Spontaneous Symmetry Breaking - Spontaneous Symmetry Breaking in Physics - Condensed Matter Physics - Continuous Symmetry

Continuous Symmetry

The ferromagnet is the canonical system which spontaneously breaks the continuous symmetry of the spins below the Curie temperature and at, where is the external magnetic field. Below the Curie temperature the energy of the system is invariant under inversion of the magnetization such that . The symmetry is spontaneously broken as when the Hamiltonian becomes invariant under the inversion transformation, but the expectation value is not invariant.

Spontaneously symmetry broken phases of matter are characterized by an order-parameter that describes the quantity which breaks the symmetry under consideration. For example, in a magnet, the order parameter is the local magnetization.

Spontaneously breaking of a continuous symmetry is inevitably accompanied by gapless (meaning that these modes do not cost any energy to excite) Nambu-Goldstone modes associated with slow long-wavelength fluctuations of the order-parameter. For example, vibrational modes in a crystal, known as phonons, are associated with slow density fluctuations of the crystal's atoms. The associated Goldstone mode for magnets are oscillating waves of spin known as spin-waves. For symmetry breaking states, whose order parameter is not a conserved quantity, Nambu-Goldstone modes are typically massless and propagate at a constant velocity.

An important theorem, due to Mermin and Wagner, states that, at finite temperature, thermally activated fluctuations of Nambu-Goldstone modes destroy the long-range order, and prevent spontaneous symmetry breaking in one- and two-dimensional systems. Similarly, quantum fluctuations of the order parameter prevent most types of continuous symmetry breaking in one-dimensional systems even at zero-temperature (an important exception is ferromagnets, whose order parameter, magnetization, is an exactly conserved quantitity and does not have any quantum fluctuations).

Other long-range interacting systems such as cylindrical curved surfaces interacting via the Coulomb potential or Yukawa potential has been shown to break translational and rotational symmetries. It was shown, in the presence of a symmetric Hamiltonian, and in the limit of infinite volume, the system spontaneously adopts a chiral configuration, i.e. breaks mirror plane symmetry.