Showing papers on "Landau theory published in 2008"
TL;DR: In this paper, the generic case of a ferroelectric solid solution is considered where different symmetry phases located at opposing ends of the diffusionless phase diagram are separated by a morphotropic boundary (MB).
Abstract: The generic case of a ferroelectric solid solution is considered wherein different symmetry phases located at opposing ends of the diffusionless phase diagram are separated by a morphotropic boundary (MB). It is shown that the Landau theory of weak first-order phase transformations automatically predicts vanishing of the anisotropy of polarization, continuity of thermodynamic properties, and a drastic decrease in domain wall energy near the MB line that results in the formation of adaptive ferroelectric nanodomain states. Low-resolution diffraction from these adaptive states acquired at the coherence lengths of elastic x-ray or neutron scattering probes will produce the same diffraction pattern as attributed to monoclinic (MA,MB,MC) phases. It is further shown that the electric- or stress-field-induced reconfiguration of these adaptive nanodomain states results in a softening of the piezoelectric, elastic, and dielectric properties near the MB line. In addition, the spherical degeneration of the polarization direction, reflecting the decoupling of the polarization from the crystal lattice at the MB, also predicts the formation of a polar glass state whose properties should be similar to the special properties of amorphous ferromagnets. In particular, the vanishing of the polarization anisotropy at the MB should result in ferroelectric domains with irregular shapes exhibiting high configurational sensitivity to external forces. The theory further predicts that two tricritical points will occur on the line of paraelectric→ferroelectric transitions and it is shown that all two-phase equilibrium lines of the diffusionless phase diagram—including the MB line—must be replaced by two-phase fields. Within these two-phase fields, the adjacent ferroelectric-ferroelectric and paraelectric-ferroelectric phases coexist. Possible topologies of the equilibrium MB phase diagram illustrating these two-phase equilibrium fields are computed and discussed.
211 citations
TL;DR: In this paper, a chain of singly charged particles, confined by a harmonic potential, exhibits a sudden transition to a zigzag configuration when the radial potential reaches a critical value, depending on the particle number.
Abstract: A chain of singly charged particles, confined by a harmonic potential, exhibits a sudden transition to a zigzag configuration when the radial potential reaches a critical value, depending on the particle number. This structural change is a phase transition of second order, whose order parameter is the crystal displacement from the chain axis. We study analytically the transition using Landau theory and find full agreement with numerical predictions by Schiffer [Phys. Rev. Lett. 70, 818 (1993)] and Piacente et al. [Phys. Rev. B 69, 045324 (2004)]. Our theory allows us to determine analytically the system's behavior at the transition point.
129 citations
TL;DR: In this paper, the problem of the choice of the order parameter of the Landau expansion of proper ferroelectrics is discussed, and the expansion with respect to the electric displacement, polarization, ferroelectric contribution to the polarization, and softmode displacements within the weak-ferroelectric approach are compared.
Abstract: The problem of the choice of the order parameter of the Landau expansion of proper ferroelectrics is discussed. The expansion with respect to the (i) electric displacement, (ii) polarization, (iii) ferroelectric contribution to the polarization, (iv) “order” parameter within the pseudo-proper approach, and (v) soft-mode displacements within the weak-ferroelectric approach are compared. The following conclusions concerning applicability of these expansions are made. The description in terms of the electric displacement is only of historical interest. This approach may lead to essentially wrong conclusions while it has no advantage over approach (ii) in terms of the simplicity of the framework. Approaches (ii) to (v) provide progressively more reliable descriptions of properties of ferroelectrics, although they are progressively more complex as well. Depending on the situation addressed, approaches (ii) to (iv) may lead to erroneous results or may be unable to describe a certain effect. In case the depolari...
125 citations
TL;DR: In this article, the authors derived thermodynamically the free energy functional of a ferroelectric under an applied field E that contains the depolarization field Ed. They found that it is crucial to recognize the relationship between the total polarization and the applied field due to the degradations of the field, and that this relationship effectively double-counts the field and thus causes serious overestimation of its effects.
Abstract: We derived thermodynamically from first principles the free energy functional of a ferroelectric under an applied field E that contains the depolarization field Ed. We found that it is crucial to recognize the relationship between the total polarization and the applied field due to the depolarization field when expressing the free energy functional in the Landau form. Ad hoc formulations of ferroelectric models that overlooked this relationship effectively double-count the depolarization field, and thus cause serious overestimation of its effects. Examples are given in which the Curie temperature and the critical thickness of ferroelectricity, etc., can be overestimated this way by orders of magnitude.
