1/N expansion for critical exponents of magnetic phase transitions in the CPN-1 model for 2<d<4 (original) (raw)

On the critical behavior of the magnetization in high-dimensional Ising models

Journal of Statistical Physics, 1986

Long-range components of the interaction in statistical mechanical systems may affect the critical behavior, raising the system's 'effective dimension'. Presented here are explicit implications to this effect of a collection of rigorous results on the critical exponents in ferromagnetic models with one-component lsing (and more generally Griffiths-Simon class) spin variables. In particular, it is established that even in dimensions d < 4 if a ferromagnetic Ising spin model has a reflection-positive pair interaction with a sufficiently slow decay, e.g. as Jx = 1/Ixl a+~ with 0 < a~< d/2, then the exponents ~, 6, ? and A 4 exist and take their mean-field values. This proves rigorously an early renormalization-group prediction of Fisher, Ma and Nickel. In the converse direction: when the decay is by a similar power law with try> 2, then the long-range part of the interaction has no effect on the existent critical exponent bounds, which coincide then with those obtained for short-range models.

Analysis of the 3d massive renormalization group perturbative expansions: a delicate case

Condensed Matter Physics, 2010

The effectiveness of the perturbative renormalization group approach at fixed space dimension d in the theory of critical phenomena is analyzed. Three models are considered: the O(N ) model, the cubic model and the antiferromagnetic model defined on the stacked triangular lattice. We consider all models at fixed d = 3 and analyze the resummation procedures currently used to compute the critical exponents. We first show that, for the O(N ) model, the resummation does not eliminate all non-physical (spurious) fixed points (FPs). Then the dependence of spurious as well as of the Wilson-Fisher FPs on the resummation parameters is carefully studied. The critical exponents at the Wilson-Fisher FP show a weak dependence on the resummation parameters. On the contrary, the exponents at the spurious FP as well as its very existence are strongly dependent on these parameters. For the cubic model, a new stable FP is found and its properties depend also strongly on the resummation parameters. It appears to be spurious, as expected. As for the frustrated models, there are two cases depending on the value of the number of spin components. When N is greater than a critical value Nc, the stable FP shows common characteristic with the Wilson-Fisher FP. On the contrary, for N < Nc, the results obtained at the stable FP are similar to those obtained at the spurious FPs of the O(N ) and cubic models. We conclude from this analysis that the stable FP found for N < Nc in frustrated models is spurious. Since Nc > 3, we conclude that the transitions for XY and Heisenberg frustrated magnets are of first order.

Effective critical behaviour of diluted Heisenberg-like magnets

Journal of Magnetism and Magnetic Materials, 2003

In agreement with the Harris criterion, asymptotic critical exponents of threedimensional (3d) Heisenberg-like magnets are not influenced by weak quenched dilution of non-magnetic component. However, often in the experimental studies of corresponding systems concentration-and temperature-dependent exponents are found with values differing from those of the 3d Heisenberg model. In our study, we use the field-theoretical renormalization group approach to explain this observation and to calculate the effective critical exponents of weakly diluted quenched Heisenberg-like magnet. Being non-universal, these exponents change with distance to the critical point T c as observed experimentally. In the asymptotic limit (at T c) they equal to the critical exponents of the pure 3d Heisenberg magnet as predicted by the Harris criterion.

O ct 2 01 5 Functional renormalization group approach to non-collinear magnets

2018

A functional renormalization group approach to d-dimensional, N-component, non-collinear magnets is performed using various truncations of the effective action relevant to study their long distance behavior. With help of these truncations we study the existence of a stable fixed point for dimensions between d = 2.8 and d = 4 for various values of N focusing on the critical value, Nc(d), that, for a given dimension d, separates a first order region for N < Nc(d) from a second order region for N > Nc(d). Our approach concludes to the absence of stable fixed point in the physical – N = 2, 3 and d = 3 – cases, in agreement with ǫ = 4 − d-expansion and in contradiction with previous perturbative approaches performed at fixed dimension and with recent approaches based on conformal bootstrap program.

Dimensional crossover in quantum critical metallic magnets

Physical Review B, 2008

Nearly magnetic metals often have layered lattice structures, consisting of coupled planes. In such a situation, physical properties will display, upon decreasing temperature or energy, a dimensional crossover from two-dimensional (2d) to three-dimensional (3d) behavior, which is particularly interesting near quantum criticality. Here we study this crossover in thermodynamics using a suitably generalized Landau-Ginzburg-Wilson approach to the critical behavior, combined with renormalization group techniques. We focus on two experimentally relevant cases: the crossover from a 2d to a 3d antiferromagnet, and the crossover from a 2d ferromagnet to a 3d antiferromagnet. As naive scaling does not apply at and above the upper critical dimension, two crossover scales arise which can be associated with separate dimensional crossovers of classical and quantum fluctuations, respectively. In particular, we find an intermediate regime with novel power laws where the quantum fluctuations still have a 2d and the classical fluctuations already have a 3d character. For the ferromagnet-to-antiferromagnet crossover, the mismatch of the dynamical exponents between the 2d and 3d regimes leads to an even richer crossover structure, with an interesting 2d non-critical regime sandwiched between two critical regimes. For all cases, we find that thermal expansion and compressibility are particularly sensitive probes of the dimensional crossover. Finally, we relate our results to experiments on the quantum critical heavy-fermion metals CeCu6−xAux, YbRh2Si2, and CeCoIn5.

