Magnetic catalysis
Encyclopedia
Magnetic catalysis is a phenomenon in quantum field theory
which explains a spontaneous breaking of flavor or chiral symmetry
, triggered by the presence of an external magnetic field.
. For gauge theories in 3+1 dimensions, such as quantum electrodynamics
and quantum chromodynamics
, the dimensional reduction leads to an effective (1+1)-dimensional low-energy dynamics as if the space-time were (1+1)-dimensional. (Here the dimensionality of space-time is written as D+1, where D stands for the number of space-like directions and 1 stands for a single time-like direction.) In simple terms, the dimensional reduction reflects the fact that the motion of charged particles is (partially) restricted in the two space-like directions perperndicular to the magnetic field. However, this orbital motion constraint alone is not sufficient (for example, there is no dimensional reduction for charged scalar particles, carrying spin 0, although their orbital motion is constrained in the same way.) It is also important that fermions have spin 1/2 and, as follows from the Atiyah–Singer index theorem
, their lowest Landau level states have an energy independent of the magnetic field. (The corresponding energy vanishes in the case of massless particles.) This is in contrast to the energies in the higher Landau levels, which are proportional to the square root of the magnetic field. Therefore, if the field is sufficiently strong, only the lowest Landau level states are dynamically accessible at low energies. The states in the higher Landau levels decouple and become almost irrelevant. The phenomenon of magnetic catalysis has applications in particle physics
, nuclear physics
and condensed matter physics
.
Recently, there was an increased activity to cross-check the effects of magnetic catalysis in the limit of a large number of colors, using the technique of AdS/CFT correspondence. Again, in this framework, the universality of the effect withstood the test.
in strong magnetic fields beyond the standard anomalous sequence at filling factors ν=4(n+½) where n is an integer. The additional quantum Hall plataues develop at ν=0, ν=±1, ν=±3 and ν=±4.
The mechanism of magnetic catalysis in a relativistic-like planar systems such as graphene is very natural. In fact, it was originally proposed for a 2+1 dimensional model, which is almost the same as the low-energy effective theory of graphene written in terms of massless Dirac fermions. In application to a single layer of graphite (i.e., graphene), magnetic catalysis triggers the breakdown of an approximate internal symmetry and, thus, lifts the 4-fold degeneracy of Landau levels.
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
which explains a spontaneous breaking of flavor or chiral symmetry
Chiral symmetry
In quantum field theory, chiral symmetry is a possible symmetry of the Lagrangian under which the left-handed and right-handed parts of Dirac fields transform independently...
, triggered by the presence of an external magnetic field.
General description
The underlying mechanism behind the magnetic catalysis is the dimensional reduction of the low-energy charged spin-1/2 particles and, as a result of such reduction, a strong enhancement of the particle-antiparticle pairing responsible for symmetry breakingSymmetry breaking
Symmetry breaking in physics describes a phenomenon where small fluctuations acting on a system which is crossing a critical point decide the system's fate, by determining which branch of a bifurcation is taken. To an outside observer unaware of the fluctuations , the choice will appear arbitrary...
. For gauge theories in 3+1 dimensions, such as quantum electrodynamics
Quantum electrodynamics
Quantum electrodynamics is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and special relativity is achieved...
and quantum chromodynamics
Quantum chromodynamics
In theoretical physics, quantum chromodynamics is a theory of the strong interaction , a fundamental force describing the interactions of the quarks and gluons making up hadrons . It is the study of the SU Yang–Mills theory of color-charged fermions...
, the dimensional reduction leads to an effective (1+1)-dimensional low-energy dynamics as if the space-time were (1+1)-dimensional. (Here the dimensionality of space-time is written as D+1, where D stands for the number of space-like directions and 1 stands for a single time-like direction.) In simple terms, the dimensional reduction reflects the fact that the motion of charged particles is (partially) restricted in the two space-like directions perperndicular to the magnetic field. However, this orbital motion constraint alone is not sufficient (for example, there is no dimensional reduction for charged scalar particles, carrying spin 0, although their orbital motion is constrained in the same way.) It is also important that fermions have spin 1/2 and, as follows from the Atiyah–Singer index theorem
Atiyah–Singer index theorem
In differential geometry, the Atiyah–Singer index theorem, proved by , states that for an elliptic differential operator on a compact manifold, the analytical index is equal to the topological index...
, their lowest Landau level states have an energy independent of the magnetic field. (The corresponding energy vanishes in the case of massless particles.) This is in contrast to the energies in the higher Landau levels, which are proportional to the square root of the magnetic field. Therefore, if the field is sufficiently strong, only the lowest Landau level states are dynamically accessible at low energies. The states in the higher Landau levels decouple and become almost irrelevant. The phenomenon of magnetic catalysis has applications in particle physics
Particle physics
Particle physics is a branch of physics that studies the existence and interactions of particles that are the constituents of what is usually referred to as matter or radiation. In current understanding, particles are excitations of quantum fields and interact following their dynamics...
, nuclear physics
Nuclear physics
Nuclear physics is the field of physics that studies the building blocks and interactions of atomic nuclei. The most commonly known applications of nuclear physics are nuclear power generation and nuclear weapons technology, but the research has provided application in many fields, including those...
and condensed matter physics
Condensed matter physics
Condensed matter physics deals with the physical properties of condensed phases of matter. These properties appear when a number of atoms at the supramolecular and macromolecular scale interact strongly and adhere to each other or are otherwise highly concentrated in a system. The most familiar...
.
Chiral symmetry breaking in quantum chromodynamics
In the theory of quantum chromodynamics, magnetic catalysis can be applied when quark matter is subject to extremely strong magnetic fields. Such strong magnetic fields can lead to more pronounced effects of chiral symmetry breaking, e.g., lead to (i) a larger value of the chiral condensate, (ii) a larger dynamical (constituent) mass of quarks, (iii) larger baryon masses, (iv) modified pion decay constant, etc.Recently, there was an increased activity to cross-check the effects of magnetic catalysis in the limit of a large number of colors, using the technique of AdS/CFT correspondence. Again, in this framework, the universality of the effect withstood the test.
Cosmology
There were a few studies suggesting that magnetic catalysis may play a crucial role in the Early Universe when there existed superstrong primodial magnetic fields. The magnetic catalysis may then affect the dynamics of the electroweak phase transition.Condensed matter physics
It seems very natural to test the ideas of magnetic catalysis in applications to many field theories used in tudies of condensed matter systems.Quantum Hall effect in graphene
The idea of magnetic catalysis can be used to explain the observation of new quantum Hall plateaux in grapheneGraphene
Graphene is an allotrope of carbon, whose structure is one-atom-thick planar sheets of sp2-bonded carbon atoms that are densely packed in a honeycomb crystal lattice. The term graphene was coined as a combination of graphite and the suffix -ene by Hanns-Peter Boehm, who described single-layer...
in strong magnetic fields beyond the standard anomalous sequence at filling factors ν=4(n+½) where n is an integer. The additional quantum Hall plataues develop at ν=0, ν=±1, ν=±3 and ν=±4.
The mechanism of magnetic catalysis in a relativistic-like planar systems such as graphene is very natural. In fact, it was originally proposed for a 2+1 dimensional model, which is almost the same as the low-energy effective theory of graphene written in terms of massless Dirac fermions. In application to a single layer of graphite (i.e., graphene), magnetic catalysis triggers the breakdown of an approximate internal symmetry and, thus, lifts the 4-fold degeneracy of Landau levels.
External links
- Articles on Magnetic catalysis at http://scholar.google.com/
- Magnetic catalysis in graphene