The critical exponents determined from a non-trivial fixed point are comparable to experimental estimates for uranium ferromagnetic superconductors. A comparison of predicted critical exponents with available experimental estimates is given in Table 6.2 [206].

## Landau Theory of Phase Transition

Since the equilibrium properties of the system are determined by the minimum free energy state, minimizing Eq. To calculate h(δφ)2i, one can use the canonical probability distribution for the above free energy configuration which is proportional to.

## Critical Exponents and the Scaling Hypothesis

Since the correlation length is the only relevant scale of the length of the problem, we can choose =ξ and write . 1.25). To resolve this discrepancy, a length scale other than the correlation length must be included in the dimensional analysis, as described in Ref.[71].

## The Renormalization-Group

If you are interested in the variation of the block spinsk (Fourier transform of sx) with ks less than the limit Λ, we can integrate all tesk with k >Λfrome-H/T so that it becomes a new probability distribution. We can see that the exponents yt or yh are related to the usual known exponents.

## Renormalization-Group Treatment of the Ginzburg-Landau Model

So the last expression that appears in Eq. 1.68) can be computed perturbatively in a cumulative expansion as. 1.71). It is important to note that the critical exponents can be calculated up to O(2) and beyond by considering higher order terms in the perturbative (cumulant) expansion.

## Critical Dynamics

For the same static universality class, different dynamical universality classes can be obtained depending on the associated conservation laws for the order parameter. It can be shown that if the above analysis is performed with the full correlation function, then.

## Experiments on Various Strongly Correlated Magnetic Systems

*Perovskite Manganites**Uranium Ferromagnetic Superconductors**Anomalous Uniaxial Ferromagnets**Hexagonal Antiferromagnets*

The kink related to the ionic radiirA of Atoms is measured in terms of tolerance factor (f) given by. This can be attributed to the fact that the spin-lattice coupling has a dominant role in such systems and it affects their critical properties.

## Theoretical Investigations

However, early theoretical work within GL phenomenology indicates that the quartic coupling term becomes nonlocal due to the presence of magnetoelastic coupling in such systems. However, the formulation of this LR model was completely different from the above theoretical developments [237,2] where later importance was given to the effect of spin-lattice interaction on compressible systems.

## Outline of the Thesis

The thesis concludes with chapter 9 which includes a discussion that elaborates on the impact of the thesis on future perspectives. The model captures well the critical exponents obtained for several perovskite manganite samples exhibiting near tricritical behavior.

## Introduction

We consider a modified Ginzburg-Landau model with a non-local mode-coupling interaction in the quartic term. The upper critical dimension is found for bedc= 4−2ρ and we obtain the critical exponents in the leading order in=dc−d.

## Nonlocal Model Hamiltonian

The same upper critical dimension is obtained from the marginal stability of the stable eigenvalue, which we shall show in the section. We thus write the bare correlation of the order parameter with respect to Gaussian weight as.

## Renormalization-Group Scheme

### Self-energy Corrections

The self-energy corrections at one-loop order, namely Σa and Σb, represented by the Feynman diagrams in Figs. 2.2(a) and Fig. 2.10). Using the form of the nonlocal coupling given by Eq. 2.6), we find that the self-energy correction Σa given by Eq.

### Vertex Corrections

By expanding the denominators of the above integrands into the same limit (qk) and carrying out integrations of the momentum, we get Taking into account that the above expressions for Πa and Πb are in the form of a bare vertex u(−k−k1) [Eq. 2.6)] appearing in the original Hamiltonian [Eq. 2.5)], so this correction is insignificant.

Rescaling and Recursion Relations

## Flow Equations and Fixed Point

The above expressions for the critical exponents depend on the dimension of the space and the model parameter ρ that comes from the nonlocal interaction term. From the above expressions for the critical exponents, we note that for the marginal case = 0, that is, at the maximum value of ρ (ρmax = 0.5) for d = 3, the values of the critical exponents exactly coincide with those of the tricritical mean field exponents[94 ], that is, α= 12, β= 14, γ= 1 and δ= 5.

