The phenomenon of colossal magnetoresistance in manganites is generally agreed to be a result of competition between crystal phases with different electronic, magnetic, and structural order; a competition which can be strong enough to cause phase separation between metallic ferromagnet and insulating charge modulated states [1,2,3,4,5]. Nevertheless, closer inspection of phase diagrams in many manganites reveals complex phases where the two order parameters of magnetism and charge modulation unexpectedly coexist [6,7].Here we show that such experiments can be naturally explained within a phenomenological Ginzburg-Landau theory.In contrast to models where phase separation originates from disorder [8] or as a strain induced kinetic phenomenon [9], we argue that magnetic and charge modulation coexist in new thermodynamic phases. This leads to a rich diagram of equilibrium phases, qualitatively similar to those seen in experiment. The success of this model argues for a fundamental reinterpretation of the nature of charge modulation in these materials from a localised to a more extended "charge density wave" picture. The same symmetry considerations that favour textured coexistance of charge and magnetic order may apply to many electronic systems with competing phases. The resulting "Electronically soft" phases of matter with incommensurate, inhomogeneous and mixed order may be general phenomena in correlated systems.The manganese perovskites (RE 3+ 1−x AE 2+ x MnO 3 , RE rare earth, AE alkaline earth) provide a laboratory to study the interplay of a variety of magnetic, electronic and structural phases of matter in a strongly correlated electronic system. As in many strongly correlated electronic systems, the basic paradigm for manganite physics is the competition between the delocalising effects of the electron kinetic energy and the localising effects of the Coulomb repulsion, aided by coupling to lattice degrees of freedom. When the kinetic energy is dominant, one finds a metallic ground state with ferromagnetic alignment of the core moments.When the localising effects preponderate, instead we see charge and/or orbitally ordered ground states with substantial local lattice distortions from the near cubic symmetry of the metal, along with insulating behaviour and antiferromagnetism. One may tune between these two phases by many external parameters, especially chemical substitution, but also lattice strain, and magnetic field. The competition between metal and insulator is famously evident in the phenomenon of bulk colossal magnetoresistance, where a magnetic field tunes the conductivity of the material, and even more clearly in the strong tendency toward phase separation and inhomogeneity and regimes of percolative transport.The origin of charge and orbital ordered phases is still the subject of debate. Charge modulation has been traditionally seen as the ordering of Mn 4+ and Mn 3+ ions [10]. More recently, the charge disproportionation of the Mn ions has been argued to be much smaller than one [11,12,13] but sti...

The time-independent bound-state perturbation theory derived for the quantum probability equation (

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Time-independent perturbation theory in the scattering domain, developed on the basis of the quantum probability equation (Milward G C and Wilkin C 2001 J. Phys. A: Math. Gen. 34 5101), is extended to study the energy dependence of the S-wave scattering phase shifts and probability density to second order in the energy change. At low energies, the Bethe formula for the effective range is reproduced and an analogous formula for the shape-dependent term is also derived. The energy dependence of the probability density at short distances is important in final state interaction theory and the method allows the investigation of some of the prescriptions used there. Results from the present approach are illustrated with soluble models.