Spontaneous symmetry breaking
Quantum field theory |
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History |
Statistical mechanics |
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Spontaneous symmetry breaking is a
Overview
By definition, spontaneous symmetry breaking requires the existence of physical laws (e.g. quantum mechanics) which are invariant under a symmetry transformation (such as translation or rotation), 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.[4]
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,
-
(1)
It is in this potential term that the symmetry breaking is triggered. An example of a potential, due to Jeffrey Goldstone[5] is illustrated in the graph at the left.
-
.
(2)
This potential has an infinite number of possible
-
.
(3)
for any real θ between 0 and 2π. The system also has an unstable vacuum state corresponding to Φ = 0. This state has a U(1) symmetry. However, once the system falls into a specific stable vacuum state (amounting to a choice of θ), 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 (vacuum) 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.[6][7]
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 (in the idealized situation where we have full equilibrium; otherwise, translational symmetry gets broken as well). 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 (the galaxies act like gas particles at cosmological scales) acts as an order parameter breaking this symmetry. Similar comments can be made about the cosmic microwave background.
- For the electroweakmodel, 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 equilibrium point at the bottom of the hoop (intuitively stable, lowest gravitational potential). At a certain critical rotational velocity, this point will become unstable and the bead will jump to one of two other newly created equilibria, equidistantfrom 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
In
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
Chiral symmetry
Chiral symmetry breaking is an example of spontaneous symmetry breaking affecting the
Higgs mechanism
The strong, weak, and electromagnetic forces can all be understood as arising from
In the standard model of particle physics, spontaneous symmetry breaking of the SU(2) × U(1) gauge symmetry associated with the electro-weak force generates masses for several particles, and separates the electromagnetic and weak forces. The
Without spontaneous symmetry breaking, the Standard Model of elementary particle interactions requires the existence of a number of particles. However, some particles (the W and Z bosons) would then be predicted to be massless, when, in reality, they are observed to have mass. To overcome this, spontaneous symmetry breaking is augmented by the Higgs mechanism to give these particles mass. It also suggests the presence of a new particle, the Higgs boson, detected in 2012.
Superconductivity of metals is a condensed-matter analog of the Higgs phenomena, in which a condensate of Cooper pairs of electrons spontaneously breaks the U(1) gauge symmetry associated with light and electromagnetism.
Dynamical symmetry breaking
Dynamical symmetry breaking (DSB) is a special form of spontaneous symmetry breaking in which the ground state of the system has reduced symmetry properties compared to its theoretical description (i.e., Lagrangian).
Dynamical breaking of a global symmetry is a spontaneous symmetry breaking, which happens not at the (classical) tree level (i.e., at the level of the bare action), but due to quantum corrections (i.e., at the level of the effective action).
Dynamical breaking of a gauge symmetry is subtler. In conventional spontaneous gauge symmetry breaking, there exists an unstable
In condensed matter physics
Most phases of matter can be understood through the lens of spontaneous symmetry breaking. For example, crystals are periodic arrays of atoms that are not invariant under all translations (only under a small subset of translations by a lattice vector). Magnets have north and south poles that are oriented in a specific direction, breaking rotational symmetry. In addition to these examples, there are a whole host of other symmetry-breaking phases of matter — including nematic phases of liquid crystals, charge- and spin-density waves, superfluids, and many others.
There are several known examples of matter that cannot be described by spontaneous symmetry breaking, including: topologically ordered phases of matter, such as fractional quantum Hall liquids, and spin-liquids. These states do not break any symmetry, but are distinct phases of matter. Unlike the case of spontaneous symmetry breaking, there is not a general framework for describing such states.[12]
Continuous symmetry
The ferromagnet is the canonical system that spontaneously breaks the continuous symmetry of the spins below the Curie temperature and at h = 0, where h is the external magnetic field. Below the Curie temperature, the energy of the system is invariant under inversion of the magnetization m(x) such that m(x) = −m(−x). The symmetry is spontaneously broken as h → 0 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.
