We argue for the logical necessity of the existence of a phase transition in a polymer system which is identical to the n=0 limit of two different magnetic systems, one possessing a continuous symmetry, the other one possessing a discrete symmetry. The phase transition occurs across a curve AC passing through the critical point, as has been argued recently by Gujrati. We accomplish this by demonstrating explicitly a phase transition across AC in the mean-field calculation of the discrete model for n1. We show that the phase below AC in this model has no polymer counterpart, and corresponds to a complete restoration of symmetry among the n components, even though the magnetic field couples to only one of the components. We discuss the nature of the magnetization and the susceptibilities on and near AC in the context of (i) the mean-field calculation with and without fluctuations and (ii) the \ensuremath{\epsilon} expansion. Within the general context of renormalization and scaling, we argue that in the scaling regime near the critical point, the longitudinal susceptibility must be positive for small magnetic fields, and the magnetization must not be less than the spontaneous magnetization, provided there is no transition across AC. Even though this result seems to be in contradiction with the \ensuremath{\epsilon} expansion result, we argue that there is no contradiction, and that our results are consistent with the \ensuremath{\epsilon} expansion results provided the existence of the phase transition is acknowledged.
We argue for the logical necessity of the existence of a phase transition in a polymer system which is identical to the n=0 limit of two different magnetic systems, one possessing a continuous symmetry, the other one possessing a discrete symmetry. The phase transition occurs across a curve AC passing through the critical point, as has been argued recently by Gujrati. We accomplish this by demonstrating explicitly a phase transition across AC in the mean-field calculation of the discrete model for n1. We show that the phase below AC in this model has no polymer counterpart, and corresponds to a complete restoration of symmetry among the n components, even though the magnetic field couples to only one of the components. We discuss the nature of the magnetization and the susceptibilities on and near AC in the context of (i) the mean-field calculation with and without fluctuations and (ii) the \ensuremath{\epsilon} expansion. Within the general context of renormalization and scaling, we argue that in the scaling regime near the critical point, the longitudinal susceptibility must be positive for small magnetic fields, and the magnetization must not be less than the spontaneous magnetization, provided there is no transition across AC. Even though this result seems to be in contradiction with the \ensuremath{\epsilon} expansion result, we argue that there is no contradiction, and that our results are consistent with the \ensuremath{\epsilon} expansion results provided the existence of the phase transition is acknowledged.
The O(n) model consisting of n-component spins $\stackrel{\ensuremath{\rightarrow}}{S}$ with the constraint ${\stackrel{\ensuremath{\rightarrow}}{S}}^{2}=\ensuremath{\lambda}$ in a magnetic field h is studied. It is shown that mathematically it is possible for the susceptibility to become negative for n<1, which implies a violation of convexity properties for n<1. In a mean-field approximation, the susceptibility ${\ensuremath{\chi}}_{n}$ and the specific heat Cn are positive near the critical temperature Tc for all $n\ensuremath{\ge}0$ in contradiction with the $\ensuremath{\epsilon}$ expansion, but they become negative at very low temperatures for n<1. It is also shown that the spontaneous magnetization is not a monotone function of temperature for n<1. Our calculation also supports the conclusion drawn by des Cloizeaux that the low-temperature phase of the O(0) model describes the semidilute regime of the polymer system as $h\ensuremath{\rightarrow}0$.
Introduction Our discussion of critical phenomena surrounding a second-order phase transition has thus far focused only on qualitative features. We have now examined three systems, fluids, magnets and random percolation, each of which displays an abruptly sharp transition from a less ordered to a more ordered phase as the respective transition point is encountered. Each shows a similar pattern of developing structure just in advance of the critical point. Fluctuations in the respective order parameter display a self-similar structure that is limited only by a single relevant length scale, the correlation length ξ, which diverges on approach to the transition point. For fluids and magnets, this structure arises from inherent fluctuations that are amplified by either a diverging compressibility or susceptibility, respectively. In this chapter, we explore more quantitative, theoretical approaches taken to understand the features of second-order phase transitions. The simplest of these are the mean field theories , in which the pairwise interaction (needed to produce a phase transition) is introduced in the form of an average field. In this approach, the effects of the growing fluctuations of the order parameter near the critical point are ignored. Although the mean field approach does meet many of the requirements and does predict divergences of certain quantities near the transition, the critical exponents predicted by the theory do not match those seen experimentally. Obtaining correct exponents requires a more advanced approach involving renormalization techniques that exploit the self-similar structure of the fluctuations near the critical point and allow all of the various critical exponents to be inter-related, such that knowledge of any two yields all the others.
