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We address a subject that could have been analyzed century ago: how does the universe of general relativity look like when it would have been filled with solid matter? Solids break spontaneously the translations and rotations of space itself. Only rather recently it was realized in various context that the order parameter of the solid has a relation to Einsteins dynamical space time which is similar to the role of a Higgs field in a Yang-Mills gauge theory. Such a "crystal gravity" is therefore like the Higgs phase of gravity. The usual Higgs phases are characterized by a special phenomenology. A case in point is superconductivity exhibiting phenomena like the Type II phase, characterized by the emergence of an Abrikosov lattice of quantized magnetic fluxes absorbing the external magnetic field. What to expect in the gravitational setting? The theory of elasticity is the universal effective field theory associated with the breaking of space translations and rotations having a similar status as the phase action describing a neutral superfluid. A geometrical formulation appeared in its long history, similar in structure to general relativity, which greatly facilitates the marriage of both theories. With as main limitation that we focus entirely on stationary circumstances -- the dynamical theory is greatly complicated by the lack of Lorentz invariance -- we will present a first exploration of a remarkably rich and often simple physics of "Higgsed gravity"., Comment: 64 pages, 22 figures. The introduction has been revised compared to the first version

Perhaps the most important aspect of symmetry in physics is the idea that a state does not need to have the same symmetries as the theory that describes it. This phenomenon is known as spontaneous symmetry breaking. In these lecture notes, starting from a careful definition of symmetry in physics, we introduce symmetry breaking and its consequences. Emphasis is placed on the physics of singular limits, showing the reality of symmetry breaking even in small-sized systems. Topics covered include Nambu-Goldstone modes, quantum corrections, phase transitions, topological defects and gauge fields. We provide many examples from both high energy and condensed matter physics. These notes are suitable for graduate students., Comment: 149 pages; matches published version

The DC Josephson effect is the flow of supercurrent across a weak link between two superconductors with a different value of their order parameter, the phase. We generalize this notion to any kind of spontaneous continuous symmetry breaking. The quantity that flows between the two systems is identified as the Noether current associated with the broken symmetry. The AC Josephson effect is identified as the oscillations due to the energy difference between the two systems caused by an imposed asymmetric chemical potential. As an example of novel physics, a Josephson effect is predicted between two crystalline solids, potentially measurable as a force periodic in the separation distance., Comment: RevTeX, 5 pages, 2 figures

In the Abelian-Higgs model, or Ginzburg-Landau model of superconductivity, the existence of an infrared stable charged fixed point ensures that there is a parameter range where the superconducting phase transition is second order, as opposed to fluctuation-induced first order as one would infer from the Coleman-Weinberg mechanism. We study the charged and neutral fixed points of a two-field generalization of the Abelian-Higgs model, where two N-component fields are coupled to two gauge fields and to each other, using the functional renormalization group. Focusing mostly on three dimensions, in the neutral case, this is a model for two-component Bose-Einstein condensation, and we confirm the fixed-point structure established in earlier works using different methods. The charged model is a dual theory of two-dimensional dislocation-mediated quantum melting. We find the existence of three charged fixed points for all N>2, while there are additional fixed points for N=2., Comment: RevTeX. 14 pages, 4 figures. Matches published version

The dislocation-mediated quantum melting of solids into quantum liquid crystals is extended from two to three spatial dimensions, using a generalization of boson-vortex or Abelian-Higgs duality. Dislocations are now Burgers-vector-valued strings that trace out worldsheets in spacetime while the phonons of the solid dualize into two-form (Kalb-Ramond) gauge fields. We propose an effective dual Higgs potential that allows for restoring translational symmetry in either one, two or three directions, leading to the quantum analogues of columnar, smectic or nematic liquid crystals. In these phases, transverse phonons turn into gapped, propagating modes while compressional stress remains massless. Rotational Goldstone modes emerge whenever translational symmetry is restored. We also consider electrically charged matter, and find amongst others that as a hard principle only two out of the possible three rotational Goldstone modes are observable using electromagnetic means., Comment: RevTex, 55 pages, 14 figures

We present a self-contained review of the theory of dislocation-mediated quantum melting at zero temperature in two spatial dimensions. The theory describes the liquid-crystalline phases with spatial symmetries in between a quantum crystalline solid and an isotropic superfluid: quantum nematics and smectics. It is based on an Abelian-Higgs-type duality mapping of phonons onto gauge bosons ("stress photons"), which encode for the capacity of the crystal to propagate stresses. Dislocations and disclinations, the topological defects of the crystal, are sources for the gauge fields and the melting of the crystal can be understood as the proliferation (condensation) of these defects, giving rise to the Anderson-Higgs mechanism on the dual side. For the liquid crystal phases, the shear sector of the gauge bosons becomes massive signaling that shear rigidity is lost. Resting on symmetry principles, we derive the phenomenological imaginary time actions of quantum nematics and smectics and analyze the full spectrum of collective modes. The quantum nematic is a superfluid having a true rotational Goldstone mode due to rotational symmetry breaking, and the origin of this 'deconfined' mode is traced back to the crystalline phase. The two-dimensional quantum smectic turns out to be a dizzyingly anisotropic phase with the collective modes interpolating between the solid and nematic in a non-trivial way. We also consider electrically charged bosonic crystals and liquid crystals, and carefully analyze the electromagnetic response of the quantum liquid crystal phases. In particular, the quantum nematic is a real superconductor and shows the Meissner effect. Their special properties inherited from spatial symmetry breaking show up mostly at finite momentum, and should be accessible by momentum-sensitive spectroscopy., Comment: Review article, 137 pages, 32 figures. Accepted version

