Optical conductivity of a two-dimensional metal at the onset of spin-density-wave order

Physical Review B - Condensed Matter and Materials Physics

Volumn 89, Issue 15, 2014, Pages

  Chubukov, Andrey V ^a   Maslov, Dmitrii L ^b   Yudson, Vladimir I ^c  

^a University of Wisconsin Madison   (United States)

^b University of Florida   (United States)

^c INSTITUTE OF SPECTROSCOPY   (Russian Federation)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84899715547     PISSN: 10980121     EISSN: 1550235X     Source Type: Journal    
DOI: 10.1103/PhysRevB.89.155126     Document Type: Article

References (81)

1. N. E. Hussey, R. A. Cooper, Xiaofeng Xu, Y. Wang, I. Mouzopoulou, B. Vignolle, and C. Proust, Philos. Trans. R. Soc. A. 369, 1626 (2011). PTRMAD 1364-503X 10.1098/rsta.2010.0196
2. See J. A. N. Bruin, H. Sakai, R. S. Perry, and A. P. Mackenzie, Science 339, 804 (2013), and references therein. SCIEAS 0036-8075 10.1126/science. 1227612
3. R. Mahajan, M. Barkeshli, and S. A. Hartnoll, Phys. Rev. B 88, 125107 (2013). PRBMDO 1098-0121 10.1103/PhysRevB.88.125107
4. See, e.g., A. A. Abrikosov, L. P. Gor'kov, and I. E. Dzyaloshinski, Methods of Quantum Field Theory in Statistical Physics (Dover, New York, 1963);
5. D. Pines and P. Nozieres, The Theory of Quantum Liquids (Addison-Wesley, Menlo Park, 1966)
6. L. Taillefer, Annu. Rev. Condens. Matter Phys. 1, 51 (2010) PNASA6 1947-5454 10.1146/annurev-conmatphys-070909-104117;
7. M. A. Tanatar, N. P. Butch, K. Jin, K. Kirshenbaum, R. L. Greene, and J. Paglione, Proc. Natl. Acad. Sci. USA 109, 8440 (2012). 0027-8424 10.1073/pnas.1120273109
8. A similar behavior with (Equation presented) observed in the helimagnet MnSi under pressure [38] is unlikely to be associated with quantum criticality, as the (Equation presented) behavior is seen in a wide range of pressures above the critical one
9. G. Oomi, T. Kagayama, and Y. Onuki, J. Alloys Compd. 271-273, 482 (1998) JALCEU 0925-8388 10.1016/S0925-8388(98)00122-4;
10. M. Nicklas, M. Brando, G. Knebel, F. Mayr, W. Trinkl, and A. Loidl, Phys. Rev. Lett. 82, 4268 (1999); 10.1103/PhysRevLett.82.4268
11. P. G. Niklowitz, F. Beckers, G. G. Lonzarich, G. Knebel, B. Salce, J. Thomasson, N. Bernhoeft, D. Braithwaite, and J. Flouquet, Phys. Rev. B 72, 024424 (2005). PRBMDO 1098-0121 10.1103/PhysRevB.72.024424
12. T. Timusk and B. Statt, Rep. Prog. Phys. 62, 61 (1999) RPPHAG 0034-4885 10.1088/0034-4885/62/1/002;
13. D. N. Basov, R. D. Averitt, D. van der Marel, M. Dressel, and K. Haule, Rev. Mod. Phys. 83, 471 (2011). RMPHAT 0034-6861 10.1103/RevModPhys.83.471
14. Recent experiment in the pseudogap region did find a (Equation presented) behavior of the optical scattering rate, reminiscent of a FL; see S. I. Mirzaei, D. Stricker, J. N. Hancock, C. Berthod, A. Georges, E. van Heumen, M. K. Chan, X. Zhao, Y. Li, M. Greven, N. Barišić, and D. van der Marel, Proc. Natl. Acad. Sci. USA 110, 5774 (2013). PNASA6 0027-8424 10.1073/pnas.1218846110
