1932
0
  1. Home
  2. A-Z Publications
  3. Annual Review of Condensed Matter Physics
  4. Volume 11, 2020
  5. Article

Annual Review of Condensed Matter Physics

Volume 11, 2020

Counting Rules of Nambu–Goldstone Modes

Abstract

When global continuous symmetries are spontaneously broken, there appear gapless collective excitations called Nambu–Goldstone modes (NGMs) that govern the low-energy property of the system. The application of this famous theorem ranges from high-energy particle physics to condensed matter and atomic physics. When a symmetry breaking occurs in systems that lack the Lorentz invariance to start with, as is usually the case in condensed matter systems, the number of resulting NGMs can be lower than that of broken symmetry generators, and the dispersion of NGMs is not necessarily linear. In this article, we review recently established formulae for NGMs associated with broken internal symmetries that work equally for relativistic and nonrelativistic systems. We also discuss complexities of NGMs originating from space-time symmetry breaking. Along the way we cover many illuminating examples from various context. We also present a complementary point of view from the Lieb–Schultz–Mattis theorem.

Keyword(s): internal symmetries, low-energy effective theory, nonrelativsitic systems, space–time symmetries, spontaneous symmetry breaking
Loading

Article metrics loading...

/content/journals/10.1146/annurev-conmatphys-031119-050644
2020-03-10
2024-04-25
Loading full text...

Full text loading...

/deliver/fulltext/conmatphys/11/1/annurev-conmatphys-031119-050644.html?itemId=/content/journals/10.1146/annurev-conmatphys-031119-050644&mimeType=html&fmt=ahah

