}}$ versus the micro-canonical temperature. This shows that, as
expected, the two plots agree in the high temperature limit but disagree in
the low temperature limit. Figure 15 gives the same superposition of plots at
$N=20$.

Figure 14:  Plot of the expectation value of the energy $U={\cal U}/N^{2}$ in
the canonical ensemble versus canonical temperature $T$, superposed upon
$E={k\over N^{2}}$ versus identification of $T_{\rm{micro}}$ in micro-
canonical ensembles : Equivalence of ensemble above the transition region.
This plot is for $N=15$. The micro-canonical data starts at $k=4$ and ends at
$k=100$.  Figure 15:  Plot of the expectation value of the energy $U={\cal
U}/N^{2}$ in the canonical ensemble versus canonical temperature $T$,
superposed upon $E={k\over N^{2}}$ versus identification of $T_{\rm{micro}}$
in micro-canonical ensembles : Equivalence of ensemble above the transition
region. This plot is for $N=20$. The micro-canonical data starts at $k=4$ and
ends at $k=100$.  Figure 16:  Plot of canonical ensemble entropy versus
canonical temperature $T$ superposed upon plot of micro-canonical entropy
versus $T_{\rm{micro}}$ in the micro-canonical : Equivalence of ensemble above
the transition region. This plot is for $N=20$. The micro-canonical data
starts at $k=4$ and ends at $k=100$.

It is also instructive to perform a similar comparison for entropies. In the
micro-canonical ensemble

$\displaystyle\displaystyle S_{\rm{micro}}(E)={1\over
N^{2}}\log\Omega(E)={1\over N^{2}}\log{\cal Z}(N,k=N^{2}E)$ (4.16)

Expressing $E$ as a function of $T_{\rm{micro}}$, we can obtain a plot of
$S_{\rm{micro}}(T_{\rm{micro}})$. As discussed around equations (3.31), we can
also obtain an entropy function $S(T)$ for the canonical ensemble. Superposing
$S_{\rm{micro}}(T)$. Superposing these plots in Figure 16 we obtain the
expected feature discussed above of ensemble in-equivalence at low temperature
followed by equivalence at high temperature.

##  5  High temperature expansion of PIMQ-Thermo and small cycle dominance

In this section we will analyse the behaviour of the canonical partition
function ${\cal Z}(N,x=e^{-\beta}=e^{-1/T})$ in the infinite temperature limit
where $x\rightarrow 1$. The behaviour in this limit is controlled by the
singularity as $x\rightarrow 1$. Given that ${\cal Z}(N,x)$ is a sum over
partitions $p$ of ${\cal Z}(N,p,x)$ which, as we show, takes the form

$\displaystyle\displaystyle{\cal
Z}(N,p,x)={1\over(1-x)^{{\rm{Deg}}(N,p)}}{\cal R}(x)$ (5.1)

where ${\rm{Deg}}(N,p)$ is an integer we will define and ${\cal R}(x)$ is a
regular as $x\rightarrow 1$. We will show that the leading singularity of
${\cal Z}(N,x)$ as $x\rightarrow 1$ comes from $p=[1^{N}]$. The next-to-
leading singularity come from $[1^{N-2},2]$, followed by other cycle
structures involving a large number of $1$-cycles and a few small cycles. The
proof that $p=[1^{N}]$ gives the leading singularity follows, while the proof
that the next-to-leading singularity comes from $p=[2,1^{N-2}]$ is given in
Appendix B.

We will start from (2.13) rewritten here for convenience

$\displaystyle\displaystyle{\cal
Z}(N,p;x)=\prod_{i}{1\over(1-x^{a_{i}})^{a_{i}p_{i}^{2}}}\prod_{i<j}{1\over(1-x^{L(a_{i},a_{j})})^{2G(a_{i},a_{j})p_{i}p_{j}}}$

Note that

$\displaystyle\displaystyle{1\over(1-x^{a})}={1\over(1-x)}{1\over(1+x+x^{2}+\cdots+x^{a-1})}\equiv{1\over(1-x)}{\cal
R}_{a}(x)$ (5.2)

The residue

$\displaystyle\displaystyle Res({1\over(1-x^{a})},x=1)={\cal
R}_{a}(x=1)=a^{-1}$ (5.3)

This means that where we have separated the terms singular at $x=1$

$\displaystyle\displaystyle{\cal Z}(N,p;x)$
$\displaystyle=\prod_{i}{1\over(1-x)^{a_{i}p_{i}^{2}}}({\cal
R}_{a_{i}}(x))^{a_{i}p_{i}^{2}}\prod_{i<j}{1\over(1-x)^{2G(a_{i},a_{j})p_{i}p_{j}}}({\cal
R}_{L(a_{i},a_{j})}(x))^{2G(a_{i},a_{j})p_{i}p_{j}}$ (5.5)
$\displaystyle={1\over(1-x)^{\sum_{i}a_{i}p_{i}^{2}+\sum_{i<j}2G(a_{i},a_{j})p_{i}p_{j}}}\prod_{i}({\cal
R}_{a_{i}}(x))^{a_{i}p_{i}^{2}}\prod_{i<j}({\cal
R}_{L(a_{i},a_{j})}(x))^{2G(a_{i},a_{j})p_{i}p_{j}}$

###  5.1  Cycle structures and the high temperature expansion

We will define the degree function

$\boxed{\hbox{ Deg
}(N,p)=\sum_{i}a_{i}p_{i}^{2}+\sum_{i<j}2G(a_{i},a_{j})p_{i}p_{j}}$ (5.7)

which is the power of $(1-x)^{-1}$ in the formula (5.5). Note that

$\displaystyle\displaystyle{\rm{Deg}}(N,[1^{N}])=N^{2}$ (5.8)

and for $p=[1^{N}]$

$\displaystyle\displaystyle{1\over{\rm{Sym}}~{}p}Z(N,p;x)={1\over
N!}{1\over(1-x)^{N^{2}}}$ (5.9)

We will prove the following proposition
Proposition 1:  The function

$\displaystyle\displaystyle\hbox{
Deg}(N,p)=\sum_{i}a_{i}p_{i}^{2}+\sum_{i<j}2G(a_{i},a_{j})p_{i}p_{j}$ (5.10)

as $p$ ranges over the set of partitions of $N$ is maximised by $p=[1^{N}]$,
where it takes values $N^{2}$.

Before proving the result in general, we work out, by simple calculations,
that the result holds for special classes of partitions. Consider partitions

$\displaystyle\displaystyle p=[2^{k},1^{N-2k}]$ (5.11)

In the notation of $p=[\\{a_{i}^{p_{i}}\\}]$

$\displaystyle a_{1}=1,p_{1}=N-2k,a_{2}=2,p_{2}=k$ (5.12) $\displaystyle
G(a_{1},a_{2})=1$ (5.13)

The degree is calculated as

$\displaystyle{\rm{Deg}}(N,[2^{k},1^{N-2k}])=(N-2k)^{2}+2k^{2}+2.1.k(N-2k)=N^{2}-4Nk+4k^{2}+2k^{2}+2kN-4k^{2}$
(5.14) $\displaystyle=N^{2}-2kN+2k^{2}$ (5.15)

For $k=1$,we have ${\rm{Deg}}(N,[2,1^{N-2}])=N^{2}-2N+2$. Note that the
coefficient of $N$ is $-2B([2^{k},1^{N-2k}])$ where $B$ is the branching
number of the partition $[[2^{k},1^{N-2k}]$. The branching number of a general
partition $p=[\\{a_{i}^{p_{i}}\\}]$ is $\sum_{i}p_{i}(a_{i}-1)$. It has an
interpretation in terms of branched covers and plays a role in the Riemann-
Hurwitz formula for the genus of covering surfaces in two dimensions. We will
find shortly that this property, that the coefficient of $N$ in the degree is
the branching number of $p$, holds for general $p$.

For $p=[1^{N-2p_{2}-\cdots-
Kp_{K}},a_{2}^{p_{2}},a_{3}^{p_{3}},\cdots,a_{K}^{p_{K}}]$ with
$p_{2},p_{3},\cdots,p_{K}>0$, $K<N$, we calculate the degree (5.7) to find

$\displaystyle{\rm{Deg}}(p,N)=N^{2}-2N(\sum_{i=2}^{K}(a_{i}-1)p_{i})+\sum_{i=2}^{K}p_{i}^{2}a_{i}(a_{i}-1)+\sum_{i<j}2p_{i}p_{j}(G(a_{i},a_{j})-a_{i}-a_{j}+a_{i}a_{j})$
(5.16)
$\displaystyle=N^{2}-2N(\sum_{i=2}^{K}(a_{i}-1)p_{i})+\sum_{i=2}^{K}p_{i}^{2}a_{i}(a_{i}-1)+\sum_{i<j}2p_{i}p_{j}\left((G(a_{i},a_{j})-1)+(a_{i}-1)(a_{j}-1)\right)$
(5.17)

We can, without loss of generality, assume $a_{2}<a_{3}<\cdots<a_{K}$.
Proof  Using the definition of the degree in (5.7) and the form of $p$ above

$\displaystyle{\rm{Deg}}(p,N)=(N-\sum_{i=2}^{K}a_{i}p_{i})^{2}+\sum_{i=2}^{K}a_{i}p_{i}^{2}+\sum_{j=2}^{K}2G(1,a_{j})(N-\sum_{i=2}^{K}a_{i}p_{i})p_{j}+\sum_{i<j}2G(a_{i},a_{j})p_{i}p_{j}$
(5.19)
$\displaystyle=N^{2}-2N\sum_{i=2}^{K}a_{i}p_{i}+(\sum_{i=2}^{K}a_{i}p_{i})^{2}+\sum_{i}a_{i}p_{i}^{2}+\sum_{j=2}^{K}2(N-\sum_{i}a_{i}p_{i})p_{j}+\sum_{i<j}2G(a_{i},a_{j})p_{i}p_{j}$
(5.20)
$\displaystyle=N^{2}-2N\sum_{i=2}^{K}(a_{i}-1)p_{i}+\sum_{i}a_{i}(a_{i}-1)p_{i}^{2}+2\sum_{i<j}a_{i}a_{j}p_{i}p_{j}$
(5.21)
$\displaystyle-2\sum_{j=2}^{K}a_{i}p_{i}^{2}-2\sum_{i<j}p_{i}p_{j}(a_{i}+a_{j})+2\sum_{i<j}G(a_{i},a_{j})p_{i}p_{j}$
(5.22)
$\displaystyle=N^{2}-2N\sum_{i=2}^{K}(a_{i}-1)p_{i}+\sum_{i=2}^{K}a_{i}(a_{i}-1)p_{i}^{2}+\sum_{i<j}2p_{i}p_{j}(G(a_{i},a_{j})+a_{i}a_{j}-a_{i}-a_{j})$
(5.23)
$\displaystyle=N^{2}-2N(\sum_{i=2}^{K}(a_{i}-1)p_{i})+\sum_{i=2}^{K}p_{i}^{2}a_{i}(a_{i}-1)+\sum_{i<j}2p_{i}p_{j}\left((G(a_{i},a_{j})-1)+(a_{i}-1)(a_{j}-1)\right)$
(5.24)

We have used $G(1,a_{j})=1$.

