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Brian P. Dolan

Publications and source records attributed to Brian P. Dolan.

At least 19 recordsLinked to original sources
arXiv2026

Fixed points of the renormalisation group running of quark and fermion mixing matrices in the Standard Model and beyond

The renormalisation group running of fermion mixing matrices in the Standard model and beyond is studied. For the massless 1-loop running with three generations six fixed points are found. Their associated anomalous dimension matrices are calculated and the nature of each fixed point, whether attractive, repulsive or mixed, is determined. An argument is given that the fixed points found at 1-loop must remain fixed points to all orders in perturbation theory and even non-perturbatively, as they are associated with certain differential geometric properties of vector fields on the space of mixing matrices. With $N_g$ dark or sterile neutrinos there are at least $N_g!$ fixed points of the fermion mixing matrix.

hep-ph↗
arXiv2025

Geometric and General Relativistic Techniques for Non-relativistic Quantum Systems

This thesis explores the application of differential geometric and general relativistic techniques to deepen our understanding of quantum mechanical systems. We focus on three systems, employing these mathematical frameworks to uncover subtle features within each. First, we examine Unruh radiation in the context of an accelerated two-state atom, determining transition frequencies for a variety of accelerated trajectories via first-order perturbation theory. For harmonic motion of the atom in a vacuum, we derive transition rates with potential experimental realizations. Next, we investigate the quantum Hall effect in a spherical geometry using the Dirac operator for non-interacting fermions in a background magnetic field generated by a Wu-Yang monopole. The Atiyah-Singer index theorem constrains the degeneracy of the ground state, and the fractional quantum Hall effect is studied using the composite fermion model, where Dirac strings associated with the monopole field supply the statistical gauge field vortices. A unique, gapped ground state emerges, yielding fractions of the form $ν= \frac{1}{2k+1}$ for large particle numbers. Finally, we examine the AdS/CMT correspondence through a bulk fermionic field in an RN-AdS$_4$ background (with a U(1) gauge field), dual to a boundary fermionic operator. Spherical and planar event horizon geometries are discussed, with the temperature of the RN black hole identified with that of the dual system on the boundary. By numerically solving for the spectral functions of the dual theory, for a spherical event horizon at zero temperature, we identify a shift in the Fermi surface from that which arises in the planar case. Preliminary evidence of a phase transition emerges upon examining these spectral functions, again for the spherical horizon, at non-zero temperature.

hep-th↗
arXiv2024

Ground state wave functions for the quantum Hall effect on a sphere and the Atiyah-Singer index theorem

The quantum Hall effect is studied in a spherical geometry using the Dirac operator for non-interacting fermions in a background magnetic field, which is supplied by a Wu-Yang magnetic monopole at the centre of the sphere. Wave functions are cross-section of a non-trivial $U(1)$ bundle, the zero point energy then vanishes and no perturbations can lower the energy. The Atiyah-Singer index theorem constrains the degeneracy of the ground state. The fractional quantum Hall effect is also studied in the composite Fermion model. Vortices of the statistical gauge field are supplied by Dirac strings associated with the monopole field. A unique ground state is attained only if the vortices have an even number of flux units and act to counteract the background field, reducing the effective field seen by the composite fermions. There is a unique gapped ground state and, for large particle numbers, fractions $ν=\frac{1}{2 k+1}$ are recovered.

hep-th↗
arXiv2021

Constrained Dynamics in the Hamiltonian formalism

These are pedagogical notes on the Hamiltonian formulation of constrained dynamical systems. All the examples are finite dimensional, field theories are not covered, and the notes could be used by students for a preliminary study before the infinite dimensional phase space of field theory is tackled. Holonomic constraints in configuration space are considered first and Dirac brackets introduced for such systems. It is shown that Dirac brackets are a projection of Poisson brackets onto the constrained phase space and the projection operator is constructed explicitly. More general constraints on phase space are then considered and exemplified by a particle in a strong magnetic field. First class constraints on phases are introduced using the example of motion on the complex projective space ${\mathbf{C P}}^{n-1}$. Motion of a relativistic particle in Minkowski space with a reparameterisation invariant world-line is also discussed. These notes are based on a short lecture course given at Bhubaneswar Indian Institute of Technology in November 2021.

