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Franco Ferrari

Publications and source records attributed to Franco Ferrari.

At least 37 records · Page 2Linked to original sources
arXiv2013

Polymer plats and multicomponent anyon gases

Anyon systems are studied in connection with several interesting applications including high $T_C$ superconductivity and topological quantum computing. In this work we show that these systems can be realized starting from directed polymers braided together to form a nontrivial link configuration belonging to the topological class of plats. The statistical sum of a such plat is related here to the partition function of a two-component anyon gas. The constraints that preserve the topological configuration of the plat are imposed on the polymer trajectories using the so-called Gauss linking number, a topological invariant that has already been well studied in polymer physics. Due to these constraints, short-range forces act on the monomers or, equivalently, on the anyon quasiparticles in a way that closely resembles the appearance of reaction forces in the constrained systems of classical mechanics. If the polymers are homogeneous, the anyon system reaches a self-dual point, in which these forces vanish exactly. A class of self-dual solutions that minimize the energy of the anyons is derived. The two anyon gas discussed here obeys an abelian statistics, while for quantum computing it is known that nonabelian anyons are necessary. However, this is a limitation due to the use of the Gauss linking invariant to impose the topological constraints, which is a poor topological invariant and is thus unable to capture the nonabelian characteristics of the braided polymer chains. A more refined treatment of the topological constraints would require more sophisticated topological invariants, but so far their application to the statistical mechanics of linked polymers is an open problem.

cond-mat.soft↗
arXiv2012

On a path integral representation of the Nekrasov instanton partition function and its Nekrasov--Shatashvili limit

In this work we study the Nekrasov--Shatashvili limit of the Nekrasov instanton partition function of Yang--Mills field theories with ${\cal N}=2$ supersymmetry and gauge group SU(N). The theories are coupled with fundamental matter. A path integral expression of the full instanton partition function is derived. It is checked that in the Nekrasov--Shatashvili (thermodynamic) limit the action of the field theory obtained in this way reproduces exactly the equation of motion used in the saddle-point calculations.

hep-th↗
arXiv2012

On a singular Fredholm-type integral equation arising in N=2 super Yang-Mills theories

In this work we study the Nekrasov-Shatashvili limit of the Nekrasov instanton partition function of Yang-Mills field theories with N=2 supersymmetry and gauge group SU(n). The theories are coupled with fundamental matter. The equation that determines the density of eigenvalues at the leading order in the saddle-point approximation is exactly solved. The dominating contribution to the instanton free energy is computed. The requirement that this energy is finite imposes quantization conditions on the parameters of the theory that are in agreement with analogous conditions that have been derived in previous works. Using methods borrowed from the theory of matrix models, a field theoretical expression of the full instanton partition function is derived. It is checked that in the Nekrasov-Shatashvili (thermodynamic) limit the action of the field theory obtained in this way reproduces exactly the equation of motion used in the saddle-point calculations.

hep-th↗
arXiv2012

A numerical technique for preserving the topology of polymer knots: The case of short-range attractive interactions

The statistical mechanics of single polymer knots is studied using Monte Carlo simulations. The polymers are considered on a cubic lattice and their conformations are randomly changed with the help of pivot transformations. After each transformation, it is checked if the topology of the knot is preserved by means of a method called pivot algorithm and excluded area (in short PAEA) and described in a previous publication of the authors. As an application of this method the specific energy, the radius of gyration and heat capacity of a few types of knots are computed. The case of attractive short-range forces is investigated. The sampling of the energy states is performed by means of the Wang-Landau algorithm. The obtained results show that the specific energy and heat capacity increase with increasing knot complexity as in the case of repulsive interactions. The data about the gyration radius allow to estimate the size of the polymer knots at different temperatures.

cond-mat.soft↗
arXiv2012

Topological effects in the thermal properties of knotted polymer rings

The topological effects on the thermal properties of several knot configurations are investigated using Monte Carlo simulations. In order to check if the topology of the knots is preserved during the thermal fluctuations we propose a method that allows very fast calculations and can be easily applied to arbitrarily complex knots. As an application, the specific energy and heat capacity of the trefoil, the figure-eight and the $8_1$ knots are calculated at different temperatures and for different lengths. Short-range repulsive interactions between the monomers are assumed. The knots configurations are generated on a three-dimensional cubic lattice and sampled by means of the Wang-Landau algorithm and of the pivot method. The obtained results show that the topological effects play a key role for short-length polymers. Three temperature regimes of the growth rate of the internal energy of the system are distinguished.