Note that the coefficient of $(-2N)$ is the branching number of $p$, denoted
$B(p)$, which obeys $0\leq B(p)\leq(N-1)$. To see this note that

$\displaystyle\displaystyle
B(p)=\sum_{i=2}^{N}(a_{i}-1)p_{i}=\sum_{i=1}^{N}(a_{i}-1)p_{i}=N-\sum_{i=1}^{N}p_{i}$
(5.26)

which is minimised when all the $a_{i}=1$ ( i.e $p=[1^{N}]$ ) and maximised
when $p_{N}=1$ with all other $p$’s zero, i.e. $p=[N]$. When $K,a_{i},p_{i}\ll
N$, the significant term is $-2NB(p)$, and it is evident that the largest
degree term is the one where the $p_{2}\cdots p_{K}$ are all zero, and $B(p)$
which means that the degree is maximised by $p=[1^{N}]$. We can also relax
this restriction $K,a_{i},p_{i}\ll N$ and prove the desired maximisation
property of $p=[1^{N}]$.

We know that

$\displaystyle\displaystyle N>\sum_{i=2}^{K}(a_{i}-1)p_{i}$ (5.27)

This holds because the number of $1$-cycles in $p$ is greater or equal to $0$
:

$\displaystyle\displaystyle N-\sum_{i=2}^{K}a_{i}p_{i}\geq 0$ (5.28)

which means

$\displaystyle\displaystyle N-\sum_{i=1}^{K}(a_{i}-1)p_{i}>0$ (5.29)

We can extend this by including $a_{1}=1$ and
$p_{1}=N-\sum_{i=2}^{K}a_{i}p_{i}$ so that

$\displaystyle\displaystyle N-\sum_{i=1}^{K}(a_{i}-1)p_{i}>0$ (5.30)

The inequality (5.27) has the geometrical interpretation that the branching
number of a branch point for an $N$-sheeted cover of a two dimensional surface
can be at most $(N-1)$. It is useful to write

$\displaystyle\displaystyle{\rm{Deg}}(p,N)=N^{2}+\tilde{X}(N,\vec{a})$ (5.31)

where $\tilde{X}$ is read off from (LABEL:DegpNrewrite) as

$\displaystyle\tilde{X}(N)=-2N(\sum_{i=2}^{K}(a_{i}-1)p_{i})+\sum_{i=2}^{K}p_{i}^{2}a_{i}(a_{i}-1)+\sum_{i<j}2p_{i}p_{j}\left((G(a_{i},a_{j})-1)+(a_{i}-1)(a_{j}-1)\right)$
(5.32)

Our desired inequality is

$\displaystyle\displaystyle\tilde{X}(N)<0$ (5.34)

Given the inequality (5.27), it follows that if we replace
$N\rightarrow\sum_{i=2}^{K}(a_{j}-1)p_{j}$ in $\tilde{X}(N)$ to give
$\tilde{X}(N,\vec{a})\rightarrow X(\vec{a})$, we have the inequality
$\tilde{X}<X$. Therefore proving $X<0$ will prove (5.34).

The explicit form of $X(\vec{a})$ is

$\displaystyle
X=-2\sum_{j=2}^{K}(a_{j}-1)p_{j}\sum_{i=2}^{K}(a_{i}-1)p_{i}+\sum_{i=2}^{K}p_{i}^{2}a_{i}(a_{i}-1)+$
(5.35) $\displaystyle\sum_{2\leq i<j\leq
K}2p_{i}p_{j}((G(a_{i},a_{j})-1)+(a_{i}-1)(a_{j}-1))$ (5.36)

To prove $X<0$, we simplify

$\displaystyle
X=-2\sum_{i}p_{i}^{2}(a_{i}-1)^{2}-2\sum_{i<j}p_{i}p_{j}\left((a_{i}-1)(a_{j}-1)-(G(a_{i},a_{j})-1)\right)$
(5.38)
$\displaystyle=-2\sum_{i}p_{i}^{2}(a_{i}-1)^{2}-2\sum_{i<j}p_{i}p_{j}(a_{i}-1)(a_{j}-1)+2\sum_{i<j}p_{i}p_{j}(G(a_{i},a_{j})-1)$
(5.39)

The GCD satisfies $G(a_{i},a_{j})\leq a_{i}$, so

$\displaystyle\displaystyle
X<-2\sum_{i}p_{i}^{2}(a_{i}-1)^{2}-2\sum_{i<j}p_{i}p_{j}(a_{i}-1)(a_{j}-1)+2\sum_{i<j}p_{i}p_{j}(a_{i}-1)\equiv
X_{1}$ (5.41)

We simplify

$\displaystyle\displaystyle
X_{1}=-2\sum_{i}p_{i}^{2}(a_{i}-1)^{2}-2\sum_{i<j}p_{i}p_{j}(a_{i}-1)(a_{j}-2)$
(5.42)

We have, by assumption, $i\geq 2$, $a_{i},a_{j}\geq 2$ and with $a_{j}>a_{i}$
we know $a_{j}\geq 3$ in the second sum above. This means

$\displaystyle\displaystyle
X_{1}\leq-2\sum_{i}p_{i}^{2}(a_{i}-1)^{2}-2\sum_{i<j}p_{i}p_{j}(a_{i}-1)\equiv
X_{2}$ (5.43)

Now it is evident that $X_{2}<0$. Combining with $\tilde{X}<X<X_{1}\leq
X_{2}<0$, we conclude $\tilde{X}<0$,

This proves the proposition.

###  5.2  Breakdown of the high $T$ expansion and the characteristic scale
$x_{c}={\log N\over N}$.

The degree function of the partitions can be easily computed in Mathematica
and shows that the second most singular term at $x=1$, after $[1^{N}]$ comes
from $[2,1^{N-2}]$. We have checked this for $N$ up to $25$ and give the proof
for general $N$ in Appendix B. We proceed here to investigate the implications
for the nature of the transition region. The two leading terms in the high
temperature expansion thus define a truncated partition function we denote
$Z_{\rm{trunc}}$.

$\displaystyle{\cal Z}_{{\rm{trunc}}}(N,x)={1\over N!(1-x)^{N^{2}}}+{1\over
2(N-2)!}{1\over(1-x)^{(N-2)^{2}}(1-x^{2})^{2N-2}}$ (5.44)
$\displaystyle={1\over N!(1-x)^{N^{2}}}\left(1+{N(N-1)\over
2}{(1-x)^{2N-2}\over(1+x)^{2N-2}}\right)$ (5.45) $\displaystyle={1\over
N!(1-x)^{N^{2}}}\left(1+{N(N-1)\over 2}(\tanh{\beta\over 2})^{2N-2}\right)$
(5.46)

To get an estimate of the breakdown scale $x_{\rm{bkdn}}$ of the high
temperature expansion, we set

$\displaystyle\displaystyle{N(N-1)\over
2}{(1-x_{\rm{bkdn}})^{2N-2}\over(1+x_{\rm{bkdn}})^{2N-2}}=a$ (5.47)

where $a$ is a small number $a<1$, but $a$ does not depend on $N$. This is
solved by

$\displaystyle\displaystyle x_{\rm{bkdn}}=\tanh({B\over 2})={B\over
2}-{B^{3}\over 24}+\cdots$ (5.48)

where

$\displaystyle\displaystyle B={1\over 2N}(1-{1\over N})\left(2\log
N+\log(1-{1\over N})+\log 2-\log a\right)$ (5.49)

The expansion of $x_{\rm{bkdn}}$ is a series in powers of the variables ${\log
N\over N},{1\over N}$ with constant coefficients. The first few terms are

$\displaystyle x_{\rm{bkdn}}={\log N\over 2N}+{1\over 4N}(\log 2-\log
a)+{1\over 4N^{2}}(\log({a\over 2})-1)-{\log N\over 2N^{2}}+{\cal O}({1\over
N^{3}})+{\cal O}({\log N\over N^{3}})$ (5.50)

Note that the first term ${\log N\over 2N}$ is independent of $a$. We will
define $x_{c}={\log N\over N}$ as the characteristic scale of the transition.
This scale has played an important role in the tables given in sections 3 and
4, where the characteristics of the transition measured in units of $x_{c}$
show good evidence of being numbers of order $1$ as $N\rightarrow\infty$.

Developing further the applications of $x_{c}$ and $x_{\rm{bkdn}}$, we present
Figure 17 where we use the data from Tables 1 and 3.2.2 and fit the
corresponding critical value of $x$ against $1/N$. The data shows that the
critical temperatures are driven to zero as expected and established by the
high temperature analysis. The figure shows that the microcanonical transition
is consistently at a lower temperature than the canonical one as expected from
Figure 15 but both go to zero at infinite $N$ as expected from the known
stable structure of the degeneracies and from our high temperature analysis
which establishes that the transition has to occur for $x\lessapprox
x_{c}=\frac{\log N}{N}$. Our fitting curves are motivated by the functional
form suggested by the first three terms in LABEL:xbkform.

Figure 17: Comparison of microcanonical and canonical values for the tansition
value of $x={\rm e}^{-\beta}$ with $\beta=\beta_{micro}(k_{crit})$ for the
microcanonical ensemble. The best fit curves, plotted against
$\nu=\frac{1}{N}$ are
$x_{crit}(\nu)=-0.508\nu\log(\nu)+0.032\nu-0.522\nu^{2}\log(\nu)$ for the
microcanonical ensemble and
$x_{max}(\nu)=-2.86538\nu-1.37557\nu\log(\nu)-5.1274\nu^{2}\log(\nu)$ for the
canonical ensemble.

##  6 Path Integral for Tensor Models

In this section we give the path integral formula for the thermal partition
function of $d$ copies of $s$-index tensors, as an application of [6]. This is
used to derive a palindromy property for the canonical partition function by
considering a transformation $x\rightarrow x^{-1}$, and to compute a number of
explicit examples of the partition functions. In the next section, we obtain
the high temperature limit from the path integral.

Let $\Phi^{a}_{i_{1}\cdots i_{d}}$, $a=1,\dots,d$ be a set of tensors with
each index transforming in the fundamental of distinct $U(N)$ groups, i.e.
they transform under $U(N)^{s}$. The Euclidean thermal action for this system
is then

$S[\overline{\Phi},\Phi]=\int_{0}^{\beta}d\tau\left(({\cal
D}_{\tau}\overline{\Phi}^{a}_{i_{1}\cdots i_{s}}({\cal
D}_{\tau}{\Phi}^{a})^{i_{1}\cdots
i_{s}}+m_{B}^{2}\overline{\Phi}^{a}_{i_{1}\cdots i_{s}}{\Phi}^{a\,i_{1}\cdots
i_{s}}\right)$ (6.1)

where ${\cal D}_{\tau}=\partial_{\tau}-iA$ is the covariant derivative with
the gauge field $A(\tau)=\sum_{r=1}^{s}A^{r}(\tau)$ being a Hermitian matrix
in the Lie algebra of $U(N)^{s}$. A subtlety associated with this system is
that though the gauge group is $U(N)^{s}$ only $SU(N)^{s}\otimes U(1)$ has a
non-trivial action on the tensors i.e. there is only a single $U(1)$ gauge
gield. The remaining $U(1)^{s-1}$ decouple completely from the system. The
reason is rather straightforward: $U(N)$ can always be factored into $SU(N)$
and an overall $U(1)$ phase. But the $s$ overall phases act as a single phase,
yielding a single diagonal $U(1)$ action. The $U(1)$ gauge fields $A_{k}$ are
proportional to the identity matrix and therefore do not rotate the tensor’s
index, therefore they combine into with their sum $A_{1}+\cdots+A_{s}$ which
is the effective $U(1)$ gauge field.