Determining whether an additional local symmetry affects the universality class of a sta-tistical model is an important issue in the theory of critical phenomena. A basic example is provided by the RP N −1 model, in which N -component spin variables at each lattice site interact through an Hamiltonian invariant under global O(N ) rotations and local spin re-versals. The local symmetry makes the difference with the usual O(N ) model and amounts to the head-tail symmetry characteristic of liquid crystals. In three dimensions, the weak first order transition observed in numerical simulations of the ferromagnetic model is con-sistent with the mean field scenario. On the other hand, in the two-dimensional case –the one we focus on in this thesis – fluctuations are stronger and minimize the reliability of mean field predictions, as illustrated by the phase transition of the three-state Potts model, which becomes continuous on planar lattices. For the RP N −1 model, the absence of spontaneous breaking of continuous symmetry in two dimensions generically suggests that criticality is limited to zero temperature, and Monte Carlo studies for T → 0 showed a fast growth of the correlation length which made particularly hard to reach the asymp-totic limit and draw conclusions. On the other hand, the possibility of finite temperature topological transitions similar to the Berezinskii-Kosterlitz-Thouless (BKT) one – which should definitely occur for RP 1 ∼ O(2) – and mediated by ”disclination” defects has also been debated in numerical studies. While two-dimensional criticality has allowed for an impressive amount of exact solutions thanks to lattice integrability and conformal field theory, models with local symmetries traditionally remained outside the range of application of these methods. In this thesis, however, we show how the renormalization group fixed points of the RP N −1 model and of its complex generalization, the CP N −1 model, can be accessed in an exact way. This is achieved in the scale invariant scattering framework which, as we review in the introductory part of the thesis, implements in the basis of particle excitations the infinite-dimensional conformal symmetry characteristic of critical points in two dimensions and has provided in the last few years new results for pure and disordered systems. In the last part of the thesis, we will exploit the generality of the scale invariant scattering method to progress with another long standing problem of two-dimensional criticality, namely that of spin clusters in Potts correlated percolation. It has been known for long time that the problem can be addressed considering two coupled Potts models, but the need to consider the number of states of one of these models as a continuous variable has severely limited the analytical or numerical study of the critical points. Also here, the ability of the scattering method to enforce conformal invariance for the internal symmetry characteristic of the universality class will allow us to obtain exact equations for the critical points and to determine their solutions in the relevant limits.
We reconsider the problem of the critical behavior of a three-dimensional O(m) symmetric magnetic system in the presence of random-anisotropy disorder with a generic trimodal random-axis distribution. By introducing n replicas to average over disorder it can be coarse grained to a ${\ensuremath{\phi}}^{4}$ theory with an $m\ifmmode\times\else\texttimes\fi{}n$ component order parameter and five coupling constants taken in the limit of $n\ensuremath{\rightarrow}0$. Using a field theory approach we renormalize the model to two-loop order and calculate the $\ensuremath{\beta}$ functions within the $\ensuremath{\varepsilon}$ expansion and directly in three dimensions. We analyze the corresponding renormalization group flows with the help of the Pad\'e-Borel resummation technique. We show that there is no stable fixed point accessible from physical initial conditions whose existence was argued in previous studies. This may indicate the absence of a long-range ordered phase in the presence of random-anisotropy disorder with a generic random-axis distribution.
We consider classical spin models of two- and three-dimensional spins with continuous symmetry and investigate the effect of a symmetry-breaking unidirectional quenched disorder on the magnetization of the system. We work in the mean-field regime. We show by perturbative calculations and numerical simulations that although the continuous symmetry of the magnetization is lost due to disorder, the system still magnetizes in specific directions, albeit with a lower value as compared to the case without disorder. The critical temperature at which the system starts magnetizing and the magnetization at low- and high-temperature limits in the presence of disorder are estimated. Moreover, we treat the SO(n)n-component spin model to obtain the generalized expressions for the near-critical scalings, which suggest that the effect of disorder in magnetization increases with increasing dimension. We also study the behavior of magnetization of the classical XY spin model in the presence of a constant magnetic field, in addition to the quenched disorder. We find that in the presence of the uniform magnetic field, disorder may enhance the component of magnetization in the direction that is transverse to the disordered field.
In this work we use field-theoretic and renormalization-group methods to study the finite-temperature ferromagnetic phase transition in the three-dimensional Hubbard model. We show that the nature of the ferromagnetic transition may strongly depend on the constraint imposed on the system. For example, if the system is allowed to exchange particles with a reservoir, the stability criterion for the occurrence of a continuous phase transition can be calculated using the mean-field approximation. On the other hand, if the total charge is conserved and charge- and spin-density fluctuations are considered in the theory, a different criterion, quite distinct from the mean-field one, is found, resulting in a fluctuation-induced renormalized Heisenberg tricritical point. The tricritical behavior is studied by calculating the one-loop renormalized free energy and using scaling analysis to determine the tricritical exponents. Finally, the exact solution of the model in the spherical limit and near the ferromagnetic transition is presented, in which case no fluctuation-induced tricritical point is found.