## Tricriticality in some Perovskite Manganites

Predictions of critical exponents derived from the current model and their comparison with experimental ones are shown in the table. Although current theoretical estimates for critical exponents near tricriticality agree well with those of the various perovskite manganite samples shown in Table 2.2, there are a significant number of other samples with β>0.375.

## Discussion and Conclusion

### Experiments on the Critical Behavior of Perovskite Manganites 56

The lattice distortions are shown to be directly related to the applied strain due to perturbations induced via changes in R, A and x. Different strain modes are shown to develop depending on whether the perturbation is due to the size distribution of R/A atoms or the change in the doping concentration x.

### Theoretical Developments

The microscopic models for the magnetic (and electrical) properties of perovskite-manganites are based on the framework of these two mechanisms, which provide a satisfactory explanation for the origin of CMR and the change in resistivity when the system passes through the PM-FM transition. Wagner[236,237] constructed an effective Hamiltonian for the compressible magnetic lattice under externally applied stress by introducing spin-lattice interactions, where the exchange integral is assumed to depend on the lattice spacing.

### A Phenomenological Model

As a result, a long-range (nonlocal) four-spin interaction term is generated in the effective Hamiltonian. The model predicted that the spontaneous magnetization exponent β was limited to the interval 0.250<β <.

## Momentum Shell RG

### Self-energy Corrections

The corresponding self-energy integrals, obtained from their amputated parts (excluding the external legs), are given by. 2π)du(−k−q)G>0(q), (3.9) where the prefactors are combinatorial factors and the integrals are limited to. These self-energy integrals yield the relevant corrections ∆en ∆cto the bare parameters r0 enc0, given by the expansion.

### Vertex Corrections

Since the integrals over q in the expressions for Πa(0,0) and Πb(0,0,0) are restricted to the large-momentum shell Λ/b6q6Λ, this implies q ∼Λ, hence, as described earlier, q m. Thus the nodal function (q2+m2)−σ appearing inside the integral for Πb is expanded in a large moment expansion in the limit qm given by z.

## RG Flow Equations and Stability

Rescaling and Flow Equations

### Fixed Point and Stability

The stability analysis associated with the condition1>0andy260 sets limits on the allowed values of σ. In the next section, we calculate the critical exponents in an expansion scheme with identification.

Critical Exponents

## Universality Classes in Perovskite Manganites

Comparing the experimental estimates with our theoretical results for the critical exponents β, γ and δ, we first agree with the value of β. Although part of the experimental work compares estimated results with those of short-range models (Ising, 3D Heisenberg, mean-field) and Fisher et al.'s LR model, our current estimates are more in line with experimental ones.

## Comparison with a Long-Range Model

The Feynman diagrams (Fig.3.1(a) and Fig.3.2(a)), which did not contribute in the previous case, now give relevant corrections and as a result the critical exponents appear to be n-dependent. This extended range is solely attributed to the presence of screening in the Hamiltonian model.

## Discussion and Conclusion

Various recent experiments show that the Mn site-doped perovskite-manganites near the critical point of paramagnetic-to-ferromagnetic phase transition exhibit widely different critical exponents, which have no theoretical explanation. This model thus offers a theoretical explanation for the critical exponents of Mn-site doped perovskite manganites.

## Nonlocal Screened Model and Critical Exponents

In Chapter 3, a nonlocal GL model was investigated, from which very different critical exponents were obtained for perovskite manganite samples for different nonlocality strengths. This model was found to satisfactorily capture the very different critical exponents measured in perovskite manganites for different chemical compositions.

## Comparison with Experiments

The critical exponents obtained for other doping levels than the above, namely y= 0.05in ref.[46]andy= 0.15in ref.[178]are away from tricritical mean field and corresponding critical exponents are also in good agreement with our theoretical estimates, as shown in Table 4.1. Similar to the experimental measurement in ref.[67], the critical exponents for La0.8Ba0.2Mn1−yFeyO3 with y obtained via the KF method are in better agreement with our analytical estimates than for the 3D Heisenberg model.