Literature Cited

  1. 1. 
    Nambu Y 1960. Phys. Rev. Lett. 4:380–82
  2. 2. 
    Nambu Y, Jona-Lasinio G 1961. Phys. Rev. 122:345–58
  3. 3. 
    Goldstone J 1961. Nuovo Cim 19:154–64
  4. 4. 
    Goldstone J, Salam A, Weinberg S 1962. Phys. Rev. 127:965–70
  5. 5. 
    Nambu Y 2009. Rev. Mod. Phys. 81:1015–18
  6. 6. 
    Weinberg S 1995. The Quantum Theory of Fields: I. Foundations, II. Applications Cambridge, UK: Cambridge Univ. Press
  7. 7. 
    Anderson PW 1984. Basic Notions of Condensed Matter Physics Boca Raton, FL: CRC
  8. 8. 
    Kittel C 2005. Introduction to Solid State Physics Hoboken, NJ: John Wiley & Sons. 8th ed.
  9. 9. 
    Ashcroft NW, Mermin ND 1976. Solid State Physics Boston: Cengage Learning
  10. 10. 
    Grosso G, Parravicini GP 2014. Solid State Physics San Diego: Elsevier. 2nd ed.
  11. 11. 
    Senthil T 2015. Annu. Rev. Condens. Matter Phys. 6:299–324
  12. 12. 
    Wen XG 2019. Science 363:6429eaal3099
  13. 13. 
    Miransky VA, Shovkovy IA 2002. Phys. Rev. Lett. 88:111601
  14. 14. 
    Schäfer T, Son D, Stephanov M, Toublan D, Verbaarschot J 2001. Phys. Lett. B 522:67–75
  15. 15. 
    Blaschke D, Ebert D, Klimenko KG, Volkov MK, Yudichev VL 2004. Phys. Rev. D 70:014006
  16. 16. 
    Ebert D, Klimenko KG, Yudichev VL 2005. Phys. Rev. C 72:015201
  17. 17. 
    He L, Jin M, Zhuang P 2006. Phys. Rev. A 74:033604
  18. 18. 
    Lange RV 1965. Phys. Rev. Lett. 14:3–6
  19. 19. 
    Lange RV 1966. Phys. Rev. 146:301–3
  20. 20. 
    Brauner T 2007. Phys. Rev. D 75:105014
  21. 21. 
    Brauner T 2010. Symmetry 2:609–57
  22. 22. 
    Watanabe H, Brauner T 2011. Phys. Rev. D 84:125013
  23. 23. 
    Watanabe H, Murayama H 2012. Phys. Rev. Lett. 108:251602
  24. 24. 
    Hidaka Y 2013. Phys. Rev. Lett. 110:091601
  25. 25. 
    Watanabe H, Murayama H 2014. Phys. Rev. X 4:031057
  26. 26. 
    Nielsen H, Chadha S 1976. Nuclear Phys. B 105:445–53
  27. 27. 
    Nambu Y 2004. J. Stat. Phys. 115:7–17
  28. 28. 
    Nozières P, Pines D 1990. Theory of Quantum Liquids: Superfluid Bose Liquids Boca Raton, FL: CRC
  29. 29. 
    Pethick CJ, Smith H 2008. Bose-Einstein Condensation in Dilute Gases Cambridge, UK: Cambridge Univ. Press
  30. 30. 
    Ueda M 2010. Fundamentals and New Frontiers of Bose-Einstein Condensation Singapore: World Scientific
  31. 31. 
    Auerbach A 2012. Interacting Electrons and Quantum Magnetism New York: Springer-Verlag
  32. 32. 
    Leutwyler H 1994. Phys. Rev. D 49:3033–43
  33. 33. 
    Burgess C 2000. Phys. Rep. 330:193–261
  34. 34. 
    Fradkin E 2013. Field Theories of Condensed Matter Physics Cambridge, UK: Cambridge Univ. Press. 2nd ed.
  35. 35. 
    Ho TL 1998. Phys. Rev. Lett. 81:742–45
  36. 36. 
    Ohmi T, Machida K 1998. J. Phys. Soc. Jpn. 67:1822–25
  37. 37. 
    Takahashi DA, Nitta M 2015. Ann. Phys. 354:101–56
  38. 38. 
    Ueda M 2012. Annu. Rev. Condens. Matter Phys. 3:263–83
  39. 39. 
    Kawaguchi Y, Ueda M 2012. Phys. Rep. 520:253–381
  40. 40. 
    Stamper-Kurn DM, Ueda M 2013. Rev. Mod. Phys. 85:1191–244
  41. 41. 
    Cazalilla MA, Rey AM 2014. Rep. Prog. Phys. 77:124401
  42. 42. 
    Marti GE, MacRae A, Olf R, Lourette S, Fang F, Stamper-Kurn DM 2014. Phys. Rev. Lett. 113:155302
  43. 43. 
    Pich A 2018. Lectures given at The 2017 Les Houches Summer School on “Effective Field Theory in Particle Physics and Cosmology, Les Houches, France, July 3–28, 2017.. IFIC/18-13, FTUV/18-0415. arXiv:1804.05664
    [Google Scholar]
  44. 44. 
    Georgi H 1993. Annu. Rev. Nuclear Part. Sci. 43:209–52
  45. 45. 
    Bernard V, Meißner UG 2007. Annu. Rev. Nuclear Part. Sci. 57:33–60
  46. 46. 
