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Denjoe O'Connor

Publications and source records attributed to Denjoe O'Connor.

At least 19 recordsLinked to original sources
arXiv2026

Negative heat capacities in spherically symmetric sectors of $d$-matrix quantum mechanics

We consider the $SO(d)$ and $O(d)$ invariant sectors of the bosonic $d$-matrix harmonic oscillator with $U(N)$ gauge symmetry. The micro-canonical degeneracy $\mathcal{Z}( N , d , k )$ for fixed energy $k$ is expressed as a pairing between an $N$-dependent vector and a $d$-dependent vector in the space of partitions of the integer $k$. This pairing formula is derived by counting invariant words in multi-matrix variables $X^i_{j,a}$, using properties of Clebsch-Gordan multiplicities (Kronecker coefficients) for the symmetric group $S_k$, Schur-Weyl duality and harmonic analysis on the homogeneous space $U(d)/SO(d)$. Analytic formulae for large $N$ and $k$ with $ k \le N $ are obtained using group integrals over $U(N)$ and $SO(d)$ (or $ O(d)$). The micro-canonical heat capacity in this regime is negative and turns positive, at a critical value $k_{\rm crit}$, due to finite $N$ modifications to the counting, thus forming what we denote as a characteristic caloric fold in the $ E $ versus $T$ curve. Data from the pairing formula is well fitted by $k_{\rm crit} \sim { N^2 \over 4 }$ for small values of $d$. A derivation of this large $N$ formula is given using a matrix model approximation and semi-classical analysis of the eigenvalue density. The large $N,d$ limit of the degeneracies reveals a key role for ribbon graph combinatorics. The caloric fold is also notably a property of black hole thermodynamics in anti-de-Sitter spaces. We propose the spherically symmetric \(SO(d)\) and \(O(d)\) invariant sectors of \(d\)-matrix quantum mechanics as tractable matrix systems for capturing key features of dual descriptions of black-hole thermodynamics.

hep-th↗
arXiv2025

Gauged permutation invariant tensor quantum mechanics, least common multiples and the inclusion-exclusion principle

We derive the canonical ensemble partition functions for gauged permutation invariant tensor quantum harmonic oscillator thermodynamics, finding surprisingly simple expressions with number-theoretic characteristics. These systems have a gauged symmetry of $S_N$, the symmetric group of all permutations of a set of $N$ objects. The symmetric group acts on tensor variables $ Φ_{ i_1, \cdots , i_s } $, where the $s$ indices each range over $ \{ 1, 2, \cdots , N \} $ and have the standard $S_N$ action of permutations. The result is a sum over partitions of $N$ and the summand is a product admitting simple expressions, which depend on the least common multiples (LCMs) of subsets of the parts of the partition. The inclusion-exclusion principle of combinatorics plays a central role in the derivation of these expressions. The behaviour of these partition functions under inversion of the Boltzmann factor $ x = e^{ - β} $ is governed by universal sequences associated with invariants of symmetric groups and alternating groups. The partition functions allow the development of a high temperature expansion analogous to the $s=2$ matrix case. The calculation of an $s$-dependent breakdown point leads to a critical Boltzmann factor $ x_c = { \log N \over sN^{ s-1}}$ as the leading large $N$ approximation.

hep-th↗
arXiv2025

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

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.

hep-th↗
arXiv2024

Gauged permutation invariant matrix quantum mechanics: Partition functions

The Hilbert spaces of matrix quantum mechanical systems with $N \times N$ matrix degrees of freedom $ X $ have been analysed recently in terms of $S_N$ symmetric group elements $U$ acting as $X \rightarrow U X U^T $. Solvable models have been constructed uncovering partition algebras as hidden symmetries of these systems. The solvable models include an 11-dimensional space of matrix harmonic oscillators, the simplest of which is the standard matrix harmonic oscillator with $U(N)$ symmetry. The permutation symmetry is realised as gauge symmetry in a path integral formulation in a companion paper. With the simplest matrix oscillator Hamiltonian subject to gauge permutation symmetry, we use the known result for the micro-canonical partition function to derive the canonical partition function. It is expressed as a sum over partitions of $N$ of products of factors which depend on elementary number-theoretic properties of the partitions, notably the least common multiples and greatest common divisors of pairs of parts appearing in the partition. This formula is recovered using the Molien-Weyl formula, which we review for convenience. The Molien-Weyl formula is then used to generalise the formula for the canonical partition function to the 11-parameter permutation invariant matrix harmonic oscillator.