The lowest-energy state of a macroscopic in which symmetry is spontaneously broken, is a very stable wavepacket centered around a spontaneously chosen, classical direction in symmetry space. However, for a Heisenberg ferromagnet the quantum groundstate is exactly the classical groundstate. This coincides with other exceptional properties of the ferromagnet, including spontaneous time-reversal symmetry breaking, a reduced number of Nambu-Goldstone modes and the absence of a thin spectrum (Anderson tower of states). Recent discoveries of other non-relativistic systems with fewer Nambu-Goldstone modes suggest these specialties apply there as well. I establish precise criteria for the absence of quantum fluctuations and all the other features. In particular, it is not sufficient that the order parameter commute with the Hamiltonian. It leads to a measurably larger coherence time of superpositions in small but macroscopic systems., Comment: 41 pages, 3 figures, published version

We study theoretically the dynamics of magnons in the presence of a single skyrmion in chiral magnets featuring Dzyaloshinskii-Moriya interaction. We show by micromagnetic simulations that the scattering process of magnons by a skyrmion can be clearly defined although both originate in the common spins. We find that (i) the magnons are deflected by a skyrmion, with the angle strongly dependent on the magnon wavenumber due to the effective magnetic field of the topological texture, and (ii) the skyrmion motion is driven by magnon scattering through exchange of the momenta between the magnons and a skyrmion: the total momentum is conserved. This demonstrates that the skyrmion has a well-defined, though highly non-Newtonian, momentum.

The Goldstone theorem states that there should be a massless mode for each spontaneously broken symmetry generator. There is no such rotational mode in crystals, however superconducting quantum nematics should carry rotational Goldstone modes. By generalization of thermal 2D defect mediated melting theory into a 2+1D quantum duality, the emergence of the rotational mode at the quantum phase transition from the solid to the nematic arises as a deconfinement phenomenon, with the unusual property that the stiffness of the rotational mode originates entirely in the dual dislocation condensate., Comment: 5 pages

The Mott insulating state formed from bosons is ubiquitous in solid He-4, cold atom systems, Josephson junction networks and perhaps underdoped high-Tc superconductors. We predict that close to the quantum phase transition to the superconducting state the Mott insulator is not at all as featureless as is commonly believed. In three dimensions there is a phase transition to a low temperature state where, under influence of an external current, a superconducting state consisting of a regular array of 'wires' that each carry a quantized flux of supercurrent is realized. This prediction of the "type-II Mott insulator" follows from a field theoretical weak-strong duality, showing that this 'current lattice' is the dual of the famous Abrikosov lattice of magnetic fluxes in normal superconductors. We argue that this can be exploited to investigate experimentally whether preformed Cooper pairs exist in high-Tc superconductors., Comment: RevTeX, 17 pages, 6 figures, published version

In condensed matter physics gauge symmetries other than the U(1) of electromagnetism are of an emergent nature. Two emergence mechanisms for gauge symmetry are well established: the way these arise in Kramers-Wannier type local-global dualities, and as a way to encode local constraints encountered in (doped) Mott insulators. We demonstrate that these gauge structures are closely related, and appear as counterparts in either the canonical or field-theoretical language. The restoration of symmetry in a disorder phase transition is due to having the original local variables subjected to a coherent superposition of all possible topological defect configurations, with the effect that correlation functions are no longer well-defined. This is completely equivalent to assigning gauge freedom to those variables. Two cases are considered explicitly: the well-known vortex duality in bosonic Mott insulators serves to illustrate the principle. The acquired wisdoms are then applied to the less familiar context of dualities in quantum elasticity, where we elucidate the relation between the quantum nematic and linearized gravity. We reflect on some deeper implications for the emergence of gauge symmetry in general., Comment: RevTeX, 12 pages, 4 figures

Electrodynamic phenomena related to vortices in superconductors have been studied since their prediction by Abrikosov, and seem to hold no fundamental mysteries. However, most of the effects are treated separately, with no guiding principle. We demonstrate that the relativistic vortex worldsheet in spacetime is the object that naturally conveys all electric and magnetic information, for which we obtain simple and concise equations. Breaking Lorentz invariance leads to down-to-earth Abrikosov vortices, and special limits of these equations include for instance dynamic Meissner screening and the AC Josephson relation. On a deeper level, we explore the electrodynamics of two-form sources in the absence of electric monopoles, in which the electromagnetic field strength itself acquires the characteristics of a gauge field. This novel framework leaves room for unexpected surprises., Comment: LaTeX, 23 pages, 5 figures

The vortex-boson (or Abelian-Higgs, XY) duality in 2+1 dimensions demonstrates that the quantum disordered superfluid is equivalent to an ordered superconductor and the other way around. Such a duality structure should be ubiquitous but in 3+1 (and higher) dimensions a precise formulation of the duality is lacking. The problem is that the topological defects become extended objects, strings in 3+1D. We argue how the condensate of such vortex strings must behave from the known physics of the disordered superfluid, namely the Bose-Mott insulator. A flaw in earlier proposals is repaired, and a more direct viewpoint, avoiding gauge fields, in terms of the physical supercurrent is laid out, that also easily generalizes to higher-dimensional and more complicated systems. Furthermore topological defects are readily identified; we demonstrate that the Bose-Mott insulator supports line defects, which may be seen in cold atom experiments., Comment: LaTeX, 25 pages, 5 figures; several revisions and additions