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Luiz C. de Albuquerque

Publications and source records attributed to Luiz C. de Albuquerque.

12 recordsLinked to original sources
arXiv2004

Quantum Topology Change and Large N Gauge Theories

We study a model for dynamical localization of topology using ideas from non-commutative geometry and topology in quantum mechanics. We consider a collection $X$ of $N$ one-dimensional manifolds and the corresponding set of boundary conditions (self-adjoint extensions) of the Dirac operator $D$. The set of boundary conditions encodes the topology and is parameterized by unitary matrices $g_N$. A particular geometry is described by a spectral triple $x(g_N)=(A_X,{\cal H}_X, D(g_N))$. We define a partition function for the sum over all $g_N$. In this model topology fluctuates but the dimension is kept fixed. We use the spectral principle to obtain an action for the set of boundary conditions. Together with invariance principles the procedure fixes the partition function for fluctuating topologies. In the simplest case the model has one free-parameter $β$ and it is equivalent to a one plaquette gauge theory. We argue that topology becomes localized at $β=\infty$ for any value of $N$. Moreover, the system undergoes a third-order phase transition at $β=1$ for large $N$. We give a topological interpretation of the phase transition by looking how it affects the topology.

hep-th↗
arXiv2004

Casimir effect for the scalar field under Robin boundary conditions: A functional integral approach

In this work we show how to define the action of a scalar field in a such a way that Robin boundary condition is implemented dynamically, i.e., as a consequence of the stationary action principle. We discuss the quantization of that system via functional integration. Using this formalism, we derive an expression for the Casimir energy of a massless scalar field under Robin boundary conditions on a pair of parallel plates, characterized by constants $c_1$ and $c_2$. Some special cases are discussed; in particular, we show that for some values of $c_1$ and $c_2$ the Casimir energy as a function of the distance between the plates presents a minimum. We also discuss the renormalization at one-loop order of the two-point Green function in the $λϕ^4$ theory submitted to Robin boundary condition on a plate.

hep-th↗
arXiv2003

Fluctuating Commutative Geometry

We use the framework of noncommutative geometry to define a discrete model for fluctuating geometry. Instead of considering ordinary geometry and its metric fluctuations, we consider generalized geometries where topology and dimension can also fluctuate. The model describes the geometry of spaces with a countable number $n$ of points. The spectral principle of Connes and Chamseddine is used to define dynamics.We show that this simple model has two phases. The expectation value $ $, the average number of points in the universe, is finite in one phase and diverges in the other. Moreover, the dimension $δ$ is a dynamical observable in our model, and plays the role of an order parameter. The computation of $<δ>$ is discussed and an upper bound is found, $<δ> < 2$. We also address another discrete model defined on a fixed $d=1$ dimension, where topology fluctuates. We comment on a possible spontaneous localization of topology.

hep-th↗
arXiv2003

Fluctuating Dimension in a Discrete Model for Quantum Gravity Based on the Spectral Principle

The spectral principle of Connes and Chamseddine is used as a starting point to define a discrete model for Euclidean quantum gravity. Instead of summing over ordinary geometries, we consider the sum over generalized geometries where topology, metric and dimension can fluctuate. The model describes the geometry of spaces with a countable number $n$ of points, and is related to the Gaussian unitary ensemble of Hermitian matrices. We show that this simple model has two phases. The expectation value $ $, the average number of points in the universe, is finite in one phase and diverges in the other. We compute the critical point as well as the critical exponent of $ $. Moreover, the space-time dimension $δ$ is a dynamical observable in our model, and plays the role of an order parameter. The computation of $<δ>$ is discussed and an upper bound is found, $<δ> < 2$.

hep-th↗
arXiv2003

Yang-Lee Zeros of the Two- and Three-State Potts Model Defined on $ϕ^3$ Feynman Diagrams

