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One way of making the transition between the quasi-long range order in a chain of S=1/2 spins coupled antiferromagnetically and the true long range order that occurs in a plane, is by assembling chains to make ladders of increasing width. Surprisingly this crossover between one and two dimensions is not at all smooth. Ladders with an even number of legs have purely short range magnetic order and a finite energy gap to all magnetic excitations. Predictions of this novel groundstate have now been verified experimentally.Holes doped into these ladders are predicted to pair, and possibly superconduct.

A wide variety of experimental results and theoretical investigations in recent years have convincingly demonstrated that several transition metal oxides and other materials have dominant states that are not spatially homogeneous. This occurs in cases in which several physical interactions-spin, charge, lattice, and/or orbital-are simultaneously active. This phenomenon causes interesting effects, such as colossal magnetoresistance, and it also appears crucial to understand the high-temperature superconductors. The spontaneous emergence of electronic nanometer-scale structures in transition metal oxides, and the existence of many competing states, are properties often associated with complex matter where nonlinearities dominate, such as soft materials and biological systems. This electronic complexity could have potential consequences for applications of correlated electronic materials, because not only charge (semiconducting electronic), or charge and spin (spintronics) are of relevance, but in addition the lattice and orbital degrees of freedom are active, leading to giant responses to small perturbations. Moreover, several metallic and insulating phases compete, increasing the potential for novel behavior.

The Springer Series in Solid-State Sciences consists of fundamental scientific books prepared by leading researchers in the field. They strive to communicate, in a systematic and comprehensive way, the basic principles as well as new developments in theoretical and experimental solid-state physics.

With the perovskite multiferroic RMnO3 (R = Gd, Tb, Dy) as guidance, we argue that the Dzyaloshinskii-Moriya interaction (DMI) provides the microscopic mechanism for the coexistence and strong coupling between ferroelectricity and incommensurate magnetism. We use Monte-Carlo simulations and zero temperature exact calculations to study a model incorporating the doubleexchange, superexchange, Jahn-Teller and DMI terms. The phase diagram contains a multiferroic phase between A and E antiferromagnetic phases, in excellent agreement with experiments.

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High-temperature superconductivity in the iron-based materials emerges from, or sometimes coexists with, their metallic or insulating parent compound states. This is surprising, as these undoped states exhibit dramatically different antiferromagnetic spin arrangements and Néel temperatures. Although there is a general consensus that magnetic interactions are important for superconductivity, much remains unknown concerning the microscopic origin of the magnetic states. In this review, we summarize the progress in this area, focusing on recent experimental and theoretical results, and their microscopic implications. We conclude that the parent compounds are in a state that is more complex than that implied by a simple Fermi surface nesting scenario, and a dual description including both itinerant and localized degrees of freedom is needed to properly describe these fascinating materials.S oon after the discovery of high critical-temperature (high-T c ) superconductivity in copper oxides 1 , neutron scattering studies revealed that the parent compounds of these superconductors have an antiferromagnetic (AF) ground state with a simple collinear spin structure ( Fig. 1a) 2,3 . Because the associated AF spin fluctuations may be responsible for electron pairing and superconductivity 4-6 , over the past 25 years a tremendous effort has focused on characterizing the interplay between magnetism and superconductivity in these materials 7 . In the undoped state, the parent compounds of copper oxide superconductors are Mott insulators and have exactly one valence fermion with spin 1/2 for each copper atom, leading to robust electronic correlations and localized magnetic moments 5,6 . Superconductivity emerges after introducing charge carriers that suppress the static AF order. Although the strong Coulomb repulsion in the parent compounds is screened by the doped charge carriers, the electronic correlations are certainly important for the physics of the doped cuprates, particularly in the underdoped regime 6 .Consider now the iron-based superconductors [8][9][10] . Several parent compounds of these materials, such as LaFeAsO, BaFe 2 As 2 , NaFeAs and FeTe, are not insulators but semimetals 11-14 . In these cases, electronic band structure calculations have revealed that their Fermi surfaces (FSs) are composed of nearly cylindrical hole and electron pockets at the (0,0) and M (1,0)/M (0,1) points, respectively 15,16 . The high density of states (DOS) resulting from the extended momentum space with nearly parallel FS between the hole and electron pockets leads to an enhancement of the particle-hole susceptibility. This suggests that FS nesting among those pockets could induce spin-density-wave (SDW) order at the in-plane AF wave vector Q AF = (1,0) with a collinear spin structure (Fig. 1b) 17 , much like the FS-nesting-induced SDW in pure chromium 18 . Neutron scattering experiments on LaFeAsO (ref. 19), BaFe 2 As 2 (ref. 20) and NaFeAs (ref. 21) have reported results compatible with the theoretically predicted AF spin structure...