By taking a lattice gauge theory approach we have shown [6] that all the gauge
group elements can be accumulated on a single link to give a holonomy element
around the thermal circle and that the continuum partition function, after
removing the zero point energy, $\prod_{a=1}^{d}x_{a}^{N^{s}}$, is given by
the Molien-Weyl form

$Z^{(s)}_{N}(x_{1},\cdots
x_{d})=\int\mu(g_{1})\cdots\mu(g_{s})\prod_{a=1}^{d}\frac{1}{{\rm\bf det}[{\bf
1}-x_{a}g_{1}\otimes\cdots\otimes g_{s}]}\frac{1}{{\rm\bf det}[{\bf
1}-x_{a}g_{1}^{-1}\otimes\cdots\otimes g_{s}^{-1}]}$ (6.2)

Here it is convenient to leave the $U(1)$ gauge fields that act trivially in
the system. They simply do not contribute as their integrals automatically
give one.

Diagonalization of a $U(N)$ group element $g$ reduces it to the Cartan torus
with eigenvalues $z_{i}={\rm e}^{i\theta_{i}}$ and the measure becomes a
product of Vandermonde determinants

$\Delta(z)={\displaystyle\prod_{1\leq i<j\leq N}}(z_{i}-z_{j})\,\hbox{and
}|\Delta(z)|^{2}=\Delta(z)\Delta(\frac{1}{z}).$ (6.3)

The partition function (6.2) then becomes a set of contour integrals so that

$Z^{(s)}_{N}(x_{1},\cdots,x_{d})=\frac{1}{(N!)^{s}}\oint\prod_{r=1}^{s}|\Delta(z^{r}))|^{2}\prod_{a=1}^{d}\prod_{i_{1}\cdots
i_{s}=1}^{N}\frac{1}{\left|1-x_{a}z^{1}_{i_{1}}\cdots
z^{s}_{i_{s}}\right|^{2}}\,.$ (6.4)

The integration is a coutour integral $\oint\frac{dz}{2\pi iz}$ for each of
the $N\times s$ Cartan torus variables $z^{s}_{i}$.

Explicitly for a set of $d$ complex vectors $\Phi^{a}_{i}$ transforming under
the group $U(N)$ we have

$Z^{(1)}_{N}(x_{1},\cdots,x_{d})=\frac{1}{(N!)}\oint|\Delta(z)|^{2}\prod_{a=1}^{d}\prod_{i=1}^{N}\frac{1}{(1-x_{a}z_{i})(1-\frac{x_{a}}{z_{i}})}$
(6.5)

and for a set of $d$ two-tensors $\Psi^{a}_{ij}$ with transformaton group
$U(N)\times U(N)$ we have

$Z^{(2)}_{N}(x_{1},\cdots,x_{d})=\frac{1}{(N!)^{2}}\oint|\Delta(z)\Delta(w)|^{2}\prod_{a=1}^{d}\prod_{i=1}^{N}\prod_{j=1}^{N}\frac{1}{(1-x_{a}z_{i}w_{j})(1-\frac{x_{a}}{z_{i}w_{j}})}$
(6.6)

and finally for the 3-tensor system $\Psi^{a}_{ijk}$ with transformation group
$U(N)\times U(N)\times U(N)$ we get

$Z^{(3)}_{N}(x_{1},\cdots,x_{d})=\frac{1}{(N!)^{3}}\oint|\Delta(z)\Delta(v)\Delta(w)|^{2}\prod_{a=1}^{d}\prod_{i=1}^{N}\prod_{j=1}^{N}\prod_{k=1}^{N}\frac{1}{(1-x_{a}z_{i}v_{j}w_{k})(1-\frac{x_{a}}{z_{i}v_{j}w_{k}})}\,.$
(6.7)

When $x^{a}=x$, so that all $x^{a}$ are equal, there is an additional $SO(d)$
symmetry and the resulting partition functions take the form

$Z^{(s)}_{N}(x;d)=\frac{P^{(s)}_{N,K}(x;d)}{Q^{(s)}_{N,L}(x;d)}$ (6.8)

with $P^{(s)}_{N,K}(x;d)$ and $Q^{(s)}_{N,L}(x;d)$ polynomials of degree $K$
and $L$ respectively. Also, assuming $x<1$ one has the inversion relation

$Z^{(s)}_{N}(\frac{1}{x};d)=(-1)^{sN-n_{0}}x^{2dN^{s}}Z_{N}^{(s)}(x;d)$ (6.9)

and the $(-1)^{sN-n_{0}}$ comes from reversing the contours, i.e. the
transformation of $\frac{dz^{-1}}{z^{-1}}=-\frac{dz}{z}$ in the Molien-Weyl
formula, when returning it to its original form and $n_{0}$ is the number of
integrations that decouple completely. For $s>1$ and $d>1$ we have
$n_{0}=s-1$. Here we see that $L-K=2dN^{s}$ is the number of oscillators in
the system. The multi-variable case satisfies a similar relation when $x_{a}$
are inverted with
$\prod_{a=1}^{d}x_{a}^{N^{s}}Z^{(3)}_{N}(x_{1},\cdots,x_{d})$ transforming
only by a sign. Due to the inversion relation, the polynomial
$P^{(s)}_{N,K}(x,d)$ has palindromic coefficients modulo signs.

The transformation of $Z^{(s)}_{N}(x_{1},\cdots,x_{d})$ follows from the path
integral (6.1), which automatically includes the zero-point energy prefactor,
by observing that $\beta m_{a}\rightarrow-\beta m_{a}$, which sends
$x_{a}\rightarrow\frac{1}{x_{a}}$ is an invariance of the thermal action. What
is less obvious is the phase factor, but this can be obtained from the high
temperature analysis below.

The expressions (6.5) (6.6) (6.7) can be evaluated exactly for small $N$ and
$d$ and we exhibit some illustrative results below.

###  6.1 Complex-Vectors

For small $N$ and $d$ the vector case can be evaluated explicitly to give for
$d=2$ and $N=1$

$Z_{1}^{(1)}(x,y)=\frac{1-x^{2}y^{2}}{(1-x^{2})(1-y^{2})(1-xy)^{2}}.$ (6.10)

One can easily work out $Z_{1}^{(1)}(x;d)$ for higher $d$ to find

$Z_{1}^{(1)}(x;d)=\frac{P^{(1)}_{1,2d-2}(x;d)}{(1-x^{2})^{2d-1}}$ (6.11)

where $P^{(1)}_{1,2d-2}(x;d)$ is a palindromic even polynomial of degree
$2d-2$, e.g. $P^{(1)}_{1,2}(x;3)=1+x^{2}$, $P^{(1)}_{1,4}(x;4)=1+4x^{2}+x^{4}$
and $P^{(1)}_{1,6}(x;3)=1+9^{2}+9x^{4}+x^{6}$.

For $N$-component complex vectors with $N>1$ we find

$Z_{N}^{(1)}(x,y)=\frac{1}{(1-x^{2})(1-y^{2})(1-xy)^{2}}\,.$ (6.12)

For $d=3$ with $U(2)$ gauge invariance we get

$Z_{2}^{(1)}(x_{1},x_{2},x_{3})=\frac{1-x_{1}^{2}x_{2}^{2}x_{3}^{2}}{(1-x_{1}^{2})(1-x_{2}^{2})(1-x_{3}^{2})(1-x_{1}x_{2})^{2}(1-x_{2}x_{3})^{2}(1-x_{3}x_{1})^{2}}$
(6.13)

a result which indicates that the $9$ invariants formed from the products of a
$\Phi^{a}_{i}$ and $\overline{\Phi}^{bi}$ are not all independent and
expanding the denominator overcounts the invariants. The overcounting is
compensated for by the subtraction indicated in the numerator. The implied
relation between invariants follows from the vanishing determinant of the
$3\times 3$ matrix of quadratic invariants.

Setting $x^{a}=0$ for $a>1$ reduces to the single vector case which can be
evaluated in general to find

$Z_{N}^{(1)}(x)=\frac{1}{1-x^{2}}\,.$ (6.14)

i.e. all invariants are formed from powers of the inner product of the vector
with its conjugate, $\overline{\Phi}.\Phi$. In the 2 vector case there are
four basic invariants $\overline{\Phi}^{1}.{\Phi}^{1}$,
$\overline{\Phi}^{2}.\Phi^{1}$, $\overline{\Phi}^{1}.\Phi^{2}$ and
$\overline{\Phi}^{2}.\Phi^{2}$ and all other invariants are polynomials in
these.

For general $d$ with $d<N$ we obtain

$Z^{(1)}_{N}(x_{1},x_{2},\cdots,x_{d})=\frac{1}{\prod_{a=1}^{d}(1-x_{a}^{2})\prod_{a<b=1}^{d}(1-x_{a}x_{b})^{2}}$
(6.15)

so that in the $SO(d)$ symmetric case with all $x_{a}=x$ one gets

$Z^{(1)}_{N}(x,d)=\frac{1}{(1-x^{2})^{d^{2}}}\,,$ (6.16)

for all $d\leq N$. For $d>N$ the situation is more complicated as is indicated
by the general scaling near $x=1$, i.e. at asymptotically large temperatures,
see 7.2.1.

###  6.2 Two-Tensors

For the two-index tensor case one can evaluate the contour integrals
explicitly for all $N$ with $d=1$ finding

$Z_{N}^{(2)}(x)=\prod_{n=1}^{N}\frac{1}{(1-x^{2n})}\,.$ (6.17)

One recognoises that the invariants are built from traces of
$X_{i}^{j}=\Psi_{ik}\overline{\Psi}^{kj}$ or equivalently the eigenvalues of
$X_{i}^{j}$ whose square roots are the singular values of $\Psi_{ij}$ and the
problem reduces to counting the invariants of a single matrix. So the
invariants are the singular values of the 2-tensor, which can be thought of as
a generic complex two matrix.

