Critical Exponent Beta Research Articles

New insight into 3-picoline—deuterium oxide (D2O) mixtures of limited miscibility with the lower critical consolute temperature

Coexistence curves in mixtures of limited miscibility with the lower critical consolute temperature (LCT), of 3-picoline with deuterium oxide (D2O), and D2O/H2O have been determined. They were tested with respect to the order parameter, the diameter of the binodal, and coordinates of the critical consolute point (critical temperature {T}_{mathrm{C}} and critical concentration {phi }_{mathrm{C}}). Studies were carried out using the innovative method based on the analysis of relative volumes occupied by coexisting phases, yielding high-resolution data. The clear violation of the Cailletet-Mathew law of rectilinear diameter for the LCT mixtures of limited miscibility is evidenced. For the order parameter, the new distortion-sensitive analysis method yielded the evidence for the model-value of the order parameter critical exponent beta =0.326pm 0.003, up to ca. 1 K from {T}_{mathrm{C}}. Finally, the simple & easy method for determining the critical concentration by testing relative volumes of coexisting phases (or alternatively fractional meniscus heights, h) is presented. The significance of the invariant value {h}_{mathrm{C}}=hleft({T}_{mathrm{C}},{phi }_{mathrm{C}}right)=1/2 is highlighted. The appearance of the milky & bluish critical opalescence is also shown.Graphical abstract

Three-state majority-vote model on small-world networks

In this work, we study the opinion dynamics of the three-state majority-vote model on small-world networks of social interactions. In the majority-vote dynamics, an individual adopts the opinion of the majority of its neighbors with probability 1-q, and a different opinion with chance q, where q stands for the noise parameter. The noise q acts as a social temperature, inducing dissent among individual opinions. With probability p, we rewire the connections of the two-dimensional square lattice network, allowing long-range interactions in the society, thus yielding the small-world property present in many different real-world systems. We investigate the degree distribution, average clustering coefficient and average shortest path length to characterize the topology of the rewired networks of social interactions. By employing Monte Carlo simulations, we investigate the second-order phase transition of the three-state majority-vote dynamics, and obtain the critical noise q_c, as well as the standard critical exponents beta /nu, gamma /nu, and 1/nu for several values of the rewiring probability p. We conclude that the rewiring of the lattice enhances the social order in the system and drives the model to different universality classes from that of the three-state majority-vote model in two-dimensional square lattices.

Entanglement Order Parameters and Critical Behavior for Topological Phase Transitions and Beyond

Topological phases are exotic quantum phases which are lacking the characterization in terms of order parameters. In this paper, we develop a unified framework based on variational iPEPS for the quantitative study of both topological and conventional phase transitions through entanglement order parameters. To this end, we employ tensor networks with suitable physical and/or entanglement symmetries encoded, and allow for order parameters detecting the behavior of any of those symmetries, both physical and entanglement ones. First, this gives rise to entanglement-based order parameters for topological phases. These topological order parameters allow to quantitatively probe topological phase transitions and to identify their universal behavior. We apply our framework to the study of the Toric Code model in different magnetic fields, which in some cases maps to the (2+1)D Ising model. We identify 3D Ising critical exponents for the entire transition, consistent with those special cases and general belief. However, we moreover find an unknown critical exponent beta=0.021. We then apply our framework of entanglement order parameters to conventional phase transitions. We construct a novel type of disorder operator (or disorder parameter), which is non-zero in the disordered phase and measures the response of the wavefunction to a symmetry twist in the entanglement. We numerically evaluate this disorder operator for the (2+1)D transverse field Ising model, where we again recover a critical exponent hitherto unknown in the model, beta=0.024, consistent with the findings for the Toric Code. This shows that entanglement order parameters can provide additional means of characterizing the universal data both at topological and conventional phase transitions, and altogether demonstrates the power of this framework to identify the universal data underlying the transition.

Three-State Majority-vote Model on Scale-Free Networks and the Unitary Relation for Critical Exponents

We investigate the three-state majority-vote model for opinion dynamics on scale-free and regular networks. In this model, an individual selects an opinion equal to the opinion of the majority of its neighbors with probability 1 − q, and different to it with probability q. The parameter q is called the noise parameter of the model. We build a network of interactions where z neighbors are selected by each added site in the system, a preferential attachment network with degree distribution k−λ, where λ = 3 for a large number of nodes N. In this work, z is called the growth parameter. Using finite-size scaling analysis, we obtain that the critical exponents beta /bar{nu } and gamma /bar{nu } associated with the magnetization and the susceptibility, respectively. Using Monte Carlo simulations, we calculate the critical noise parameter qc as a function of z for the scale-free networks and obtain the phase diagram of the model. We find that the critical exponents add up to unity when using a special volumetric scaling, regardless of the dimension of the network of interactions. We verify this result by obtaining the critical noise and the critical exponents for the two and three-state majority-vote model on cubic lattice networks.