hep-th↗
arXiv2021

Duality and modular symmetry in the quantum Hall effect from Lifshitz holography

The temperature dependence of quantum Hall conductivities is studied in the context of the AdS/CMT paradigm using a model with a bulk theory consisting of (3+1)-dimensional Einstein-Maxwell action coupled to a dilaton and an axion, with a negative cosmological constant. We consider a solution which has a Lifshitz like geometry with a dyonic black-brane in the bulk. There is an $Sl(2,R)$ action in the bulk corresponding to electromagnetic duality, which maps between classical solutions, and is broken to $Sl(2,Z)$ by Dirac quantisation of dyons. This bulk $Sl(2,Z)$ action translates to an action of the modular group on the 2-dimensional transverse conductivities. The temperature dependence of the infra-red conductivities is then linked to modular forms via gradient flow and the resulting flow diagrams show remarkable agreement with existing experimental data on the temperature flow of both integral and fractional quantum Hall conductivities.

hep-th↗
arXiv2021

A renormalisation group equation for transport co-efficients in (2+1)-dimensions derived from the AdS/CMT correspondence

Within the framework of the AdS/CMT correspondence asymptotically anti-de Sitter black holes in four space-time dimensions can be used to analyse transport properties in two space dimensions. A non-linear renormalisation group equation for the conductivity in two dimensions is derived in this model and, as an example of its application, both the Ohmic and Hall DC and AC conductivities are studied in the presence of a magnetic field, using a bulk dyonic solution of the Einstein-Maxwell equations in asymptotically AdS$_4$ space-time. The ${\cal Q}$-factor of the cyclotron resonance is shown to decrease as the temperature is increased and increase as the charge density is increased in a fixed magnetic field. Likewise the dissipative Ohmic conductivity at resonance increases as the temperature is decreased and as the charge density is increased. The analysis also involves a discussion of the piezoelectric effect in the context of the AdS/CMT framework.

hep-th↗
arXiv2020

On the group generated by $\mathbf C$, $\mathbf{P}$ and $\mathbf T$: $\mathbf {I^2 = T^2 = P^2 = I T P= -1}$, with applications to pseudo-scalar mesons

We study faithful representations of the discrete Lorentz symmetry operations of parity $\mathbf P$ and time reversal $\mathbf T$, which involve complex phases when acting on fermions. If the phase of $\mathbf P$ is a rational multiple of $π$ then $\mathbf P^{2 n}=1$ for some positive integer $n$ and it is shown that, when this is the case, $\mathbf P$ and $\mathbf T$ generate a discrete group, a dicyclic group (also known as a generalised quaternion group) which are generalisations of the dihedral groups familiar from crystallography. Charge conjugation $\mathbf C$ introduces another complex phase and, again assuming rational multiples of $π$ for complex phases, $\mathbf T \mathbf C$ generates a cyclic group of order $2 m$ for some positive integer $m$.There is thus a doubly infinite series of possible finite groups labelled by $n$ and $m$. Demanding that $\mathbf C$ commutes with $\mathbf P$ and $\mathbf T$ forces $n=m=2$ and the group generated by $\mathbf P$ and $\mathbf T$ is uniquely determined to be the quaternion group. Neutral pseudo-scalar mesons can be simultaneous $\mathbf C$ and $\mathbf P$ eigenstates. $\mathbf T$ commutes with $\mathbf P$ and $\mathbf C$ when acting on fermion bi-linears so neutral pseudo-scalar mesons can also be $\mathbf T$ eigenstates. The $\mathbf T$-parity should therefore be experimentally observable and the $\mathbf{CPT}$ theorem dictates that $T= C P$.

