Abstract
The low-energy and low-momentum dynamics of systems with a spontaneously broken continuous symmetry is dominated by the ensuing Nambu-Goldstone bosons. It can be conveniently encoded in a model-independent effective field theory whose structure is fixed by symmetry up to a set of effective coupling constants. We construct the most general effective Lagrangian for the Nambu-Goldstone bosons of spontaneously broken global internal symmetry up to fourth order in derivatives. Rotational invariance and spatial dimensionality of one, two or three are assumed in order to obtain compact explicit expressions, but our method is completely general and can be applied without modifications to condensed matter systems with a discrete space group as well as to higher-dimensional theories. The general low-energy effective Lagrangian for relativistic systems follows as a special case. We also discuss the effects of explicit symmetry breaking and classify the corresponding terms in the Lagrangian. Diverse examples are worked out in order to make the results accessible to a wide theoretical physics community.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
H. Georgi, Effective field theory, Ann. Rev. Nucl. Part. Sci. 43 (1993) 209 [INSPIRE].
A.V. Manohar, Effective field theories, in Perturbative and Nonperturbative Aspects of Quantum Field Theory: Proceedings of the 35 th Internationale Universitätswochen für Kernund Teilchenphysik, H. Latal and W. Schweiger eds., Schladming, Austria, March 2-9 (1996), Lect. Notes Phys, 479 (1997) 311 [hep-ph/9606222] [INSPIRE].
S.R. Coleman, J. Wess and B. Zumino, Structure of phenomenological Lagrangians. 1., Phys. Rev. 177 (1969) 2239 [INSPIRE].
C.G. Callan Jr., S.R. Coleman, J. Wess and B. Zumino, Structure of phenomenological Lagrangians. 2., Phys. Rev. 177 (1969) 2247 [INSPIRE].
S. Weinberg, Phenomenological Lagrangians, Physica A 96 (1979) 327 [INSPIRE].
J. Gasser and H. Leutwyler, Chiral Perturbation Theory to One Loop, Annals Phys. 158 (1984) 142 [INSPIRE].
J. Gasser and H. Leutwyler, Chiral Perturbation Theory: Expansions in the Mass of the Strange Quark, Nucl. Phys. B 250 (1985) 465 [INSPIRE].
H. Leutwyler, Nonrelativistic effective Lagrangians, Phys. Rev. D 49 (1994) 3033 [hep-ph/9311264] [INSPIRE].
J.M. Román and J. Soto, Effective Field Theory Approach to Ferromagnets and Antiferromagnets in Crystalline Solids, Int. J. Mod. Phys. B 13 (1999) 755 [cond-mat/9709298].
J.M. Román and J. Soto, Spin waves in canted phases: An application to doped manganites, Phys. Rev. B 62 (2000) 3300.
C.P. Hofmann, Spontaneous Magnetization of an Ideal Ferromagnet: Beyond Dyson’s Analysis, Phys. Rev. B 84 (2011) 064414 [arXiv:1103.4110] [INSPIRE].
C.P. Hofmann, Thermodynamics of Two-Dimensional Ideal Ferromagnets - Three-Loop Analysis, Phys. Rev. B 86 (2012) 184409 [arXiv:1207.5937] [INSPIRE].
C.P. Hofmann, Thermodynamics of Ferromagnetic Spin Chains in a Magnetic Field: Impact of the Spin-Wave Interaction, Physica B: Cond. Mat. 442 (2014) 81 [arXiv:1306.0600] [INSPIRE].
C.P. Hofmann, Effective analysis of the O(N) antiferromagnet: Low temperature expansion of the order parameter, Phys. Rev. B 60 (1999) 406 [hep-ph/9706418] [INSPIRE].
F. Kämpfer, M. Moser and U.-J. Wiese, Systematic low-energy effective theory for magnons and charge carriers in an antiferromagnet, Nucl. Phys. B 729 (2005) 317 [cond-mat/0506324] [INSPIRE].
C. Brügger, F. Kämpfer, M. Moser, M. Pepe and U.-J. Wiese, Two-Hole Bound States from a Systematic Low-Energy Effective Field Theory for Magnons and Holes in an Antiferromagnet, Phys. Rev. B 74 (2006) 224432 [cond-mat/0606766] [INSPIRE].
C.P. Hofmann, Thermodynamics of O(N) Antiferromagnets in 2+1 Dimensions, Phys. Rev. B 81 (2010) 014416 [arXiv:0909.5239] [INSPIRE].
H. Leutwyler, On the foundations of chiral perturbation theory, Annals Phys. 235 (1994) 165 [hep-ph/9311274] [INSPIRE].
H. Watanabe, T. Brauner and H. Murayama, Massive Nambu-Goldstone Bosons, Phys. Rev. Lett. 111 (2013) 021601 [arXiv:1303.1527] [INSPIRE].
H. Watanabe and H. Murayama, The effective Lagrangian for nonrelativistic systems, arXiv:1402.7066 [INSPIRE].
J. Bijnens, Chiral perturbation theory beyond one loop, Prog. Part. Nucl. Phys. 58 (2007) 521 [hep-ph/0604043] [INSPIRE].