hep-th↗
arXiv2024

Gauged permutation invariant matrix quantum mechanics: Path Integrals

We give a path integral construction of the quantum mechanical partition function for gauged finite groups. Our construction gives the quantization of a system of $d$, $N\times N$ matrices invariant under the adjoint action of the symmetric group $S_N$. The approach is general to any discrete group. For a system of harmonic oscillators, i.e. for the non-interacting case, the partition function is given by the Molien-Weyl formula times the zero-point energy contribution. We further generalise the result to a system of non-square and complex matrices transforming under arbitrary representations of the gauge group.

hep-th↗
arXiv2022

Backreacted D0/D4 background

We construct a supergravity background corresponding to a backreacted D0/D4-brane system. The background is holographically dual to the Venecianno limit of the Berkoos-Douglas matrix model. It is known that the localized D0/D4 system is unstable when the D0-branes are within the D4-branes. To circumvent this difficulty we separate the D4s from the D0s, which are placed at the origin, and restore the symmetry of the combined system by distributing the D4-branes on a spherical shell around the D0-branes. The backreacted solution is first obtained perturbatively in N_f/N_c and displayed analytically to 1st order. A non-perturbative numerical solution is then presented.

hep-th↗
arXiv2022

The nonperturbative phase diagram of the bosonic BMN matrix model

We study the thermal phase transition of the bosonic BMN model which is a mass deformed version of the bosonic part of the BFSS model. Our results connect the massless region of the phase diagram described by the bosonic BFSS model with the large-mass region, where the model is analytically solvable. We observe that at finite value of the matrix size $N$, the critical region is smeared over a small temperature range. The model has a single critical temperature, which arises as the large $N$ limit of two apparent transitions at finite $N$. We emphasise the vital role played by finite $N$ corrections in the confined phase and illustrate this with a novel treatment of the noninteracting Gaussian model.

hep-th↗
arXiv2020

The Confining Transition in the Bosonic BMN Matrix Model

We study the confining/deconfining phase transition in the mass deformed Yang-Mills matrix model which is obtained by the dimensional reduction of the bosonic sector of the four-dimensional maximally supersymmetric Yang-Mills theory compactified on the three sphere, i.e. the bosonic BMN model. The $1/D$ (with $D$ the number of matrices) expansion suggests that the model may have two closely separated transitions. However, using a second order lattice formulation of the model we find that for the small value of the mass parameter, $μ=2$, those two apparent critical temperatures merge at large $N$, leaving only a single weakly first-order phase transition, in agreement with recent numerical results for $μ=0$ (the bosonic BFSS model).

hep-th↗
arXiv2018

A Computer Test of Holographic Flavour Dynamics II

We study the second derivative of the free energy with respect to the fundamental mass (the mass susceptibility) for the Berkooz-Douglas model as a function of temperature and at zero mass. The model is believed to be holographically dual to a D0/D4 intersection. We perform a lattice simulation of the system at finite temperature and find excellent agreement with predictions from the gravity dual.

hep-th↗
arXiv2018

The non-perturbative phase diagram of the BMN matrix model

We study the maximally supersymmetric plane wave matrix model (the BMN model) at finite temperature, $T$, and locate the high temperature phase boundary in the $(μ,T)$ plane, where $μ$ is the mass parameter. We find the first transition, as the system is cooled from high temperatures, is from an approximately $SO(9)$ symmetric phase to one where three matrices expand to form fuzzy spheres. For $μ> 3.0$ there is a second distinct transition at a lower temperature. The two transitions approach one another at smaller $μ$ and merge in the vicinity of $μ=3.0$. The resulting single transition curve then approaches the gauge/gravity prediction as $μ$ is further decreased. We find a rough estimate of the transition, for all $μ$, is given by a Padé resummation of the large-$μ$, 3-loop perturbative, predictions. We find evidence that the transition at small $μ$ is to an M5-brane phase of the theory.

hep-th↗
arXiv2018

Triple Point of a Scalar Field Theory on a Fuzzy Sphere

The model of a scalar field with quartic self-interaction on the fuzzy sphere has three known phases: a uniformly ordered phase, a disordered phase and a non-uniformly ordered phase, the last of which has no classical counterpart. These three phases are expected to meet at a triple point. By studying the infinite matrix size limit, we locate the position of this triple point to within a small triangle in terms of the parameters of the model. We find the triple point is closer to the coordinate origin of the phase diagram than previous estimates but broadly consistent with recent analytic predictions.

