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Quantum criticality in the SO(5) theory of antiferromagnetism and superconductivity
Physica C 387 (2003) 65–68 www.elsevier.com/locate/physc
Quantum criticality in the SO(5) theory of antiferromagnetism and superconductivity T.K. Kopec *, T.A. Zaleski Institute of Low Temperature and Structure Research of the Polish Academy of Sciences, 50-950 Wrocław, Poland
Abstract The quantum critical point scenario is studied within the unified theory of superconductivity and antiferromagentism based on the SO(5) symmetry. The quantum-critical scaling function for the dynamic spin susceptibility is obtained from the lattice SO(5) quantum nonlinear r-model in three dimensions, revealing that in the quantum-critical region the frequency scale for spin excitations is simply set by the absolute temperature. Implications for the non-Fermi liquid behavior of the normal-state resistivity due to spin fluctuations in the quantum-critical region are also noted. Ó 2003 Elsevier Science B.V. All rights reserved. Keywords: Quantum criticality; SO(5) group; Antiferromagnetism; High-Tc theory
1. Introduction The principal idea behind the universality at phase transitions––that the physics at the transition point is ruled only by interaction at macroscopic length scales––leads to critical behavior, where classical thermal fluctuations govern the phase transition scenario. However, there is an ample class of systems for which quantum effects dominate the phase transition, resulting in a critical point accessible by tuning the relevant variable other than the temperature [1]. Recently, there have been a number of detailed dynamic scattering measurements on superconducting cuprates [2] showing that the overall frequency x scale of the spin excitation spectrum is given simply by the
*
Corresponding author. E-mail address: [email protected] (T.K. Kopec´).
absolute temperature kB T . Interestingly, this x=kB T scaling is consistent with an existence of the nearby quantum critical point (QCP). The possibility of a quantum critical behavior governing the physical properties of the superconducting cuprates has attracted great attention in recent years motivated by the fact that quantum criticality affords a controlled study of the non-Fermi liquid behavior [3]. The proximity of magnetism and superconductivity (and competition between these phases) is one of the most important hallmarks in the physics of high-Tc superconductors. In a recent theoretical framework antiferromagnetic (AF) and superconducting (SC) order parameters form a five-dimensional supervector n and the SO(5) rotation turns AF into SC states and vice versa [4]. With increasing quantum fluctuations ordering in some phases can be suppressed opening the possibility to a QCP where the AF state goes into SC
0921-4534/03/$ - see front matter Ó 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0921-4534(03)00642-7
66
T.K. Kopec, T.A. Zaleski / Physica C 387 (2003) 65–68
state through a direct second-order phase transition [4]. In this paper we characterize the quantum critical behavior which develops in the vicinity of this special point.
2. The model We begin by writing down the SO(5) Hamiltonian for a simple cubic (s.c.) three-dimensional (3D) lattice: H¼
X X 1 X 2 Li V ðnÞ 2l L15 Jij ni nj i 2u i i i
where i ¼ 1; . . . ; N (N is the number of lattice sites). The locally defined five-component superspin vector ni ¼ ðn1 ; n2 ; n3 ; n4 ; n5 Þi describes AF ðn2 ; n3 ; n4 Þi and SC ðn1 ; n5 Þi degrees of freedom. In the ZhangÕs formulation [4] these are treated as mutually commuting coordinates and their dynamics is given by their conjugate momenta: pli ¼ io=onli and ½nl ; pm ¼ idlm . Here, Llm i ¼ nli pmi nmi pli are the generators of SO(5) algebra and L15 i is a charge operator coupled to the chemical-potential l (measured from the half-filling). The parameter u in turn regulate the quantum fluctuation effects by controlling the kinetic energy of rotors (u is simply the moment of inertia of the rigid rotor). Furthermore, Jij ( J for nearest neighbors and 0 otherwise) corresponds to the stiffness in the charge and spin channel. In the presence of SO(5) symmetry breaking, P a quadratic term of the form V ðni Þ ¼ ðg=2Þ i ðn22i þ n23i þ n24i Þ is also allowed. The anisotropy constant g selects either the ‘‘easy plane’’ in the SC space, or an ‘‘easy sphere’’ in the AF space, depending on the sign of g. Given H , one can simply perform a Legendre transformation to obtain the corresponding Euclidean Lagrangian in the Matsubara ‘‘imaginary time’’ s formulation (0 6 s 6 1=kB T b): L½p; n ¼ ipðsÞ