Assessment of the critical behavior near the FM to PM phase transition in nano-crystalline La0.7Ca0.3Mn1−xNixO3 (x = 0, 0.02, 0.07, 0.10) samples synthesized by auto-combustion

The structural and critical exponent analysis along with the universal behavior of the magnetocaloric effect in nano-crystalline La0.7Ca0.3Mn1−xNixO3 (x = 0.0, 0.02, 0.07, 0.10), manganites, exhibiting second order phase transition, are experimentally studied. Structural study using Rietveld refinement of X-ray diffraction patterns shows orthorhombic structure with Pnma space group. The Curie temperature could be tuned over a wide temperature range upon variation in the Ni2+ content. The nature of the transition was found to be of second order for all Ni2+-concentrations. Various techniques such as modified Arrott plots, Kouvel-Fisher method and critical isotherm were used to analyze the magnetic-field dependence of magnetization. Although the critical exponents beta (β), gamma (γ) and delta (δ) were analyzed with various theoretical models such as 3D-Ising model and 3D-Heisenberg model, the estimated critical exponents, obtained for different values of X, seemed to be consistent with those predicted by the mean-field theory. The critical exponents were further confirmed using the single scaling equation M(H,ε) = εβf ± (H/εγ+β). The achieved results indicated that the obtained values of the critical exponents were reliable and reasonably accurate.

One-dimensional backreacting holographic p-wave superconductors

We analytically and numerically study the properties of one-dimensional holographic p-wave superconductors in the presence of backreaction. We employ the Sturm–Liouville eigenvalue problem for the analytical calculation and the shooting method for the numerical investigations. We apply the hbox {AdS}_{{3}}/hbox {CFT}_{{2}} correspondence and determine the relation between the critical temperature T_{c} and the chemical potential mu for different values of the mass m of a charged spin-1 field rho _{mu } and backreacting parameters. We observe that the data of both analytical and numerical studies are in good agreement. We find that increasing the backreaction and the mass parameter causes the greater values for {T_{c}}/{mu }. Thus, it makes the condensation harder to form. In addition, the analytical and numerical approaches show that the value of the critical exponent beta is 1 / 2, which is the same as in the mean field theory. Moreover, both methods confirm the existence of a second order phase transition.

Critical behavior of magnetization in URhAl: Quasi-two-dimensional Ising system with long-range interactions

The critical behavior of dc magnetization in the uranium ferromagnet URhAl with the hexagonal ZrNiAl-type crystal structure has been studied around the ferromagnetic transition temperature T_C. The critical exponent beta for the temperature dependence of the spontaneous magnetization below T_C, gamma for the magnetic susceptibility, and delta for the magnetic isotherm at T_C have been obtained with a modified Arrott plot, a Kouvel-Fisher plot, the critical isotherm analysis and the scaling analysis. We have determined the critical exponents as beta = 0.287 +- 0.005, gamma = 1.47 +- 0.02, and delta = 6.08 +- 0.04 by the scaling analysis and the critical isotherm analysis. These critical exponents satisfy the Widom scaling law delta=1+gamma/beta. URhAl has strong uniaxial magnetic anisotropy, similar to its isostructural UCoAl that has been regarded as a three-dimensional (3D) Ising system in previous studies. However, the universality class of the critical phenomenon in URhAl does not belong to the 3D Ising model (beta = 0.325, gamma = 1.241, and delta = 4.82) with short-range exchange interactions between magnetic moments. The determined exponents can be explained with the results of the renormalization group approach for a two-dimensional (2D) Ising system coupled with long-range interactions decaying as J(r)~r^-(d+sigma) with sigma = 1.44. We suggest that the strong hybridization between the uranium 5f and rhodium 4d electrons in the U-Rh_I layer in the hexagonal crystal structure is a source of the low dimensional magnetic property. The present result is contrary to current understandings of the physical properties in a series of isostructural UTX uranium ferromagnets (T: transition metals, X: p-block elements) based on the 3D Ising model.