hep-th↗
arXiv2020

Shaking photons from the vacuum: acceleration radiation from vibrating atoms

Acceleration radiation - or Unruh radiation - the thermal radiation observed by an ever accelerating observer or detector, although having similarities to Hawking radiation, so far has proved extremely challenging to observe experimentally. One recent suggestion is that, in the presence of a mirror, constant acceleration of an atom in its ground state can excite the atom while at the same time cause it to emit a photon in an Unruh-type process. In this work we show that merely by shaking the atom, in simple harmonic motion for example, can have the same effect. We calculate the transition rate for this in first order perturbation theory and consider harmonic motion of the atom in the presence of a stationary mirror, or within a cavity or just in empty vacuum. For the latter we propose a circuit-QED potential implementation that yields transition rates of $\sim 10^{-4}\,{\rm Hz}$, which may be detectable experimentally.

quant-ph↗
arXiv2016

Where is the PdV term in the first law of black hole thermodynamics?

Traditional treatments of the first law of black hole thermodynamics do not include a discussion of pressure and volume. We give an overview of recent developments proposing a definition of volume that can be used to extend the first law to include these appropriately. New results are also presenting relating to the critical point and the associated second order phase transition for a rotating black-hole in four-dimensional space-time which is asymptotically anti-de Sitter. In line with known results for a non-rotating charged black-hole, this phase transition is shown to be of Van der Waals type with mean field exponents.

gr-qc↗
arXiv2015

The intrinsic curvature of thermodynamic potentials for black holes with critical points

The geometry of thermodynamic state space is studied for asymptotically anti-de Sitter black holes in D-dimensional space times. Convexity of thermodynamic potentials and the analytic structure of the response functions is analysed. The thermodynamic potentials can be used to define a metric on the space of thermodynamic variables and two commonly used such metrics are the Weinhold metric, derived from the internal energy, and the Ruppeiner metric, derived from the entropy. The intrinsic curvature of these metrics is calculated for charged and for rotating black holes and it is shown that the curvature diverges when heat capacities diverge but, contrary to general expectations, the singularities in the Ricci scalars do not reflect the critical behaviour. When a cosmological constant is included as a state space variable it can be interpreted as a pressure and the thermodynamically conjugate variable as a thermodynamic volume. The geometry of the resulting extended thermodynamic state space is also studied, in the context of rotating black holes, and there are curvature singularities when the heat capacity at constant angular velocity diverges and when the black hole is incompressible. Again the critical behaviour is not visible in the singularities of the thermodynamic Ricci scalar.

gr-qc↗
arXiv2014

Black holes and Boyle's law -- the thermodynamics of the cosmological constant

When the cosmological constant, $Λ$, is interpreted as a thermodynamic variable in the study of black hole thermodynamics a very rich structure emerges. It is natural to interpret $Λ$ as a pressure and define the thermodynamically conjugate variable to be the thermodynamic volume of the black hole (which need not bear any relation to the geometric volume). Recent progress in this new direction for black hole thermodynamics is reviewed.

gr-qc↗
arXiv2014

Thermodynamic stability of asymptotically anti-de Sitter rotating black holes in higher dimensions

Conditions for thermodynamic stability of asymptotically anti-de Sitter rotating black holes in D-dimensions are determined. Local thermodynamic stability requires not only positivity conditions on the specific heat and the moment of inertia tensor but it is also necessary that the adiabatic compressibility be positive. It is shown that, in the absence of a cosmological constant, neither rotation nor charge is sufficient to ensure full local thermodynamic stability of a black hole. Thermodynamic stability properties of anti-de Sitter Myers-Perry black holes are investigated for both singly spinning and multi-spinning black holes. Simple expressions are obtained for the specific heat and moment of inertia tensor in any dimension. An analytic expression is obtained for the boundary of the region of parameter space in which such space-times are thermodynamically stable.

gr-qc↗
arXiv2014

Isolated critical point from Lovelock gravity

For any K(=2k+1)th-order Lovelock gravity with fine-tuned Lovelock couplings, we demonstrate the existence of a special isolated critical point characterized by non-standard critical exponents in the phase diagram of hyperbolic vacuum black holes. In the Gibbs free energy this corresponds to a place wherefrom two swallowtails emerge, giving rise to two first-order phase transitions between small and large black holes. We believe that this is a first example of a critical point with non-standard critical exponents obtained in a geometric theory of gravity.