For a set of two tensors $\Psi^{1}_{ij}$ and $\Psi^{2}_{ij}$ one can again
evaluate the partition function finding

$Z_{2}^{(2)}(x,y)=\frac{(1+x^{2}y^{2})(1+xy^{3}+2x^{2}y^{2}+x^{3}y+x^{4}y^{4})}{(1-x^{2})(1-x^{4})(1-y^{2})(1-y^{4})(1-xy)^{2}(1-x^{2}y^{2})(1-xy^{3})(1-x^{3}y)}\,,$
(6.18)

which when reduced to the rotationally invariant case

$Z_{2}^{(2)}(x;2)=\frac{(1+x^{4})(1+4x^{4}+x^{8})}{(1-x^{2})^{4}(1-x^{4})^{5}}\,.$
(6.19)

For three and more tensors the multi-variable results become increasingly more
complicated and we quote only the rotationally invariant case for $d=3$ where
we find

$Z_{2}^{(2)}(x;3)=\frac{P^{(2)}_{2,28}(x;3)}{(1-x^{2})^{8}(1-x^{4})^{9}}$
(6.20)

and $P^{(2)}_{2,28}(x;3)$ is the palindromic polynomial

$\displaystyle P^{(2)}_{2,28}(x;3)$ $\displaystyle=$ $\displaystyle
1+x^{2}+37x^{4}+56x^{6}+353x^{8}+389x^{10}+1037x^{14}+704x^{14}$ (6.22)
$\displaystyle\qquad+1037x^{16}+389x^{18}+353x^{20}+56x^{22}+37x^{24}+x^{26}+x^{28}$

while for $U(3)$ we get

$Z_{3}^{(2)}(x;2)=\frac{P^{(2)}_{3,48}(x;2)}{(1-x^{2})^{3}(1-x^{4})^{9}(1-x^{6})^{7}}$
(6.23)

where $P_{3,48}(x;2)$ is again an even palindromic polynomial of degree 48
given by

$\displaystyle P^{(2)}_{3,48}(x;2)$ $\displaystyle=$ $\displaystyle
1+x^{2}+2x^{4}+19x^{6}+45x^{8}+78x^{10}+208x^{12}+426x^{14}$ (6.25)
$\displaystyle\qquad+621x^{16}+911x^{18}+1328x^{20}+1507x^{22}+1490x^{24}+\cdots$

###  6.3 Three-Tensors

The 3-tensor case becomes significantly more complicated and we only provide
explicit partition functions for $N=2$ where we find

$Z_{2}^{(3)}(x)=\frac{1-x^{4}+x^{8}}{(1-x^{2})(1-x^{4})^{4}(1-x^{6})}$ (6.26)

Expanding in $x$ gives

$Z_{2}^{(3)}(x)=1+x^{2}+4x^{4}+5x^{6}+12x^{8}+15x^{10}+\cdots$ (6.27)
$Z_{2}^{(3)}(x_{1},x_{2})=\frac{P^{(3)}_{2,50}(x,y)}{Q^{(3)}_{2,50}(x_{1},x_{2})}$
(6.28)

where $P^{(3)}_{2,50}(x_{1},x_{2})$ has 847 terms. and

$Q^{(3)}_{2,50}(x_{1},x_{2})=\prod_{i=1}^{2}(1-x_{i}^{2})(1-x_{i}^{4})^{4}(1-x_{i}^{6})\prod_{n=1}^{5}(1-x_{1}^{6-n}x_{2}^{n})(1-x_{1}x_{2})^{3}(1-x_{1}^{2}x_{2}^{2})^{4}(1-x_{1}x_{2}^{3})^{4}(1-x_{1}^{3}x_{2})^{4}$
(6.29)
$Z_{2}^{(3)}(x;2)=\frac{P^{(3)}_{2,60}(x;2)}{(1-x^{2})^{4}(1-x^{4})^{12}(1-x^{6})^{6}}$
(6.30)

where the even palindromic polynomial
$P^{(3)}_{2,60}(x;2)=1+18x^{4}+90x^{6}+487x^{8}+1844x^{10}+6523x^{12}+18546x^{14}+46581x^{16}+100536x^{18}+192179x^{20}+321634x^{22}+480212x^{24}+635840x^{26}+753583x^{28}+795508x^{30}+\cdots$
which has the expansion

$Z_{2}^{(3)}(x;2)=1+4x^{2}+40x^{4}+236x^{6}+1500x^{8}+7844x^{10}+37976x^{12}+162984x^{14}+\cdots$
(6.31)

###  6.4 Hermitian Matrices with adjoint $U(N)$ action.

For a system of $d$ Hermitian matrices $X^{a}$, $a=1,\cdots,d$ the thermal
action is

$S[X]=\int_{0}^{\beta}d\tau\;{\bf tr}\left(\frac{1}{2}({\cal
D}_{\tau}X^{a})^{2}+\frac{1}{2}m^{2}(X^{a})^{2}\right)$ (6.32)

where the covariant derivative is now ${\cal
D}_{\tau}=\partial_{\tau}-i[A,\,\cdot\,]$ and the gauge field $A(\tau)$ is an
$N\times N$ hermitian matrix acting by commutation. The diagonal $U(1)$ acts
trivially so that the gauge group is in fact $SU(N)$ but it is convenient to
leave this harmless $U(1)$ in the resulting partition function is given by the
zero-point energy $\prod_{a=1}^{d}x_{a}^{N^{2}}$ times the normal ordered
Molien-Weyl expression:

$Z_{N}(x_{1},\cdots,x_{d})=\frac{1}{N!}\oint\prod_{i=1}^{N}\frac{dz_{i}}{2\pi
iz_{i}}\Delta(z)\Delta(z^{-1})\prod_{a=1}^{d}\prod_{i=1}^{N}\prod_{j=1}^{N}\frac{1}{1-x_{a}z_{i}z_{j}^{-1}}\,.$
(6.33)

Again one can evaluate partition functions for small $N$ and $d$.

Explicitly we find

$Z_{2}(x,y)=\frac{1}{1-xy}\prod_{n=1}^{2}\frac{1}{(1-x^{n})(1-y^{n})}$ (6.34)
$Z_{3}(x,y)=\frac{1+x^{3}y^{3}}{(1-xy)(1-x^{2}y)(1-xy^{2})(1-x^{2}y^{2})}\prod_{n=1}^{3}\frac{1}{(1-x^{n})(1-y^{n})}$
(6.35)

with similar more complicated expressions for $N=4,5$ and $6$ where

$Z_{2}(x;2)=\frac{1}{(1-x)(1-x^{2})^{3}}$ (6.36)
$Z_{3}(x;2)=\frac{1+x^{4}+x^{8}}{(1-x)^{2}(1-x^{2})^{3}(1-x^{3})^{4}(1-x^{6})}$
(6.37)
$Z_{4}(x;2)=\frac{(1-x^{12})(1+2x^{5}+x^{6}+2x^{7}+4x^{8}+4x^{9}+4x^{10}+2x^{11}+x^{12}+2x^{13}+x^{18})}{(1-x)^{2}(1-x^{2})^{3}(1-x^{3})^{4}(1-x^{4})^{6}(1-x^{6})^{3}}$
(6.38)

The full two parameter expressions for $Z_{N}(x;2)$ for $N=5,6$ and $7$ can be
found in can be found in [46] while only the one parameter expression for
$N=7$ has been evaluated [10].

$Z_{3}(x_{1},x_{2},x_{3})=\frac{P_{8}(x_{1},x_{2},x_{3})}{\prod_{n=1}^{3}\prod_{i=1}^{3}(1-x_{i}^{n})\prod_{j>i=1}^{3}(1-x_{i}x_{j})^{2}(1-x_{i}^{2}x_{j})(1-x_{i}x_{j}^{2})}$
(6.39)

with $P_{8}=1-x_{1}x_{2}-x_{1}x_{3}-x_{2}x_{3}+x_{1}^{2}x_{2}^{2}+\cdots-
x_{1}^{8}x_{2}^{8}x_{3}^{8}$ a polynimial of degree $8$ in each of the $x_{i}$
which has $158$ terms.

$Z_{3}(x;3)=\frac{1-x-2x^{2}+6x^{3}+6x^{4}-9x^{5}+x^{6}+17x^{7}+x^{8}-9x^{9}+6x^{10}+6x^{11}-2x^{12}-x^{13}+x^{14}}{(1-x)^{4}(1-x^{2})^{8}(1-x^{3})^{7}}$
(6.40)
$Z_{3}(x_{1},x_{2},x_{3},x_{4})=\frac{P_{3,15}(x_{1},x_{2},x_{3},x_{4})}{\prod_{n=1}^{3}\prod_{i=1}^{4}(1-x_{i}^{n})\prod_{j>i=1}^{4}(1-x_{i}x_{j})^{2}(1-x_{i}^{2}x_{j})(1-x_{i}x_{j}^{2})}$
(6.41)

with
$P_{3,15}(x;4)=1-x_{1}x_{2}-\cdots+x_{1}^{15}x_{2}^{15}x_{3}^{15}x_{4}^{15}$ a
degree $8$ polynomisal in $x_{i}$ of degree $15$ with a total of 16106 terms.

$Z_{3}(x;4)=\frac{P_{3,24}(x;4)}{(1-x)^{6}(1-x^{2})^{12}(1-x^{3})^{10}}$
(6.42)

$P_{3,24}(x;4)=g_{3,11}(x;4)-76x^{12}+g_{3,11}(\frac{1}{x};4)x^{24}$ where
$g_{3,11}(x;4)=1-2x-x^{2}+18x^{3}+6x^{4}-30x^{5}+75x^{6}+150x^{7}-30x^{8}+30x^{9}+401x^{10}+238x^{11}$.

###  6.5 The Adjoint Matrix system with $d=2$ and the $U(1)$-Charge-0 Sector

A system of pairs of Hermitian matrices $X,Y$ transforming under the adjoint
action of $U(N)$ can be used to define a complex $\Phi=X+iY$ which transforms
in the adjoint. This system can be restricted to the charge neutral sector by
imposing an additional $U(1)$ gauge invariance with an additional $U(1)$
gaugefield so that $\Phi\rightarrow{\rm e}^{i\theta(\tau)}\Phi$ and
$\overline{\Phi}\rightarrow{\rm e}^{-i\theta(\tau)}\overline{\Phi}$. The
covariant derivative in (6.32) then involves an additional $U(1)$ gauge field
and the path integral includes integration over this field.

The relevant Molien-Weyl result can then be obtained from the more general
complex case given in [6] or equivalently from the Hermitian case by replacing
$x$ with $zx$ and in the first determinant and $y$ by $x/z$ thereby including
the further integral over the additional $z$ corresponding to the $U(1)$.

In the 2-matrix case where the two matrices transform under the adjoint
representation of $U(N)$ we then have

$Z^{0}_{N}(x_{1},\cdots,x_{d})=\oint\frac{dz}{2\pi
iz}\frac{1}{N!}\oint\prod_{i=1}^{N}\frac{dz_{i}}{2\pi
iz_{i}}\Delta(z)\Delta(z^{-1})\prod_{i=1}^{N}\prod_{j=1}^{N}\frac{1}{1-zxz_{i}z_{j}^{-1}}\frac{1}{1-z^{-1}xz_{i}z_{j}^{-1}}$
(6.43)

These expressions can be evaluated using the known small $N$ results and one
finds

$Z^{0}_{2}(x)=\frac{1+x^{4}}{(1-x^{2})^{2}(1-x^{4})^{2}}$ (6.44)
$Z^{0}_{3}(x)=\frac{1+3x^{4}+6x^{6}+9x^{8}+6x^{10}+12x^{12}+6x^{14}+9x^{16}+6x^{18}+3x^{20}+x^{24}}{(1-x^{2})^{2}(1-x^{4})^{3}(1-x^{6})^{3}(1-x^{8})}$
(6.45)
$Z^{0}_{4}(x)=\frac{P^{0}_{72}(x)}{(1-x^{2})^{2}(1-x^{4})^{3}(1-x^{6})^{4}(1-x^{8})^{4}(1-x^{10})^{2}(1-x^{12})}$
(6.46)

where $P^{0}_{72}(x)$ is a 36th order palindromic polynomial in $x^{2}$. The
denominator order minus the numerator order is $2N^{2}$ as required from our
earlier arguments which require

$Z^{0}_{N}(\frac{1}{x})=(-1)^{N}x^{2N^{s}}Z^{0}(x)\,.$ (6.47)

The universal large $N$ low temperature expression for the charge-zero
partition function can be obtained from the known expression [50, 76] (8.5)
for this limit in the two matrix model and is given by

$Z^{0}_{\infty}(x)=\oint\frac{dz}{2\pi
iz}\prod_{n=1}^{\infty}\frac{1}{1-(z^{n}+z^{-n})x^{n}}=\int\frac{d\theta}{2\pi}\prod_{n=1}^{\infty}\frac{1}{1-2\cos(n\theta)x^{n}}$
(6.48)

This expression is well approximated by the first term in the product and on
doing the integral one finds

$Z^{0}_{\infty}(x)\simeq\frac{1}{\phi(\frac{1}{2})}\frac{1}{\sqrt{1-4x^{2}}}$
(6.49)

with $\phi(x)=(x,x)_{\infty}$ the Euler function and $(a,q)_{\infty}$ is the
q-Pochhammer symbol. This latter expression reproduces the asymptotic counting
of charge-zero states as found in [52].

