Randomly placed impurities that alter the local exchange couplings, but do not generate random fields or destroy the long-range order, roughen domain walls in Ising systems for dimensionality T & d & 5. They also pin (localize) the walls in energetically favorable positions. This drastically slows down the kinetics of ordering. The pinned domain wall is a new critical phenomenon governed by a zero-temperature fixed point. For d =2, the critical exponents for domain-wall pinning energies and roughness as a function of length scale are estimated from numerically generated ground states.PACS numbers: 75.10.Hk, 05.50.+q, 75.70.Kw Let us consider an Ising ferromagnet (or unfrustrated antiferromagnet) with randomly placed impurities at a temperature below its ordering temperature, T, .The impurities are assumed to generate random exchange couplings, but not random fields. If the effects of the impurities are sufficiently weak, the system will still order ferromagnetically and we may consider a domain wall separating two domains of predominantly "up" and "down" magnetized spins, respectively.The impurities break the translational symmetry of the system and will tend to pin such a domain wall in certain favorable locations where the exchange couplings are weaker than average.The Hamiltonian of our system may be written as H=H"""+H; ", where&tJ& " is the Hamiltonian of a pure, nonrandom Ising system and H;~contains the effects due to the random impurities. The local equilibrium position of a domain wall in such a random magnet is determined by a compromise between H,"", which tries to minimize the total (d -1)-dimensional area of domain wall, and 0' p which wants the domain wall to deviate from flatness in order to pass through the locations where it has the lowest local energy. The impurity part of our Hamiltonian may be written as H; p -g EJiJs, sJ,where the jb Jij) are randomly distributed. Our results will apply for any distribution of the random couplings 4J&J provided that the disorder has only short-range correlations and is not so strong as to destroy the ferromagnetic or antiferromagnetic ordering at low temperatures. This includes the cases of dilution and substitution of negative couplings. In this paper we address the following questions about domain walls in such random-exchange Ising systems: How rough are they? Do the impurities succeed in pinning them? If so, what are the energy barriers hindering their motion? The last question is important in the understanding of the kinetics of domain growth, which is a process that can be studied experimentally and is discussed at the end of this paper. Some answers to these questions have recently been obtained for domain walls in random field Ising -systems, ' where Hi "= Xi h;s; and each h; is random.Here we are not considering random-field or other impurities which couple directly to the local order parameter, but only impurities that couple to the local energy and therefore preserve the up-down Ising symmetry.An example of such a system is a dilute antiferromag...
The "Coulomb phase" is an emergent state for lattice models (particularly highly frustrated antiferromagnets), which have local constraints that can be mapped to a divergence-free "flux." The coarse-grained versions of this flux or polarization behave analogously to electric or magnetic fields; in particular, defects at which the local constraint is violated behave as effective charges with Coulomb interactions. I survey the derivation of the characteristic power-law correlation functions and the pinch points in reciprocal space plots of diffuse scattering, as well as applications to magnetic relaxation, quantum-mechanical generalizations, phase transitions to long-range-ordered states, and the effects of disorder.
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