## A Discussion on Universality

This low momentum expansion is invalid in the self-energy and vertex integrals, which require a high momentum expansion (instead of a low momentum expansion), because the internal momenta in the loop integrals are limited in the high momentum band Λ/b 6q6 Λ. This shows that the non-local model does not belong to the universality class of the Φ4 model and gives the same critical exponents as in Ref.[205] which has been used in this chapter to obtain critical exponents consistent with the experimental exponents for Mn-site-doped manganese perovskites.

## Conclusion

These studies show the existence of different classes of universality in such materials in the sense that they exhibit very different values of static critical indices near the critical point. Interestingly, the linewidth exponent $ is found to be almost constant ($≈0.5), although the critical static exponents vary with the nonlocal exponent σ in the range -0.56σ60.

## Nonlocal Unconserved Dynamics

This suggests that the non-local model Hamiltonian is a viable model for the critical behavior of PM-FM phase transition in perovskite manganites.

## Dynamic Renormalization-Group Calculation

### Self-energy Corrections

Decimation of scales from the impulse shell that lies in the range Λb 6k6Λ (Λ is the ultraviolet cutoff) gives the equation of motion in the reduced range 0< k < Λb as. Γ0+r0+c0q2 has one pole at the top half smooth and another at the bottom half given by.

### Vertex Corrections

We note that the noise amplitude Γ0 does not get any correction in this order of calculation because ∂(−iω)∂Σ. We note that Υc does not have the same momentum dependence as in Eq. 5.5) for a nonlocal vertex and therefore does not contribute to ∆λ.

### Rescaling

Using this expression in Eqs. 5.25) and (5.26) and performing moment integrations, we calculate the correction to the four-point nonlocal peak as We note that Υc does not share the same momentum dependence as in Eq. 5.5) for the nonlocal peak and therefore does not contribute to ∆λ. 5.39) With rescaling noise and coupling constants the above equation becomes.

## Flow Equations and Fixed Point

5.54) Assuming b=eδin the above recurrence relations in the limit ofδl→0, we construct the continuous RG flow equations as. These RG flow equations suggest the existence of a non-trivial fixed point r→r∗, λ→λ∗,c→c∗=c, andΓ0→Γ∗0= Γ0, corresponding to.

## Critical Exponents

It can be noted that, similar to the original model A dynamics with SR Φ4 potential, the current LR model does not give any relevant RG corrections to the noise amplitudeΓ0 at one-loop order. However, due to the incorporation of non-local coupling functions u(k), we obtain a non-zero correction toη at one-loop order in the leading order of.

## Comparison with Experiments

Based on the closeness of this experimental value to the prediction of the 3D Heisenberg model, they concluded that the critical behavior of both samples must be governed by the 3D Heisenberg model. Consequently, finite size effects are expected to play some role in determining the critical behavior in the case of thin films.

## Discussions and Conclusions

### Magnetoelastic Coupling

In single crystal URhGe[12], the ferromagnetic phase appears below Tc = 9.5 K, and it becomes superconducting with the transition temperature 0.27 K. Similarly, UIr orders ferromagnetically below Tc≈45K, and superconductivity appears at high pressure in the ferromagnetic phase][ 7,121.

### Theoretical Investigations

By introducing pressure effects in compressible magnetic systems, Imry[100] showed, after integrating the elastic modes, that the magnetoelastic coupling term gives rise to an effective Hamiltonian with a modified quartic (φ4) interaction term. Bergman and Halperin[21] considered a one-component GL model with the lattice degrees of freedom and proposed a non-local quartic interaction term by integrating out the elastic variables.

### Static Critical Behavior

The ranges for the critical exponents shown for the 'FMN' model correspond to the model parameter range 1.56σ61.981 with a single component order parameter in three dimensions.

### Dynamic Critical Behavior

On the other hand, recent theoretical studies [39,38] on ferromagnetic superconductors show that although the total spin of localized fermions is not a conserved quantity, in the presence of itinerant electrons complete spin conservation is considered. It has also been suggested that inelastic neutron scattering experiments [97,187] picked up the contribution of localized electrons only, leaving out the contribution of traveling electrons.

### Present Motivation

Due to the same reason as for Σa, Υa does not contribute to the large long-scale. We see that the static critical exponents, calculated to the leading order of , are in very good agreement with the experimental estimates.