    Furnstahl RJ, Rupak G, Schäfer T 2008. Annu. Rev. Nuclear Part. Sci. 58:1–25
  47. 47. 
    Hayata T, Hidaka Y 2015. Phys. Rev. D 91:056006
  48. 48. 
    Minami Y, Hidaka Y 2018. Phys. Rev. E 97:012130
  49. 49. 
    Coleman S, Wess J, Zumino B 1969. Phys. Rev. 177:2239–47
  50. 50. 
    Callan CG, Coleman S, Wess J, Zumino B 1969. Phys. Rev. 177:2247–50
  51. 51. 
    Weinberg S 1979. Phys. A: Stat. Mech. Appl. 96:327–40
  52. 52. 
    Pekker D, Varma C 2015. Annu. Rev. Condens. Matter Phys. 6:269–97
  53. 53. 
    Leutwyler H 1994. Ann. Phys. 235:165–203
  54. 54. 
    Hohenberg PC, Halperin BI 1977. Rev. Mod. Phys. 49:435–79
  55. 55. 
    Kapustin A 2012. arXiv:1207.0457
  56. 56. 
    Hohenberg PC 1967. Phys. Rev. 158:383–86
  57. 57. 
    Mermin ND, Wagner H 1966. Phys. Rev. Lett. 17:1133–36
  58. 58. 
    Coleman S 1973. Commun. Math. Phys. 31:259–64
  59. 59. 
    Anderson PW 1990. Phys. Today 43:117
  60. 60. 
    Landau LD, Lifshitz EM 1959. Course of Theoretical Physics, Vol. 7: Theory and Elasticity London: Pergamon
  61. 61. 
    Chaikin PM, Lubensky TC, Witten TA 1995. Principles of Condensed Matter Physics Cambridge, UK: Cambridge Univ. Press
  62. 62. 
    Fukuyama H 1975. Solid State Commun. 17:1323–26
  63. 63. 
    Watanabe H, Murayama H 2014. Phys. Rev. D 89:101701
  64. 64. 
    Nagaosa N, Tokura Y 2013. Nat. Nanotechnol. 8:899–911
  65. 65. 
    Fert A, Reyren N, Cros V 2017. Nat. Rev. Mater. 2:17031
  66. 66. 
    Stone M 1996. Phys. Rev. B 53:16573–78
  67. 67. 
    Zang J, Mostovoy M, Han JH, Nagaosa N 2011. Phys. Rev. Lett. 107:136804
  68. 68. 
    Watanabe H, Murayama H 2014. Phys. Rev. Lett. 112:191804
  69. 69. 
    Watanabe H, Parameswaran SA, Raghu S, Vishwanath A 2014. Phys. Rev. B 90:045145
  70. 70. 
    Kobayashi M, Nitta M 2014. Phys. Rev. D 90:025010
  71. 71. 
    Boninsegni M, Prokof'ev NV 2012. Rev. Mod. Phys. 84:759–76
  72. 72. 
    Chan MHW, Hallock RB, Reatto L 2013. J. Low Temp. Phys. 172:317–63
  73. 73. 
    Son DT 2005. Phys. Rev. Lett. 94:175301
  74. 74. 
    Watanabe H, Brauner T 2012. Phys. Rev. D 85:085010
  75. 75. 
    Kunimi M, Kato Y 2012. Phys. Rev. B 86:060510
  76. 76. 
    Kobayashi M, Nitta M 2014. Phys. Rev. Lett. 113:120403
  77. 77. 
    Salam A, Strathdee J 1969. Phys. Rev. 184:1760–68
  78. 78. 
    Low I, Manohar AV 2002. Phys. Rev. Lett. 88:101602
  79. 79. 
    De Gennes P, Prost J 1993. The Physics of Liquid Crystals Oxford, UK: Oxford Univ. Press. 2nd ed.
  80. 80. 
    Grinstein G, Pelcovits RA 1981. Phys. Rev. Lett. 47:856–59
  81. 81. 
    Grinstein G, Pelcovits RA 1982. Phys. Rev. A 26:915–25
  82. 82. 
    Ivanov EA, Ogievetskii VI 1975. Theor. Math. Phys. 25:1050–59
  83. 83. 
    Brauner T, Watanabe H 2014. Phys. Rev. D 89:085004
  84. 84. 
    Hidaka Y, Noumi T, Shiu G 2015. Phys. Rev. D 92:045020
  85. 85. 
    Rothstein IZ, Shrivastava P 2018. J. High Energy Phys. 2018:14
  86. 86. 
    Watanabe H, Murayama H 2013. Phys. Rev. Lett. 110:181601
  87. 87. 
    Hayata T, Hidaka Y 2014. Phys. Lett. B 735:195–99
  88. 88. 
    Oganesyan V, Kivelson SA, Fradkin E 2001. Phys. Rev. B 64:195109
  89. 89. 
    Fradkin E, Kivelson SA, Lawler MJ, Eisenstein JP, Mackenzie AP 2010. Annu. Rev. Condens. Matter Phys. 1:153–78
  90. 90. 
    Watanabe H, Vishwanath A 2014. PNAS 111:16314–18
  91. 91. 
    Ruhman J, Berg E 2014. Phys. Rev. B 90:235119
  92. 92. 
    Bahri Y, Potter AC 2015. Phys. Rev. B 92:035131
  93. 93. 
    Chua V, Assawasunthonnet W, Fradkin E 2017. Phys. Rev. B 96:035110
  94. 94. 
    Wilczek F 2012. Phys. Rev. Lett. 109:160401
  95. 95. 
    Bruno P 2013. Phys. Rev. Lett. 111:070402