##  7 High Temperature Limits from path integrals

In the high temperature limit the thermal circle in (6.1) and (6.32)
approaches zero circumference and all of the non-zero Matsubara frequencies
are driven to up to infinity and decouple. The result is the reduction to a
zero dimensional model.

At limiting high temperature (6.1) and (6.32) give classical tensor and matrix
models with two terms in their potentials. The first comes from the gauge
field interacting with the field $\Psi$ and the second comes from the
potential itself. For the finite group case one only has the potential so the
scaling at high temperature comes from scaling the temperature dependence and
one gets

$Z^{(s)}_{N}(x;d)\sim\frac{1}{(m\beta)^{2dN^{s}}}=\frac{1}{(1-x)^{2dN^{s}}}$
(7.1)

while the adjoint matrix models with discrete gauge group give
$Z_{N}(x;d)\sim(1-x)^{dN^{2}}$.

The key point here is that in the high temperature limit only the constant
$\tau$ independent mode survives. In continuous group case the gauged case the
gauge field removes degrees of freedom due to the constraint it implements.

###  7.1 Matrix Models under Adjoint Action of $U(N)$

For multi-matrix quantum mechanical models (6.32) with $d$ matrices and $d>1$
the generic counting follows from the Euclidean path integral by first
dimensional reduction to time independent model at large $N$. The reduced path
integral is

$Z_{N}(x;d)\sim\int[dX][dA]{\rm e}^{\frac{1}{2}{\bf
tr}([A,X^{a}]^{2})-\frac{1}{2}\beta^{2}m^{2}{\bf tr}((X^{a})^{2})}$ (7.2)

where the only temperature dependence is as shown. Then rescaling
$X^{a}={(m\beta)}^{-1}\overline{X}^{a}$ and $A=(m\beta)\overline{A}$ all the
temperature dependence is extracted to an overall scale so that asymptotically
we have

$Z_{N}(x;d)\sim{(m\beta)}^{(d-1)N^{2}+1}\int[dX][dA]{\rm e}^{\frac{1}{2}{\bf
tr}([\overline{A},\overline{X}^{a}]^{2})-\frac{1}{2}{\bf
tr}((\overline{X}^{a})^{2})}\sim\frac{1}{(1-x)^{(d-1)N^{2}+1}}$ (7.3)

in agreement with known exact expressions at small $N$ and $d>1$.

The case of $d=1$ is special since a single Hermitian matrix gauged under the
adjoint action of $U(N)$ can be diagonalized to its eigenvalues which are
invariants. The residual phases of the Cartan leave the diagonal matrix
invariant and form its stability group. The counting is therefore
$N^{N^{2}-(N^{2}-N)}$ and the asymptotic behaviour is

$Z_{N}(x)\sim\frac{1}{(1-x)^{N}}$ (7.4)

for two matrices these phases act as $Y_{ij}\rightarrow{\rm
e}^{i(\theta_{i}-\theta_{j})}X_{ij}$ so we can use these phases to remove $N$
of the off diagonal phases of the matrix removing $N-1$ degrees of freedom.
For a third or more matrices there is no freedom to further gauge fix. Hence
the counting for $d>1$ Hermitian matrices can be understood as

$Z_{N}(x,d)\sim\frac{1}{(1-x)^{N+(N^{2}-(N-1))+(d-2)N^{2}}}=\frac{1}{(1-x)^{dN^{2}-(N^{2}-1)}}$
(7.5)

in agreement with the path integral scaling argument. Here $dN^{2}$ came from
rescaling the $d$-Hermitian matrices and the $N^{2}-1$ came from rescaling the
generators of $U(N)$ for which only $SU(N)$ contributed since the overall
$U(1)$ phase cancels due to the adjoint action of the gauge group on the
matrices.

For the zero charge sector the additional integration over the $U(1)$ removes
an additional phase and the singularity near $x=1$ gives

$Z^{0}_{N}(x)\sim(1-x)^{-N^{2}}\,.$ (7.6)

###  7.2 Tensor Models

We now consider the tensor models in more detail. Once the high temperature
dimensional reduction is performed the resulting path integral, with any
temperature dependence removed from the measure, becomes a pure tensor
potential model i.e.

$Z_{N}^{(3)}(x,d)\sim\int\prod_{k=1}^{s}[dA^{k}]\prod_{a=1}^{d}[d\Psi^{a}][d\overline{\Psi}^{a}]\,{\rm
e}^{-\sum_{a=1}^{d}\left(|(A^{(1)}_{i_{1}i_{1}^{\prime}}+\cdots
A^{(s)}_{i_{s}i_{s}^{\prime}})\Psi^{a}_{i_{1}^{\prime}\dots
i_{s}^{\prime}}|^{2}+\beta^{2}m^{2}|\Psi^{a}_{i_{1},\dots,i_{s}}|^{2}\right)}$
(7.7)

where the only integration is over the constant modes and the gauge fields. We
can extract the temperature and mass dependence from this by rescaling the
fields $\Psi^{a}_{i,j,k}\rightarrow{(\beta m_{a})}^{-1}\Psi^{a}_{ijk}$ this
induces temperature dependence in the gauge filed first term which can be
cancelled by rescaling $A_{a}\rightarrow(\beta m)A_{a}$. We should remember
that there is only a single $U(1)$ gauge field. The resulting integration over
the tensor and gauge fields has no temperature or mass dependence and is a
pure number hence the asymptotic form of the partition function is given by

$Z_{N}^{(s)}(x;d)\sim(\beta
m)^{s(N^{2}-1)+1-2dN^{s}}\sim\frac{1}{(1-x)^{2dN^{s}-s(N^{2}-1)+1}}\,.$ (7.8)

a result which agrees with explicit calculations for the both the 2- and
3-tensor special cases we have calculated. This tells us that the total number
of degrees of freedom as distinct from the number of original oscillators is
given by

$N_{\rm{phys}}=2dN^{s}-s(N^{2}-1)+1$ (7.9)

This scaling gives the number of degrees of freedom that behave like free
oscillators at high temperature. Since our model has no interactions, only
constraints from the gauge invariance, which impose singlet conditions, the
high temperature scaling counts the total number of degrees of freedom in the
system as opposed to the number of oscillators which is larger being $2dN^{s}$
and can be accessed by the inversion discussed earlier.

Again for a finite group the gauge field drops out at high temperature since
there are no Lie algebra valued connections $A^{s}$. The high temperature
scaling is therefore always $(1-x)^{2dN^{s}}$ and $N_{\rm{phys}}=2dN^{s}$ in
the complex case (the exponent is $dN^{s}$ in the real case).

An alternative understanding of two tensors acted on by $U(N)\otimes U(N)$ is
to view them as matrices transforming as

$\Psi_{ij}\rightarrow
U_{ii^{\prime}}W_{jj^{\prime}}\Psi_{i^{\prime}j^{\prime}}=U_{ii^{\prime}}\Psi_{i^{\prime}j^{\prime}}W^{T}{j^{\prime}j}=(U\Psi
V^{\dagger})_{ij}\hbox{ with }W_{j,j^{\prime}}=V^{*}_{j,j^{\prime}}$ (7.10)

Under such a transformation the tensor can be brought into singular value from
with diagonal real positive semi-definite entries. For a single tensor these
form the $N$ invariants of the system. The stability subgroup that leaves the
singular values invariant is the set of the diagonal $U(1)$’s. If there is a
second tensor $\Phi_{ij}$ we can use the residual action to remove $N$ of its
complex phases but for a 3rd tensor there is no further freedom. For $d$
2-tensors one can therefore understand the counting for $d>1$ as

$Z^{(2)}_{N}(x;d)\sim\frac{1}{(1-x)^{N+(2N^{2}-(N-1)))+(d-2)2N^{2}}}=\frac{1}{(1-x)^{2(d-1)N^{2}+1}}$
(7.11)

The case of complex vectors requires more special cases and the general result
(7.9) only applies for $d\geq N$.

####  7.2.1 Counting Complex Vector Invariants

A complex vector acted on by $U(N)$ can be rotated to a single real component
with a residual $U(N-1)$ symmetry. A second vector can be reduced to two
components by rotating the normal to the first vector leaving it with $3$ real
components. One can proceed brought in this fashion till all the freedom
associated with $U(N)$ is exhausted for a system of $N$ complex vectors. There
is no additional freedom to remove components from more than $N$ such vectors.
The first $k$ vectors then give a total of $1+3+\cdots+2k-1=k^{2}$ degrees of
freedom and the counting for general $d$ and $U(N)$ is

$\displaystyle\begin{matrix}N_{\rm{phys}}&=&d^{2}&\hskip 32.0ptd\leq N\\\
&=&2Nd-N^{2}&\hskip 32.0ptd\geq N\end{matrix}$ (7.12)

We therefore have the asymptotic scaling for complex vectors as

$\displaystyle\begin{matrix}Z^{(1)}_{N}(x;d)&\sim&{\kern-25.0pt}(1-x)^{d^{2}}&\hskip
32.0pt&d\leq N\\\ &\sim&(1-x)^{2Nd-N^{2}}&\hskip 32.0pt&d\geq
N\,.\end{matrix}$ (7.13)

##  8  Hagedorn transitions in $U(N)$ invariant models and negative SHC in
tensor models

In this section, we will present a comparison of the thermodynamic properties
of the $S_{N}$ invariant permutation invariant matrix harmonic oscillator
thermodynamics we have described in earlier sections and the thermodynamics of
$U(N)$ invariant matrix or tensor harmonic oscillator systems. For
concreteness, we will discuss the comparison with three cases:

(I) the quantum mechanics of the $U(N)$ invariant sector of the quantum
mechanics of a hermitian matrix under the influence of a harmonic oscillator
potential.

(II) the $U(N)$ invariant sector of two hermitian matrices under the influence
of a harmonic oscillator potential.

(III) the $U(N)\times U(N)\times U(N)$ invariant sector of the quantum
mechanics of a complex $3$-index tensor $\Phi_{ijk}$ with a harmonic
oscillator potential.