115 citations
TL;DR: In this article, the evolution of the structure of SrMoO{sub 4} upon compression was established using synchrotron X-ray diffraction and a diamond-anvil cell.
90 citations
TL;DR: In this paper, a general power counting scheme for the infrared limit of Landau gauge SU (N ) Yang-Mills theory in arbitrary dimensions was developed. But the results were limited to the case of 2, 3 and 4.
85 citations
TL;DR: It is demonstrated that the radial degree of freedom of strings of trapped ions in the quantum regime may be prepared and controlled accurately through the variation of the external trapping potential while at the same time its properties are measurable with high spatial and temporal resolution.
Abstract: Since Hund's work on the ammonia molecule, the double well potential has formed a key paradigm in physics. Its importance is further underlined by the central role it plays in the Landau theory of phase transitions. Recently, the study of entanglement properties of many-body systems has added a new angle to the study of quantum phase transitions of discrete and continuous degrees of freedom, i.e., spin and harmonic chains. Here we show that control of the radial degree of freedom of trapped ion chains allows for the simulation of linear and non-linear Klein-Gordon fields on a lattice, in which the parameters of the lattice, the non-linearity and mass can be controlled at will. The system may be driven through a phase transition creating a double well potential between different configurations of the ion crystal. The dynamics of the system are controllable, local properties are measurable and tunnelling in the double well potential would be observable.
82 citations
TL;DR: In this paper, a soliton solution of one dimensional modified complex Ginzburg Landau equation is reported and the parametric region where such soliton solutions are possible is also identified.
Abstract: In this paper we have reported soliton solution of one dimensional modified complex Ginzburg Landau equation. The parametric region where such soliton solution is possible is also identified.
79 citations
TL;DR: In this article, the magnon and optical phonon spectrum of an incommensurate multiferroic was considered in the framework of a phenomenological Landau theory, and it was shown that the optical response of a multi-finerroic reveals much more about its magnetic excitations than previously anticipated on the basis of simpler models.
Abstract: The magnon and optical phonon spectrum of an incommensurate multiferroic such as $\mathrm{Bi}\mathrm{Fe}{\mathrm{O}}_{3}$ is considered in the framework of a phenomenological Landau theory. The resulting spin wave spectrum is quite distinct from commensurate substances due to soft mode anisotropy and magnon zone folding. The former allows electrical control of spin wave propagation via reorientation of the spontaneous ferroelectric moment. The latter gives rise to multiple magnetodielectric resonances due to the coupling of optical phonons at zero wave vector to magnons at integer multiples of the cycloid wave vector. These results show that the optical response of a multiferroic reveals much more about its magnetic excitations than previously anticipated on the basis of simpler models.
78 citations
TL;DR: Using the unrestricted Hartree-Fock approximation and Landau theory, a new phase is identified which has the same structure factor as the former but in which magnetic moments at nearest-neighbor sites are at right angles making electrons acquire a nontrivial phase when circulating a plaquette at strong coupling.
Abstract: Using the unrestricted Hartree-Fock approximation and Landau theory we identify possible phases competing with superconductivity in FeAs layers. We find that close to half-filling the transition from the paramagnet to the magnetically ordered phase is first order, making anharmonicities relevant and leading to a rich phase diagram. Between the already known one-dimensionally modulated magnetic stripe phase and the paramagnet we find a new phase which has the same structure factor as the former but in which magnetic moments at nearest-neighbor sites are at right angles making electrons acquire a nontrivial phase when circulating a plaquette at strong coupling. Another competing phase has magnetic and charge order and may be stabilized by charged impurities.
73 citations
TL;DR: The solid solubility between LaNbO4 and LaTaO4 was investigated by X-ray diffraction, and a two-phase region was observed in the composition region La Nb1−xTaxO4 where 0.4⩽x-0.8 was shown to increase with increasing Ta-content as discussed by the authors.
TL;DR: In this paper, the authors extend the Janovec-Privratska theory to an analytic model of magnitudes (local magnetization and polarization) within the domain-walls and discuss the wall thickness and domain widths as a function of those magnitudes.