Spontaneous 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 quantity and does not have any quantum fluctuations.)
Other long-range interacting systems, such as cylindrical curved surfaces interacting via the
Generalisation and technical usage
For spontaneous symmetry breaking to occur, there must be a system in which there are several equally likely outcomes. The system as a whole is therefore
When a theory is symmetric with respect to a symmetry group, but requires that one element of the group be distinct, then spontaneous symmetry breaking has occurred. The theory must not dictate which member is distinct, only that one is. From this point on, the theory can be treated as if this element actually is distinct, with the proviso that any results found in this way must be resymmetrized, by taking the average of each of the elements of the group being the distinct one.
The crucial concept in physics theories is the
The symmetry group can be discrete, such as the
Nobel Prize
On October 7, 2008, the
See also
- Autocatalytic reactions and order creation
- Catastrophe theory
- Chiral symmetry breaking
- CP-violation
- Fermi ball
- Gauge gravitation theory
- Goldstone boson
- Grand unified theory
- Higgs mechanism
- Higgs boson
- Higgs field (classical)
- Irreversibility
- Magnetic catalysis of chiral symmetry breaking
- Mermin–Wagner theorem
- Norton's dome
- Second-order phase transition
- Spontaneous absolute asymmetric synthesis in chemistry
- Symmetry breaking
- Tachyon condensation
- 1964 PRL symmetry breaking papers
Notes
- ^ Note that (as in fundamental Higgs driven spontaneous gauge symmetry breaking) the term "symmetry breaking" is a misnomer when applied to gauge symmetries.
References
- ISBN 9810215584.
- ISBN 9781402017452.
- ISBN 9780521426732.
- ^ Edelman, Gerald M. (1992). Bright Air, Brilliant Fire: On the Matter of the Mind. New York: BasicBooks. p. 203.
- S2CID 120409034.
- S2CID 119131875.
- S2CID 119290021.
- ISBN 978-0-307-78786-6.
- .
- ^ A Brief History of Time, Stephen Hawking, Bantam; 10th anniversary edition (1998). pp. 73–74.[ISBN missing]
- ^
PMID 10012522.
- S2CID 14593420.
- ^
Kohlstedt, K.L.; Vernizzi, G.; Solis, F.J.; Olvera de la Cruz, M. (2007). "Spontaneous Chirality via Long-range Electrostatic Forces". S2CID 37983980.
- ^ The Nobel Foundation. "The Nobel Prize in Physics 2008". nobelprize.org. Retrieved January 15, 2008.
External links
- For a pedagogic introduction to electroweak symmetry breaking with step by step derivations, not found in texts, of many key relations, see http://www.quantumfieldtheory.info/Electroweak_Sym_breaking.pdf
- Spontaneous symmetry breaking
- Physical Review Letters – 50th Anniversary Milestone Papers
- In CERN Courier, Steven Weinberg reflects on spontaneous symmetry breaking
- Englert–Brout–Higgs–Guralnik–Hagen–Kibble Mechanism on Scholarpedia
- History of Englert–Brout–Higgs–Guralnik–Hagen–Kibble Mechanism on Scholarpedia
- The History of the Guralnik, Hagen and Kibble development of the Theory of Spontaneous Symmetry Breaking and Gauge Particles
- International Journal of Modern Physics A: The History of the Guralnik, Hagen and Kibble development of the Theory of Spontaneous Symmetry Breaking and Gauge Particles
- Guralnik, G S; Hagen, C R and Kibble, T W B (1967). Broken Symmetries and the Goldstone Theorem. Advances in Physics, vol. 2 Interscience Publishers, New York. pp. 567–708 ISBN 0-470-17057-3
- Spontaneous Symmetry Breaking in Gauge Theories: a Historical Survey
- The Royal Society Publishing: Spontaneous symmetry breaking in gauge theories
- University of Cambridge, David Tong: Lectures on Quantum Field Theory for masters level students.