In all these systems with the number of invariant states with energy $k$,
denoted ${\cal Z}(N,k)$, has a universal form for $k\leq N$ :

$\displaystyle\displaystyle{\cal Z}(k,N)={\cal Z}(k,M)\equiv{\cal
Z}(k,\infty)~{}~{}\hbox{ if }~{}~{}M\geq N\geq k$ (8.1)

We refer to this region of the parameters $k,N$ as the stable region. As
discussed earlier, there is similar feature in the GPIMQM, with the minor
difference that the threshold is at $N=2k$ :

$\displaystyle\displaystyle Z(k,N)=Z(k,M)\equiv Z(k,\infty)~{}~{}~{}\hbox{ if
}M\geq N\geq 2k$ (8.2)

The universal forms in all these cases define positive integer sequences of
numbers counting combinatorial objects, which can be defined without reference
to $N$. In all instances considered in this paper, negative SHC arises from
the properties of at large $k$ of the counting functions in the stable limit.

The case (I), the invariant states are polynomials in a matrix oscillator
$A^{\dagger}_{ij}$ which are invariant under $U(N)$. The space of states is
isomorphic to the space of polynomials of a hermitian matrix invariant under
$U(N)$. The vector space of invariants is spanned by traces of the matrix and
products of traces (i.e. multi-traces). For a fixed degree $k$, the number of
linearly independent multi-traces is the number of partitions of $k$, a number
which is independent of $N$. This number grows as $e^{\sqrt{k}}$. The
canonical partition function converges for all finite temperatures since
$e^{-\beta k+\sqrt{k}}$ vanishes rapidly at large $k$, for all finite $\beta$.
This thermodynamics has been discussed in [17] in connection with toy models
of AdS/CFT. There is no Hagedorn phase transition in the large $N$ limit. When
$k>N$, there are finite $N$ relations between the traces which allow for
example the expression of ${\rm{tr}}X^{N+1}$ as a polynomial involving
products of lower traces.

The quantisation of the complex harmonic oscillator and gauged complex
harmonic oscillator has been discussed in the AdS/CFT context in [16, 55].
There is an equivalent free fermion description of these systems which plays
an important role in the dual space of half-BPS supergravity solution [56].
Aspects of the fermion description are discussed in [16, 17, 55, 57].
Information theoretic perspectives on the gravitational thermodynamics for
AdS/CFT have been developed using this toy model [58, 59].

The case (II) is a 2-matrix generalisation of (I). The 2-matrix harmonic
oscillator is described by a quantum mechanical Lagrangian for two hermitian
matrices $X,Y$ with a standard kinetic term and quadratic potential
proportional to ${\rm{tr}}X^{2}+{\rm{tr}}Y^{2}$. An identical counting problem
arises for the holomorphic sector of a model with two complex matrices which
is relevant to the quarter BPS sector $N=4$ SYM (see [50, 51]). The polynomial
functions invariant under unitary transformations $X\rightarrow
UXU^{\dagger},Y\rightarrow UYU^{\dagger}$ can be organised according to the
degrees $(k_{1},k_{2})$ in the two matrices. For fixed $(k_{1},k_{2})$
invariant functions include traces with $k_{1}$ copies of $X$ and $k_{2}$
copies of $Y$. This is a counting of necklaces with beads of two colours. The
general invariant polynomials at degrees $(k_{1},k_{2})$ include include
multi-traces. Let ${\cal Z}(k_{1},k_{2};N)$ be the dimension of the space of
invariant polynomials of degrees $(k_{1},k_{2})$ for matrices of size $N$.
This counting has a stable form for $k_{1}+k_{2}\leq N$, i.e. there is a
function ${\cal Z}(k_{1},k_{2})$ independent of $N$ such that

$\displaystyle\displaystyle{\cal Z}(k_{1},k_{2})={\cal Z}(k_{1},k_{2};N)\hbox{
for all }N\geq(k_{1}+k_{2})$ (8.3)

The generating function ${\cal Z}(x,y)$ defined as

$\displaystyle\displaystyle{\cal Z}(x,y)=\sum_{k_{1},k_{2}=0}^{\infty}{\cal
Z}(k_{1},k_{2})x^{k_{1}}y^{k_{2}}$ (8.4)

was shown in [50, 76] to be

$\displaystyle\displaystyle{\cal
Z}(x,y)=\prod_{i=1}^{\infty}{1\over(1-x^{i}-y^{i})}$ (8.5)

The uncolored specialisation of this function ${\cal
Z}(k)=\sum_{k_{1}=0}^{k}{\cal Z}(k_{1},k-k_{1})$ has a generating function

$\displaystyle\displaystyle{\cal Z}(x)=\sum_{k}{\cal
Z}(k)x^{k}=\prod_{i=1}^{\infty}{1\over(1-2x^{i})}$ (8.6)

From the singularity of the $i=1$ term in the generating function, it is
deduced that there is a Hagedorn transition at $x=e^{-\beta}=1/2$, i.e.
$\beta=\log 2$ [8]. The asymptotic form of the counting ${\cal Z}(k)$ has the
form ${\cal Z}(k)\sim 2^{r}$. Another interesting asymptotic formula which we
will turn to in section 9.2 gives ${\cal Z}(r,r)\sim{4^{r}\over\sqrt{r}}$
[52].

In case (III) we have the invariant theory problem of describing for a complex
tensor $\Phi_{ijk}$, with $1\leq i,j,k\leq N$, the problem of counting
invariant functions constructed from $k$ copies of $\Phi$ and $k$ copies of
$\bar{\Phi}$. This problem also has applications in zero-dimensional tensor
integration models. Key initial references are [36][37] while recent overviews
of the subject are in [60][61]. A detailed treatment of the counting of the
tensor invariants using permutation methods was developed in [38] (see also
[62, 64, 63]). The counting function ${\cal Z}(n,N)$ has a stable form ${\cal
Z}(n)$ which is valid for all $N$ obeying $N\geq n$. The universal form counts
bi-partite ribbon graphs with $n$ edges and any number of nodes and can also
be expressed as a sum of squares of Kronecker coefficients (i.e. Clebsch-
Gordan multiplicities for symmetric groups). The tensor counting function was
recognised to have an asymptotic growth as $n!$ and this was interpreted in
terms of Hagedorn temperature vanishing at large $N$. Indeed in the infinite
$N$ limit, $n!e^{-\beta n}$ grows as $n\rightarrow\infty$ for all finite
$\beta$. This means that the generating function of invariants in the stable
limit ${\cal Z}(x)=\sum{\cal Z}(n)x^{n}$ has a vanishing radius of
convergence. The all-orders asymptotic formula for ${\cal Z}(n)$ was developed
in [39].

In the present case of the permutation invariant harmonic oscillator, as we
have argued in section 3, there is also a very rapid factorial growth of
degeneracies and this is responsible for the negative specific heat capacity
on the low-temperature side of a finite $N$ cross-over transition. This
naturally raises the question of whether the tensor model also shows negative
specific heat capacity in the micro-canonical ensemble, as we found for the
GPIMQM in section 4. In tne next sub-section, we will find computational
evidence for the negative SHC and will argue, using the results on the high-
temperature scaling of the partitition function from section 7, that the
3-index tensor harmonic oscillator has the same thermodynamic features as the
GPIMQM.

###  8.1  3-index complex tensor model : Phase structure and computational
evidence.

In this section we will consider the quantum mechanics of the a complex
$3$-index tensor variable with harmonic oscillator potential and gauged $U(N)$
symmetry. The complex tensor $\Phi_{ijk}$ transforms in the $V_{N}\otimes
V_{N}\otimes V_{N}$ representation of $U(N)$, where $V_{N}$ is the
fundamental. The action and path integral of the gauged quantum mechanics were
described in section 6 and the high temperature behaviour of the canonical
partition function was derived in 7. As discussed at the beginning of this
section, the stable limit of the counting ${\cal Z}(n)$ valid when $n\leq N$
has a large $n$ limit which goes like $n!$. This leads directly to a negative
SHC in the micro-canonical ensemble. The high temperature behaviour of the
canonical partition function is just that of $(2N^{3}-3N^{2}+2)$ harmonic
oscillators, this has positive SHC. We expect therefore a turn-over from
negative SHC to positive SHC in the micro-canonical ensemble with a breakdown
of the equivalence between the canonical and micro-canonical ensemble at low
temperatures, all features we have seen in the partition functions of the
GPIMQM. In this section, we provide evidence for this picture by using known
group-theoretic formulae for ${\cal Z}(n,N)$ and demonstrating the turnover
from negative SHC to positive SHC for $N=3$ and $N=4$.

For energy $n$, and gauge group $U(N)$, the dimension of the space of
invariant states is

$\boxed{{\cal Z}(n,N)=\sum_{\begin{subarray}{c}R_{1},R_{2},R_{3}\vdash n\\\
l(R_{i})\leq N\end{subarray}}C(R_{1},R_{2},R_{3})^{2}}$ (8.7)

where $C(R_{1},R_{2},R_{3})$ is the Kronecker coefficient, i.e. the
multiplicity of the trivial representation in the tensor product
$V_{R_{1}}\otimes V_{R_{2}}\otimes V_{R_{3}}$ of three irreducible
representations $R_{1},R_{2},R_{3}$ of $S_{n}$.

The computation of the Kronecker coefficients is conveniently done in SAGE
[65] using Schur symmetric functions [66]. The short code needed to produce
the sums in 8.7 is displayed below:

    s = SymmetricFunctions(QQ).s()
    def Z (n , N ) :
        L = len ( Partitions(n).list() )
        S = 0
        P = Partitions(n).list()
        for i in range(L):
            for j in range (L):
                for k in range (L):
                    if len( Partitions(n).list()[i]) < N+1  :
                        if len( Partitions(n).list()[j] ) < N+1  :
                            if  len( Partitions(n).list()[k] ) < N+1  :
                                S = S + (s(P[i]).itensor(s(P[j])).scalar (   s( P[k]) ) )^2
        return S

Following this, producing the output for fixed $N$ is done by one-line
commands. Going to high $n$ becomes expensive in memory. The computations for
$N=2,3$ for $n$ up to $11$ and $14$ respectively are illustrated below

    [In] [ Z ( i , 2)  for i in range (12)]
    [Out] [1, 1, 4, 5, 12, 15, 30, 37, 65, 80, 128, 156]
    [In] [  Z ( i +1 , 3 ) for  i in range (14) ]
    [Out] [ 1, 4, 11, 31, 92, 327, 1042, 3479, 11136, 34669, 104038,302494, 848113, 2303667  ]

Figure 18: Micro-canonical energy versus temperature for 3-index tensor at
$N=3$ with $k$ equals $3$ to $13$ using the symmetric $D_{\rm{sym}}$ discrete
derivative  Figure 19: Micro-canonical energy versus temperature for $3$-index
tensor $N=4$ with $k$ equals $3$ to $12$ using the symmetric $D_{\rm{sym}}$
discrete derivative. Note the curve turns around, i.e. SHC become positive at
higher energy.

The data ${\cal Z}(n,N)$ is copied to Mathematica and used to plot the micro-
canonical energy $n$ versus the micro-canonical tempertature. The cases
$N=3,4$ are shown in Figures 18 and 19. They demonstrate the turn-over from
negative SHC to positive SHC. In the GPIMQM case, with the more explicit
formula (2.13) in hand we were able to demonstrate this behaviour for $N$ up
to $40$. Producing the data at higher $N$ and providing detailed evidence that
this system is, as argued here, in the same universality class as the GPIMQM
is an interesting challenge.