cond-mat.soft↗
arXiv2012

Liouville theory, N=2 gauge theories and accessory parameters

The correspondence between the semiclassical limit of the DOZZ quantum Liouville theory and the Nekrasov-Shatashvili limit of the N = 2 (Omega-deformed) U(2) super-Yang-Mills theories is used to calculate the unknown accessory parameter of the Fuchsian uniformization of the 4-punctured sphere. The computation is based on the saddle point method. This allows to find an analytic expression for the N_f = 4, U(2) instanton twisted superpotential and, in turn, to sum up the 4-point classical block. It is well known that the critical value of the Liouville action functional is the generating function of the accessory parameters. This statement and the factorization property of the 4-point action allow to express the unknown accessory parameter as the derivative of the 4-point classical block with respect to the modular parameter of the 4-punctured sphere. It has been found that this accessory parameter is related to the sum of all rescaled column lengths of the so-called 'critical' Young diagram extremizing the instanton 'free energy'. It is shown that the sum over the 'critical' column lengths can be rewritten in terms of a contour integral in which the integrand is built out of certain special functions closely related to the ordinary Gamma function.

hep-th↗
arXiv2011

From constrained stochastic processes to the nonlinear sigma model. Two old problems revisited

In this work a method is presented to derive the generating functional in path integral form for a system with an arbitrary number of degrees of freedom and constrained by general conditions. The method is applied to the case of the dynamics of an inextensible chain subjected to external forces. Next, the generating functional of the inextensible chain is computed assuming that the interactions are switched off. Finally, the generating functional of a two dimensional nonlinear sigma model with O(3) symmetry is derived exploiting its similarities with the model describing the dynamics of the inextensible chain.

cond-mat.stat-mech↗
arXiv2011

Dynamical aspects of inextensible chains

In the present work the dynamics of a continuous inextensible chain is studied. The chain is regarded as a system of small particles subjected to constraints on their reciprocal distances. It is proposed a treatment of systems of this kind based on a set Langevin equations in which the noise is characterized by a non-gaussian probability distribution. The method is explained in the case of a freely hinged chain. In particular, the generating functional of the correlation functions of the relevant degrees of freedom which describe the conformations of this chain is derived. It is shown that in the continuous limit this generating functional coincides with a model of an inextensible chain previously discussed by one of the authors of this work. Next, the approach developed here is applied to a inextensible chain, called the freely jointed bar chain, in which the basic units are small extended objects. The generating functional of the freely jointed bar chain is constructed. It is shown that it differs profoundly from that of the freely hinged chain. Despite the differences, it is verified that in the continuous limit both generating functionals coincide as it is expected.

cond-mat.stat-mech↗
arXiv2010

A gaussian model of the dynamics of an inextensible chain

In this work an approximated path integral model describing the dynamics of a inextensible chain is presented. To this purpose, the nonlinear constraints which enforce the property of inextensibility of the chain are relaxed and are just imposed in an average sense. This strategy, which has been originally proposed for semi-flexible polymers in statistical mechanics, is complicated in the case of dynamics by the extra dependence on the time variable and by the presence of nontrivial boundary conditions. Despite these complications, the probability function of the chain, which measures the probability to pass to a given initial conformation to a final one, is computed exactly. The Lagrange multiplier imposing the relaxed condition satisfies a complicated nonlinear equation, which has been solved assuming that the chain is very long.

cond-mat.stat-mech↗
arXiv2010

On a possible approach to general field theories with nonpolynomial interactions

In this work a class of massive scalar field theories with self-interactions described by a general potential is studied. Under the sole condition that the potential admits the Fourier representation, it is shown that such theories may be mapped into a standard field theory, in which the interaction of the new fields is a polynomial of fourth degree. With some restrictions, this mapping allows the perturbative treatment of models that are otherwise intractable with standard field theoretical methods. A nonperturbative approach to these theories is attempted. The original scalar field is integrated out exactly at the price of introducing auxiliary vector fields. The latter are treated in a mean field theory approximation. The singularities that arise after the elimination of the auxiliary fields are cured using the dimensional regularization. The expression of the counterterms to be subtracted is computed.

