Abstract The two-dimensional case occupies a special position in the theory of critical phenomena due to the exact results provided by lattice solutions and, directly in the continuum, by the infinite-dimensional character of the conformal algebra. However, some sectors of the theory, and most notably criticality in systems with quenched disorder and short-range interactions, have appeared out of reach of exact methods and lacked the insight coming from analytical solutions. In this article, we review recent progress achieved implementing conformal invariance within the particle description of field theory. The formalism yields exact unitarity equations whose solutions classify critical points with a given symmetry. It provides new insight in the case of pure systems, as well as the first exact access to criticality in presence of short range quenched disorder. Analytical mechanisms emerge that in the random case allow the superuniversality of some critical exponents and make explicit the softening of first-order transitions by disorder. Graphic abstract
Abstract The two-dimensional case occupies a special position in the theory of critical phenomena due to the exact results provided by lattice solutions and, directly in the continuum, by the infinite-dimensional character of the conformal algebra. However, some sectors of the theory, and most notably criticality in systems with quenched disorder and short-range interactions, have appeared out of reach of exact methods and lacked the insight coming from analytical solutions. In this article, we review recent progress achieved implementing conformal invariance within the particle description of field theory. The formalism yields exact unitarity equations whose solutions classify critical points with a given symmetry. It provides new insight in the case of pure systems, as well as the first exact access to criticality in presence of short range quenched disorder. Analytical mechanisms emerge that in the random case allow the superuniversality of some critical exponents and make explicit the softening of first-order transitions by disorder. Graphic abstract
This article outlines a tutorial review of some recent progress in the theory of phase transitions and criticality. Topics mentioned are: |φ|6 field theories in (d=3)-dimensional space which correspond to the limits of tricriticality in spherical models; the desirability of finding real systems displaying critical endpoints, etc., in a magnetic field; scaling theories for first-order magnetic transitions in block and cylinder geometries, including spin-wave contributions, ϵ=4 - d expansions for finite-size effects at and near criticality; the universal correlation length amplitude for critical-point correlations in a finite cylinder and its evaluation in d = 2 dimensions on the basis of conformal covariance; the predictions of conformal covariance for other finite-size and surface critical-point properties; the implications, stemming from field theory, of conformal covariance and unitarity for general criticality in two dimensions, including infinitely many sets of exact critical exponents for Ising-like, three-state Potts-like, … systems; the discovery of new, exactly soluble two-dimensional, “restricted solid-on-solid” models which display all the predicted conformally covariant multicritical points and a second infinite set as well!
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.
Phase transitions of second order play an important role in solid state physics. It is in their nature that they occur only in systems that are composed from an infinite number of components since, only that way, the necessary conditions for this (for example, scale invariance) are granted. Such a system at a second order phase transition is called critical. The infinite number of particles poses a particular challenge, even for the description of classical systems. Quantum mechanical systems, however, are distinctly more difficult to treat in this context, since the dimension of their state space grows exponentially with the number of their particles, not to mention the fact that only a few – and usually only one-dimensional – quantum systems are exactly solvable. One of these exactly solvable one-dimensional quantum systems is the SU(N) Haldane-Shastry spin chain that may be regarded as the archetype of long-range spin chains. Moreover, it is critical in the continuum limit and its effective low-energy theory is the so-called SU(N)_1 Wess-Zumino-Witten model. It is a quantum field theory which is not only conformally invariant, so, in particular, scale invariant, but, furthermore, exhibits additional symmetry in the shape of an infinite extension of the Lie algebra su(N) associated with SU(N). Recent studies show that, from these very structures of the SU(N)_1 WZW model, one can, in turn, derive spin systems, whose arrangement is not necessarily the one of a spin chain, but even two-dimensional distributions of the spins in the plane are possible. These systems are again characterized by long-range interactions, comparable to those of the Haldane-Shastry spin chain, which is also obtained as a result of an appropriate choice of parameters. \n \nIn this thesis, we extend the construction already known for the SU(N) case to the supersymmetric case GL(m|n). Here, we construct explicitly both, a special quantum state as well as a Hamiltonian that projects this quantum state to zero. We also discuss the Hamiltonian in the special case of the GL(1|1) spin chain and compare it to the respective GL(1|1) Haldane-Shastry spin chain on a bipartite state space. Both are critical and we identify the corresponding conformal field theories. Subsequently, we describe a generalization of this system in terms of two parameters and explain how its spectrum was found. It is then analyzed and its continuum limit is determined. In doing so, it shows that the system displays criticality only for generic values of one of the two parameters.
The aim of this contribution is to draw attention to some recent developments in the theory of critical phenomena and phase transitions that may be of interest to the magnetism community in that magnetic systems have provided a fertile testing ground for modern theoretical ideas concerning ordering and phase transitions while, on the other hand, many aspects of observed magnetic phase behavior have required sophisticated theory for their proper elucidation. The new works in question have been published or will appear shortly: accordingly, only an outline is presented here together with references to the main literature sources which, in turn, include fuller references to earlier work not explicitly mentioned. Four linked topics are addressed: (i) finite-size effects at first-order transitions; (ii) asymptotic degeneracy in transfer matrices, which is intimately related to long-range order; (iii) some consequences of the conformal covariance of fluctuations at a critical point in bulk and in finite systems, especially in two spatial dimensions; (iv) the discovery of ‘‘exactly soluble’’ two-dimensional models exhibiting two infinite classes of distinct critical and multicritical points.
Recently, a concept of generalized multifractality, which characterizes fluctuations and correlations of critical eigenstates, was introduced and explored for all 10 symmetry classes of disordered systems. Here, by using the nonlinear sigma-model ($\mathrm{NL}\ensuremath{\sigma}\mathrm{M}$) field theory, we extend the theory of generalized multifractality to boundaries of systems at criticality. Our numerical simulations on two-dimensional systems of symmetry classes A, C, and AII fully confirm the analytical predictions of pure-scaling observables and Weyl symmetry relations between critical exponents of surface generalized multifractality. This demonstrates the validity of the $\mathrm{NL}\ensuremath{\sigma}\mathrm{M}$ for the description of Anderson-localization critical phenomena, not only in the bulk but also on the boundary. The critical exponents strongly violate generalized parabolicity, in analogy with earlier results for the bulk, corroborating the conclusion that the considered Anderson-localization critical points are not described by conformal field theories. We further derive relations between generalized surface multifractal spectra and linear combinations of Lyapunov exponents of a strip in quasi-one-dimensional geometry, which hold under the assumption of invariance with respect to a logarithmic conformal map. Our numerics demonstrate that these relations hold with an excellent accuracy. Taken together, our results indicate an intriguing situation: the conformal invariance is broken but holds partially at critical points of Anderson localization.
Abstract We obtain the exact scale invariant scattering solutions for two-dimensional field theories with replicated permutational symmetry S q . After sending to zero the number of replicas they correspond to the renormalization group fixed points of the q -state Potts model with quenched disorder. We find that all solutions with non-zero disorder possess q -independent sectors, pointing to superuniversality (i.e. symmetry independence) of some critical exponents. The solution corresponding to the random bond ferromagnet, for which disorder vanishes as q → 2 , allows for superuniversality of the correlation length exponent ν (Delfino 2017 Phys. Rev. Lett . 118 250601). Of the two solutions which are strongly disordered for all values of q , one is completely q -independent and accounts for the zero-temperature percolation fixed point of the randomly bond diluted ferromagnet. The other is the main candidate to describe the Nishimori-like fixed point of the Potts model with ± J disorder, and leaves room for superuniversality of the magnetic exponent η , a possibility not yet excluded by available numerical data.