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Theory of strongly phase fluctuating d-wave superconductors and the spin response in underdoped cuprates
Physica C 408–410 (2004) 414–415 www.elsevier.com/locate/physc
Theory of strongly phase fluctuating d-wave superconductors and the spin response in underdoped cuprates Igor F. Herbut
*
Physics Department, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6
Abstract General theory of d-wave quasiparticles coupled to phase fluctuations of the superconducting order parameter is discussed. In the charge sector the superfluid density is found to conform to the Uemura scaling. The spin susceptibility exhibits four distinct regimes with increasing frequency, and scales with the superconducting Tc , as observed. Ó 2004 Elsevier B.V. All rights reserved. Keywords: Vortex fluctuations; Underdoped cuprates; Magnetic response
The intimate relationship between antiferromagnetism and d-wave superconductivity (dSC) has been one of the central themes in the physics of cuprates. Recently a new connection between the two has been put forward [1,2], by which the antiferromagnetism is the result of the dynamical breaking of an approximate ‘‘chiral’’ symmetry of the standard dSC. The relevant interaction between the low-energy quasiparticles is provided by the quantum vortex fluctuations, which are assumed to increase with underdoping. The low energy theory of d-wave quasiparticles and the phase fluctuations of the superconducting order parameter at T ¼ 0 Rand in two dimensions may be written as [3] S ¼ d3 xL, L ¼ Lw þ Lgauge þ L/ , where Lw ¼
2 X
Wi cl ðol ial ÞWi ;
ð1Þ
L/ ¼
2 X b jðr iAn ÞUn j2 þ l2 jUn j2 þ jUn j4 : 2 n¼1
Wi describe the neutral spin-1/2 quasiparticles (spinons) near the four nodes at K1;2 [2], complex U1;2 create the fluctuating vortex loops associated with spin up and down, and a, v [4] and A1;2 ¼ Aþ A are the auxiliary gauge fields that facilitate the spinon-vortex coupling. Jc is the charge current, and A an electromagnetic gauge potential. The two quasiparticle velocities have been set to vF ¼ vD ¼ 1 in (1). K is the bare superfluid density. When hU1 i ¼ hU2 i 6¼ 0, the above theory reduces to the QED3 [4, 2], with the insulating spin density wave (SDW) as the ground state [5]. Here I focus on the superconducting state, in which hUn i ¼ 0. Integrating out~ v and the virtual vortex fluctuations yields [3]
i¼1
Lgauge
*
1 1 ¼ a ðr A Þ þ v ðr Aþ Þ þ 2Kðv þ AÞ2 p p þ iJc ðv þ AÞ; ð2Þ
Fax: +1-604-291-3592. E-mail address: [email protected] (I.F. Herbut).
ð3Þ
L ! Lw þ
6m 2 qsf 2 a þ A þ iZJc A þ OðJ2c Þ; p 2
ð4Þ
with qsf ¼ 48mK=ð12m þ 4pKÞ. Z ¼ 12m=ð12m þ 4pKÞ is the charge renormalization factor. m2 ¼ l2 þ OðbÞ is the uniform vortex susceptibility, assumed to be proportional to doping. Since on dimensional grounds m Tc , for m K, qsf Tc , in qualitative agreement with the Uemura scaling. Furthermore, spin may be considered separated from the charge in Eq. (4). Upon integration over the massive ~ a, the spin sector becomes
0921-4534/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2004.03.021
I.F. Herbut / Physica C 408–410 (2004) 414–415
the 2+1 dimensional Thirring model. In the RPA approximation the spin susceptibility becomes [6] pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 12m N Hðx2 q2 Þ x2 q2 : ð5Þ v00 ð2Ki q; xÞ ¼ p ð96m=pÞ2 þ x2 q2 This describes the soft spin mode in the dSC, the condensation of which on the insulating side yields the SDW [1], [2]. Eq. (5) implies that for x Tc , the maximum values of v00 are located at four diagonally incommensurate wave vectors 2K1;2 . Due to their low intensity ( x) and narrowness, these ‘mother’ peaks should be very hard to observe, and indeed have not yet been detected. As the frequency increases, however, both the intensity and the width of the peaks grow. Restoring vF vD in (5), and shifting the peaks to the vicinity of ðp; pÞ, one finds the ‘mother’ peaks overlapping first at four parallel incommensurate positions, independent of frequency. With the further increase in frequency, the initial peaks at some point begin also to overlap at ðp; pÞ, and for a while the commensurate response dominates. Assuming K1 ¼ 0:57ðp; pÞ [7], and vF ¼ 1:2 eV A the energy of the ‘resonance’ may be estimated to 60 meV, not far from 40 meV seen in slightly underdoped cuprates. Finally,
415
at even larger x the maximum in Eq. (5) shifts to a q 6¼ 0, which implies a weak redistribution of the commensurate peak to four parallel incommensurate positions, but this time with upward dispersion. The evolution of the spin response is in agreement with the observations in underdoped YBCO [8].
Acknowledgements This work has been supported by NSERC of Canada and the Research Corporation.
References [1] [2] [3] [4] [5] [6] [7] [8]
I.F. Herbut, Phys. Rev. Lett. 88 (2002) 047006. I.F. Herbut, Phys. Rev. B 66 (2002) 094504. I.F. Herbut, cond-mat/0302058. M. Franz, Z. Tesanovic, Phys. Rev. Lett. 87 (2001) 257003. B.H. Seradjeh, I.F. Herbut, Phys. Rev. B 66 (2002) 184507. I.F. Herbut, D.J. Lee, Phys. Rev. B 68 (2003) 104518. H.A. Mook et al., Nature 395 (1998) 580. M. Arai et al., Phys. Rev. Lett. 83 (1999) 608.
