Order Parameter Dimension Research Articles

Model A of critical dynamics: 5-loop [formula omitted] expansion study

We have calculated the five-loop RG expansions of the n-component A model of critical dynamics in dimensions d=4−ɛ within the Minimal Subtraction scheme. This is made possible by using the advanced diagram reduction method and the Sector Decomposition technique adapted to the problems of critical dynamics. The ɛ expansions for the critical dynamic exponent z for an arbitrary value of the order parameter dimension n are derived. Based on these series, the numerical estimates of z for different universality classes are extracted and compared with the results obtained within different theoretical and experimental methods.

Marginal dimensions for multicritical phase transitions

The field-theoretical model describing multicritical phenomena with two coupled order parameters with n_{||} and n_{\perp} components and of O(n_{||}) \oplus O(n_{\perp}) symmetry is considered. Conditions for realization of different types of multicritical behaviour are studied within the field-theoretical renormalization group approach. Surfaces separating stability regions for certain types of multicritical behaviour in parametric space of order parameter dimensions and space dimension d are calculated using the two-loop renormalization group functions. Series for the order parameter marginal dimensions that control the crossover between different universality classes are extracted up to the fourth order in \varepsilon=4-d and to the fifth order in a pseudo-\varepsilon parameter using the known high-order perturbative expansions for isotropic and cubic models. Special attention is paid to a particular case of O(1) \oplus O(2) symmetric model relevant for description of anisotropic antiferromagnets in an external magnetic field.

RANDOM FIELDS IN RELAXOR FERROELECTRICS — A JUBILEE REVIEW

Substitutional charge disorder as in PbMg1/3Nb2/3O3 , structural cation vacancies as in Sr x Ba 1-x Nb 2 O 6 and isovalent substitution of off-centered cations as in BaTi 1-x Sn x O 3 and BaTi 1-x Zr x O 3 give rise to quenched electric random-fields (RF s ), which we proposed to be at the origin of the peculiar behavior of relaxor ferroelectrics 20 years ago. These are, e.g. a strong frequency dispersion of the dielectric response and an apparent lack of macroscopic symmetry breaking in the low temperature phase. Both are related to mesoscopic RF-driven phase transitions, which give rise to irregularly shaped quasi-stable polar nanoregions below the characteristic temperature T*, but above the global transition temperature Tc. Their co-existence with the paraelectric parent phase can be modeled by time-dependent field equations under the control of quenched RF s and stress-free strain (in the case of order parameter dimension n ≥ 2). Transitions into global polar order at Tc may occur in uniaxial relaxors as observed on the uniaxial relaxor ferroelectric Sr0.8Ba0.2Nb2O6 and come close to RF Ising model criticality. Re-entrant relaxor transitions as observed in solid solutions of Ba2Pr0.6Nd0.4(FeNb4)O15 are proposed to evidence the coexistence of distinct normal and relaxor ferroelectric phases within the framework of percolation theory.

Random fields in dipolar glasses and relaxors

In dipolar glasses and relaxors both site and substitutional charge disorder generally occur simultaneously albeit with different weight. While random interactions control dipolar glassy behavior of K 1− x Li x TaO 3 ( x<0.02), quenched random-fields (RFs) due to charge disorder dominate the uniaxial relaxor Sr 1− x Ba x Nb 2O 6 (SBN, x≈0.4). RFs smear the glass transition of Rb 1− x (DH 4) x H 2PO 4 (DRADP, x≈0.5) as described by a random-bond random-field (RBRF) Ising model, while cubic relaxor glasses like PbMg 1/3Nb 2/3O 3 (PMN) rather obey spherical RBRF modeling. In the case of relaxors spatial fluctuations of the RFs correlate the dipolar fluctuations and give rise to polar nanoregions in the paraelectric regime. The order parameter dimension decides upon whether the ferroelectric phase transition is destroyed (PMN) or modified towards random-field Ising model behavior (SBN).

Random Fields at Transitions from Relaxor to Glassy and Ferroelectric States

In dipolar glasses and relaxors both site and substitutional charge disorder generally occur simultaneously albeit with different weight. While random interactions control dipolar glassy behavior of K 1 m x Li x TaO 3 ( x < 0.02), quenched random-fields ( RF s) due to charge disorder dominate the uniaxial relaxor Sr 1 m x Ba x Nb 2 O 6 (SBN, x , 0.4). RF s smear the glass transition of Rb 1 m x (DH 4 ) x H 2 PO 4 (DRADP, x , 0.5) as described by a random-bond random-field ( RBRF ) Ising model, while cubic relaxor glasses like PbMg 1/3 Nb 2/3 O 3 (PMN) obey spherical RBRF modeling. Spatial fluctuations of the RF s correlate the dipolar fluctuations of relaxors and give rise to polar nanoregions in the paraelectric regime. The order parameter dimension decides upon whether the ferroelectric phase transition is destroyed (PMN) or modified towards random-field Ising model behavior (SBN).

A marginal dimension of a weakly diluted quenched m-vector model

We calculate a marginal order parameter dimension $m_c$ which in a weakly diluted quenched $m$-vector model controls the crossover from a universality class of a ``pure'' model ($m>m_c$) to a new universality class ($m<m_c$). Exploiting the Harris criterion and the field-theoretical renormalization group approach allows us to obtain $m_c$ as a five-loop $\epsilon$-expansion as well as a six-loop pseudo-$\epsilon$ expansion. In order to estimate the numerical value of $m_c$ we process the series by precisely adjusted Pad\'e--Borel--Leroy resummation procedures. Our final result $m_c=1.912\pm0.004<2$ stems from the longer and more reliable pseudo-$\epsilon$ expansion, suggesting that a weak quenched disorder does not change the values of $xy$-model critical exponents as it follows from the experiments on critical properties of ${\rm He}^4$ in porous media.

