Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userCritical exponents for Gaussian fixed point of renormalization
738
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We present mathematical details of derivation of the critical exponents for the free energy and magnetization in the vicinity of the Gaussian fixed point of renormalization. We treat the problem in general terms and do not refer to particular models of interaction energy. We discuss the case of arbitrary dispersion of the fixed point.
rate research
Read More
The paramagnetic-to-ferromagnetic phase transition is believed to proceed through a critical point, at which power laws and scaling invariance, associated with the existence of one diverging characteristic length scale -- the so called correlation length -- appear. We indeed observe power laws and scaling behavior over extraordinarily many decades of the suitable scaling variables at the paramagnetic-to-ferromagnetic phase transition in ultrathin Fe films. However, we find that, when the putative critical point is approached, the singular behavior of thermodynamic quantities transforms into an analytic one: the critical point does not exist, it is replaced by a more complex phase involving domains of opposite magnetization, below as well as $above$ the putative critical temperature. All essential experimental results are reproduced by Monte-Carlo simulations in which, alongside the familiar exchange coupling, the competing dipole-dipole interaction is taken into account. Our results imply that a scaling behavior of macroscopic thermodynamic quantities is not necessarily a signature for an underlying second-order phase transition and that the paramagnetic-to-ferromagnetic phase transition proceeds, very likely, in the presence of at least two long spatial scales: the correlation length and the size of magnetic domains.
The orientation fluctuations of the director of a liquid crystal are measured, by a sensitive polarization interferometer, close to the Freedericksz transition, which is a second order transition driven by an electric field. We show that near the critical value of the field the spatially averaged order parameter has a generalized Gumbel distribution instead of a Gaussian one. The latter is recovered away from the critical point. The relevance of slow modes is pointed out. The parameter of generalized Gumbel is related to the effective number of degrees of freedom.
Renormalization group theory does not restrict the from of continuous variation of critical exponents which occurs in presence of a marginal operator. However, the continuous variation of critical exponents, observed in different contexts, usually follows a weak universality scenario where some of the exponents (e.g., $beta, gamma, u$) vary keeping others (e.g., $delta , eta$) fixed. Here we report a ferromagnetic phase transition in (Sm$_{1-y}$Nd$_{y}$)$_{0.52}$Sr$_{0.48}$MnO$_3$ $(0.5le yle1)$ single crystal where all critical exponents vary with $y.$ Such variation clearly violates both universality and weak universality hypothesis. We propose a new scaling theory that explains the present experimental results, reduces to the weak universality as a special case, and provides a generic route leading to continuous variation of critical exponents and multicriticality.
A modified three-dimensional mean spherical model with a L-layer film geometry under Neumann-Neumann boundary conditions is considered. Two spherical fields are present in the model: a surface one fixes the mean square value of the spins at the boundaries at some $rho > 0$, and a bulk one imposes the standard spherical constraint (the mean square value of the spins in the bulk equals one). The surface susceptibility $chi_{1,1}$ has been evaluated exactly. For $rho =1$ we find that $chi_{1,1}$ is finite at the bulk critical temperature $T_c$, in contrast with the recently derived value $gamma_{1,1}=1$ in the case of just one global spherical constraint. The result $gamma_{1,1}=1$ is recovered only if $rho =rho_c= 2-(12 K_c)^{-1}$, where $K_c$ is the dimensionless critical coupling. When $rho > rho_c$, $chi_{1,1}$ diverges exponentially as $Tto T_c^{+}$. An effective hamiltonian which leads to an exactly solvable model with $gamma_{1,1}=2$, the value for the $nto infty $ limit of the corresponding O(n) model, is proposed too.
A nonconventional renormalization-group (RG) treatment close to and below four dimensions is used to explore, in a unified and systematic way, the low-temperature properties of a wide class of systems in the influence domain of their quantum critical point. The approach consists in a preliminary averaging over quantum degrees of freedom and a successive employment of the Wilsonian RG transformation to treat the resulting effective classical Ginzburg-Landau free energy functional. This allows us to perform a detailed study of criticality of the quantum systems under study. The emergent physics agrees, in many aspects, with the known quantum critical scenario. However, a richer structure of the phase diagram appears with additional crossovers which are not captured by the traditional RG studies. In addition, in spite of the intrinsically static nature of our theory, predictions about the dynamical critical exponent, which parametrizes the link between statics and dynamics close to a continuous phase transition, are consistently derived from our static results.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