Abstract: V. Janovec and J. Privratska developed [Ferroelectrics 204, 321 (1997)] a group theoretical model for the existence of pyromagnetism (or pyroelectricity) within the domain walls of an antiferromagnet (or antiferroelectric). However, their approach did not permit any estimate of magnitudes, or even whether it would be observable. In the present work we extend the Janovec-Privratska theory to an analytic model of magnitudes (local magnetization and polarization) within the domain-walls and discuss the wall thickness and domain widths as a function of those magnitudes. Specifically, it is shown that both the wall thickness and the domain width of magnetoelectric multiferroics are bigger than for pure ferroelectrics and smaller than for pure ferromagnets. It is also shown how ferroelectric domain walls in magnetoelectric materials may have net magnetization even when the domains themselves are still paramagnetic. Possible implications for magnetoelectric BiFeO 3 are indicated.
TL;DR: In this paper, the spin wave spectra of multiferroic BiFeO3 films are calculated using a phenomenological Landau theory that includes magnetostatic effects, and the lowest frequency magnon dispersion is shown to be quite sensitive to the angle between spin wave propagation vector and the Neel moment.
Abstract: The spin wave spectra of multiferroic BiFeO3 films is calculated using a phenomenological Landau theory that includes magnetostatic effects. The lowest frequency magnon dispersion is shown to be quite sensitive to the angle between spin wave propagation vector and the Neel moment. Since electrical switching of the Neel moment has recently been demonstrated in this material, the sensitivity of the magnon dispersion permits direct electrical switching of spin wave propagation. This effect can be used to construct spin wave logical gates without current pulses, potentially allowing reduced power dissipation per logical operation.
TL;DR: In this paper, a tricritical point is observed in the Nd1−xSrxMnO3 (NSMO) (x=0.33) manganites at x = 0.33 which separates the first-order transition in NSMO-0.3 and second order transition of NSMO -0.4, and the contribution to the free energy from the presence of correlated clusters are strongly influencing the magnetocaloric effect.
Abstract: A tricritical point is observed in the Nd1−xSrxMnO3 (NSMO) (x=0.3, 0.33, and 0.4) manganites at x=0.33 which separates the first-order transition in NSMO-0.3 and second order transition in NSMO-0.4. The ferromagnetic transition of these compounds is further investigated by measuring magnetocaloric effect (MCE) and by applying a theoretical model based on Landau theory of phase transitions. Results indicate that the contributions to the free energy from the presence of correlated clusters are strongly influencing the MCE by coupling with the order parameter around the Curie temperature.
TL;DR: In this article, the temperature-field phase diagrams for perovskite-type ferroelectrics are discussed within the Landau theory for each of the cubic-tetragonal-orthorhombic-rhombohedral phases.
Abstract: The temperature–field phase diagrams are discussed within the Landau theory for perovskite-type ferroelectrics, which undergo successive transitions of first order. The sequences of the cubic–tetragonal–orthorhombic–rhombohedral phases and the cubic–rhombohedral–orthorhombic–tetragonal phases are considered, and the field is applied along the [100], [110], or [111] direction. The topology of the phase diagrams obtained are found to depend much on the degree of the anisotropy of the Landau potential functions in the order parameter space. When the free energy function is nearly isotropic, the critical end points associated with all the possible ferroelectric phases, stabilized successively with decreasing temperature, appear at a certain field and temperature. Only the one associated with the highest temperature ferroelectric phase does that at a fairly high field. It is pointed out that the critical end points experimentally observed in the relaxor ferroelectrics like Pb(Mg 1/3 Nb 2/3 )O 3 suggests that t...
TL;DR: For the Schrodinger and Pauli operators with constant magnetic field, the spectrum is perturbed if a perturbation by a compactly supported magnetic field is performed as discussed by the authors.
Abstract: For the Schrodinger and Pauli operators with constant magnetic field it is investigated how the spectrum is perturbed if a perturbation by a compactly supported magnetic field is performed
TL;DR: In this article, a phenomenological theory for the homogeneous phases of nematic liquid crystals constituted by biaxial molecules is presented, and a general polynomial potential in two macroscopic order parameter tensors that reproduces the mean-field phase diagram confirmed by Monte Carlo simulations is proposed.
