Magnetic Catalysis: A Review

Abstract

We give an overview of the magnetic catalysis phenomenon. In the framework of quantum field theory, magnetic catalysis is broadly defined as an enhancement of dynamical symmetry breaking by an external magnetic field. We start from a brief discussion of spontaneous symmetry breaking and the role of a magnetic field in its a dynamics. This is followed by a detailed presentation of the essential features of the phenomenon. In particular, we emphasize that the dimensional reduction plays a profound role in the pairing dynamics in a magnetic field. Using the general nature of underlying physics and its robustness with respect to interaction types and model content, we argue that magnetic catalysis is a universal and model-independent phenomenon. In support of this claim, we show how magnetic catalysis is realized in various models with short-range and long-range interactions. We argue that the general nature of the phenomenon implies a wide range of potential applications: from certain types of solid state systems to models in cosmology, particle and nuclear physics. We finish the review with general remarks about magnetic catalysis and an outlook for future research.

Appendix: Fermion Propagator in a Magnetic Field

Let us start from the discussion of the Dirac fermion propagator in a magnetic field in 3+1 dimensions. It is formally defined by the following expression:

$$ S\bigl(x,x^\prime\bigr) = i \bigl[ i \gamma^0 \partial_t - (\boldsymbol{\pi}_{\perp}\cdot\boldsymbol{\gamma }_{\perp})-\pi^{3}\gamma^3-m \bigr]^{-1} \delta^{4}\bigl(x-x^\prime\bigr), $$

(2.80)

where x≡(x ⁰,x ¹,x ²,x ³)=(t,r). By definition, the spatial components of the canonical momenta are π ⁱ≡−i∂ _i −eA ⁱ, where i=1,2,3. (The perpendicular components are i=1,2.) Here we assume that e is the fermion electric charge (i.e., one should take e<0 in the case of the electron) and use the Landau gauge A=(0,Bx ¹,0), where B is the magnetic field pointing in the x ³-direction. By definition, the components of the usual three-dimensional vectors A (vector potential) and r (position vector) are identified with the contravariant components A ⁱ and x ⁱ, respectively.

In the Landau gauge used, it is convenient to perform a Fourier transform in the time (t−t′) and the longitudinal (x ³−x ^′ 3) coordinates. Then, we obtain

(2.81)

where r _⊥ is the position vector in the plane perpendicular to the magnetic field.

In order to obtain a Landau level representation for the propagator (2.81), it is convenient to utilize the complete set of eigenstates of the operator $\boldsymbol{\pi}_{\perp}^{2}$. This operator has the eigenvalues (2k+1)|eB|, where k=0,1,2,… is the quantum number associated with the orbital motion in the perpendicular plane. The corresponding normalized wave functions read

$$ \psi_{k p_2}(\mathbf{r}_{\perp})=\frac{1}{\sqrt{2\pi\ell}} \frac {1}{\sqrt{2^k k!\sqrt{\pi}}} H_k \biggl(\frac{x^1}{\ell}+p_2\ell \biggr)e^{-\frac{1}{2\ell ^2}(x^1+p_2\ell^2)^2} e^{-i s_{\perp}x^2 p_2}, $$

(2.82)

where H _k (z) are the Hermite polynomials [94], $\ell=1/\sqrt{|e B |}$ is the magnetic length, and s _⊥≡sign(eB). The wave functions satisfy the conditions of normalizability and completeness,

(2.83)

(2.84)

respectively.

By making use of the spectral expansion of the δ-function in (2.84), as well as the following identities:

(2.85)

(2.86)

with being the spin projectors onto the direction of the magnetic field, we can rewrite the propagator in (2.81) as follows:

(2.87)

The Schwinger phase is given by

$$ \varPhi\bigl(\mathbf{r}_{\perp},\mathbf{r}_{\perp}^{\prime} \bigr) = s_{\perp} \frac {(x^1+x^{\prime\, 1})(x^2-x^{\prime\, 2})}{2\ell^2}, $$

(2.88)

and the translationary invariant part of the propagator reads

(2.89)

(2.90)

where we used the short-hand notation

$$ \xi= \frac{(\mathbf{r}_{\perp}-\mathbf{r}_{\perp}^{\prime})^2}{2\ell^2}. $$

(2.91)

In order to integrate over the quantum number p ₂ in (2.87), we took into account the following table integral [94]:

$$ \int\limits _{-\infty}^\infty e^{-x^2}H_m(x+y)H_n(x+z)dx =2^n\pi^{1/2}m!z^{n-m}L_m^{n-m}(-2yz), $$

(2.92)

which is valid when m≤n. Here $L^{\alpha}_{n}$ are the generalized Laguerre polynomials, and $L_{n} \equiv L^{0}_{n}$.

Here a short remark is in order regarding the general structure of the Dirac propagator in a magnetic field. It is not a translationally invariant function, but has the form of a product of the Schwinger phase factor $e^{i\varPhi(\mathbf{r}_{\perp},\mathbf {r}_{\perp}^{\prime})}$ and a translationally invariant part. The Schwinger phase spoils the translational invariance. From a physics viewpoint, this reflects a simple fact that the fermion momenta in the two spatial directions perpendicular to the field are not conserved quantum numbers.

The Fourier transform of the translationary invariant part of the propagator (2.89) reads

$$ \tilde{S}\bigl(\omega,p^3;\mathbf{p}_{\perp}\bigr) = 2 i e^{-p_{\perp}^2 \ell^2 } \sum_{n=0}^{\infty} \frac{(-1)^n D_n(\omega,p^3;\mathbf{p}_{\perp})}{\omega^2-2n|eB| -(p^3)^2-m^2}, $$

(2.93)

where

(2.94)

Taking into account the earlier comment that the perpendicular momenta of charged particles are not conserved quantum numbers, this representation may appear surprising. However, one should keep in mind that the result in (2.93) is not a usual momentum representation of the propagator, but the Fourier transform of its translationary invariant part only.

In some applications, it is convenient to make use of the so-called proper-time representation [168], in which the sum over Landau levels is traded for a proper-time integration. This is easily derived from (2.93) by making the following substitution:

$$ \frac{i}{\omega^2-2n|eB| -(p^3)^2-m^2+i 0} = \int_{0}^{\infty} ds e^{i s [\omega^2-2n|eB| -(p^3)^2-m^2+i 0 ]}. $$

(2.95)

Then, the sum over Landau levels can be easily performed with the help of the summation formula for Laguerre polynomials [94],

$$ \sum_{n=0}^{\infty}L^{\alpha}_n(x)z^n= (1-z)^{-(\alpha+1)}\exp \biggl(\frac{xz}{z-1} \biggr). $$

(2.96)

The final expression for the propagator in the proper-time representation reads

(2.97)

where (γ⋅p)≡(γ _⊥⋅p _⊥)+γ ³ p ³.

Using the same method, one can also derive the Dirac fermion propagator in a magnetic field in 2+1 dimensions. It has the same structure as the propagator in Eqs. (2.87), (2.88), (2.89), and (2.90), but with p ³=0, i.e.,

(2.98)

where

(2.99)

The Fourier transform of the translationally invariant part is

(2.100)

Finally, the proper-time representation reads

(2.101)