15. A. V. Puchkov J. Phys. Chem. Solids 59, 1907 (1998) JPCSAW 0022-3697 10.1016/S0022-3697(98)00146-2;
16. D. N. Basov, R. Liang, B. Dabrovski, D. A. Bonn, W. N. Hardy, and T. Timusk, Phys. Rev. Lett. 77, 4090 (1996) PRLTAO 0031-9007 10.1103/PhysRevLett. 77.4090;
17. A. Puchkov, D. Basov, and T. Timusk, J. Phys.: Condens. Matter 8, 10049 (1996) JCOMEL 0953-8984 10.1088/0953-8984/8/48/023;
18. R. Haslinger, A. V. Chubukov, and Ar. Abanov, Phys. Rev. B 63, 020503 (2000) PRBMDO 0163-1829 10.1103/PhysRevB.63.020503;
19. Ar. Abanov, A. V. Chubukov, and J. Schmalian, Phys. Rev. B 63, 180510 (2001). PRBMDO 0163-1829 10.1103/PhysRevB.63.180510
20. A. El Azrak, R. Nahoum, N. Bontemps, M. Guilloux-Viry, C. Thivet, A. Perrin, S. Labdi, Z. Z. Li, and H. Raffy, Phys. Rev. B 49, 9846 (1994). PRBMDO 0163-1829 10.1103/PhysRevB.49.9846
21. D. van der Marel, H. J. A. Molegraaf, J. Zaanen, Z. Nussinov, F. Carbone, A. Damascelli, H. Eisaki, M. Greven, P. H. Kes, and M. Li, Nature (London) 425, 271 (2003). NATUAS 0028-0836 10.1038/nature01978
22. M. R. Norman and A. V. Chubukov, Phys. Rev. B 73, 140501 (2006). PRBMDO 1098-0121 10.1103/PhysRevB.73.140501
23. J. S. Dodge, C. P. Weber, J. Corson, J. Orenstein, Z. Schlesinger, J. W. Reiner, and M. R. Beasley, Phys. Rev. Lett. 85, 4932 (2000). PRLTAO 0031-9007 10.1103/PhysRevLett.85.4932
24. Y. S. Lee, J. Yu, J. S. Lee, T. W. Noh, T. H. Gimm, H.-Y. Choi, and C. B. Eom, Phys. Rev. B 66, 041104 (R) (2002). PRBMDO 0163-1829 10.1103/PhysRevB.66. 041104
25. S. Kamal, D. M. Kim, C. B. Eom, and J. S. Dodge, Phys. Rev. B 74, 165115 (2006). PRBMDO 1098-0121 10.1103/PhysRevB.74.165115
26. F. P. Mena, D. van der Marel, A. Damascelli, M. Fäth, A. A. Menovsky, and J. A. Mydosh, Phys. Rev. B 67, 241101 (2003). PRBMDO 0163-1829 10.1103/PhysRevB.67.241101
27. See E. Abrahams and C. Varma, Proc. Natl. Acad. Sci. USA 97, 5714 (2000), and references therein. PNASA6 0027-8424 10.1073/pnas.100118797
28. R. Hlubina and T. M. Rice, Phys. Rev. B 51, 9253 (1995). PRBMDO 0163-1829 10.1103/PhysRevB.51.9253
29. See, e.g., A. Shekhter and C. M. Varma, Phys. Rev. B 79, 045117 (2009), and references therein. PRBMDO 1098-0121 10.1103/PhysRevB.79.045117
30. S. A. Hartnoll, D. M. Hofman, M. A. Metlitski, and S. Sachdev, Phys. Rev. B 84, 125115 (2011). PRBMDO 1098-0121 10.1103/PhysRevB.84.125115
31. A "generic FS" means that it corresponds to a nonquadratic anisotropic dispersion for the 3D case, and is either concave or multiply connected for the 2D case; see Refs. [24,25]
32. See, e.g., A. J. Schofield, Contemp. Phys. 40, 95 (1999). CTPHAF 0010-7514 10.1080/001075199181602
33. A. Rosch and P. C. D. Howell, Phys. Rev. B 72, 104510 (2005) PRBMDO 1098-0121 10.1103/PhysRevB.72.104510;
34. A. Rosch, Ann. Phys. (NY) 15, 526 (2006). ANPYA2 0003-3804 10.1002/andp.200510203
35. D. L. Maslov, V. I. Yudson, and A. V. Chubukov, Phys. Rev. Lett. 106, 106403 (2011) PRLTAO 0031-9007 10.1103/PhysRevLett.106.106403;
36. H. K. Pal, V. I. Yudson, and D. L. Maslov, Phys. Rev. B 85, 085439 (2012) PRBMDO 1098-0121 10.1103/PhysRevB.85.085439;