hep-th↗
arXiv2014

Vacuum energy and the latent heat of AdS-Kerr black holes

Phase transitions for rotating asymptotically anti-de Sitter black holes in four dimensions are described in the $P-T$ plane, in terms of the Hawking temperature and the pressure provided by the cosmological constant. The difference between constant angular momentum and constant angular velocity is highlighted, the former has a second order phase transition while the latter does not. If the angular momentum is fixed there a line of first order phase transitions terminating at a critical point with a second order phase transition and vanishing latent heat, while if the angular velocity is fixed there is a line of first order phase transitions terminating at a critical point with infinite latent heat. For constant angular velocity the analytic form of the phase boundary is determined, latent heats derived and the Clapeyron equation verified.

gr-qc↗
arXiv2014

Bose condensation and branes

When the cosmological constant is considered to be a thermodynamical variable in black hole thermodynamics, analogous to a pressure, its conjugate variable can be thought of as a thermodynamic volume for the black hole. In the AdS/CFT correspondence this interpretation cannot be applied to the CFT on the boundary but, from the point of view of the boundary $SU(N)$ gauge theory, varying the cosmological constant in the bulk is equivalent to varying the number of colors in the gauge theory. This interpretation is examined in the case of $AdS_5\times S^5$, for ${\cal N}=4$ SUSY Yang-Mills at large $N$, and the variable thermodynamically conjugate to $N$, a chemical potential for color, is determined. It is shown that the chemical potential in the high temperature phase of the Yang-Mills theory is negative and decreases as temperature increases, as expected. For spherical black holes in the bulk the chemical potential approaches zero as the temperature is lowered below the Hawking-Page temperature and changes sign at a temperature that is within one part in a thousand of the temperature at which the heat capacity diverges.

hep-th↗
arXiv2014

On the thermodynamic stability of rotating black holes in higher dimensions -- a comparison of thermodynamic ensembles

Thermodynamic potentials relevant to the micro-canonical, the canonical and the grand-canonical ensembles, associated with rotating black holes in D-dimensions, are analysed and compared. Such black holes are known to be thermodynamically unstable, but the instability is a consequence of a subtle interplay between specific heats and the moments of inertia and it manifests itself differently in the different ensembles. A simple relation between the product of the specific heat and the determinant of the moment of inertia in both the canonical and the grand-canonical ensembles is derived. Myers-Perry black holes in arbitrary dimension are studied in detail. All temperature extrema in the micro-canonical ensemble are determined and classified. The specific heat and the moment of inertia tensor are evaluated in both the canonical and the grand-canonical ensembles in any dimension. All zeros and poles of the specific heats, as a function of the angular momenta, are determined and the eigenvalues of the isentropic moment of inertia tensor are also found and classified. It is further shown that many of the thermodynamic properties of a Myers-Perry black hole in D-2 dimensions can be obtained from those of a black hole in D dimensions by sending one of the angular momenta to infinity.

gr-qc↗
arXiv2013

The compressibility of rotating black holes in D-dimensions

Treating the cosmological constant as a pressure, in the context of black hole thermodynamics, a thermodynamic volume for the black hole can be defined as being the thermodynamic variable conjugate to the pressure, in the sense of a Legendre transform. The thermodynamic volume is explicitly calculated, as the Legendre transform of the pressure in the enthalpy for a rotating asymptotically anti-de Sitter Myers-Perry black hole in D space-time dimensions. The volume obtained is shown to agree with previous calculations using the Smarr relation. The compressibility is calculated and shown to be non-negative and bounded. Taking the limit of zero cosmological constant, the compressibility of a rotating black hole in asymptotically flat space-times is determined and the corresponding speed of sound computed. The latter is bounded above and has an elegant expression purely in terms of the angular momenta, in the form of quartic and quadratic Casimirs of the rotation group, SO(D-1).

gr-qc↗