Effect of rare locally ordered regions on a disordered itinerant quantum antiferromagnet with cubic anisotropy

We study the quantum phase transition of an itinerant antiferromagnet with cubic anisotropy in the presence of quenched disorder, paying particular attention to the locally ordered spatial regions that form in the Griffiths region. We derive an effective action where these rare regions are described in terms of static annealed disorder. A one loop renormalization group analysis of the effective action shows that for order parameter dimensions $p<4$ the rare regions destroy the conventional critical behavior. For order parameter dimensions $p>4$ the critical behavior is not influenced by the rare regions, it is described by the conventional dirty cubic fixed point. We also discuss the influence of the rare regions on the fluctuation-driven first-order transition in this system.

Degenerate Hopf bifurcation in the presence of symmetry

We investigate the bifurcation of time-periodic states from a stationary state destabilized by the undamping of a set of modes associated with a degenerate pair of complex-conjugate frequencies. This problem is of particular interest for bifurcations in driven systems with symmetry whose order-parameter dimension n is even and n ≥ 4. For this case of a degenerate Hopf bifurcation a star of symmetry-equivalent limit cycles bifurcates in analogy to the star of symmetry-related domains arising at a symmetry-breaking phase transition in equilibrium systems. We illustrate this fact by analyzing a concrete example with n = 4. Within the framework of an amplitude expansion, we explicitly construct the time-periodic states and discuss their stability. In particular, it is shown that fairly general conclusions for the bifurcation behaviour can be drawn on the sole basis of the knowledge of the order-parameter symmetry.

Symmetry aspects of instabilities in driven systems

The authors investigate bifurcations initiated by either soft-mode or hard-mode instabilities in autonomous dynamical systems, with strong emphasis on symmetry considerations. The analysis is performed within the Landau theory of phase transitions extended to driven systems. The bifurcation behaviour is described in terms of an n-component order parameter which transforms either as an irreducible representation of the symmetry group of the system (symmetry-induced bifurcations) or as a reducible representation consisting of an irreducible representation occurring twice (coupling induced bifurcations). Symmetry criteria are derived for bifurcations to stationary and to time-periodic structures. The types of bifurcation may be separated into different classes characterised by the distinct matrix groups which are induced in the order parameter space. They state a number of general stability criteria, and give a complete classification of bifurcations occurring for order parameter dimensions n=1, 2 and 3.

Critical phenomena of random spin systems: Second order ϵ-expansion

The critical behaviour of quenched random spin systems is studied within a renormalization group approach. The critical exponents for the broken scaling laws are calculated in second order in ε =4− d. It can be shown that in the region of small concentration of the magnetic atoms α → − 1, β → 0.5, γ → 2 and δ → 5. These exponents turn out to be independent of the order parameter dimension n. The exponents of the non-leading singularities for small ε are in this order of ε no longer negligible in three dimensions ( ε = 1) and the related singularities may become dominant, if n is small enough (Ising magnet!).

Nonferroic phase transitions

Nonferroic phase transitions are defined as the structural transitions occurring with a breaking of translational symmetry within the same crystal class. They involve no new macroscopic tensor components below the transition point, and they are generally identified experimentally through the onset of superlattice reflections denoting the multiplication of the crystal's unit cell. A theoretical analysis of these transitions is presented, based on Landau's symmetry criteria for continuous transitons, of the order-parameter symmetries, space-group changes, and free-energy expansions. We establish that the order parameters of such transitions are necessarily related to one-dimensional (real or complex) small representations of the group of the $\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}$ vector. Their symmetry characteristics are, in general, simpler than those of other types of structural transitions. Most of them possess a one-component order parameter inducing a doubling of the unit cell. The remaining ones are associated with order-parameter dimensions as high as six, and unit-cell multiplications up to thirty-two. The coupling of the order parameter to macroscopic quantities, illustrated by the example of dielectric ones, is shown to belong to two possible schemes. The relation between nonferroics and antiferroelectrics is discussed. The theoretical results are compared to the available experimental data which pertain mainly to organic compounds and metallic alloys.

Non-ferroic phase transitions

Abstract A systematic investigation is presented of non-ferroic phase transitions, i.e.of transitions which break the translational symmetry but preserve the crystal class. Possible space group changes, unit cell multiplications, order parameter dimensions and associated free energies are given in table form for the transitions predicted in the framework of the Landau theory by the use of group theoretical methods. Illustrative examples of non-ferroic compounds are indicated.

The Landau-Ginzburg-Wilson model in 2 and 2+ϵ dimensions at low temperatures

By introducing a collective variable the two-dimensional Landau-Ginzburg-Wilson model (classical order parameter of non-rigid magnitude) may, if the order parameter dimension exceeds one, be solved at absolute zero. A low temperature expansion about this solution is developed. The low temperature expansion supports the hypothesis that d = 2 systems with a two-component order parameter will have a non-zero critical point, below which the susceptibility diverges, although no symmetry breaking occurs. The leading order temperature dependence of η is determined. In addition an ϵ expansion for systems of spatial dimension 2 + ϵ is developed. The critical exponents, calculated here to lowest order in ϵ, agree with those found for stiff spins.

Absence of long-range order in certain random systems

It is proved that a classical X-Y model where the field conjugate to the order parameter is static and random shows no long-range order at any temperature for spatial dimensions d ⩽ 4. The proof can be extended to systems with higher order parameter dimension n.

Electronic properties of disordered systems as derived by analytic continuation from renormalization group theory

The averaged single particle Green function for electrons moving in a gaussian random potential is calculated by analytic continuation from the order parameter susceptibility of a thermodynamic system with order parameter dimension O. New results are obtained and discussed.