Abstract
We classify condensed matter systems in terms of the spacetime symmetries they spontaneously break. In particular, we characterize condensed matter itself as any state in a Poincare-invariant theory that spontaneously breaks Lorentz boosts while preserving at large distances some form of spatial translations, time-translations, and possibly spatial rotations. Surprisingly, the simplest, most minimal system achieving this symmetry breaking pattern - the framid - does not seem to be realized in Nature. Instead, Nature usually adopts a more cumbersome strategy: that of introducing internal translational symmetries - and possibly rotational ones - and of spontaneously breaking them along with their space-time counterparts, while preserving unbroken diagonal subgroups. This symmetry breaking pattern describes the infrared dynamics of ordinary solids, fluids, superfluids, and - if they exist - supersolids. A third, extra-ordinary, possibility involves replacing these internal symmetries with other symmetries that do not commute with the Poincar, group, for instance the galileon symmetry, supersymmetry or gauge symmetries. Among these options, we pick the systems based on the galileon symmetry, the galileids, for a more detailed study. Despite some similarity, all different patterns produce truly distinct physical systems with different observable properties. For instance, the low-energy 2 --> 2 scattering amplitudes for the Goldstone excitations in the cases of framids, solids and galileids scale respectively as E-2, E-4, and E-6. Similarly the energy momentum tensor in the ground state is trivial for framids (rho + p = 0), normal for solids (rho + p > 0) and even inhomogenous for galileids.
Highlights
Sometimes it can be useful to keep in mind that — to the best of our knowledge — the fundamental laws of physics are Lorentz invariant, and that real-world condensed matter systems emerge as particular Lorentz-violating states subject to such fundamentally relativistic laws
We leave open the possibility that the unbroken translational and rotational symmetries featured by a given condensed matter system — those governing the collective excitations, or quasi-particles — may not be those originally appearing in the Poincare group
Despite the difficulties encountered in realizing symmetry breaking patterns 3 and 5, we should make clear that these problems do not affect all extra-ordinary systems — i.e., all those systems whose additional symmetries do not commute with Poincare
Summary
We are interested in classifying all the symmetry breaking patterns that can be associated with a static, homogeneous, and isotropic medium in a relativistic theory. In the usual condensed matter jargon, P0 is the (usually, non-relativistic) Hamiltonian of the quasi-particles or collective excitations of the system.] This structure can be complicated at will by the addition of internal symmetries, both broken and unbroken. Often these constraints can be interpreted as gauge fixing conditions that eliminate a redundancy in the parametrization of the Goldstone excitations; for certain systems though, this interpretation is not available and imposing inverse Higgs constraints amounts to integrating out gapped modes [11, 13, 14] Regardless of their interpretation, the criterion for when inverse Higgs constraints can be imposed goes as follows: whenever the commutator between some unbroken translation Pand a multiplet of broken generators Q contains another multiplet of broken generators Q , i.e. one can impose some inverse Higgs constraints and solve them to express the Goldstones of Q in terms of derivatives of those of Q. One obtains another nonlinear realization of the same symmetry breaking pattern with fewer Goldstone fields
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