Bosonization

In theoretical condensed matter physics and particle 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 C. Mattis and Alan Luther in 1975.[1]

In particle physics, however, the boson is interacting, cf, the Sine-Gordon model, and notably through topological interactions,[2] cf. Wess–Zumino–Witten model.

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.[3] Bosonization is an effective field theory that focuses on low-energy excitations.[4]

Mathematical descriptions

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 ):

[5]

while the inverse map is given by

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

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

Examples

In particle physics

The standard example in particle physics, for a Dirac field in (1+1) dimensions, is the equivalence between the massive Thirring model (MTM) and the quantum Sine-Gordon model. Sidney Coleman showed the Thirring model is S-dual to the sine-Gordon model. The fundamental fermions of the Thirring model correspond to the solitons (bosons) of the sine-Gordon model.[6]

In condensed matter

The Luttinger liquid model, proposed by Tomonaga and reformulated by J.M. Luttinger, describes electrons in one-dimensional electrical conductors under second-order interactions. Daniel C. Mattis and Elliot H. Lieb, proved in 1965,[7] that electrons could be modeled as bosonic interactions. The response of the electron density to an external perturbation can be treated as plasmonic waves. This model predicts the emergence of spin–charge separation.

References

Gogolin, Alexander O. (2004). Bosonization and Strongly Correlated Systems. Cambridge University Press. ISBN 978-0-521-61719-2.
Coleman, S. (1975). "Quantum sine-Gordon equation as the massive Thirring model" Physical Review D11 2088; Witten, E. (1984). "Non-abelian bosonization in two dimensions", Communications in Mathematical Physics 92 455-472. online
Sénéchal, David (1999). An Introduction to Bosonization. Theoretical Methods for Strongly Correlated Electrons. CRM Series in Mathematical Physics. Springer. pp. 139–186. arXiv:cond-mat/9908262. Bibcode:2004tmsc.book..139S. doi:10.1007/0-387-21717-7_4. ISBN 978-0-387-00895-0.
Sohn, Lydia (ed.) (1997). Mesoscopic electron transport. Springer. pp. cond–mat/9610037. arXiv:cond-mat/9610037. Bibcode:1996cond.mat.10037F. ISBN 978-0-7923-4737-8.CS1 maint: extra text: authors list (link)
In actuality, there is a cocycle prefactor to give correct (anti-)commutation relations with other fields under consideration.
Coleman, S. (1975). "Quantum sine-Gordon equation as the massive Thirring model". Physical Review D. 11 (8): 2088. Bibcode:1975PhRvD..11.2088C. doi:10.1103/PhysRevD.11.2088.
Mattis, Daniel C.; Lieb, Elliot H. (February 1965). Exact solution of a many-fermion system and its associated boson field. Journal of Mathematical Physics. 6. pp. 98–106. Bibcode:1994boso.book...98M. doi:10.1142/9789812812650_0008. ISBN 978-981-02-1847-8.

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[gogolin-1] Gogolin, Alexander O. (2004). Bosonization and Strongly Correlated Systems. Cambridge University Press. ISBN 978-0-521-61719-2.

[2] Coleman, S. (1975). "Quantum sine-Gordon equation as the massive Thirring model" Physical Review D11 2088; Witten, E. (1984). "Non-abelian bosonization in two dimensions", Communications in Mathematical Physics 92 455-472. online

[senechal-3] Sénéchal, David (1999). An Introduction to Bosonization. Theoretical Methods for Strongly Correlated Electrons. CRM Series in Mathematical Physics. Springer. pp. 139–186. arXiv:cond-mat/9908262. Bibcode:2004tmsc.book..139S. doi:10.1007/0-387-21717-7_4. ISBN 978-0-387-00895-0.

[glazman-4] Sohn, Lydia (ed.) (1997). Mesoscopic electron transport. Springer. pp. cond–mat/9610037. arXiv:cond-mat/9610037. Bibcode:1996cond.mat.10037F. ISBN 978-0-7923-4737-8.CS1 maint: extra text: authors list (link)

[5] In actuality, there is a cocycle prefactor to give correct (anti-)commutation relations with other fields under consideration.

[Coleman-6] Coleman, S. (1975). "Quantum sine-Gordon equation as the massive Thirring model". Physical Review D. 11 (8): 2088. Bibcode:1975PhRvD..11.2088C. doi:10.1103/PhysRevD.11.2088.

[7] Mattis, Daniel C.; Lieb, Elliot H. (February 1965). Exact solution of a many-fermion system and its associated boson field. Journal of Mathematical Physics. 6. pp. 98–106. Bibcode:1994boso.book...98M. doi:10.1142/9789812812650_0008. ISBN 978-981-02-1847-8.