• Padé-Borel
• SU(N) antiferromagnets

Phase transitions and critical phenomena have fascinated and challenged
many generations of physicists and still constitute a considerable part of modern
research in theoretical condensed-matter physics.

The main ideas at the base of the description of phase transitions are the
existence of an order parameter, a physical quantity whose mean value distinguishes different phases of the system, and that this quantity is the only one significant near the critical point. More specifically in the case of continuous
transitions, at the critical point fluctuations of the order parameter are correlated over the whole system, namely they have a diverging correlation length and the basic assumption is that this is the only cause of the observed singularities.

So it is possible to integrate out short distance degrees of freedom and study long distance properties of an effective Landau-Ginzburg-Wilson (LGW) φ^4 field theory, where the number of components of the field φ, which corresponds to the order parameter, and the kind of terms contained in the effective LGW action
only depends on the symmetries of the system. This task is then performed by
means of renormalization-group (RG) transformations. One considers the RG
flow in Hamiltonian space. The critical behaviour is then determined by stable
fixed points of the RG flow.

Besides classical thermal phase transitions, recently much interest has been
focused on quantum phase transitions, which are driven by quantum fluctuations. These transitions involve the ground state properties of a system and are obtained by varying coupling parameters in the Hamiltonian. In recent years
an experimental realization of quantum phase transitions has become possible
thanks to improvements in cooling techniques, magnetic trapping and optical
lattices. The same method used to describe classical thermal transitions can
also be exploited for continuous quantum phase transitions. The partition function of a quantum system in d dimensions can be expressed in a path integral representation and directly related to a corresponding classical system in d + 1
dimensions by identifying the quantum imaginary time direction with an additional euclidean dimension.

Various physical systems can be modeled by SU(N) Heisenberg spins residing on the sites of a lattice, with antiferromagnetic interactions. A single SU(N) spin is an effective quantum degree of freedom for the states of a small number
of electrons or atoms. Antiferromagnetic SU(2) spin systems are relevant for the description of many insulator materials and high-Tc superconductors. In particular there are physical systems in which the spins are naturally dimerized, that is each spin has a stronger interaction with a single partner on an adjacent
site, such as the insulator TlCuCl3 or double layer configurations in high-Tc superconductors.

SU(N) antiferromagnetic Heisenberg spin systems with N > 2
serve to model many physical systems ranging from spin-orbit coupled transition metal compounds, to ultracold atoms in optical lattice potentials, with N as large as 10.
SU(N) antiferromagnets on 2d bipartite lattices, where spins on different
sublattices transform under rotations with relative conjugate representations of SU(N), are described in the continuum semiclassical limit by the CP^(N-1) model plus an additional Berry phase term. In systems in which the spins
are dimerized, because of the natural pairing of the spins, contributions to the
Berry phase term can be grouped into mutually canceling terms, so that they are described by the simple CP^(N-1) model in three dimensions.
In these systems we observe a quantum phase transition when we vary a
parameter g in the Hamiltonian. For g < g_c we have a Néel state, where spins
on the same sublattice are aligned and antiparallel to the spins on the other
sublattice. For g > g_c long range magnetic order is destroyed and the spins
tend to form independent pairs. This state is called valence-bond-liquid (VBL),
and in this state all symmetries of the system are preserved.
In this work we study the nature of the phase transitions described by the CP^(N-1) model. In our analysis we consider a LGW effective action for the CP^(N-1) model. Taking into account only the SU(N) symmetry of the original
action, we should include a cubic interaction term in the effective action. However we observe that there is an extra Z_2 symmetry that we should consider, due to the symmetry of the system for an exchange of the even sublattice with the odd one. This extra symmetry forbids the inclusion of odd terms in the
action, therefore we also study the theory without these terms.
The contents of the work are divided as follow:
•Chapter 1 contains a brief summary of classical theory of phase transitions.
After a phenomenological introduction, Landau theory and the mean field
approximation are presented.
•In chapter 2 we introduce some concepts of renormalization theory in
quantum field theory, which will be useful for the RG method. The renormalization procedure within the MS-scheme with dimensional regularization is outlined.
•In chapter 3 we present the RG approach to critical phenomena. After an
abstract description we show how it can be implemented through quantum
field theory techniques.
•In chapter 4 we outline basic properties of quantum phase transitions.
Then we focus on generalized SU(N) antiferromagnets and we show how
their critical behaviour can be described by the continuum CP^(N-1) model.
•In chapter 5 we begin our study of the critical properties of the CP^(N-1) model. After a brief analysis within the mean field approximation, we
compute the RG β-functions and the fixed points of the RG flow at 1-loop
order in the framework of the ε-expansion.
•Chapter 6 contains a non perturbative analysis of the RG flow within the
3d-MS scheme. We then present our final results.

We briefly summarize here our results. The case
N = 2 is well known, the CP^1 model is equivalent to an O(3) non linear sigma model, therefore no further study is needed. For N = 3, in the theory without odd terms we find an enlarged
O(8) symmetry. The system may undergo a continuous phase transition which is
in the O(8) universality class. This is a remarkable result and it is in agreement
with the prediction of a continuous transition supported by other authors by
means of Monte Carlo simulations. For N ≥4 we do not find stable fixed points
after an analysis of the RG flow and we conclude that the transition should be
discontinuous.