Abstract: We present a phenomenological theory for the homogeneous phases of nematic liquid crystals constituted by biaxial molecules. We propose a general polynomial potential in two macroscopic order parameter tensors that reproduces the mean-field phase diagram confirmed by Monte Carlo simulations [De Matteis et al. in Phys Rev E 72:041706 (2005)] and recently recognized to be universal [Bisi et al. in Phys Rev E 73:051709 (2006)] for dispersion force molecular pair-potentials enjoying the D 2h symmetry. The requirement that the phenomenological theory comply uniquely with this phase diagram reduces considerably the admissible phenomenological coefficients, both in their number and in the ranges where they can vary.
TL;DR: In this paper, the authors derived a phenomenological Landau theory that predicts the orientation of the spins within incommensurate spiral ordered states and described how the spins reorient in a magnetic field, and how they may undergo a low temperature "lock-in" transition to a commensurate state.
Abstract: Insulating spinel materials, with the chemical formula AB2X4, behave as diamond lattice antiferromagnets when only the A-site atom is magnetic. Many exhibit classic signatures of frustration, induced not geometrically but by competing first and second neighbor exchange interactions. In this paper, we further develop a theory 1 of the magnetism of these materials, focusing on the physics observable within the ordered state. We derive a phenomenological Landau theory that predicts the orientation of the spins within incommensurate spiral ordered states. It also describes how the spins reorient in a magnetic field, and how they may undergo a low temperature “lock-in” transition to a commensurate state. We discuss microscopic mechanisms for these magnetic anisotropy effects. The reduction of the ordered moment by quantum fluctuations is shown to be enhanced due to frustration. Our results are compared to experiments on MnSc2S4 , the best characterized of such A-site spinels, and more general implications are discussed. One prediction is that magnetically-induced ferroelectricity is generic in these materials, and a detailed description of the relation of the electric polarization to the magnetism is given.
TL;DR: It is shown using Landau theory that quenched dislocations can facilitate the supersolid to normal solid transition, making it possible for the transition to occur even if it does not in a dislocation-free crystal.
Abstract: I show using Landau theory that quenched dislocations can facilitate the supersolid to normal solid transition, making it possible for the transition to occur even if it does not in a dislocation-free crystal. I make detailed predictions for the dependence of the supersolid to normal solid transition temperature T_{c}(L), superfluid density rho_{S}(T,L), and specific heat C(T,L) on temperature T and dislocation spacing L, all of which can be tested against experiments. The results should also be applicable to an enormous variety of other systems, including, e.g., ferromagnets.
TL;DR: In this article, the magnetic properties of fine grained perovskite manganites of the type La0.7Ba0.3−zNazMnO3 have been investigated.
TL;DR: In this paper, the Green's function was used to estimate the longitudinal susceptibility of the spin-1/2 low-dimensional Heisenberg ferromagnet in a magnetic field.
Abstract: Longitudinal susceptibility of the spin-1/2 low-dimensional Heisenberg ferromagnet in a magnetic field, is studied by the Green's function method within the random phase approximation. The static and dynamic longitudinal susceptibilities are calculated in the low- and high-field regions. Power laws for the position and height of the static susceptibility maximum are shown not to support the predictions of Landau theory.
TL;DR: In this paper, a self-consistent scheme was proposed to study the effects of dislocations on microstructure, particularly twinned domain walls, in an Fe-Pd alloy undergoing a martensitic transformation.
Abstract: In dislocation-free martensites the components of the elastic strain tensor are constrained by the Saint-Venant compatibility condition which guarantees continuity of the body during external loading However, in dislocated materials the plastic part of the distortion tensor introduces a displacement mismatch that is removed by elastic relaxation The elastic strains are then no longer compatible in the sense of the Saint-Venant law and the ensuing incompatibility tensor is shown to be proportional to the gradients of the Nye dislocation density tensor We demonstrate that the presence of this incompatibility gives rise to an additional long-range contribution in the inhomogeneous part of the Landau energy functional and to the corresponding stress fields Competition among the local and long-range interactions results in frustration in the evolving order parameter (elastic) texture We show how the Peach-Koehler forces and stress fields for any distribution of dislocations in arbitrarily anisotropic media can be calculated and employed in a Fokker-Planck dynamics for the dislocation density This approach represents a self-consistent scheme that yields the evolutions of both the order parameter field and the continuous dislocation density We illustrate our method by studying the effects of dislocations on microstructure, particularly twinned domain walls, in an Fe-Pd alloy more » undergoing a martensitic transformation « less
TL;DR: The results of optical texture, birefringence, and the tilt angle suggest that a part of the Sm-A* phase is of de Vries type, since an increase in the tilt angles with decreasing temperature results in a reduction in the value of the bireFringence in the smectic-A-Sm-C* phase.
