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Quantized massive collective modes and the T-dependence of the Fermi arc in underdoped cuprates
ARTICLE IN PRESS
Physica B 359–361 (2005) 515–517 www.elsevier.com/locate/physb
Quantized massive collective modes and the T-dependence of the Fermi arc in underdoped cuprates I. Kanazawa Department of Physics, Tokyo Gakugei University, 4-1-1 Nukuikitamachi, Koganeishi, Tokyo 184-8501, Japan
Abstract It has been proposed that the T-evolution of the Fermi arc is derived naturally from the restoration of the spontaneously broken symmetry in underdoped cuprates. r 2005 Published by Elsevier B.V. PACS: 75.10.Nr; 75.10.Jm; 74.70.Xy Keywords: High-T c cuprates; Pseudogap; Superconductivity
1. Introduction Angle-resolved photoemission (ARPES) in underdoped cuprates is quite unusual, where the Fermi surface changes topology. The pseudogap suppression first opens up near ðp; 0Þ and progressively gaps out larger portions of the Fermi contour, leading to gapless arcs which shrink with decreasing T. That is, as the temperature is lowered, the Fermi surface is progressively destroyed [1]. Recently, Ono and Ando [2] have observed that the magnitude of Hall coefficient drastically decreases with increasing temperature in lightly doped LaSrCuO. This result suggests that the effective carrier density becomes larger Tel.: +81 42 329 7548; fax:+84 42 329 7491.
E-mail address: [email protected] (I. Kanazawa). 0921-4526/$ - see front matter r 2005 Published by Elsevier B.V. doi:10.1016/j.physb.2005.01.132
than the nominal hole density at elevated temperatures, which is related to the T-evolution of the Fermi surface. In this paper, we will introduce the T-evolution of the Fermi surface in underdoped cuprates, taking into account the restoration of the spontaneously broken symmetry with increasing temperature, by expanding the previous formula [3,4].
2. Restoration of the spontaneously broken symmetry Taking into account that the symmetry in the undoped ð2 þ 1Þ-dimensional quantum antiferromagnet is invariant under local SU(2), we think that the perturbing gauge fields Aam introduced by
ARTICLE IN PRESS I. Kanazawa / Physica B 359– 361 (2005) 515–517
516
the hole has a local SU(2) symmetry. Then it is assumed that SU(2) gauge fields Aam are spontaneously broken through the Anderson–Higgs mechanism in a way similar to the breaking of the antiferromagnetic symmetry around the hole. After the symmetry breaking h0jfa j0i ¼ h0; 0; mðk^F Þi; we can obtain the effective Lagrangian density, Leff ; at small doping of holes [5,6]. That is, h0jf3 j0i can be regarded as a kind of the disorder parameter [7]. The value, mðk^F Þ; of the symmetry breaking depends strongly on the direction of Fermi momentum, k^F ; on the Fermi surface. Furthermore, the valve mðk^F Þ; is much correlated to the gap energy of the high-energy pseudogap. If the value of the high-energy pseudogap is related to the strength of the antiferromagnetic short-range order [8], the distortion, which is induced by the doped hole, becomes larger as the hole is doped in the state of the larger gap energy of the high-energy pseudogap. Since the value, mðk^F Þ; means the strength of the distortion induced by the doped hole, the value, mðk^F Þ; is higher around the hot spot: Leff ¼