Classical Monte Carlo Study for Antiferro Quadrupole Orders in a Diamond Lattice

We investigate antiferro quadrupole orders in a diamond lattice under magnetic fields by Monte Carlo simulations for two types of classical effective models. One is an XY model with Z_3 anisotropy, and the other is a two-component phi^4 model with a third-order anisotropy. We confirm that the universality class of the zero-field transition is that for the three-dimensional XY model. Magnetic field corresponds to a Z_3 field in the effective model, and under this field, we find that collinear and canted antiferro-quadrupole orders compete. Each phase is characterized by symmetry breaking in the sector of (sublattice Z_2)x(reflection Z_2 for the order parameter). When Z_3 anisotropy and magnetic field vary, it turns out that this system is a good playground for various multicritical points; bicritical and tetracritical points emerge in a finite field. Another important finding is about the scaling of parasitic ferro quadrupole order at the zero-field critical point. This is the secondary order parameter induced by the primary antiferro order, and its critical exponent beta'=0.815 clearly differs from the expected value that is twice the value for the primary order parameter. The corresponding correlation length exponent is also different, nu'=0.597(12). We also discuss relation of the present effective quadrupole models with the 3-state Potts model as well as implication to undertanding of orbital orders in Pr-based 1-2-20 compounds.

Chiral matrix model of the semi-QGP in QCD

A chiral matrix model applicable to QCD with 2+1 flavors is developed. This requires adding a SU(3)_L x SU(3)_R x Z(3)_A nonet of scalar fields, with both parities, and coupling these to quarks through a Yukawa coupling, y. Treating the scalar fields in mean field approximation, the effective Lagrangian is computed by integrating out quarks to one loop order. In addition to the usual symmetry breaking term, linear in the current quark mass m_qk, at a nonzero temperature T it is necessary to add a new term, ~ m_qk T^2. The parameters of the gluon part of the matrix model are identical to that for the pure glue theory without quarks. The parameters in the chiral matrix model are fixed by the values, at zero temperature, of the pion decay constant and the masses of the pions, kaons, eta, and eta'. The temperature for the chiral crossover at T_chi = 155 MeV is determined by adjusting the Yukawa coupling y. We find reasonable agreement with the results of numerical simulations on the lattice for the pressure and related quantities. In the chiral limit, besides the divergence in the chiral susceptibility there is also a milder divergence in the susceptibility between the Polyakov loop and the chiral order parameter, with critical exponent beta - 1. We compute derivatives with respect to a quark chemical potential to determine the susceptibilities for baryon number, the chi_2n. Especially sensitive tests are provided by chi_4 - chi_2 and by chi_6, which changes in sign about T_chi. The behavior of the susceptibilities in the chiral matrix model strongly suggests that as the temperature increases from T_chi, that the transition to deconfinement is significantly quicker than indicated by the measurements of the (renormalized) Polyakov loop on the lattice.

Phase diagram structure of topological Mott transition for zero-gap semiconductors beyond conventional Landau-Ginzburg-Wilson scenario

We show that a wide class of unconventional quantum criticality emerges when orbital currents cause quantum phase transitions from zero-gap semiconductors such as Dirac fermions to topological insulator (TI) or Chern insulator (CI). Changes in Fermi surface topology concomitant with (SU(2) or time reversal) symmetry breakings generate quantum critical lines (QCL) even beyond the quantum critical point. This QCL running at temperature $T=0$ separates two distinct topological phases. This is in contrast to the simple termination of the finite temperature critical line at the quantum critical point without any extension of it at $T=0$. Topology change causes the unconventionality beyond the concept of simple spontaneous symmetry breaking assumed in the conventional Landau-Ginzburg-Wilson (LGW) scenario. The unconventional universality implied by mean-field critical exponents beta>1/2 and delta<3 is protected by the existence of the quantum critical line. It emerges for several specific lattice models including the honeycomb, kagome, diamond and pyrochlore lattices. We also clarify phase diagrams of the topological phases in these lattices at finite temperatures.

Intrinsically limited critical temperatures of highly dopedGa1−xMnxAsthin films

Ga1-xMnxAs films with exceptionally high saturation magnetizations of approximate to 100 emu/cm(3) corresponding to effective Mn concentrations of x(eff)approximate to 0.10 still have a Curie temperature T-C smaller than 195 K contradicting mean-field predictions. The analysis of the critical exponent beta of the remnant magnetization-beta = 0.407(5)-in the framework of the models for disordered/amorphous ferromagnets suggests that this limit on T-C is intrinsic and due to the short range of the ferromagnetic interactions resulting from the small mean-free path of the holes. This result questions the perspective of room-temperature ferromagnetism in highly doped GaMnAs.