We present both analytic and numerical results on the position of the partition function zeros on the complex magnetic field plane of the $q=2$ (Ising) and $q=3$ states Potts model defined on $ϕ^3 $ Feynman diagrams (thin random graphs). Our analytic results are based on the ideas of destructive interference of coexisting phases and low temperature expansions. For the case of the Ising model an argument based on a symmetry of the saddle point equations leads us to a nonperturbative proof that the Yang-Lee zeros are located on the unit circle, although no circle theorem is known in this case of random graphs. For the $q=3$ states Potts model our perturbative results indicate that the Yang-Lee zeros lie outside the unit circle. Both analytic results are confirmed by finite lattice numerical calculations.

cond-mat.stat-mech↗
arXiv2002

When the Casimir energy is not a sum of zero-point energies

We compute the leading radiative correction to the Casimir force between two parallel plates in the $λΦ^4$ theory. Dirichlet and periodic boundary conditions are considered. A heuristic approach, in which the Casimir energy is computed as the sum of one-loop corrected zero-point energies, is shown to yield incorrect results, but we show how to amend it. The technique is then used in the case of periodic boundary conditions to construct a perturbative expansion which is free of infrared singularities in the massless limit. In this case we also compute the next-to-leading order radiative correction, which turns out to be proportional to $λ^{3/2}$.

hep-th↗
arXiv2001

Uniaxial Lifshitz Point at $O(ε_L^2)$

The critical exponents $ν_{L2}, η_{L2}$ and $γ_{L2}$ of a uniaxial Lifshitz point are calculated at two-loop level using renormalization group and $ε_{L}$-expansion techniques. We found a new constraint involving the loop momenta along the competition axis, which allows to solve the two-loop integrals. The exponent $γ_{L2}$ obtained using our method is in good agreement with numerical estimates based on Monte Carlo simulations.

cond-mat↗
arXiv2001

Reply on "Lifshitz-point critical behaviour to $O(ε_L^2)$"

We reply to a recent comment by H. W. Diehl and M. Shpot (cond-mat/0106502) criticizing our paper J. Phys. A: Math. Gen. 34 (2001) L327-332. We show that the approximation we use for evaluating higher-loop integrals is consistent with homogeneity. A new renormalization group approach is presented in order to compare the two methods with high-precision numerical data concerning the uniaxial case. We stress that the isotropic behaviour cannot be obtained from the anisotropic one.

cond-mat.stat-mech↗
arXiv2001

Anisotropic Lifshitz Point at $O(ε_L^2)$

We present the critical exponents $ν_{L2}$, $η_{L2}$ and $γ_{L}$ for an $m$-axial Lifshitz point at second order in an $ε_{L}$ expansion. We introduced a constraint involving the loop momenta along the $m$-dimensional subspace in order to perform two- and three-loop integrals. The results are valid in the range $0 \leq m < d$. The case $m=0$ corresponds to the usual Ising-like critical behavior.

cond-mat.stat-mech↗
arXiv2000

Yang-Lee Zeros of the Ising model on Random Graphs of Non Planar Topology

We obtain in a closed form the 1/N^2 contribution to the free energy of the two Hermitian N\times N random matrix model with non symmetric quartic potential. From this result, we calculate numerically the Yang-Lee zeros of the 2D Ising model on dynamical random graphs with the topology of a torus up to n=16 vertices. They are found to be located on the unit circle on the complex fugacity plane. In order to include contributions of even higher topologies we calculated analytically the nonperturbative (sum over all genus) partition function of the model Z_n = \sum_{h=0}^{\infty} \frac{Z_n^{(h)}}{N^{2h}} for the special cases of N=1,2 and graphs with n\le 20 vertices. Once again the Yang-Lee zeros are shown numerically to lie on the unit circle on the complex fugacity plane. Our results thus generalize previous numerical results on random graphs by going beyond the planar approximation and strongly indicate that there might be a generalization of the Lee-Yang circle theorem for dynamical random graphs.

hep-th↗
arXiv1998

Renormalization Ambiguities in Casimir Energy

Some questions were recently raised about the equivalence of two methods commonly used to compute the Casimir energy: the mode summation approach and the one-loop effective potential. In this respect, we argue that the scale dependence induced by renormalization effects, displayed by the effective potential approach, also appears in the MS method.

hep-th↗