Abstract: Mixtures of different compositions of an antiferroelectric liquid crystal compound that exhibits direct smectic-A*(Sm-A*)-smectic-C*(A) (Sm-C*(A)) transition with a ferroelectric liquid crystal compound that exhibits Sm-A*-smectic-C*(Sm-C*) transition are studied using electro-optics and dielectric spectroscopy. The results of optical texture, birefringence, and the tilt angle suggest that a part of the Sm-A* phase is of de Vries type, since an increase in the tilt angle with decreasing temperature results in a reduction in the value of the birefringence in the Sm-A* phase, whereas the birefringence at Sm-A* to Sm-C* transition goes up by 12.7%. The soft mode relaxation strength, the Landau coefficient of the temperature dependent term, and the other related parameters of the de Vries-type Sm-A-Sm-C*(A) and Sm-A*-Sm-C* transitions are determined using the Landau theory of the second-order phase transition. For the Sm-A*-Sm-C* transition, we find that the soft mode relaxation strength decreases, the Landau coefficient increases, and the Curie-Weiss temperature range decreases with an increased ferroelectric composition in the mixture. These observations can be explained by assuming that with increased ferroelectric composition in the mixture, the layer shrinkage at the de Vries Sm-A*-Sm-C* transition increases. On comparing the results of de Vries-type Sm-A* to Sm-C*(A) and Sm-C* transitions, we find that the soft mode dielectric strength and the other related Landau parameters of the de Vries Sm-A* phase are of the same order of magnitude for transitions from Sm-A* to Sm-C* and to Sm-C*(A) except for the composition of the mixture where both Sm-C* and Sm-C*(A) transitions are stable and the phase diagram shows phase sequence Sm-A* to Sm-C* to Sm-C*(A).
01 Jan 2008
TL;DR: The boundary controllability of the Ginzburg-Landau equation was investigated in this paper, where a new Carleman estimate and an analysis based upon the theory of sectorial operators were derived.
Abstract: The paper investigates the boundary controllability, as well as the internal controllability, of the complex Ginzburg–Landau equation. Zero-controllability results are derived from a new Carleman estimate and an analysis based upon the theory of sectorial operators.
TL;DR: In this paper, a new dynamic transition classi cation scheme and a Ginzburg-Landau theory for modeling equilibrium phase transitions are introduced, and the analysis for the PVT system in this article leads to a few physical predications, which are otherwise unclearfrom the physical point of view.
Abstract: Department of Mathematics, Indiana University, Bloomington, IN 47405(Dated: February 4, 2008)The main objective of this article are two-fold. First, we introduce some general principles on phasetransition dynamics, including a new dynamic transition classi cation scheme, and a Ginzburg-Landau theory for modeling equilibrium phase transitions. Second, apply the general principlesand the recently developed dynamic transition theory to study dynamic phase transitions of PVTsystems. In particular, we establish a new time-dependent Ginzburg-Landau model, whose dynamictransition analysis is carried out. It is worth pointing out that the new dynamic transition theory,along with the dynamic classi cation scheme and new time-dependent Ginzburg Landau models forequilibrium phase transitions can be used in other phase transition problems, including e.g. theferromagnetism and superuidity, which will be reported elsewhere. In addition, the analysis forthe PVT system in this article leads to a few physical predications, which are otherwise unclearfrom the physical point of view.
TL;DR: In this paper, the authors investigated the fate of a zero-temperature transition upon coupling of the electronic to lattice degrees of freedom, and found that the Lifshitz transition tends to merge with the Kondo volume collapse and hence becomes first order, whereas the kondo breakdown transition remains of second order except for very soft lattices.
