arXiv subjects

Denjoe O'Connor

Publications and source records attributed to Denjoe O'Connor.

At least 55 records · Page 3Linked to original sources

arXiv2007

A Multitrace Matrix Model from Fuzzy Scalar Field Theory

We present the analytical approach to scalar field theory on the fuzzy sphere which has been developed in arXiv:0706.2493 [hep-th]. This approach is based on considering a perturbative expansion of the kinetic term in the partition function. After truncating this expansion at second order, one arrives at a multitrace matrix model, which allows for an application of the saddle-point method. The results are in agreement with the numerical findings in the literature.

hep-th↗

arXiv2007

Fuzzy Scalar Field Theory as a Multitrace Matrix Model

We develop an analytical approach to scalar field theory on the fuzzy sphere based on considering a perturbative expansion of the kinetic term. This expansion allows us to integrate out the angular degrees of freedom in the hermitian matrices encoding the scalar field. The remaining model depends only on the eigenvalues of the matrices and corresponds to a multitrace hermitian matrix model. Such a model can be solved by standard techniques as e.g. the saddle-point approximation. We evaluate the perturbative expansion up to second order and present the one-cut solution of the saddle-point approximation in the large N limit. We apply our approach to a model which has been proposed as an appropriate regularization of scalar field theory on the plane within the framework of fuzzy geometry.

hep-th↗

arXiv2006

Monte Carlo Simulation of a NC Gauge Theory on The Fuzzy Sphere

We find using Monte Carlo simulation the phase structure of noncommutative U(1) gauge theory in two dimensions with the fuzzy sphere S^2_N as a non-perturbative regulator. There are three phases of the model. i) A matrix phase where the theory is essentially SU(N) Yang-Mills reduced to zero dimension . ii) A weak coupling fuzzy sphere phase with constant specific heat and iii) A strong coupling fuzzy sphere phase with non-constant specific heat. The order parameter distinguishing the matrix phase from the sphere phase is the radius of the fuzzy sphere. The three phases meet at a triple point. We also give the theoretical one-loop and 1/N expansion predictions for the transition lines which are in good agreement with the numerical data. A Monte Carlo measurement of the triple point is also given.

hep-lat↗

arXiv2006

Simulating the scalar field on the fuzzy sphere

The properties of the phi^4 scalar field theory on a fuzzy sphere are studied numerically. The fuzzy sphere is a discretization of the sphere through matrices in which the symmetries of the space are preserved. This model presents three different phases: uniform and disordered phases, as in the usual commutative scalar field theory, and a non-uniform ordered phase related to UV-IR mixing like non-commutative effects. We have determined the coexistence lines between phases, their triple point and their scaling.

hep-lat↗

arXiv2005

Field Theory Simulations on a Fuzzy Sphere - an Alternative to the Lattice

We explore a new way to simulate quantum field theory, without introducing a spatial lattice. As a pilot study we apply this method to the 3d λϕ^4 model. The regularisation consists of a fuzzy sphere with radius R for the two spatial directions, plus a discrete Euclidean time. The fuzzy sphere approximates the algebra of functions of the sphere with a matrix algebra, and the scalar field is represented by a Hermitian N x N matrix at each time site. We evaluate the phase diagram, where we find a disordered phase and an ordered regime, which splits into phases of uniform and non-uniform order. We discuss the behaviour of the model in different limits of large N and R, which lead to a commutative or to a non-commutative λϕ^4 model in flat space.

hep-lat↗

arXiv2004

The Random Walk in Generalized Quantum Theory

One can view quantum mechanics as a generalization of classical probability theory that provides for pairwise interference among alternatives. Adopting this perspective, we ``quantize'' the classical random walk by finding, subject to a certain condition of ``strong positivity'', the most general Markovian, translationally invariant ``decoherence functional'' with nearest neighbor transitions.

gr-qc↗

arXiv2003

Fuzzy Complex Quadrics and Spheres

A matrix algebra is constructed which consists of the necessary degrees of freedom for a finite approximation to the algebra of functions on the family of orthogonal Grassmannians of real dimension 2N, known as complex quadrics. These matrix algebras contain the relevant degrees of freedom for describing truncations of harmonic expansions of functions on N-spheres. An Inonu-Wigner contraction of the quadric gives the co-tangent bundle to the commutative sphere in the continuum limit. It is shown how the degrees of freedom for the sphere can be projected out of a finite dimensional functional integral, using second-order Casimirs, giving a well-defined procedure for construction functional integrals over fuzzy spheres of any dimension.

hep-th↗

arXiv2003

A Fuzzy Three Sphere and Fuzzy Tori

A fuzzy circle and a fuzzy 3-sphere are constructed as subspaces of fuzzy complex projective spaces, of complex dimension one and three, by modifying the Laplacians on the latter so as to give unwanted states large eigenvalues. This leaves only states corresponding to fuzzy spheres in the low energy spectrum (this allows the commutative algebra of functions on the continuous sphere to be approximated to any required degree of accuracy). The construction of a fuzzy circle opens the way to fuzzy tori of any dimension, thus circumventing the problem of power law corrections in possible numerical simulations on these spaces.

hep-th↗

arXiv2002

Scalar Field Theory on Fuzzy S^4

Scalar fields are studied on fuzzy $S^4$ and a solution is found for the elimination of the unwanted degrees of freedom that occur in the model. The resulting theory can be interpreted as a Kaluza-Klein reduction of CP^3 to S^4 in the fuzzy context.