  96. 96. 
    Watanabe H, Oshikawa M 2015. Phys. Rev. Lett. 114:251603
  97. 97. 
    Khemani V, Lazarides A, Moessner R, Sondhi SL 2016. Phys. Rev. Lett. 116:250401
  98. 98. 
    Else DV, Bauer B, Nayak C 2016. Phys. Rev. Lett. 117:090402
  99. 99. 
    Yao NY, Potter AC, Potirniche ID, Vishwanath A 2017. Phys. Rev. Lett. 118:030401
  100. 100. 
    Choi S, Choi J, Landig R, Kucsko G, Zhou H et al. 2017. Nature 543:221–25
  101. 101. 
    Zhang J, Hess PW, Kyprianidis A, Becker P, Lee A et al. 2017. Nature 543:217–20
  102. 102. 
    Sacha K, Zakrzewski J 2017. Rep. Progress Phys. 81:016401
  103. 103. 
    Else DV, Monroe C, Nayak C, Yao NY 2020.Annu. Rev. Condens. Matter Phys. 11:467–99
  104. 104. 
    Hongo M, Kim S, Noumi T, Ota A 2019. J. High Energy Phys. 2019:131
  105. 105. 
    Hayata T, Hidaka Y 2018. RIKEN-QHP-380, RIKEN-iTHEMS-Report-18. arXiv:1808.07636
    [Google Scholar]
  106. 106. 
    Sieberer LM, Buchhold M, Diehl S 2016. Rep. Progress Phys. 79:096001
  107. 107. 
    Lieb E, Schultz T, Mattis D 1961. Ann. Phys. (N.Y.) 16:407–66
  108. 108. 
    Affleck I, Lieb EH 1986. Lett. Math. Phys. 12:57–69
  109. 109. 
    Yamanaka M, Oshikawa M, Affleck I 1997. Phys. Rev. Lett. 79:1110–13
  110. 110. 
    Oshikawa M 2000. Phys. Rev. Lett. 84:1535–38
  111. 111. 
    Hastings MB 2004. Phys. Rev. B 69:104431
  112. 112. 
    Parameswaran SA, Turner AM, Arovas DP, Vishwanath A 2013. Nat. Phys. 9:299–303
  113. 113. 
    Watanabe H, Vishwanath A 2014.PNAS 111:16314–18
  114. 114. 
    Watanabe H 2018. Phys. Rev. B 97:165117
  115. 115. 
    Wen XG 2004. Quantum Field Theory of Many-Body Systems Oxford, UK: Oxford Univ. Press
  116. 116. 
    Oshikawa M, Senthil T 2006. Phys. Rev. Lett. 96:060601
  117. 117. 
    Weinberg S 1972. Phys. Rev. Lett. 29:1698–701
  118. 118. 
    Uchino S, Kobayashi M, Nitta M, Ueda M 2010. Phys. Rev. Lett. 105:230406
  119. 119. 
    Nitta M, Takahashi DA 2015. Phys. Rev. D 91:025018
  120. 120. 
    Volkov D, Akulov V 1973. Phys. Lett. B 46:109–10
  121. 121. 
    Salam A, Strathdee J 1974. Phys. Lett. B 49:465–67
  122. 122. 
    Yu Y, Yang K 2008. Phys. Rev. Lett. 100:090404
  123. 123. 
    Blaizot JP, Hidaka Y, Satow D 2015. Phys. Rev. A 92:063629
  124. 124. 
    Sannomiya N, Katsura H, Nakayama Y 2016. Phys. Rev. D 94:045014
  125. 125. 
    Higgs PW 1964. Phys. Rev. Lett. 13:508–9
  126. 126. 
    Englert F, Brout R 1964. Phys. Rev. Lett. 13:321–23
  127. 127. 
    Hama Y, Hatsuda T, Uchino S 2011. Phys. Rev. D 83:125009
  128. 128. 
    Watanabe H, Murayama H 2014. Phys. Rev. D 90:121703
  129. 129. 
    Gongyo S, Karasawa S 2014. Phys. Rev. D 90:085014
  130. 130. 
    Gaiotto D, Kapustin A, Seiberg N, Willett B 2015. J. High Energy Phys. 2015:172
  131. 131. 
    Lake E 2018. arXiv:1802.07747
  132. 132. 
    Yamamoto N 2016. Phys. Rev. D 93:085036
  133. 133. 
    Ozaki S, Yamamoto N 2017. J. High Energy Phys. 2017:98
  134. 134. 
    Sogabe N, Yamamoto N 2019. Phys. Rev. D 99:125003
/content/journals/10.1146/annurev-conmatphys-031119-050644
Loading
Counting Rules of Nambu–Goldstone Modes
Annual Review of Condensed Matter Physics 11, 169 (2020); https://doi.org/10.1146/annurev-conmatphys-031119-050644
/content/journals/10.1146/annurev-conmatphys-031119-050644
/content/journals/10.1146/annurev-conmatphys-031119-050644
Loading

Data & Media loading...

  • Article Type: Review Article

Most Cited Most Cited RSS feed

This is a required field
Please enter a valid email address
Approval was a Success
Invalid data
An Error Occurred
Approval was partially successful, following selected items could not be processed due to error