The magnetic critical exponent in the three-state three-dimensional Potts model

Physics Letters B, 1990

We perform a numerical study of the Ports model q= 3 in three dimensions with nearest neighbour and next to nearest neighbout couplings by means of the finite-size renormalization group method. The analysis of lhe magnetic critical exponents is complementary to the one of the thermal critical exponent already presented by us and confirms our conclusions that the transition from the disordered phase to the low-temperature ordered phase is first order.

Critical dynamics of the antiferromagnetic O(3) nonlinear sigma model with conserved magnetization

Physical Review E

We study the near-equilibrium critical dynamics of the O(3) nonlinear sigma model describing isotropic antiferromagnets with a nonconserved order parameter reversibly coupled to the conserved total magnetization. To calculate response and correlation functions, we set up a description in terms of Langevin stochastic equations of motion, and their corresponding Janssen-De Dominicis response functional. We find that in equilibrium, the dynamics is well-separated from the statics, at least to one-loop order in a perturbative treatment with respect to the static and dynamical nonlinearities. Since the static nonlinear sigma model must be analyzed in a dimensional d = 2 + ε expansion about its lower critical dimension d lc = 2, whereas the dynamical modecoupling terms are governed by the upper critical dimension d c = 4, a simultaneous perturbative dimensional expansion is not feasible, and the reversible critical dynamics for this model cannot be accessed at the static critical renormalization group fixed point. However, in the coexistence limit addressing the long-wavelength properties of the low-temperature ordered phase, we can perform an = 4 − d expansion near d c. This yields anomalous scaling features induced by the massless Goldstone modes, namely subdiffusive relaxation for the conserved magnetization density with asymptotic scaling exponent z = d − 2, which may be observable in neutron scattering experiments. Intriguingly, if initialized near the critical point, the renormalization group flow for the effective dynamical exponents recovers their universal critical values z c = d/2 in an intermediate crossover region.

Extended Universal Finite-T Renormalization of Excitations in a Class of One-Dimensional Quantum Magnets

Physical Review Letters, 2008

Temperature dependencies of gap energies and magnon lifetimes are measured in the quasi-onedimensional S 1=2 gapped quantum magnets CH 3 2 CHNH 3 CuCL 3 (IPA-CuCl 3 , where IPA denotes isopropyl ammonium) and Cu 2 Cl 4 D 8 C 4 SO 2 (Sul-Cu 2 Cl 4 ) using inelastic neutron scattering. The results are compared to those found in literature for S 1 Haldane spin chain materials and to theoretical calculations for the O3-and ON-quantum nonlinear -models. It is found that when the T 0 energy gap is used as the temperature scale, all experimental and theoretical curves are identical to within system-dependent but temperature-independent scaling factors of the order of unity. This quasiuniversality extends over a surprising broad T range, at least up to T 1:5.

Effective and asymptotic criticality of structurally disordered magnets

arXiv (Cornell University), 2022

Changes in magnetic critical behaviour of quenched structurally-disordered magnets are usually exemplified in experiments and in MC simulations by diluted systems consisting of magnetic and non-magnetic components. By our study we aim to show, that similar effects can be observed not only for diluted magnets with non-magnetic impurities, but may be implemented, e.g., by presence of two (and more) chemically different magnetic components as well. To this end, we consider a model of the structurally-disordered quenched magnet where all lattice sites are occupied by Ising-like spins of different length L. In such random spin length Ising model the length L of each spin is a random variable governed by the distribution function p(L). We show that this model belongs to the universality class of the site-diluted Ising model. This proves that both models are described by the same values of asymptotic critical exponents. However, their effective critical behaviour differs. As a case study we consider a quenched mixture of two different magnets, with values of elementary magnetic moments L 1 = 1 and L 2 = s, and of concentration c and 1 − c, correspondingly. We apply field-theoretical renormalization group approach to analyze the renormalization group flow for different initial conditions, triggered by s and c, and to calculate effective critical exponents further away from the fixed points of the renormalization group transformation. We show how the effective exponents are governed by difference in properties of the magnetic components.