hep-th↗
arXiv2017

Dimer geometry, amoebae and a vortex dimer model

We present a geometrical approach for studying dimers. We introduce a connection for dimer problems on bipartite and non-bipartite graphs. In the bipartite case the connection is flat but has non-trivial ${\bf Z}_2$ holonomy round certain curves. This holonomy has the universality property that it does not change as the number of vertices in the fundamental domain of the graph is increased. It is argued that the K-theory of the torus, with or without punctures, is the appropriate underlying invariant. In the non-bipartite case the connection has non-zero curvature as well as non-zero Chern number. The curvature does not require the introduction of a magnetic field. The phase diagram of these models is captured by what is known as an amoeba. We introduce a dimer model with negative edge weights that give rise to vortices. The amoebae for various models are studied with particular emphasis on the case of negative edge weights which corresponds to the presence of vortices. Vortices gives rise to new kinds of amoebae with certain singular structures which we investigate. On the amoeba of the vortex full hexagonal lattice we find the partition function corresponds to that of a massless Dirac doublet.

hep-th↗
arXiv2017

Finite Size Scaling in 2d Causal Set Quantum Gravity

We study the $N$-dependent behaviour of $\mathrm{2d}$ causal set quantum gravity. This theory is known to exhibit a phase transition as the analytic continuation parameter $β$, akin to an inverse temperature, is varied. Using a scaling analysis we find that the asymptotic regime is reached at relatively small values of $N$. Focussing on the $\mathrm{2d}$ causal set action $S$, we find that $β\langle S\rangle $ scales like $ N^ν$ where the scaling exponent $ν$ takes different values on either side of the phase transition. For $β> β_c$ we find that $ν=2$ which is consistent with our analytic predictions for a non-continuum phase in the large $β$ regime. For $β<β_c$ we find that $ν=0$, consistent with a continuum phase of constant negative curvature thus suggesting a dynamically generated cosmological constant. Moreover, we find strong evidence that the phase transition is first order. Our results strongly suggest that the asymptotic regime is reached in $\mathrm{2d}$ causal set quantum gravity for $N \gtrsim 65$.

gr-qc↗
arXiv2017

The Flavoured BFSS Model at High Temperature

We study the high temperature series expansion of the Berkooz-Douglas matrix model which describes the D0/D4--brane system. At high temperature the model is weakly coupled and we develop the series to second order. We check our results against the high temperature regime of the bosonic model (without fermions) and find excellent agreement. We track the temperature dependence of the bosonic model and find backreaction of the fundamental fields lifts the zero temperature adjoint mass degeneracy. In the low temperature phase the system is well described by a gaussian model with three masses $m^t_A=1.964 \pm 0.003$, $m^l_A=2.001 \pm 0.003$ and $m_f=1.463 \pm 0.001$, the adjoint longitudional and transverse masses and the mass of the fundamental fields respectively.

hep-th↗
arXiv2016

A Computer Test of Holographic Flavour Dynamics

We perform computer simulations of the Berkooz-Douglas (BD) matrix model, holographically dual to the D0/D4-brane intersection. We generate the fundamental condensate versus bare mass curve of the theory both holographically and from simulations of the BD model. Our studies show excellent agreement of the two approaches in the deconfined phase of the theory and significant deviations in the confined phase. We argue the discrepancy in the confined phase is explained by the embedding of the D4-brane which yields stronger $α'$ corrections to the condensate in this phase.

hep-th↗
arXiv2016

The BFSS model on the lattice

We study the maximally supersymmetric BFSS model at finite temperature and its bosonic relative. For the bosonic model in $p+1$ dimensions, we find that it effectively reduces to a system of gauged Gaussian matrix models. The effective model captures the low temperature regime of the model including one of its two phase transitions. The mass becomes $p^{1/3}λ^{1/3}$ for large $p$, with $λ$ the 'tHooft coupling. Simulations of the bosonic-BFSS model with $p=9$ give $m=(1.965\pm .007)λ^{1/3}$, which is also the mass gap of the Hamiltonian. We argue that there is no `sign' problem in the maximally supersymmetric BFSS model and perform detailed simulations of several observables finding excellent agreement with AdS/CFT predictions when $1/α'$ corrections are included.

hep-th↗