Density Functional Theory Integrated with Renormalization Group Theory for Criticality of Nanoconfined Fluids

Nanoconfined fluids exhibit complicated behavior such as phase transitions, prewetting, wetting transitions, universal critical properties, and critical point shifts, attributed to the cooperative interactions of short- and long-range density fluctuations. In this work, a theoretical approach, an integrated density functional (DF) and renormalization group (RG) theory, is developed to investigate the critical properties of fluids confined in slit pores or bounded by single wall surfaces. The approach can take into account both the local and the long-range density fluctuations of confined fluids self-consistently. The predicted critical order parameter exponent beta is about 1/8, and the wetting exponent beta(s) is 1.056 +/- 0.011. These values are quite close to the two-dimensional Ising values, demonstrating that the new theoretical approach is reliable as a description of criticality in nanoconfined fluids. Correspondingly, the global phase transitions and surface wetting transitions are analyzed.

Critical properties of the unconventional spin-Peierls system TiOBr

We have performed detailed x-ray scattering measurements on single crystals of the spin-Peierls compound TiOBr in order to study the critical properties of the transition between the incommensurate spin-Peierls state and the paramagnetic state at Tc2 ~ 48 K. We have determined a value of the critical exponent beta which is consistent with the conventional 3D universality classes, in contrast with earlier results reported for TiOBr and TiOCl. Using a simple power law fit function we demonstrate that the asymptotic critical regime in TiOBr is quite narrow, and obtain a value of beta_{asy} = 0.32 +/- 0.03 in the asymptotic limit. A power law fit function which includes the first order correction-to-scaling confluent singularity term can be used to account for data outside the asymptotic regime, yielding a more robust value of beta_{avg} = 0.39 +/- 0.05. We observe no evidence of commensurate fluctuations above Tc1 in TiOBr, unlike its isostructural sister compound TiOCl. In addition, we find that the incommensurate structure between Tc1 and Tc2 is shifted in Q-space relative to the commensurate structure below Tc1.

Percolation in polymer-solvent systems: A Monte Carlo study

In this study we investigated the percolation in the system containing long flexible polymer chains. The system also contained explicit solvent molecules. The polymer chains were represented by linear sequences of lattice points restricted to a two-dimensional triangular lattice. The Monte Carlo simulations were performed applying the cooperative motion algorithm. The percolation thresholds and the critical exponents of chains and solvent molecules were determined. The influence of the chain length on the percolation was discussed. It was shown that the percolation threshold decreased strongly with the chain length, which is closely connected to changes in chains' structure with the decreasing polymer concentration. The critical exponent beta for all chains under consideration and for solvent molecules was found almost constant and close to the theoretical value 5/36.

The isotropic-to-nematic transition in a two-dimensional fluid of hard needles: a finite-size scaling study

The isotropic-to-nematic transition in a two-dimensional fluid of hard needles is studied using grand canonical Monte Carlo simulations, multiple histogram reweighting, and finite size scaling. The transition is shown to be of the Kosterlitz-Thouless type, via a direct measurement of the critical exponents eta and beta, of the susceptibility and order parameter, respectively. At the transition, eta=1/4 and beta=1/8 are observed, in excellent agreement with Kosterlitz-Thouless theory. Also the shift in the chemical potential of the nematic susceptibility maximum with system size is in good agreement with theoretical expectations. Some evidence of singular behavior in the density fluctuations is observed, but no divergence, consistent with a negative specific heat critical exponent. At the transition, a scaling analysis assuming a conventional critical point also gives reasonable results. However, the apparent critical exponent beta_eff obtained in this case is not consistent with theoretical predictions.