##  9  Negative SHCs in matrix quantum mechanical models and AdS/CFT

We have seen in section 4 and 8 that super-exponential dependence of the
degeneracies $\Omega(k)$ on the non-negative integer energies $k$ leads to
negative specific heat capacity in the micro-canonical ensemble. These
sequences have an interpretation in terms of different graph counting problems
and arise as asymptotic forms in the limit $N\gg k\gg 1$. When we go beyond
this regime, and consider $k$ comparable to or larger than $N$, we get finite
$N$ corrections which become significant at sufficiently high $k$. In the
GPIMQM case, we gave computational evidence that this happens at $k\sim{N\log
N\over 2}$. In this section we will look, in some generality, at different $k$
dependences which lead to negative SHC and discuss these in connection with
gauge/gravity duality and black holes, while putting this discussion in the
context of other literature on negative SHC in statistical physics.

In section 9.1 we describe classes of super-exponential as well sub-
exponential degeneracies which lead to negative specific heat capacities. The
key condition is the failure of concavity of $S(k)$, which has been discussed
in the context of equivalence/in-equivalence of the micro- and macro-canonical
ensembles [44]. An important feature of the GPIMQM is that there is a simple
transition between negative SHC and positive SHC in the micro-canonical
ensemble, where the transition region occurs on scales of order $x\sim{\log
N\over N}$. This behaviour is particularly interesting in the context of
AdS5/CFT4, notably the duality between $N=4$ SYM with $U(N)$ gauge group and
string theory on $AdS_{5}\times S^{5}$, where the bulk AdS contains large
black holes with positive SHC and small black holes with negative SHC [67, 4].
Of course the GPIMQM employs $S_{N}$ gauge symmetry, unlike the CFT4. We have
also given evidence for similar negative SHC to positive SHC transition in
large $N$ systems in the context of $U(N)$ gauge symmetry, in the presence of
tensor degrees of freedom. The entropy of small black holes in AdS raises the
question of getting negative SHC in multi-matrix systems of the kind that
appear in the $N=4$ SYM. We discuss this for the charge zero complex matrix
model in section 9.2. In section 9.3 we discuss an example using multi-matrix
counting of $U(N)$ invariants where the number of matrices is taken to scale
with the energy. It is useful to put the negative SHCs of GPIMQM in the
context of wider discussions of this phenomenon in gravitational systems and
generally in statistical physics. We make contact with this wider literature
in section 9.3.

Finally, as a point of mathematical interest, the discussion of large $N$
GPIMQM along with matrix/tensor models with $S_{N}$ or $U(N)$ symmetry has led
us to consider a variety of positive integer number sequences having
combinatorial interpretations alongside their finite $N$ regularisations
offered by natural matrix/tensor systems. We expect that the purely
mathematical study of integer sequences $\Omega(k)$ can be enriched by
thermodynamic considerations, particularly (but perhaps even more generally)
where they arise as stable limits of matrix/tensor systems having a parameter
$N$, of the kind we have studied in some generality in this paper. As a modest
step to indicate the potential of this direction, we present Appendix C where
we give an interpretation of the convergence of number sequences and
associated sequences of successive ratios in terms of micro-canonical
temperatures and specific heat capacities associated with the number sequence.

###  9.1  Super-exponential and sub-exponential growths of combinatorial
sequences and negative specific heat capacities

We start by describing some simple functional forms of degeneracies
$\Omega(k)$ associated with positive and negative specific heat capacities.

Example 1: Super-exponential growth - power-correction to linear entropy and
negative SHC

Consider $\Omega(k)=e^{ak^{b}}$ for constants $a$ and $b$, with $a>0$. The
integer $k$ is identified as energy. It is easy to show that if $b>1$ then the
specific heat capacity is negative. The entropy is

$\displaystyle\displaystyle S(k)=\log\Omega(k)=ak^{b}$ (9.1)

The micro-canonical inverse temperature

$\displaystyle\beta_{\rm{micro}}=T_{\rm{micro}}^{-1}={\partial
S(k)\over\partial k}=abk^{b-1}$ (9.2) $\displaystyle
T_{\rm{micro}}=a^{-1}b^{-1}k^{1-b}$ (9.3) $\displaystyle{\partial
T_{\rm{micro}}\over\partial k}=a^{-1}b^{-1}(1-b)k^{-b}$ (9.4) $\displaystyle
C_{{\rm{sh}};\rm{micro}}=\left({\partial T_{\rm{micro}}\over\partial
k}\right)^{-1}={ab\over(1-b)}k^{b}$ (9.5)

It is clear that for $a>0$, and $b>1$, we have negative SHC. The range
$a>0,b>1$ corresponds to super-exponential degeneracies, $b<1$ is sub-
exponential.

Example 2: Weakly super-exponential and negative SHC

We consider

$\displaystyle\displaystyle\Omega(k)=e^{ak(\log k)^{b}}$ (9.6)
$\displaystyle\beta_{\rm{micro}}(k)=ab\log^{b-1}(k)+a\log^{b}(k)$ (9.7)
$\displaystyle{\partial\beta_{\rm{micro}}\over\partial
k}=abk^{-1}\log^{b-2}(k)(b+\log(k)-1)$ (9.8)

This give a micro-canonical heat capacity

$\displaystyle\displaystyle
C_{\rm{hc}}(k)=-\beta_{\rm{micro}}(k)^{2}({\partial\beta_{\rm{micro}}\over\partial
k})^{-1}=-\frac{ak\log^{b}(k)(b+\log(k))^{2}}{b(b+\log(k)-1)}$ (9.9)

for large positive integer $k\gg 1$ and $b$ order $1$, we have

$\displaystyle\displaystyle C_{\rm{hc}}(k)\sim-{ak\over b}\log^{b+1}(k)$
(9.10)

We are interested in large positive integer $k$. In this regime it is evident
that $C_{{\rm{sh}}}(k)$ is negative for the super-exponential degeneracies
with $a>0,b>0$, while it is positive for near-exponential but sub-exponential
case $a>0,b<0$.

Along similar lines, for

$\displaystyle\displaystyle\Omega(k)=e^{ak(\log(\log k))^{b}}$ (9.11)

the micro-SHC is

$\displaystyle\displaystyle
C_{\rm{hc};\rm{micro}}(k)=-\frac{ak\log^{b}(\log(k))(b+\log(k)\log(\log(k)))^{2}}{b(b+(\log(k)-1)\log(\log(k))-1)}$
(9.12)

and at large $k$,

$\displaystyle\displaystyle C_{\rm{hc};\rm{micro}}(k)\sim{-ak\over
b}(\log(\log(k))^{b+1}\log(k)$ (9.13)

At large integer $k$, we get negative SHC for the super-exponential growths
$a>0,b>0$ while for the near-exponential but sub-exponential growths, we have
positive SHC.

Example 3: Power-law corrections to large $k$ exponential growth

Consider for $d>1$, the degeneracies which have near-exponential growth as a
function of $k$

$\displaystyle\displaystyle\Omega(k)=d^{k}k^{a}=e^{k\log d+a\log k}$ (9.14)

The micro-canonical $\beta_{\rm{micro}}(k)$ is

$\displaystyle\displaystyle\beta_{\rm{micro}}(k)=\log d+{a\over k}$ (9.15)

In the large $k$ limit, which is relevant to convergence of the partition
function

$\displaystyle\displaystyle\beta_{\rm{micro}}(k\rightarrow\infty)=\log d$
(9.16)

The derivative

$\displaystyle\displaystyle{\partial\beta_{\rm{micro}}(k)\over k}=-{a\over
k^{2}}$ (9.17)

The micro-canonical heat capacity ${}_{C}{\rm{hc};\rm{micro}}(k)$ is

$\displaystyle\displaystyle C_{\rm{hc};\rm{micro}}(k)$ $\displaystyle=$
$\displaystyle{k^{2}\over a}(\log d+{a\over k})^{2}$ (9.18)

In the large $k$ limit,

$\displaystyle\displaystyle C_{\rm{hc};\rm{micro}}(k\rightarrow\infty)$
$\displaystyle=$ $\displaystyle{k^{2}\log d\over a}$ (9.19) $\displaystyle<$
$\displaystyle 0\hbox{ for }a<0$ (9.20) $\displaystyle>$ $\displaystyle
0\hbox{ for }a>0$ (9.21)

This has the interesting consequence that these are simple functional forms of
degeneracies which can be sub-exponential, while having negative SHC,

$\displaystyle\displaystyle C_{\rm{hc};\rm{micro}}(k)<0~{}~{}~{}\hbox{ for
}\Omega(k)\sim d^{k}k^{a}\hbox{ with }a<0$ (9.22)

and super-exponential while having positive SHC

$\displaystyle\displaystyle C_{\rm{hc};\rm{micro}}(k)>0~{}~{}~{}\hbox{ for
}\Omega(k)\sim d^{k}k^{a}\hbox{ with }a>0$ (9.23)

This is to be contrasted with the examples 1 and 2, where the super-
exponential forms have positive SHC and the sub-exponential forms have
negative SHC.

###  9.2  Negative SHCs from zero-charge sector of complex matrix quantum
mechanics and AdS/CFT

There is an interesting instance of the Example 3 above, in the case of the
2-matrix model where the asymptotics of 2-matrix invariants was derived in
[52] using multi-variate asymptotic methods of [68]. It is interesting to
express the result for the case of complex matrices $Z,Z^{\dagger}$ with $r$
copies of $Z$ and $s$ copies of $Z^{\dagger}$, $N\gg r,s\gg 1$. The case $r=s$
can be interpreted as a zero charge non-BPS system which, for large enough
$r$, may be interpreted in terms of brane/anti-brane systems (see e.g. [69]).
Specialising the asymptotics to $r=s$, the coefficients take the form [52]

$\displaystyle\displaystyle\Omega(r)=a_{rr}\sim{G({1\over 2},{1\over
2})\over\sqrt{\pi}}{4^{r}\over\sqrt{r}}$ (9.24)

where

$\displaystyle\displaystyle G({1\over 2},{1\over
2})=\prod_{i=2}^{\infty}(1-2^{-i+1})$ (9.25)

is an inverse QPochammer function $(a,q)_{n}$ specialised to
$a=1/2,q=1/2,n=\infty$. This is an instance of the discussion in Example 3
above with $d=4,a={-1\over 2}$ (the overall constant
$G(1/2,1/2)\over\sqrt{\pi}$ does not affect the temperature or heat capacity).
In the large $N$ system at hand, the thermodynamic quantities which are finite
in the $N\rightarrow\infty$ limit are obtained by dividing with $N^{2}$. The
SHC thus defined

$\displaystyle\displaystyle
C_{{\rm{sh}};\rm{micro}}={C_{\rm{hc};\rm{micro}}\over N^{2}}$ (9.26)

approaches $0$ from the negative side on the low energy branch of the $E-T$
curves as $N\rightarrow\infty$. The heat capacity becomes infinite at the low
temperature end of the $E-T$ curve. Based on the similarities with the
2-matrix quantum mechanics without zero charge condition [49] we expect that
the energy at this transition scales like $N^{2}$ and for energies
sufficiently close to this threshold the specific heat capacity will be finite
and negative.