1 ðqi N jc g1 eabc ejik Abi N ka Þ2 2 þ cþ ðiq0 g2 T a Aa0 Þc 1 þ c ðir g2 T a Aaðma0Þ Þ2 c 2m 1 ðqn Aam qm Aan þ g3 eabc Abm Acn Þ2 4 1 þ ðqm fa g4 eabc Abm fc Þ2 2 1 þ m21 ½ðA1m Þ2 þ ðA2m Þ2 2 þ m1 ½A1m qm f2 A2m qm f1 h þ g4 m1 f3 ððA1m Þ2 þ ðA2m Þ2 Þ i m2 A3m ½f1 A1m þ f2 A2m 2 ðf3 Þ2 2 2 2 m22 g4 m g f ðf Þ2 2 24 ðfa fa Þ2 ; 2m1 3 a 8m1
scribes two massive vector field A1m and A2m ; and one massless U(1) gauge field A3m : The thermodynamic potential is derived from the partition function, that is, OðT; fa fa Þ ¼ T ln Z=V : In the mean field and high-temperature approximations, the thermodynamic potential can be introduced as follows: OðT; fa fa Þ ¼ l2 ðfa fa Þ2 þ ð2lðk^F Þ2 þ 14g24 T 2 1 2 4 þ 13l2 T 2 Þfa fa 10 p T :
ð2Þ
From Eq. (2), it is seen that the thermodynamic potential is a function of fa fa and temperature T. As temperature increases, the minimum of the thermodynamic potential shifts to smaller values of fa fa ; and the minimum becomes less deep. The location of the minimum is ðfa fa Þmin mðk^F ; TÞ2eff 2 g4 1 2 2 ^ ¼ mðkF Þ þ T : 8l2 6
ð3Þ
From Eq. (3), the effective mass m1;eff ðk^F ; TÞ of and A2m is introduced as
A1m
m1;eff ðk^F ; TÞ ¼ g4 mðk^F ; TÞeff 2 1=2 g4 1 2 2 ^ ¼ g4 mðkF Þ þ T : 8l2 6
ð4Þ
According to the previous theory [4], the anomalous collision integral Iðk^F ; ; TÞ of the Fermi momentum k^F is given as follows: 2 1=2 g4 1 2 2 ^ ^ IðkF ; ; TÞ / g4 mðkF Þ þ T g22 8l2 6 Z Z h i dq do Im G A ðk^F þ q; e þ oÞ RT ðk^F ; q; e; oÞ;
ð5Þ
RT ðk^F ; q; e; oÞ ¼ ½1 þ N T ðoÞ f ðk^F þ q; e þ oÞ ½1 f ðk^F ; oÞ N T ðoÞ ½1 f ðk^F þ q; e þ oÞ f ðk^F ; oÞ; ð1Þ
ð6Þ
where is the spin parameter, pffiffiffi c Fermi field of the hole, m1 ¼ mg4 ; m2 ¼ 2 2lm; and T a are the SU(2) generators. The effective Lagrangian de-
cothðo=2TÞ ; and f and G A where N T ðoÞ ¼ are the hole distribution function and advanced inplane Green’s function of the hole, respectively. When temperature T increases above the restora-
N ia
12½1
ARTICLE IN PRESS I. Kanazawa / Physica B 359– 361 (2005) 515–517
tion temperature T res ðk^F Þ mðk^F Þððg24 =8l2 Þ þ ð1=6ÞÞ1=2 ; the spontaneously broken symmetry is restored. As a result, the effective mass, m1;eff ðk^F ; TÞ; of A1m and A2m annihilates and the anomalous collision integral annihilates above the restoration temperature T res ðk^F Þ: Since the restoration temperature T res ðk^F Þ increases gradually in the Fermi momentum from the cold spot ðp=2; p=2Þ to the hot spot ðp; 0Þ; we can explain the T-evolution of the Fermi surface naturally from the standpoint of the restoration of the spontaneous symmetry breaking.
3. Conclusion We have presented that the T-evolution of the Fermi surface is much related to the restoration of
517
the spontaneous symmetry breaking in underdoped cuprates.
References [1] J.C. Campuzano, H. Ding, M.R. Norman, M. Randeria, Physics and Chemistry of Transition-Metal Oxides, Springer, Berlin, 1999, p. 152. [2] S. Ono, Y. Ando, in: Abstracts of the Third International Workshop on Novel Quantum Phenomena in Transition Metal Oxides, Sendai, 2003, p. 82. [3] I. Kanazawa, Int. J. Mod. Phys. B 15 (2001) 4013. [4] I. Kanazawa, J. Phys. A 36 (2003) 9371. [5] I. Kanazawa, The Physics and Chemistry of Oxide Superconductors, Springer, Berlin, 1992, p. 481. [6] I. Kanazawa, Physica C 185–189 (1991) 1703. [7] G. ’t Hooft, Nucl. Phys. B 138 (1978) 1. [8] R.M. Dipasupil, M. Oda, N. Momono, M. Ito, J. Phys. Soc. Japan 71 (2002) 1535.
