Spontaneous symmetry breaking
Spontaneous symmetry breaking is a spontaneous process of symmetry breaking, by which a physical system in a symmetric state spontaneously ends up in an asymmetric state. In particular, it can describe systems where the equations of motion or the Lagrangian obey symmetries, but the lowest-energy vacuum solutions do not exhibit that same symmetry. When the system goes to one of those vacuum solutions, the symmetry is broken for perturbations around that vacuum even though the entire Lagrangian retains that symmetry.
Overview
The spontaneous symmetry breaking cannot happen in quantum mechanics that describes finite dimensional systems, due to Stone-von Neumann theorem. So spontaneous symmetry breaking can be observed only in infinite dimensional theories, as quantum field theories.By definition, spontaneous symmetry breaking requires the existence of physical laws which are invariant under a symmetry transformation, so that any pair of outcomes differing only by that transformation have the same probability distribution. For example if measurements of an observable at any two different positions have the same probability distribution, the observable has translational symmetry.
Spontaneous symmetry breaking occurs when this relation breaks down, while the underlying physical laws remain symmetrical.
Conversely, in explicit symmetry breaking, if two outcomes are considered, the probability distributions of a pair of outcomes can be different. For example in an electric field, the forces on a charged particle are different in different directions, so the rotational symmetry is explicitly broken by the electric field which does not have this symmetry.
Phases of matter, such as crystals, magnets, and conventional superconductors, as well as simple phase transitions can be described by spontaneous symmetry breaking. Notable exceptions include topological phases of matter like the fractional quantum Hall effect.
Typically, when spontaneous symmetry breaking occurs, the observable properties of the system change in multiple ways. For example the density, compressibility, coefficient of thermal expansion, and specific heat will be expected to change when a liquid becomes a solid.
Examples
Sombrero potential
Consider a symmetric upward dome with a trough circling the bottom. If a ball is put at the very peak of the dome, the system is symmetric with respect to a rotation around the center axis. But the ball may spontaneously break this symmetry by rolling down the dome into the trough, a point of lowest energy. Afterward, the ball has come to a rest at some fixed point on the perimeter. The dome and the ball retain their individual symmetry, but the system does not.Image:Mexican hat potential polar.svg|270px|thumb|left|Graph of Goldstone's "sombrero" potential function.
In the simplest idealized relativistic model, the spontaneously broken symmetry is summarized through an illustrative scalar field theory. The relevant Lagrangian of a scalar field, which essentially dictates how a system behaves, can be split up into kinetic and potential terms,
It is in this potential term that the symmetry breaking is triggered. An example of a potential, due to Jeffrey Goldstone is illustrated in the graph at the left.
This potential has an infinite number of possible minima given by
for any real θ between 0 and 2π. The system also has an unstable vacuum state corresponding to. This state has a U symmetry. However, once the system falls into a specific stable vacuum state, this symmetry will appear to be lost, or "spontaneously broken".
In fact, any other choice of θ would have exactly the same energy, and the defining equations respect the symmetry but the ground state of the theory breaks the symmetry, implying the existence of a massless Nambu–Goldstone boson, the mode running around the circle at the minimum of this potential, and indicating there is some memory of the original symmetry in the Lagrangian.
Other examples
- For ferromagnetic materials, the underlying laws are invariant under spatial rotations. Here, the order parameter is the magnetization, which measures the magnetic dipole density. Above the Curie temperature, the order parameter is zero, which is spatially invariant, and there is no symmetry breaking. Below the Curie temperature, however, the magnetization acquires a constant nonvanishing value, which points in a certain direction. The residual rotational symmetries which leave the orientation of this vector invariant remain unbroken, unlike the other rotations which do not and are thus spontaneously broken.
- The laws describing a solid are invariant under the full Euclidean group, but the solid itself spontaneously breaks this group down to a space group. The displacement and the orientation are the order parameters.
- General relativity has a Lorentz symmetry, but in FRW cosmological models, the mean 4-velocity field defined by averaging over the velocities of the galaxies acts as an order parameter breaking this symmetry. Similar comments can be made about the cosmic microwave background.
- For the electroweak model, as explained earlier, a component of the Higgs field provides the order parameter breaking the electroweak gauge symmetry to the electromagnetic gauge symmetry. Like the ferromagnetic example, there is a phase transition at the electroweak temperature. The same comment about us not tending to notice broken symmetries suggests why it took so long for us to discover electroweak unification.
- In superconductors, there is a condensed-matter collective field ψ, which acts as the order parameter breaking the electromagnetic gauge symmetry.
- Take a thin cylindrical plastic rod and push both ends together. Before buckling, the system is symmetric under rotation, and so visibly cylindrically symmetric. But after buckling, it looks different, and asymmetric. Nevertheless, features of the cylindrical symmetry are still there: ignoring friction, it would take no force to freely spin the rod around, displacing the ground state in time, and amounting to an oscillation of vanishing frequency, unlike the radial oscillations in the direction of the buckle. This spinning mode is effectively the requisite Nambu–Goldstone boson.
- Consider a uniform layer of fluid over an infinite horizontal plane. This system has all the symmetries of the Euclidean plane. But now heat the bottom surface uniformly so that it becomes much hotter than the upper surface. When the temperature gradient becomes large enough, convection cells will form, breaking the Euclidean symmetry.
- Consider a bead on a circular hoop that is rotated about a vertical diameter. As the rotational velocity is increased gradually from rest, the bead will initially stay at its initial equilibrium point at the bottom of the hoop. At a certain critical rotational velocity, this point will become unstable and the bead will jump to one of two other newly created equilibria, equidistant from the center. Initially, the system is symmetric with respect to the diameter, yet after passing the critical velocity, the bead ends up in one of the two new equilibrium points, thus breaking the symmetry.
- The two-balloon experiment is an example of spontaneous symmetry breaking when both balloons are initially inflated to the local maximum pressure. When some air flows from one balloon into the other, the pressure in both balloons will drop, making the system more stable in the asymmetric state.
In particle physics
An actual measurement reflects only one solution, representing a breakdown in the symmetry of the underlying theory. "Hidden" is a better term than "broken", because the symmetry is always there in these equations. This phenomenon is called spontaneous symmetry breaking because nothing breaks the symmetry in the equations. By the nature of spontaneous symmetry breaking, different portions of the early Universe would break symmetry in different directions, leading to topological defects, such as two-dimensional domain walls, one-dimensional cosmic strings, zero-dimensional monopoles, and/or textures, depending on the relevant homotopy group and the dynamics of the theory. For example, Higgs symmetry breaking may have created primordial cosmic strings as a byproduct. Hypothetical GUT symmetry-breaking generically produces monopoles, creating difficulties for GUT unless monopoles are expelled from our observable Universe through cosmic inflation.