Using the asymptotic form of the partition function in the canonical partition
function we have the sum

$\displaystyle\displaystyle\sum_{r}{e^{2r\log 2-\beta r}\over{\sqrt{r}}}$
(9.27)

which diverges at $\beta=2\log 2\equiv\beta_{H}$ due to the near-exponential
growth of degeneracies at large $r$. This is thus a Hagedorn transition
similar to the 2-matrix harmonic oscillator. The heat capacity at finite
temperatures will be positive as required by the standard general argument
reviewed in section 3. After rescaling by $N^{2}$, the specific heat capacity
at fixed temperatures below $\beta_{H}$ approaches zero from positive values
as $N$ approaches infinity. There is an in-equivalence of ensembles at finite
$N$, which in an appropriate physical sense, tends to zero as
$N\rightarrow\infty$.

Nevertheless, as explained above, for a small range of temperatures near the
minimum in the micro-canonical ensemble, the SHC is negative at large $N$. It
will be interesting to investigate the bulk interpretation of this negative
SHC further in the context of long-range attractive forces which are known to
produce negative SHC [70, 44, 71].

We will present computational evidence that there indeed is negative heat
capacity in the micro-canonical ensemble which turns to positive heat
capacity. The computations are based on the finite $N$ formula for multi-
matrix invariants in terms of Littlewood-Richardson coefficients. For
polynomial functions of two matrices $Z$ and $Z^{\dagger}$, invariant under
$Z\rightarrow UZU^{\dagger}$ and of degree $(m,n)$ in $(Z,Z^{\dagger})$, the
dimension of the space of polynomials is

$\displaystyle\displaystyle\sum_{R\vdash m,S\vdash
n}\sum_{\begin{subarray}{c}T\vdash(m+n)\\\ l(T)\leq
N\end{subarray}}(g(R,S,T)^{2}$ (9.28)

where $R,S,T$ are Young diagrams with $m,n,(m+n)$ boxes respectively ;
$g(R,S,T)$ is the Littlewood-Richardson coefficient for the triple ; and $T$
is restricted to have no more than $N$ rows. This formula plays a role in the
construction of orthogonal restricted Schur bases for multi-matrix operators
in free field theory [72, 73, 76]. We expect that this positive heat capacity
branch extends to the high temperature regime derived using the quantum
mechanical path integral in section 3.3.

Figure 20 gives the micro-canonical energy versus temperature plot for $N=13$,
showing a short positive branch before the near-exponential asymptotic form of
the stable degeneracy function sets in. This is followed by a negative heat
capacity branch over a range of temperatures which is expected to grow in size
with $N$. This negative SHC branch does not exist for $N\leq 11$, just about
appears for $N=12$ but is is clearly visible for $N=13$. The negative heat
capacity branch extends over a larger range of temperatures for $N=15,17$ and
is expected to extend over an increasing range of temperatures in the large
$N$ limit. We expect the negative HC branch to reach a minimum temperature and
turn around to a positive SHC branch which connects with the high temperature
limit.

The sequence of degeneracies for $N=13$ is calculated in SAGE using the
formula

$\boxed{~{}~{}~{}~{}~{}\sum_{R,S\vdash n}\sum_{T\vdash
2n}(g(R,S,T))^{2}~{}~{}~{}~{}~{}}$ (9.29)

which specialises (9.28) to $m=n$. The output for $n=1$ to $n=13$ is

$\displaystyle\\{2,10,38,158,602,2382,9141,35477,136790,529258,2045921,7921783,30675577,$
(9.30) $\displaystyle
118850945,460430464,1783233892,6901543295,26683631076\\}$ (9.31)

The sequence for $N=15$ going up to $n=16$ is

$\displaystyle\\{2,10,38,158,602,2382,9142,35491,136921,530258,2052698,$
(9.32) $\displaystyle 7964239,30925953,120260841,468079803,1823504895\\}$
(9.33)

This value of $n$ is not high enough to see the start of the negative SHC
branch at $N=15$.

The SAGE code giving these outputs is shown in Appendix D.

Figure 20: Micro-canonical energy versus temperature for zero charge complex
matrix system $N=13$ with $k$ equals $3$ to $18$ using the symmetric
$D_{\rm{sym}}$ discrete derivative. Note the curve has a short positive SHC
branch, a negative SHC branch (expected to grow in size with $N$) and a
positive SHC branch expected to connect to extend to the high temperature
limit.

###  9.3  Negative SHC with super-exponential growth of degeneracies in multi-
matrix models

There is a group theoretic formula for counting the $U(N)$ invariant
polynomials of degree $k$ in $d$ matrix variables of size $N$ which has been
used in the construction of orthogonal bases of gauge invariant operators [9,
50, 74, 75, 76]

$\displaystyle\displaystyle Z(n,d,N)=\sum_{\begin{subarray}{c}R\vdash k\\\
l(R)\leq N\end{subarray}}\sum_{\begin{subarray}{c}\Lambda\vdash k\\\
l(\lambda)\leq d\end{subarray}}C(R,R,\Lambda){\rm Dim}_{U(d)}\Lambda$ (9.34)

The sum is over Young diagrams $R$ with $k$ boxes, constrained to have no more
than $N$ rows, and Young diagrams $\Lambda$ with $k$ boxes and no more than
$d$ rows. $C(R,R,\Lambda)$ is the number of trivial representations of $S_{k}$
in the decomposition of the tensor product $R\otimes R\otimes\Lambda$ into
irreducible representations of the diagonal $S_{k}$, i.e. the Kronecker
coefficient for the triple of Young diagrams $(R,R,\Lambda)$. ${\rm
Dim}_{U(d)}\Lambda$ is the dimension of the $U(d)$ representation
corresponding to Young diagram $\Lambda$. The formula simplifies in the case
$d,N\geq n$ :

$\displaystyle\displaystyle Z(k,d,N)=Z(k,d)=\sum_{l=0}^{k}d^{l}P(k,l)$ (9.35)

$P(k,l)$ is the number of partitions of $k$ with $l$ parts. The leading term
is $d^{k}$, coming from the partition $[1^{k}]$.

A simple and surprisingly useful observation is that since ${\rm
Dim}_{U(d)}\Lambda$ in (9.34) is zero for $l(\Lambda)>d$, we can extend the
summation to all partitions $\Lambda$ of $k$. This sum over $\Lambda$ can be
done using character orthogonality to simplify the expression

$\displaystyle\displaystyle Z(k,d,N;d\geq k)$
$\displaystyle=\sum_{\begin{subarray}{c}R\vdash k\\\ l(R)\leq
N\end{subarray}}\sum_{\Lambda\vdash k}C(R,R,\Lambda){\rm{Dim}}_{U(d)}\Lambda$
(9.38) $\displaystyle={1\over(k!)^{2}}\sum_{\begin{subarray}{c}R\vdash k\\\
l(R)\leq N\end{subarray}}\sum_{\Lambda\vdash
k}\sum_{\sigma}\chi^{R}(\sigma)\chi^{R}(\sigma)\chi^{\Lambda}(\sigma)\sum_{\tau\in
S_{k}}\chi^{\Lambda}(\tau)d^{C_{\tau}}$
$\displaystyle=\sum_{\begin{subarray}{c}R\vdash k\\\ l(R)\leq
N\end{subarray}}\sum_{\Lambda\vdash k}\sum_{p\vdash k}\sum_{q\vdash
k}{1\over{\rm{Sym}}p}{1\over{\rm{Sym}}q}\chi^{R}_{p}\chi^{R}_{p}\sum_{\Lambda}\chi^{\Lambda}_{p}\chi^{\Lambda}_{q}d^{C_{q}}$

We used the expression for the Kronecker coefficient as a sum of characters.
In the last line, we have converted the sum over $\sigma$ to a sum over
partitions $p$ and the sum over $\tau$ to a sum over partitions $q$.
$\chi^{R}_{p}$ is the character of a permutation $\sigma_{p}$ in conjugacy
class $p$ for the irreducible representation $\Lambda$. We can do the sum over
$\Lambda$ using character orthogonality.

$\displaystyle\sum_{\Lambda}\chi^{\Lambda}(\sigma_{p})\chi^{\Lambda}(\sigma_{q})=\delta_{p,q}~{}~{}{\rm{Sym}}~{}p$
(9.39)

Therefore

$\boxed{~{}~{}~{}~{}~{}~{}Z(n,d,N)=\sum_{p\vdash
k}{d^{C_{p}}\over{\rm{Sym}}~{}p}\sum_{\begin{subarray}{c}R\vdash n\\\ l(R)\leq
N\end{subarray}}\chi^{R}_{p}\chi^{R}_{p}~{}~{}~{}~{}~{}~{}}$ (9.40)

This formula is conveniently coded in SAGE (see Appendix E). A closely related
formula is discussed in [77] in the context of unitary matrix integrals.

Figure 21: Micro-canonical energy versus temperature for ($d=n$)-scaled multi-
matrix model with $N=4$ and $n$ between $2$ and $15$ : negative slope
evidences the negative SHC

The formulae here allow us to investigate an interesting limit of the multi-
matrix counting, namely one where $d=k$, i.e we are taking the number of
matrices to increase as the energy increases. This can be heuristically
motivated by the counting of free field operators in $N=4$ SYM : at higher
energies, i.e. dimensions of local operators, we have more matrix fields, e.g.
derivatives of the six hermitian matrix scalars. The present counting (9.34)
treats all the $d$ matrices as having dimension $1$ so is by no means a
precise reflection of the free field ${\cal N}=4$ counting, but can be viewed
as a tractable toy model for investigating the effects of increasing the
number of matrices as the dimension increases - which has not yet been done in
attempts to explore how the physics of small black holes arises from multi-
matrix combinatorics in ${\cal N}=4$ SYM. The leading term in (9.35)
$d^{k}\sim k^{k}$ has the negative SHC property. A somewhat exotic scaling of
$d$ with $k$, namely $d\sim e^{k^{1/7}}$ produces, from $d^{k}$ an entropy
$S(k)=\log\Omega(k)\sim k^{8/7}$ which matches that of ten-dimensional
Schwarzschild black holes, which is a natural scaling to look for in
connection with small black holes in $AdS_{5}\times S^{5}$ (see an earlier
discussion of an attempt towards this scaling from a different point of view
in [78] and more recent attempts from a similar perspective [53][54] [82]). We
leave a more systematic investigation of negative SHC in the context well-
motivated multi-matrix constructions emulating the physics of small black
holes in $AdS_{5}\times S^{5}$ for the future. Understanding the physics of
small black holes in $AdS_{5}\times S^{5}$ from the point of view of CFT4
remains a fascinating open problem. The negative specific heat capacities
found in the context of tractable group-theoretic counting problems associated
with gauge invariants in well defined simple quantum mechanical models of the
kind considered here may be expected to be a useful ingredient in this quest.

Another reason for exploring negative SHCs in multi-matrix models with $d$
scaling as $n$ is that novel scalings of $d$ in multi-matrix models are known
[79] to reproduce certain aspects of the physics of tensor models, in
particular the existence of limits dominated by melonic interactions. We have
in section 8 that tensor models display negative SHCs at finite $N$, so it is
natural to explore this feature in multi-matrix models with novel scalings.

Figure 22:  Micro-canonical SHC versus temperature for ($d=n$)-scaled multi-