E. Witten, Global Aspects of Current Algebra, Nucl. Phys. B 223 (1983) 422 [INSPIRE].
J. Gasser and H. Leutwyler, Thermodynamics of Chiral Symmetry, Phys. Lett. B 188 (1987) 477 [INSPIRE].
K. Splittorff, D. Toublan and J.J.M. Verbaarschot, QCD with two colors at finite baryon density at next-to-leading order, Nucl. Phys. B 620 (2002) 290 [hep-ph/0108040] [INSPIRE].
K. Splittorff, D. Toublan and J.J.M. Verbaarschot, Thermodynamics of chiral symmetry at low densities, Nucl. Phys. B 639 (2002) 524 [hep-ph/0204076] [INSPIRE].
T. Brauner and S. Moroz, Topological interactions of Nambu-Goldstone bosons in quantum many-body systems, arXiv:1405.2670 [INSPIRE].
P. Cvitanović, Group theory: Birdtracks, Lie’s, and Exceptional Groups, Princeton University Press, Princeton, NJ, U.S.A. (2008)
S.R. Coleman, There are no Goldstone bosons in two-dimensions, Commun. Math. Phys. 31 (1973) 259 [INSPIRE].
H. Georgi, Lie Algebras in Particle Physics, Frontiers in Physics, Perseus Books, Reading, MA, U.S.A. (1999).
S. Scherer, Introduction to chiral perturbation theory, Adv. Nucl. Phys. 27 (2003) 277 [hep-ph/0210398] [INSPIRE].
H.W. Fearing and S. Scherer, Extension of the chiral perturbation theory meson Lagrangian to order p 6, Phys. Rev. D 53 (1996) 315 [hep-ph/9408346] [INSPIRE].
J. Fröhlich and U.M. Studer, Gauge invariance and current algebra in nonrelativistic many body theory, Rev. Mod. Phys. 65 (1993) 733 [INSPIRE].
G.E. Volovik, Linear momentum in ferromagnets, J. Phys. C: Solid State Phys. 20 (1987) L83.
C.P. Hofmann, Spontaneous magnetization of the O(3) ferromagnet at low temperatures, Phys. Rev. B 65 (2002) 094430 [cond-mat/0106492] [INSPIRE].
H. Watanabe and H. Murayama, Unified Description of Nambu-Goldstone Bosons without Lorentz Invariance, Phys. Rev. Lett. 108 (2012) 251602 [arXiv:1203.0609] [INSPIRE].
Y. Hidaka, Counting rule for Nambu-Goldstone modes in nonrelativistic systems, Phys. Rev. Lett. 110 (2013) 091601 [arXiv:1203.1494] [INSPIRE].
T. Brauner, Spontaneous Symmetry Breaking and Nambu-Goldstone Bosons in Quantum Many-Body Systems, Symmetry 2 (2010) 609 [arXiv:1001.5212] [INSPIRE].
Y. Nambu, Spontaneous Breaking of Lie and Current Algebras, J. Statist. Phys. 115 (2004) 7 [INSPIRE].
H.B. Nielsen and S. Chadha, On How to Count Goldstone Bosons, Nucl. Phys. B 105 (1976) 445 [INSPIRE].
C.P. Burgess, Goldstone and pseudoGoldstone bosons in nuclear, particle and condensed matter physics, Phys. Rept. 330 (2000) 193 [hep-th/9808176] [INSPIRE].
C.P. Hofmann, Low-Temperature Properties of Ferromagnetic Spin Chains in a Magnetic Field, Phys. Rev. B 87 (2013) 184420 [arXiv:1212.4774] [INSPIRE].
V.A. Miransky and I.A. Shovkovy, Spontaneous symmetry breaking with abnormal number of Nambu-Goldstone bosons and kaon condensate, Phys. Rev. Lett. 88 (2002) 111601 [hep-ph/0108178] [INSPIRE].
T. Schäfer, D.T. Son, M.A. Stephanov, D. Toublan and J.J.M. Verbaarschot, Kaon condensation and Goldstone’s theorem, Phys. Lett. B 522 (2001) 67 [hep-ph/0108210] [INSPIRE].
M. Nakahara, Geometry, Topology and Physics, Institute of Physics Publishing, Bristol, U.K. (2003).
J. Bijnens, G. Colangelo and G. Ecker, The Mesonic chiral Lagrangian of order p 6, JHEP 02 (1999) 020 [hep-ph/9902437] [INSPIRE].
E. D’Hoker and S. Weinberg, General effective actions, Phys. Rev. D 50 (1994) 6050 [hep-ph/9409402] [INSPIRE].
E. D’Hoker, Invariant effective actions, cohomology of homogeneous spaces and anomalies, Nucl. Phys. B 451 (1995) 725 [hep-th/9502162] [INSPIRE].
J.A. de Azcárraga, A.J. Macfarlane and J.C. Pérez Bueno , Effective actions, relative cohomology and Chern Simons forms, Phys. Lett. B 419 (1998) 186 [hep-th/9711064] [INSPIRE].
A. Altland and B. Simons, Condensed Matter Field Theory, Cambridge University Press, Cambridge, U.K. (2010).
I.J.R. Aitchison, Berry Phases, Magnetic Monopoles and Wess-Zumino Terms or How the Skyrmion Got Its Spin, Acta Phys. Polon. B 18 (1987) 207 [INSPIRE].
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1406.3439
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Andersen, J.O., Brauner, T., Hofmann, C.P. et al. Effective Lagrangians for quantum many-body systems. J. High Energ. Phys. 2014, 88 (2014). https://doi.org/10.1007/JHEP08(2014)088
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP08(2014)088