Abstract: The unconventional critical behavior near magnetic quantum phase transitions in various heavy-fermion metals, apparently inconsistent with the standard spin-density-wave scenario, has triggered proposals on the breakdown of the Kondo effect at the critical point. Here, we investigate, within one specific scenario, the fate of such a zero-temperature transition upon coupling of the electronic to lattice degrees of freedom. We study a Kondo\char21{}Heisenberg model with volume-dependent Kondo coupling\char22{}this model displays both Kondo volume collapse and Kondo-breakdown transitions, as well as Lifshitz transitions associated with a change in the Fermi-surface topology. Within a large-$N$ treatment, we find that the Lifshitz transition tends to merge with the Kondo volume collapse and hence becomes first order, whereas the Kondo breakdown transition remains of second order except for very soft lattices. Interesting physics emerges at the zero-temperature endpoint of the Kondo volume collapse: For electrons in two space dimensions, this endpoint is located at the Lifshitz line for a large range of parameters, thus two continuous quantum phase transitions coincide without fine tuning. We analyze the effective Landau theory for such a situation and discuss critical exponents. Finally, we relate our findings to current heavy-fermion experiments.
TL;DR: The Landau potential of the martensitic phase transformation in Ni(46.8)Ti(50)Fe(3.2) was determined using high resolution x-ray diffraction to measure the spontaneous strain and calorimetric measurements to determine the excess specific heat of the R phase.
Abstract: The Landau potential of the martensitic phase transformation in Ni46.8Ti50Fe3.2 was determined using high resolution x-ray diffraction to measure the spontaneous strain and calorimetric measurements to determine the excess specific heat of the R phase. The spontaneous strain is proportional to the square of the order parameter which is tested by the relation of the excess entropy and the order parameter. The parameters of the Landau free energy were determined by fitting the temperature evolution of the order parameter and using the scaling between the excess entropy and the order parameter. The double well potential at absolute zero temperature was calculated and the interface energy and domain wall thickness were estimated.
TL;DR: This model predicts the possibility of a nonmontonic temperature dependence of birefringence and investigates each of the three types of Sm-A-Sm-C transitions near tricriticality, showing that for each type of transition, small orientational order implies de Vries behavior in the layer spacing, an unusually small layer contraction.
Abstract: We show that a generalized Landau theory for the smectic-A-smectic-C (Sm- A -Sm- C ) phases exhibits a biaxiality induced Sm- A -Sm- C tricritical point. Proximity to this tricritical point depends on the degree of orientational order in the system; for sufficiently large orientational order the Sm-A-Sm-C transition is three-dimensional XY -like, while for sufficiently small orientational order, it is either tricritical or first order. We investigate each of the three types of Sm-A-Sm-C transitions near tricriticality and show that for each type of transition, small orientational order implies de Vries behavior in the layer spacing, an unusually small layer contraction. This result is consistent with, and can be understood in terms of, the "diffuse cone" model of de Vries. Additionally, we show that birefringence grows upon entry to the Sm-C phase. For a continuous transition, this growth is more rapid the closer the transition is to tricriticality. Our model also predicts the possibility of a nonmontonic temperature dependence of birefringence.
TL;DR: In this paper, the elastic properties of magnetoelectric compound CuFeO2 in the vicinity of the magnetic and structural phase transitions near 14 K were examined. And a Landau free energy was proposed and analyzed in order to understand the elastic property of the material.
Abstract: Ultrasonic velocity measurements are used to examine the elastic properties of the magnetoelectric compound CuFeO2 in the vicinity of the magnetic and structural phase transitions near 14 K. Strong softening of modes related to C66 occurs at the transition from the high-temperature rhombohedral R3 m to lowertemperature monoclinic C2 /m symmetry. A Landau free energy is proposed and analyzed in order to understand the elastic properties of CuFeO2. The principal features of sound velocity modes vs temperature are reproduced by the model. Our analysis suggests that the transition at 14 K is primary pseudoproper ferroelastic, with the spin acting as a secondary order parameter.
TL;DR: In this article, the dynamics of phase separation through the process of spinodal decomposition in a Fermi superfluid with population imbalance is discussed. But the spinodality is not considered in this paper.
Abstract: We discuss the dynamics of phase separation through the process of spinodal decomposition in a Fermi superfluid with population imbalance. We discuss this instability first in terms of a phenomenological Landau theory. Working within the mean-field description at zero temperature, we then find the spinodal region in the phase diagram of polarization versus interaction strength and the spectrum of unstable modes in this region. After a quench, the spinodal decomposition starts from the Sarma state, which is a minimum of the free energy with respect to the order parameter at fixed density and polarization and a maximum at fixed chemical potentials. The possibility of observing nontrivial domain structures in current experiments with trapped atomic gases is discussed.