37. H. K. Pal, V. I. Yudson, and D. L. Maslov, Lith. J. Phys. 52, 142 (2012). 1648-8504 10.3952/lithjphys.52207
38. S.-S. Lee, Phys. Rev. B 80, 165102 (2009) PRBMDO 1098-0121 10.1103/PhysRevB.80.165102;
39. S.-S. Lee, Y.-B. Kim, and A. V. Chubukov (unpublished)
40. M. A. Metlitski and S. Sachdev, Phys. Rev. B 82, 075128 (2010) PRBMDO 1098-0121 10.1103/PhysRevB.82.075128;
41. M. A. Metlitski and S. Sachdev, New J. Phys. 12, 105007 (2010). NJOPFM 1367-2630 10.1088/1367-2630/12/10/105007
42. Ar. Abanov, A. V. Chubukov, and J. Schmalian, Adv. Phys. 52, 119 (2003). ADPHAH 0001-8732 10.1080/0001873021000057123
43. See, e.g., B. P. Stojkovic and D. Pines, Phys. Rev. Lett. 76, 811 (1996) PRLTAO 0031-9007 10.1103/PhysRevLett.76.811;
44. B. P. Stojkovic and D. Pines, Phys. Rev. B 55, 8576 (1997) PRBMDO 0163-1829 10.1103/PhysRevB.55.8576;
45. A. T. Zheleznyak, V. M. Yakovenko, H. D. Drew, and I. I. Mazin, Phys. Rev. B 57, 3089 (1998) PRBMDO 0163-1829 10.1103/PhysRevB.57.3089;
46. L. B. Ioffe and A. J. Millis, Phys. Rev. B 58, 11631 (1998) PRBMDO 0163-1829 10.1103/PhysRevB.58.11631;
47. J. Kokalj, N. E. Hussey, and R. H. McKenzie, Phys. Rev. B 86, 045132 (2012). PRBMDO 1098-0121 10.1103/PhysRevB.86.045132
48. E. Abrahams, J. Schmalian, and P. Wölfle, arXiv:1303.3926
49. To be more precise, the initial and the final states belong to the opposite hot spots. Once the fermions at hot spots 2 and (Equation presented) are integrated out, the effective scattering process can be labeled as (Equation presented), i.e., the fermions located initially at spots 1 and (Equation presented) remain in the vicinity of their respective hot spots. In the terminology of Ref. [34], such a process can be viewed either as a (Equation presented) part of the "backscattering process" with an amplitude given by the difference of the charge and spin components of the scattering amplitude, or as a special case of a Cooper process, in which the initial fermions with the total momentum close to zero retain their initial momenta
50. D. F. Mross, J. McGreevy, H. Liu, and T. Senthil, Phys. Rev. B 82, 045121 (2010). PRBMDO 1098-0121 10.1103/PhysRevB.82.045121
51. G. Y. Chitov and A. J. Millis, Phys. Rev. B 64, 054414 (2001). PRBMDO 0163-1829 10.1103/PhysRevB.64.054414
52. A. V. Chubukov, D. L. Maslov, S. Gangadharaiah, and L. I. Glazman, Phys. Rev. B 71, 205112 (2005). PRBMDO 1098-0121 10.1103/PhysRevB.71.205112
53. A. V. Chubukov and D. L. Maslov, Phys. Rev. B 68, 155113 (2003). PRBMDO 0163-1829 10.1103/PhysRevB.68.155113
54. A. V. Chubukov, D. L. Maslov, and A. J. Millis, Phys. Rev. B 73, 045128 (2006). PRBMDO 1098-0121 10.1103/PhysRevB.73.045128
55. I. L. Aleiner and K. B. Efetov, Phys. Rev. B 74, 075102 (2006). PRBMDO 1098-0121 10.1103/PhysRevB.74.075102
56. H. v. Löhneysen, A. Rosch, M. Vojta, and P. Wölfle, Rev. Mod. Phys. 79, 1015 (2007). RMPHAT 0034-6861 10.1103/RevModPhys.79.1015
57. H. Meier, C. Pèpin, and K. B. Efetov, Phys. Rev. B 84, 205131 (2011). PRBMDO 1098-0121 10.1103/PhysRevB.84.205131