Physica B 359–361 (2005) 515–517 www.elsevier.com/locate/physb
Quantized massive collective modes and the T-dependence of the Fermi arc in underdoped cuprates I. Kanazawa Department of Physics, Tokyo Gakugei University, 4-1-1 Nukuikitamachi, Koganeishi, Tokyo 184-8501, Japan
Abstract It has been proposed that the T-evolution of the Fermi arc is derived naturally from the restoration of the spontaneously broken symmetry in underdoped cuprates. r 2005 Published by Elsevier B.V. PACS: 75.10.Nr; 75.10.Jm; 74.70.Xy Keywords: High-T c cuprates; Pseudogap; Superconductivity
1. Introduction Angle-resolved photoemission (ARPES) in underdoped cuprates is quite unusual, where the Fermi surface changes topology. The pseudogap suppression first opens up near ðp; 0Þ and progressively gaps out larger portions of the Fermi contour, leading to gapless arcs which shrink with decreasing T. That is, as the temperature is lowered, the Fermi surface is progressively destroyed [1]. Recently, Ono and Ando [2] have observed that the magnitude of Hall coefficient drastically decreases with increasing temperature in lightly doped LaSrCuO. This result suggests that the effective carrier density becomes larger Tel.: +81 42 329 7548; fax:+84 42 329 7491.
E-mail address: [email protected] (I. Kanazawa). 0921-4526/$ - see front matter r 2005 Published by Elsevier B.V. doi:10.1016/j.physb.2005.01.132
than the nominal hole density at elevated temperatures, which is related to the T-evolution of the Fermi surface. In this paper, we will introduce the T-evolution of the Fermi surface in underdoped cuprates, taking into account the restoration of the spontaneously broken symmetry with increasing temperature, by expanding the previous formula [3,4].
2. Restoration of the spontaneously broken symmetry Taking into account that the symmetry in the undoped ð2 þ 1Þ-dimensional quantum antiferromagnet is invariant under local SU(2), we think that the perturbing gauge fields Aam introduced by
ARTICLE IN PRESS I. Kanazawa / Physica B 359– 361 (2005) 515–517
516
the hole has a local SU(2) symmetry. Then it is assumed that SU(2) gauge fields Aam are spontaneously broken through the Anderson–Higgs mechanism in a way similar to the breaking of the antiferromagnetic symmetry around the hole. After the symmetry breaking h0jfa j0i ¼ h0; 0; mðk^F Þi; we can obtain the effective Lagrangian density, Leff ; at small doping of holes [5,6]. That is, h0jf3 j0i can be regarded as a kind of the disorder parameter [7]. The value, mðk^F Þ; of the symmetry breaking depends strongly on the direction of Fermi momentum, k^F ; on the Fermi surface. Furthermore, the valve mðk^F Þ; is much correlated to the gap energy of the high-energy pseudogap. If the value of the high-energy pseudogap is related to the strength of the antiferromagnetic short-range order [8], the distortion, which is induced by the doped hole, becomes larger as the hole is doped in the state of the larger gap energy of the high-energy pseudogap. Since the value, mðk^F Þ; means the strength of the distortion induced by the doped hole, the value, mðk^F Þ; is higher around the hot spot: Leff ¼
