1. Critical properties in the 2-dimensional spin models

A relevant part of my research activity deals with the study of the critical properties of non-linear sigma models in 2 dimensions with O(N) symmetry. By assuming some hypotheses concerning the percolation properties of the model, several authors have argued that it would undergo a Kosterlitz-Thouless (KT)-type phase transition at finite temperature and that the model has no mass-gap, contradicting exact (but not rigorous) analytical results. In Nucl. Phys. B500 513 (1997) a deep analysis of the model is done by high precision simulations (O(10⁷) decorrelated statistics) with standard and improved actions (Symanzik 0-loop) and with effective schemes with perturbative corrections to the universal scaling up to 4 loops for the standard action and 3 loops for the Symanzik action. Moreover two different effective schemes are used for checking purposes. The computer code for the simulation is written in c language in order to efficiently construct the clusters of the Wolff algorithm. A modification of this algorithm allows to use it for the Symanzik action.

The mass gap obtained from the simulation shows an excellent agreement (within 4%) with the analytical calculation for the O(3) Heisenberg model and within 5‰ for the O(8) model. Also the magnetic susceptibility is in excellent agreement with large N calculations. These results support the scenario where the spin models have a single UV fixed point, as predicted by perturbation theory.

Moreover it is shown that the Monte Carlo data do not follow the scaling laws predicted by a KT phase transition. It is possible to reach this conclusion thanks to the extremely good precision data obtained from the high statistics simulation. In Phys. Rev. D59 067703 (1999) further studies for the standard action O(3) model are shown. Again the data follow the behaviour predicted by perturbation theory much better than by a KT transition.

In Phys. Rev. Lett. 83 3669 (1999) it is shown which of the hypotheses used to prove that the model cannot have a mass gap can fail. That hypothesis assumes that a type of clusters (made of spins which are almost perpendicular to a fixed unit vector) do not percolate. Our numerical simulations clearly indicate that instead such clusters percolate, at least at temperatures as low as T=0.5.

The results of the Phys. Rev. Lett. paper have been questioned by A. Patrascioiu and E. Seiler in their paper J. Statist. Phys. 106 811 (2002). Unfortunately these authors perform their simulations at an incredibly low temperature which, considering that they use lattice sizes of order L∼ 1000, makes their results insensible. To do that, they actually do not make use of the standard formulation of the O(3) model on the lattice. Instead a Lipschitz-type condition is imposed: neighbour spins cannot be more separated than a prefixed angle. This angle is directly related to the real temperature of the system (what usually is written as 1/T in the Boltzmann weight) in a complicated manner. However their choice for the angle drives the system to such a low temperature that the correlation length turns out to be many orders of magnitude larger than the lattice size itself. Thus, their results are meaningless.

2. Perturbation theory on the lattice

I have extensively worked on perturbation theory on the lattice. Such calculations, that are particularly heavy due to the lengthy expressions for vertices and propagators, have been partially simplified by creating a software which automates some of the algebraic operations needed to develop the perturbation theory, see Nucl. Phys. B413 553 (1994). Both the software and the analytical calculations have been developed independently by the other authors and only the final results were compared for obvious checking purposes.

In Phys. Lett. B249 490 (1990) and Phys. Rev. D44 513 (1991) the multiplicative renormalization Z for the topological charge, the perturbative tail and mixing coefficients of the topological susceptibility with gluon condensates for two different operators on the lattice have been calculated at lowest order. These results are essential to obtain the correct (and unique) physical result for the topological susceptibility from the two different operators.

In Phys. Lett. B268 241 (1991) the origin of the multiplicative renormalization Z is shown. On the other hand, in reference Phys. Lett. B350 70 (1995) the corrections from dynamical quarks to the value of Z are calculated.

In Nucl. Phys. B413 553 (1994) the first 3-loop calculation of the perturbative tail for the topological susceptibility in QCD is shown. The first calculation of the third term in the free energy of pure Yang-Mills is done in Phys. Lett. B324 433 (1994). Such a result proves to be very useful to improve the precision in the extraction of the quenched gluon condensate and to define effective schemes on the lattice (see for instance Phys. Lett. B100 485 (1981) and references inside).

In Nucl. Phys. B491 498 (1997) the third coefficient of the lattice beta function in pure Yang-Mills is calculated. It is the first correction to the universal scaling on the lattice. A similar result had already been obtained by M. Lüscher and P. Weisz a few months before but it was convenient to check it by an independent calculation. For instance in our paper the integrals are evaluated with a different technique from that used by M. Lüscher and P. Weisz.

In Phys. Lett. B426 361 (1998), Erratum-ibid. B553 337 (2003) the calculation of reference Phys. Lett. B324 433 (1994) is corrected by adding the fermionic contributions up to 3 loops. This allows a better precision in using effective schemes in Monte Carlo simulations with Wilson fermions.

In references Nucl. Phys. B500 513 (1997), Nucl. Phys. B563 213 (1999), Erratum-ibid. B576 658 (2000), Nucl. Phys. B562 581 (1999) and also the proceeding Nucl. Phys. (Proc. Suppl.) 83 709 (2000) a great effort was done to renormalize up to 4 loops the spin O(N) models in 2 dimensions on the lattice with standard action and with Symanzik 0-loop and 1-loop improved actions. These results are extremely useful to extract physical information from numerical simulations of the theory, such as mass gap or magnetic susceptibility. The main motivation is to check the validity of the Symanzik programme as well as to understand the critical properties of spin models (see above).

In the paper Nucl. Phys. B437 627 (1995) a study of the continuum limit in field theories defined on random lattices was performed. A positive example was carried through for the case of the O(3) model in 2 dimensions showing that in effect the theory is correctly represented on the random lattice. Several levels of randomness were taken into account. The necessary loop integrals were performed by using a novel method expressly introduced for the case at hand.

3. Lattice calculation of the topological susceptibility at the transition

In Phys. Lett. B412 119 (1997) a lattice simulation was performed to understand the behaviour of the topological susceptibility in pure gauge theory with SU(2) gauge group at the finite temperature transition. The drop of the signal after the transition is less steep than in the SU(3) case (see Most cited articles) as shown in the figure.

In the paper Phys. Lett. B483 139 (2000) an analogous study was performed for full QCD with gauge group SU(3) at finite temperature. Again the signal drops at the phase transition. The study was repeated obtaining similar results for full QCD with gauge group SU(2) at finite temperature and finite density in Nucl. Phys. B752 124 (2006). In this case a figure was prepared to show that the three observables, topology, chirality and confinement (for topology and confinement, it would be more adequate to speak about "quantities related to" instead of observables, see paper) undergo a transition at the same value for the density (or chemical potential):

In this figure a is the lattice spacing, P is the Polyakov loop, μ is the chemical potential and 〈ΨΨ〉 is the chiral condensate.