Phase transition, crystal structure, and magnetic order in VOCl

VOCl develops magnetic order below ${T}_{\text{N}}=80.3(3)\text{ }\text{K}$. Employing single-crystal x-ray diffraction and powder neutron diffraction it is shown that the magnetic phase transition is accompanied by a small, temperature-dependent monoclinic lattice distortion with $\ensuremath{\gamma}=90.212(2)\ifmmode^\circ\else\textdegree\fi{}$ at $T=3.2\text{ }\text{K}$. The phase transition presumably is of second order with a critical exponent $\ensuremath{\beta}=0.099(12)$. The monoclinic distortion lifts the frustration of interchain interactions along $\frac{1}{2}\mathbf{a}\ifmmode\pm\else\textpm\fi{}\frac{1}{2}\mathbf{b}$ of the orthorhombic room-temperature structure with the result that chains of antiferromagnetic order exist along $\frac{1}{2}\mathbf{a}+\frac{1}{2}\mathbf{b}$ while ferromagnetic chains exist along $\frac{1}{2}\mathbf{a}\ensuremath{-}\frac{1}{2}\mathbf{b}$. Both antiferromagnetic interchain coupling (if stronger along $\frac{1}{2}\mathbf{a}+\frac{1}{2}\mathbf{b}$ than along $\frac{1}{2}\mathbf{a}\ensuremath{-}\frac{1}{2}\mathbf{b}$) and ferromagnetic interchain coupling (if stronger along $\frac{1}{2}\mathbf{a}\ensuremath{-}\frac{1}{2}\mathbf{b}$ than along $\frac{1}{2}\mathbf{a}+\frac{1}{2}\mathbf{b}$) can explain the observed magnetic superstructure with magnetic space group $F{2}^{\ensuremath{'}}/{d}^{\ensuremath{'}}$ on the $2a\ifmmode\times\else\texttimes\fi{}2b\ifmmode\times\else\texttimes\fi{}2c$ supercell (standard setting $C{2}^{\ensuremath{'}}/{c}^{\ensuremath{'}}$) and with magnetic moments parallel to $\mathbf{a}$. Remarkably, the lattice distortion is not accompanied by a structural distortion so that the driving force for the lattice distortion is completely governed by the magnetic interactions.

Direct Measurement of the Percolation Probability in Carbon Nanofiber-Polyimide Nanocomposites

We present the first experimental measurement of the geometric critical exponent beta associated with the percolation probability, the probability a metallic filler belongs to the conducting network, of an electrical composite. The technique employs conducting-tip atomic force microscopy to obtain a conducting areal density, and is demonstrated on polyimide nanocomposites containing different concentrations of carbon nanofibers. We find beta approximately 1 and t (the exponent for bulk conductivity) approximately 3. These values are consistent with the predictions for the Bethe lattice and larger than the values predicted in the 3D lattice percolation model. Hence, this electrical composite likely belongs to the same universality class as the Bethe lattice. The ability to measure geometric and transport critical exponents on the same material is critical to drawing this conclusion.

Coexistence of Ferromagnetism and Paramagnetism in a Ferromagnetic Monolayer

The ferromagnetic-paramagnetic phase transition in an Fe monolayer grown on a 2 ML Au/W(110) substrate and capped with a 1 ML Au layer was studied with spin polarized low electron energy microscopy. When the data are analyzed as in previous studies, not only the Curie temperature T(C) but also the effective critical exponent beta depends upon the terrace width. Taking finite size effects into account gives the terrace width-independent two-dimensional Ising model value beta=1/8.

Orbital-Ordering Transition inSr2VO4

Temperature-dependent dc susceptibility and x-ray powder diffraction on a pure tetragonal sample of Sr(2)VO(4) show an antiferromagnetic orbital-ordering transition at T(oo) = 97 K, in which the occupied orbitals lie along the c axis. The unusual broadening of the x-ray Bragg peaks throughout the orbital-ordering transition temperature region indicates that this process occurs in stages, and the onset of short-range orbital ordering occurs at T(1) approximately 122 K. The study of the order parameter associated with this transition by analyzing the spontaneous strain results in a critical exponent beta = 0.35(2) consistent with 3D Heisenberg behavior. These results experimentally confirm the orbital-ordering state in Sr(2)VO(4) predicted by first-principles calculations using combinations of the local-density approximation and the GW method.

Supercritical series expansion for the contact process in heterogeneous and disordered environments

The supercritical series expansion of the survival probability for the one-dimensional contact process in heterogeneous and disordered lattices is used for the evaluation of the loci of critical points and critical exponents beta . The heterogeneity and disorder are modeled by considering binary regular and irregular lattices of nodes characterized by different recovery rates and identical transmission rates. Two analytical approaches based on nested Padé approximants and partial differential approximants were used in the case of expansions with respect to two variables (two recovery rates) for the evaluation of the critical values and critical exponents. The critical exponents in heterogeneous systems are very close to those for the homogeneous contact process thus confirming that the contact process in periodic heterogeneous environment belongs to the directed percolation universality class. The disordered systems, in contrast, seem to have continuously varying critical exponents.