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Critical Behavior in Fluids
Published in Ian L. Spain, Jac Paauwe, High Pressure Technology, 2017
The idea of universality of critical behavior has arisen out of the study of the Ising model and other models of ferromagnetism. With the development of refined approximation methods, it became possible to estimate critical exponents in two- and three-dimensional Ising and other models with precision. Then, it became clear that critical exponents were sensitive to only a few parameters in the system’s Hamiltonian, while they did not seem to depend on others. A relevant parameter for instance is the system’s dimensionality: two- and three-dimensional Ising models have vastly different critical exponents. The specific lattice structure and the size of the spin appear to be irrelevant parameters that do not affect the values of critical exponents. The notion of universality implies that systems can be grouped into universality classes so that systems within each class differ only with respect to irrelevant parameters; as a consequence, they all have the same critical exponents. It is expected that systems in the same class will also obey the same scaled equation of state except for at most two system-dependent constants. Critical exponents vary discontinuously from class to class. Thus, in the definition (35) of a scaled equation of state, all substances in one universality class would have the same exponents β and δ, the same functional form for h(x/xo), but in principle different values of D and xo.
Chapter 5 Metal-Insulator Transition in Correlated Two-Dimensional Systems with Disorder
Published in Sergey Kravchenko, Strongly Correlated Electrons in Two Dimensions, 2017
such that the conductivity exponent μ = x(zν). Thus the behavior observed in the T = 0limit (Eq. 5.4) also provides a stringent test of the scaling behavior observed at T ≠ 0 (Eqs. 5.2 and 5.3). One of the goals of the scaling analysis is to reveal critical exponents that characterize the transition and correspond to specific universality classes.
Magnetic Properties of Nanostructures in Non-Integer Dimensions
Published in Jean-Claude Levy, Magnetic Structures of 2D and 3D Nanoparticles, 2018
Let us firstly recall the main features of such a transition in the framework of the standard critical phenomena theory [1] in the particular case of translationally invariant systems. The spontaneous symmetry breaking at the critical point lies in the heart of such a theory: The fundamental state of a system at zero temperature described by the Ising Hamiltonian is twice degenerated, since the spins are either all up or all down; an external field, regardless of how small it is, lifts the degeneracy since a potential barrier must be crossed in order to flip the whole set of spins. Such a symmetry breaking at the transition is closely linked to the characteristic size of the fluctuations. The main point is the emergence of a collective behavior when getting close to the critical temperature: The correlation length ξ, extracted from the two spins correlation function increases in such a way that fluctuations occur over larger and larger scales until ξ diverges right at the critical point. Hence, it is easy to understand that these fluctuations will drive the emergence of a macroscopic phenomenon, so that microscopic details are not relevant to the description of the critical behavior. Such an observation supports the striking notion of universality. In the thermodynamical limit, the singular behaviors of the order parameter m, the specific heat per spin C (thermal fluctuations of the energy), the susceptibility χ (thermal fluctuations of the order parameter), and the correlation-length ξ at the critical point are respectively described by the critical exponents β, α, γ, ν. Calling t=T−TcTc the relative deviation of the temperature T from the critical one Tc, they can be written in the following way: |m(t)| ~ |t|β with t < 0, C (t) ~ |t|−α, χ (t) ~ |t|−γ . ξ(t) ~ |t|−ν. The external field h is equal to 0, t is assumed to be small and the sign ~ is understood as an asymptotic behavior when t → 0. A universality class is characterized by a set of critical exponents {α, β, γ, ν}. In the case of translationally invariant systems, a universality class depends only upon the space dimension d, the symmetrical properties of the order parameter and the interaction range. It can be shown that these exponents are not independent but fulfill scaling laws [1]: The first one can be deduced from equilibrium thermodynamical properties (α + 2β + γ = 2), and the second one, called hyperscaling law, involves the space dimensionality (γ/ν + 2β/ν = d).
Two-dimensional quantum-spin-1/2 XXZ magnet in zero magnetic field: Global thermodynamics from renormalisation group theory
Published in Philosophical Magazine, 2019
From the RG flow diagram (see Figure 2), we can calculate the phase diagram of the system (see Figure 3). Under successive RG transformations, a point in the interaction parameters space flows to a sink. In the anisotropic XXZ model, the RG flows happen in the -space. Under successive RG transformations, the parameter always grows to infinity, since integrating more and more spins into a single renormalised spin adds more and more entropy associated with the fluctuations of the integrated spin degrees of freedom. We take for an original system, and in the -plane, different phases are characterised by flows to different sinks (see Table 1). Each transition between different phases is controlled by a corresponding fixed point (see Table 2). This critical fixed point determines the universality class of the transition.
Ising-type critical exponents of the fully frustrated spin-1/2 Heisenberg FM/AF square bilayer at a critical magnetic field
Published in Phase Transitions, 2019
To summarize, we have displayed in Figure 4 the predicted finite-temperature phase diagram of the fully frustrated spin-1/2 Heisenberg FM/AF square bilayer in the plane. It should be stressed that the investigated quantum spin model exhibits two different ground states: singlet-dimer phase and ferromagnetic phase (see the ground-state phase diagram shown in Figure 7(b) of [5]), which coexist together at zero temperature. At finite temperatures the singlet-dimer and ferromagnetic phases differ from each other in the number of singlet and polarized triplet states on vertical dimers, which implies that difference of the mean value of singlet and triplet states on vertical bonds may serve as the order parameter of the spin-1/2 Heisenberg FM/AF square bilayer up to the critical temperature where both phases become indistinguishable. This result suggests that a continuous (second-order) phase transition between two phases occurs at the critical temperature , whereas two phases apparently differ from each other in a density (total number) of singlet and triplet states below the critical temperature. It could be thus concluded that a continuous phase transition emergent at the critical field and critical temperature behaves according to the universality class of two-dimensional Ising model. Moreover, it is worthwhile to remark that the singlet-dimer and polarized triplet phases of the fully frustrated spin-1/2 Heisenberg FM/AF square bilayer can be accordingly considered in analogy as a liquid and gaseous phase of the same substance.
Monte Carlo study of the phase transitions in the classical XY ferromagnets with random anisotropy
Published in Phase Transitions, 2023
Olivia Mallick, Muktish Acharyya
Experimentally, the BKT transition was observed [16] in ultracold Fermi gas. The numerical simulation was performed to study [17] the structure and dynamics of two-dimensional XY ferromagnets in the presence of an applied in-plane magnetic field. An extensive Monte Carlo simulation was employed [18, 19] to study the critical behaviour of a three-dimensional XY ferromagnet. The critical exponents are estimated and the universality class is found. Very recently, the phase transition was studied [20] in the three-dimensional XY layered antiferromagnet.