hep-th↗
arXiv2009

Diffusion of Brownian particles and Liouville field theory

In this work we review a recently proposed transformation which is useful in order to simplify non-polynomial potentials given in the form of an exponential. As an application, it is shown that the Liouville field theory may be mapped into a field theory with a polynomial interaction between two scalar fields and a massive vector field. The used methodology is illustrated with the help of the simple case of a particle performing a random walk in a delta function potentials.

hep-th↗
arXiv2009

Dynamics of a three-dimensional inextensible chain

In the first part of this work the classical and statistical aspects of the dynamics of an inextensible chain in three dimensions are investigated. In the second part the special case of a chain admitting only fixed angles with respect to the $z-$axis is studied using a path integral approach. It is shown that it is possible to reduce this problem to a two-dimensional case, in a way which is similar to the reduction of the statistical mechanics of a directed polymer to the random walk of a two-dimensional particle.

cond-mat.stat-mech↗
arXiv2008

On the connection of the generalized nonlinear sigma model with constrained stochastic dynamics

The dynamics of a freely jointed chain in the continuous limit is described by a field theory which closely resembles the nonlinear sigma model. The generating functional $Ψ[J]$ of this field theory contains nonholonomic constraints, which are imposed by inserting in the path integral expressing $Ψ[J]$ a suitable product of delta functions. The same procedure is commonly applied in statistical mechanics in order to enforce topological conditions on a system of linked polymers. The disadvantage of this method is that the contact with the stochastic process governing the diffusion of the chain is apparently lost. The main goal of this work is to reestablish this contact. To this purpose, it is shown here that the generating functional $Ψ[J]$ coincides with the generating functional of the correlation functions of the solutions of a constrained Langevin equation. In the discrete case, this Langevin equation describes as expected the Brownian motion of beads connected together by links of fixed length.

cond-mat.stat-mech↗
arXiv2008

Applications of a generalization of the nonlinear sigma model with O(d) group of symmetry to the dynamics of a constrained chain

Subject of this work are the applications of a field theoretical model, called here generalized nonlinear sigma model or simply GNLSM,to the dynamics of a chain subjected to constraints. Chains with similar properties and constraints have been discussed in a seminal paper of Edwards and Goodyear using an approach based on the Langevin equation. The GNLSM has been proposed in a previous publication in order to describe the dynamics of a two dimensional chain. In this paper the model is extended to d dimensions and a bending energy term is added to its action. As an application, two observables are computed in the case of a very stiff chain. The first observable is the dynamical form factor of a ring shaped chain. The second observable is a straightforward generalization to dynamics of the static form factor. This observable is relevant in order to estimate the average distance between two arbitrary points of the chain. Finally, a variant of the GNLNM is presented, in which the topological conditions which constrain the motion of two linked chains are imposed with the help of the Gauss linking invariant.

cond-mat.soft↗
arXiv2008

A path integral approach to the dynamics of a random chain with rigid constraints

In this work the dynamics of a freely jointed random chain which fluctuates at constant temperature in some viscous medium is studied. The chain is regarded as a system of small particles which perform a brownian motion and are subjected to rigid constraints which forbid the breaking of the chain. For simplicity, all interactions among the particles have been switched off and the number of dimensions has been limited to two. The problem of describing the fluctuations of the chain in the limit in which it becomes a continuous system is solved using a path integral approach, in which the constraints are imposed with the insertion in the path integral of suitable Dirac delta functions. It is shown that the probability distribution of the possible conformations in which the fluctuating chain can be found during its evolution in time coincides with the partition function of a field theory which is a generalization of the nonlinear sigma model in two dimensions. Both the probability distribution and the generating functional of the correlation functions of the positions of the beads are computed explicitly in a semiclassical approximation for a ring-shaped chain.

cond-mat.stat-mech↗