symmetry manifested in symmetry breaking

Spontaneous symmetry breaking has become a fundamental notion of condensed matter physics as well as high energy physics. This concept says, the low temperature properties of a physical system may not acquire the same symmetry as its basic Hamiltonian. The reason is simple: for a system with symmetry, there is always a degeneracy occurring to its ground state, i.e., it has a number of ground states with the same energy, and at low energy scales, no perturbation suffices to turn this system in one of its ground state to another, therefore, its physical properties shall bear features special to that particular ground state it is in. Let us point out that, the perturbation stems from its interaction with all the rest of our universe.

As this concept has been corroborated, people tend to skip an important thing, which is that, many consequences of this symmetry breaking actually reveal the original symmetry. One such consequence is the formation of domain structures. Roughly speaking, a domain is a region where the system is found in one of its degenerate ground states. Now that there are many equally possible (in the absence of external field) ground states, the system, when its symmetry becomes broken, falls in a state with domains that each realizes one particular ground state. So, one can factually find almost reminiscent of every ground state in this symmetry broken state.
Therefore, as a whole, this system actually respects its symmetry rather than simply break it ! Of course, domain walls are high energy regions which would dismiss the domain formation but for two factors: (1)inter-domain interaction and (2)ergodicity broken.

An often cited example is ferromagnets. No natural ferromagnet (such as iron) can be found magnetic at all, because its domains cancel each other, as a result no global magnetism found, though with very small probes (like STM) local magnetism can be detected.

Domain walls are current active research areas. They display very aberrant properties. For example, scientists found conducting domain walls in Bismuth Ferrite, despite that the material itself is insulating in bulk state,

Nature Materials 8, 229 - 234 (2009)
Published online: 25 January 2009 | doi:10.1038/nmat2373

Subject Categories: Electronic materials | Magnetic materials

Conduction at domain walls in oxide multiferroics

J. Seidel^1,2,10, L. W. Martin^2,3,10, Q. He¹, Q. Zhan², Y.-H. Chu^2,3,4, A. Rother⁵, M. E. Hawkridge², P. Maksymovych⁶, P. Yu¹, M. Gajek¹, N. Balke¹, S. V. Kalinin⁶, S. Gemming⁷, F. Wang¹, G. Catalan⁸, J. F. Scott⁸, N. A. Spaldin⁹, J. Orenstein^1,2 & R. Ramesh^1,2,3

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Abstract

Domain walls may play an important role in future electronic devices, given their small size as well as the fact that their location can be controlled. Here, we report the observation of room-temperature electronic conductivity at ferroelectric domain walls in the insulating multiferroic BiFeO₃. The origin and nature of the observed conductivity are probed using a combination of conductive atomic force microscopy, high-resolution transmission electron microscopy and first-principles density functional computations. Our analyses indicate that the conductivity correlates with structurally driven changes in both the electrostatic potential and the local electronic structure, which shows a decrease in the bandgap at the domain wall. Additionally, we demonstrate the potential for device applications of such conducting nanoscale features.