Theory of strongly phase fluctuating d-wave superconductors and the spin response in underdoped cuprates Igor F. Herbut
*
Physics Department, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6
Abstract General theory of d-wave quasiparticles coupled to phase fluctuations of the superconducting order parameter is discussed. In the charge sector the superfluid density is found to conform to the Uemura scaling. The spin susceptibility exhibits four distinct regimes with increasing frequency, and scales with the superconducting Tc , as observed. Ó 2004 Elsevier B.V. All rights reserved. Keywords: Vortex fluctuations; Underdoped cuprates; Magnetic response
The intimate relationship between antiferromagnetism and d-wave superconductivity (dSC) has been one of the central themes in the physics of cuprates. Recently a new connection between the two has been put forward [1,2], by which the antiferromagnetism is the result of the dynamical breaking of an approximate ‘‘chiral’’ symmetry of the standard dSC. The relevant interaction between the low-energy quasiparticles is provided by the quantum vortex fluctuations, which are assumed to increase with underdoping. The low energy theory of d-wave quasiparticles and the phase fluctuations of the superconducting order parameter at T ¼ 0 Rand in two dimensions may be written as [3] S ¼ d3 xL, L ¼ Lw þ Lgauge þ L/ , where Lw ¼
2 X
Wi cl ðol ial ÞWi ;
ð1Þ
L/ ¼
2 X b jðr iAn ÞUn j2 þ l2 jUn j2 þ jUn j4 : 2 n¼1
Wi describe the neutral spin-1/2 quasiparticles (spinons) near the four nodes at K1;2 [2], complex U1;2 create the fluctuating vortex loops associated with spin up and down, and a, v [4] and A1;2 ¼ Aþ A are the auxiliary gauge fields that facilitate the spinon-vortex coupling. Jc is the charge current, and A an electromagnetic gauge potential. The two quasiparticle velocities have been set to vF ¼ vD ¼ 1 in (1). K is the bare superfluid density. When hU1 i ¼ hU2 i 6¼ 0, the above theory reduces to the QED3 [4, 2], with the insulating spin density wave (SDW) as the ground state [5]. Here I focus on the superconducting state, in which hUn i ¼ 0. Integrating out~ v and the virtual vortex fluctuations yields [3]
i¼1
Lgauge
*
1 1 ¼ a ðr A Þ þ v ðr Aþ Þ þ 2Kðv þ AÞ2 p p þ iJc ðv þ AÞ; ð2Þ
Fax: +1-604-291-3592. E-mail address: [email protected] (I.F. Herbut).
ð3Þ
L ! Lw þ
6m 2 qsf 2 a þ A þ iZJc A þ OðJ2c Þ; p 2
ð4Þ
with qsf ¼ 48mK=ð12m þ 4pKÞ. Z ¼ 12m=ð12m þ 4pKÞ is the charge renormalization factor. m2 ¼ l2 þ OðbÞ is the uniform vortex susceptibility, assumed to be proportional to doping. Since on dimensional grounds m Tc , for m K, qsf Tc , in qualitative agreement with the Uemura scaling. Furthermore, spin may be considered separated from the charge in Eq. (4). Upon integration over the massive ~ a, the spin sector becomes
0921-4534/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2004.03.021
I.F. Herbut / Physica C 408–410 (2004) 414–415
the 2+1 dimensional Thirring model. In the RPA approximation the spin susceptibility becomes [6] pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 12m N Hðx2 q2 Þ x2 q2 : ð5Þ v00 ð2Ki q; xÞ ¼ p ð96m=pÞ2 þ x2 q2 This describes the soft spin mode in the dSC, the condensation of which on the insulating side yields the SDW [1], [2]. Eq. (5) implies that for x Tc , the maximum values of v00 are located at four diagonally incommensurate wave vectors 2K1;2 . Due to their low intensity ( x) and narrowness, these ‘mother’ peaks should be very hard to observe, and indeed have not yet been detected. As the frequency increases, however, both the intensity and the width of the peaks grow. Restoring vF vD in (5), and shifting the peaks to the vicinity of ðp; pÞ, one finds the ‘mother’ peaks overlapping first at four parallel incommensurate positions, independent of frequency. With the further increase in frequency, the initial peaks at some point begin also to overlap at ðp; pÞ, and for a while the commensurate response dominates. Assuming K1 ¼ 0:57ðp; pÞ [7], and vF ¼ 1:2 eV A the energy of the ‘resonance’ may be estimated to 60 meV, not far from 40 meV seen in slightly underdoped cuprates. Finally,
415
at even larger x the maximum in Eq. (5) shifts to a q 6¼ 0, which implies a weak redistribution of the commensurate peak to four parallel incommensurate positions, but this time with upward dispersion. The evolution of the spin response is in agreement with the observations in underdoped YBCO [8].
Acknowledgements This work has been supported by NSERC of Canada and the Research Corporation.
References [1] [2] [3] [4] [5] [6] [7] [8]
I.F. Herbut, Phys. Rev. Lett. 88 (2002) 047006. I.F. Herbut, Phys. Rev. B 66 (2002) 094504. I.F. Herbut, cond-mat/0302058. M. Franz, Z. Tesanovic, Phys. Rev. Lett. 87 (2001) 257003. B.H. Seradjeh, I.F. Herbut, Phys. Rev. B 66 (2002) 184507. I.F. Herbut, D.J. Lee, Phys. Rev. B 68 (2003) 104518. H.A. Mook et al., Nature 395 (1998) 580. M. Arai et al., Phys. Rev. Lett. 83 (1999) 608.