1 ðqi N jc g1 eabc ejik Abi N ka Þ2 2 þ cþ ðiq0 g2 T a Aa0 Þc 1 þ c ðir g2 T a Aaðma0Þ Þ2 c 2m 1 ðqn Aam qm Aan þ g3 eabc Abm Acn Þ2 4 1 þ ðqm fa g4 eabc Abm fc Þ2 2 1 þ m21 ½ðA1m Þ2 þ ðA2m Þ2 2 þ m1 ½A1m qm f2 A2m qm f1 h þ g4 m1 f3 ððA1m Þ2 þ ðA2m Þ2 Þ i m2 A3m ½f1 A1m þ f2 A2m 2 ðf3 Þ2 2 2 2 m22 g4 m g f ðf Þ2 2 24 ðfa fa Þ2 ; 2m1 3 a 8m1
scribes two massive vector field A1m and A2m ; and one massless U(1) gauge field A3m : The thermodynamic potential is derived from the partition function, that is, OðT; fa fa Þ ¼ T ln Z=V : In the mean field and high-temperature approximations, the thermodynamic potential can be introduced as follows: OðT; fa fa Þ ¼ l2 ðfa fa Þ2 þ ð2lðk^F Þ2 þ 14g24 T 2 1 2 4 þ 13l2 T 2 Þfa fa 10 p T :
ð2Þ
From Eq. (2), it is seen that the thermodynamic potential is a function of fa fa and temperature T. As temperature increases, the minimum of the thermodynamic potential shifts to smaller values of fa fa ; and the minimum becomes less deep. The location of the minimum is ðfa fa Þmin mðk^F ; TÞ2eff 2 g4 1 2 2 ^ ¼ mðkF Þ þ T : 8l2 6
ð3Þ
From Eq. (3), the effective mass m1;eff ðk^F ; TÞ of and A2m is introduced as
A1m
m1;eff ðk^F ; TÞ ¼ g4 mðk^F ; TÞeff 2 1=2 g4 1 2 2 ^ ¼ g4 mðkF Þ þ T : 8l2 6
ð4Þ
According to the previous theory [4], the anomalous collision integral Iðk^F ; ; TÞ of the Fermi momentum k^F is given as follows: 2 1=2 g4 1 2 2 ^ ^ IðkF ; ; TÞ / g4 mðkF Þ þ T g22 8l2 6 Z Z h i dq do Im G A ðk^F þ q; e þ oÞ RT ðk^F ; q; e; oÞ;
ð5Þ
RT ðk^F ; q; e; oÞ ¼ ½1 þ N T ðoÞ f ðk^F þ q; e þ oÞ ½1 f ðk^F ; oÞ N T ðoÞ ½1 f ðk^F þ q; e þ oÞ f ðk^F ; oÞ; ð1Þ
ð6Þ
where is the spin parameter, pffiffiffi c Fermi field of the hole, m1 ¼ mg4 ; m2 ¼ 2 2lm; and T a are the SU(2) generators. The effective Lagrangian de-
cothðo=2TÞ ; and f and G A where N T ðoÞ ¼ are the hole distribution function and advanced inplane Green’s function of the hole, respectively. When temperature T increases above the restora-
N ia
12½1
ARTICLE IN PRESS I. Kanazawa / Physica B 359– 361 (2005) 515–517
tion temperature T res ðk^F Þ mðk^F Þððg24 =8l2 Þ þ ð1=6ÞÞ1=2 ; the spontaneously broken symmetry is restored. As a result, the effective mass, m1;eff ðk^F ; TÞ; of A1m and A2m annihilates and the anomalous collision integral annihilates above the restoration temperature T res ðk^F Þ: Since the restoration temperature T res ðk^F Þ increases gradually in the Fermi momentum from the cold spot ðp=2; p=2Þ to the hot spot ðp; 0Þ; we can explain the T-evolution of the Fermi surface naturally from the standpoint of the restoration of the spontaneous symmetry breaking.
3. Conclusion We have presented that the T-evolution of the Fermi surface is much related to the restoration of
517
the spontaneous symmetry breaking in underdoped cuprates.
References [1] J.C. Campuzano, H. Ding, M.R. Norman, M. Randeria, Physics and Chemistry of Transition-Metal Oxides, Springer, Berlin, 1999, p. 152. [2] S. Ono, Y. Ando, in: Abstracts of the Third International Workshop on Novel Quantum Phenomena in Transition Metal Oxides, Sendai, 2003, p. 82. [3] I. Kanazawa, Int. J. Mod. Phys. B 15 (2001) 4013. [4] I. Kanazawa, J. Phys. A 36 (2003) 9371. [5] I. Kanazawa, The Physics and Chemistry of Oxide Superconductors, Springer, Berlin, 1992, p. 481. [6] I. Kanazawa, Physica C 185–189 (1991) 1703. [7] G. ’t Hooft, Nucl. Phys. B 138 (1978) 1. [8] R.M. Dipasupil, M. Oda, N. Momono, M. Ito, J. Phys. Soc. Japan 71 (2002) 1535.