58. We thank M. Metlitski for clarifying this point for us
59. Yu. A. Bychkov, L. P. Gor'kov, and I. E. Dzyaloshisnkii, Zh. Eksp. Teor. Fiz. 50, 738 (1966)
60. Yu. A. Bychkov, L. P. Gor'kov, and I. E. Dzyaloshisnkii, [Sov. Phys.-JETP 23, 489 (1966)]
61. See, e.g., D. L. Maslov, in Nanophysics: Coherence and Transport, edited by H. Bouchiat, Y. Gefen, G. Montambaux, and J. Dalibard, Les Houches, Session LXXXI, 2004 (Elsevier, Amsterdam, 2005), p. 1 and references therein
62. A. Imambekov, T. L. Schmidt, and L. I. Glazman, Rev. Mod. Phys. 84, 1253 (2012). RMPHAT 0034-6861 10.1103/RevModPhys.84.1253
63. G. Schwiete and K. B. Efetov, Phys. Rev. B 74, 165108 (2006) PRBMDO 1098-0121 10.1103/PhysRevB.74.165108;
64. A. Shekhter and A. M. Finkelstein, Phys. Rev. B 74, 205122 (2006) PRBMDO 1098-0121 10.1103/PhysRevB.74.205122;
65. A. V. Chubukov and D. L. Maslov, Phys. Rev. B 76, 165111 (2007). PRBMDO 1098-0121 10.1103/PhysRevB.76.165111
66. We neglect the dependence of the self-energy on (Equation presented), so that the velocity renormalization factor (Equation presented) coincides with the (Equation presented) factor
67. The reduction of the transport scattering rate compared with the quasiparticle decay rate due to a small (Equation presented) factor has recently been emphasized in X. Deng, J. Mravlje, R. Zitko, M. Ferrero, G. Kotliar, and A. Georges, Phys. Rev. Lett. 110, 086401 (2013) PRLTAO 0031-9007 10.1103/PhysRevLett.110.086401;
68. W. Xu, K. Haule, and G. Kotliar, Phys. Rev. Lett. 111, 036401 (2013). PRLTAO 0031-9007 10.1103/PhysRevLett.111.036401
69. E. M. Lifshitz and L. P. Pitaevskii, Physical Kinetics (Pergamon, Oxford, 1981);
70. J. Rammer and H. Smith, Rev. Mod. Phys. 58, 323 (1986). RMPHAT 0034-6861 10.1103/RevModPhys.58.323
71. A. I. Larkin and A. V. Varlamov, Theory of fluctuations in Superconductors (Oxford University Press, Oxford, UK, 2005)
72. In general, the current vertex contains not the bare velocity but rather the "charge" one, which differs from the bare velocity by a factor of (Equation presented) [52-54]. In our case, however, this derivative can be neglected (see comment [45]), and thus the charge velocity is almost equal to the bare one
73. K. Ueda, J. Phys. Soc. Jpn. 43, 1497 (1977) JUPSAU 0031-9015 10.1143/JPSJ.43.1497;
74. A. Rosch, Phys. Rev. Lett. 82, 4280 (1999) PRLTAO 0031-9007 10.1103/PhysRevLett.82.4280;
75. S. V. Syzranov and J. Schmalian, Phys. Rev. Lett. 109, 156403 (2012). PRLTAO 0031-9007 10.1103/PhysRevLett.109.156403
76. S. Sachdev, C. Buragohain, and M. Vojta, Science 286, 2479 (1999). SCIEAS 0036-8075 10.1126/science.286.5449.2479
77. G. M. Eliashberg, Zh. Eksp. Teor. Fiz. 41, 1241 (1961)
78. G. M. Eliashberg, [Sov. Phys.-JETP 14, 886 (1962)]
79. K. Michaeli and A. M. Finkel'stein, Phys. Rev. B 80, 115111 (2009). PRBMDO 1098-0121 10.1103/PhysRevB.80.115111
80. D. L. Maslov and A. V. Chubukov, Phys. Rev. B 86, 155137 (2012). PRBMDO 1098-0121 10.1103/PhysRevB.86.155137
81. Maslov D.L.,Chubukov A.V., Phys. Rev. B, 2012, 86., 155137