hep-th↗

arXiv2002

Matrix models on the fuzzy sphere

Field theory on a fuzzy noncommutative sphere can be considered as a particular matrix approximation of field theory on the standard commutative sphere. We investigate from this point of view the scalar $ϕ^4$ theory. We demonstrate that the UV/IR mixing problems of this theory are localized to the tadpole diagrams and can be removed by an appropiate (fuzzy) normal ordering of the $ϕ^4$ vertex. The perturbative expansion of this theory reduces in the commutative limit to that on the commutative sphere.

hep-th↗

arXiv2002

Dimensional Crossover in the Non-Linear Sigma Model

We consider dimensional crossover for an O(N) model on a d-dimensional layered geometry of thickness L, in the sigma-model limit, using ``environmentally friendly'' renormalization. We show how to derive critical temperature shifts, giving explicit results to one loop. We also obtain expressions for the effective critical exponents delta and beta effective that interpolate between their characteristic fixed point values assocliated with a d and (d-1)-dimensional system in the limits T -> T_c(L), with L(T-T_c(L))^ν->infty, and T->T_c(L) for L fixed respectively, where T_c(L) is the L-dependent critical temperature of the system.

cond-mat.stat-mech↗

arXiv2001

Fuzzy Complex Projective Spaces and their Star-products

We derive an explicit expression for an associative *-product on fuzzy complex projective spaces. This generalises previous results for the fuzzy 2-sphere and gives a discrete non-commutative algebra of functions on fuzzy complex projective spaces, represented by matrix multiplication. The matrices are restricted to ones whose dimension is that of the totally symmetric representations of SU(N). In the limit of infinite dimensional matrices we recover the commutative algebra of functions on ordinary projective space. Derivatives on the fuzzy projective space are also expressed as matrix commutators.

hep-th↗

arXiv2001

Matrix $ϕ^4$ Models on the Fuzzy Sphere and their Continuum Limits

We demonstrate that the UV/IR mixing problems found recently for a scalar $ϕ^4$ theory on the fuzzy sphere are localized to tadpole diagrams and can be overcome by a suitable modification of the action. This modification is equivalent to normal ordering the $ϕ^4$ vertex. In the limit of the commutative sphere, the perturbation theory of this modified action matches that of the commutative theory.

hep-th↗

arXiv2000

Fuzzy Actions and their Continuum Limits

Previously matrix model actions for ``fuzzy'' fields have been proposed using non-commutative geometry. They retained ``topological'' properties extremely well, being capable of describing instantons, $θ$--states, the chiral anomaly, and even chiral fermions with no ``doubling''. Here, we demonstrate that the standard scalar and spinor actions on a $d$--dimensional manifold are recovered from such actions in the limit of large matrices if their normalizations are correctly scaled as the limit is taken.

hep-th↗

arXiv1997

Universal Scaling Properties of Superconductors in Magnetic Fields

Based on renormalization group arguments we establish that for a superconductor in the presence of a weak external magnetic field, $B$, the dependence on $B$ and the deviation from the critical temperature, $τ$, of a thermodynamic quantity, $P$, takes the scaling form $P=t^θX({B\overΦ_0}τ^{-2ν},qτ^{-νω_e})$, where $θ$ and $ν$ are XY exponents, $q$ is the scaled electromagnetic coupling and $νω_e$ is the associated crossover exponent. For $q/τ^{νω_e}\ll1$, the experimentally accessible region in high-$T_c$ superconductors, there is a reduction to one-variable scaling plus small corrections. In this region we find the shift in the specific heat maximum is given by $Δ=x_0{(B/Φ_0)}^{1/2ν}$ and that the singular part of the free energy at the critical temperature takes the form $F_{sing}=c(d){(B/Φ_0)}^{d/2}$ where $c(d)$ is a universal amplitude. A one loop approximation in three dimensions gives $c(3)\sim0.22$. The results presented here should have equal applicability to the nematic to smectic-A transition.

cond-mat.supr-con↗

arXiv1996

Field Theory Entropy, the $H$-theorem and the Renormalization Group

We consider entropy and relative entropy in Field theory and establish relevant monotonicity properties with respect to the couplings. The relative entropy in a field theory with a hierarchy of renormalization group fixed points ranks the fixed points, the lowest relative entropy being assigned to the highest multicritical point. We argue that as a consequence of a generalized $H$ theorem Wilsonian RG flows induce an increase in entropy and propose the relative entropy as the natural quantity which increases from one fixed point to another in more than two dimensions.

hep-th↗

arXiv1996

Finite-Temperature Renormalization Group Predictions: The Critical Temperature Exponents, and Amplitude Ratios

$λφ^4$ theory at finite temperature suffers from infrared divergences near the temperature at which the symmetry is restored. These divergences are handled using renormalization group methods. Flow equations which use a fiducial mass as flow parameter are well adapted to predicting the non-trivial critical exponents whose presence is reflected in these divergences. Using a fiducial temperature as flow parameter, we predict the critical temperature, at which the mass vanishes, in terms of the zero-temperature mass and coupling. We find some universal amplitude ratios which connect the broken and symmetric phases of the theory which agree well with those of the three-dimensional Ising model obtained from high- and low-temperature series expansions.

hep-th↗

arXiv1996

Dimensional Crossover in the Large N Limit

We consider dimensional crossover for an $O(N)$ Landau-Ginzburg-Wilson model on a $d$-dimensional film geometry of thickness $L$ in the large $N$-limit. We calculate the full universal crossover scaling forms for the free energy and the equation of state. We compare the results obtained using ``environmentally friendly'' renormalization with those found using a direct, non-renormalization group approach. A set of effective critical exponents are calculated and scaling laws for these exponents are shown to hold exactly, thereby yielding non-trivial relations between the various thermodynamic scaling functions.

cond-mat↗