Bosonization

In theoretical condensed matter physics, Bosonization is a mathematical procedure by which a system of interacting fermions in (1+1) dimensions can be transformed to a system of massless, non-interacting bosons.^[1] The method of bosonization was conceived independently by particle physicists Sidney Coleman and Stanley Mandelstam; and condensed matter physicists Daniel Mattis and Alan Luther in 1975.^[1]

The basic physical idea behind bosonization is that particle-hole excitations are bosonic in character. However, it was shown by Tomonaga in 1950 that this principle is only valid in one-dimensional systems.^[2] Bosonization is an effective field theory that focuses on low-energy excitations.^[3] This is done for Luttinger liquid theory.

Two complex fermions $\psi ,{\bar \psi }$ are written as functions of a boson $\phi$

${\bar \psi }_{-}\psi _{+}=:\exp(i\phi ):,\qquad {\bar \psi }_{-}\psi _{+}=:\exp(-i\phi ):$^[4]

while the inverse map is given by

$\partial \phi =:{\bar \psi }\psi :$

All equations are normal-ordered. The changed statistics arises from anomalous dimensions of the fields.

See also

• Luttinger Liquid

References

1. Gogolin, Alexander O. (2004). Bosonization and Strongly Correlated Systems. Cambridge University Press. ISBN 0-521-61719-7.
2. Sénéchal, David (1999). "An Introduction to Bosonization". Theoretical Methods for Strongly Correlated Electrons. CRM Series in Mathematical Physics. doi:10.1007/0-387-21717-7_4.
3. Sohn, Lydia (ed.) (1997). Mesoscopic electron transport. Springer. arXiv:cond-mat/9610037. ISBN 0-7923-4737-4.
4. In actuality, there is a cocycle prefactor to give correct (anti-)commutation relations with other fields under consideration.