d nðsÞ þ H ðn; pÞ: ds
ð2Þ
The partition function Z ¼ TreH =kB T of the system is then given by
Z¼
Z Y
½Dni
Z Y Dp i
2p i i Rb dsL½p;n :
dð1 n2i Þdðni pi Þe 0
ð3Þ
Due to the rigid nature of the quantum rotors one must be cautious and integrate only over the momenta transverse to the supervector (pi ni ¼ 0) with fixed length (n2i ¼ 1). To this end, we cast the problem in terms of the exactly soluble quantum spherical model [5] by taking the superspin components as continuous variables, i.e., 1 < ni ðsÞ < 1, but constrained (on average) to have unit length. The spherical closure amounts Q relation 2 then to the replacement: dð1 n Þ ! dðN i i P 2 n Þ. i i The convenient way to enforce the spherical constraint is to use the functional analog of the R þ1 Dirac-d––function representation dðxÞ ¼ 1 ðdk= 2pÞeikx which introduces the Lagrange multiplier kðsÞ [5]. In the thermodynamic limit N ! 1 the method of steepest descents becomes exact and at the saddle point kðsÞ k0 will satisfy 1 X 1¼ ½2Gs ðk; x‘ Þ þ 3Ga ðk; x‘ Þ ð4Þ bN k;x ‘
where Ga ðk; x‘ Þ ¼ 1=½2k0 J ðkÞ þ ux2‘ g Gs ðk; x‘ Þ ¼ 1=½2k0 J ðkÞ þ uðx‘ þ 2ilÞ2
ð5Þ
are the (order parameter) susceptibilities in the AF and SC sector, respectively. The summation in Eq. (4) is performed over all wave vector components k (pp=Na 6 ka 6 pp=Na, where a ¼ x; y; z; p are integers 0 6 p 6 N 1 and a is the lattice spacing). Furthermore, x‘ ¼ 2p‘=b ð‘ ¼ 0; 1; 2; . . .Þ are the (Bose) Matsubara frequencies arising from the restriction to periodic functions along the imaginary time direction. Finally, J ðkÞ ¼ J eðkÞ where eðkÞ ¼ cosðakx Þ þ cosðaky Þ þ cosðakz Þ is the structure factor for the sc lattice in 3D. At the critical lines for the AF and SC phases the Lagrange multiplier k assumes values at which corresponding uniform and static order parameter susceptibilities diverge: G1 a ðk ¼ 0; x‘ ¼ 0Þ ¼ 0 and G1 ðk ¼ 0; x ¼ 0Þ ¼ 0, which yields ka0 ¼ ð3J þ ‘ s s gÞ=2 for AF and k0 ¼ ð3J þ 4l2 uÞ=2 for SC state,
T.K. Kopec, T.A. Zaleski / Physica C 387 (2003) 65–68
respectively. On examining of the saddle point equation (4) one finds that g > 0 favors the AF state whereas the l > 0 term prefers p theffiffiffiffiffiffiffiffiffiffi SC state. For ka0 ¼ ks0 (which happens for lc ¼ g=4u) both AF and SC transition lines merge at the bicritical point at (Tbc ; lc ). However, for sufficiently small parameter u quantum fluctuations will drive Tbc to zero giving rise to a quantum critical point.
v00 ðxÞ ¼ d3 Qv00 ðQ; xÞ––an important quantity for verification of the x=kB T scaling in the QC region. To establish the connection with our SO(5) calculations we consider ‘‘imaginary-time’’ spin–spin correlation function Gij ðs s0 Þ ¼ hSi ðsÞ Sj ðs0 Þi where h i denotes the ensemble averaging with the Hamiltonian (1). Utilizing the relation between spin operators and SO(5) Lie algebra generators [4] we can write: 43 0 42 42 0 Gij ðs s0 Þ ¼ hL43 i ðsÞLj ðs Þi þ hLi ðsÞLj ðs Þi
3. QC region
32 0 þ hL32 i ðsÞLj ðs Þi:
An especially interesting regime appears in the vicinity of the QCP where kB T is significantly larger than any energy scale which measures deviations of the coupling constants from their zero-temperature critical values. In this so-called quantum critical (QC) region [6] the system feels the finite value of the temperature before becoming sensitive to the deviations of l from lc (i.e. the leading behavior of the system will be described at all scales by the T ¼ 0 critical point and its universal response to a finite temperature). To quantify this observation we introduce the deviation of k from the QCP value dk ¼ k k0 (which plays the role of the critical ‘‘mass’’ vanishing at QCP; note that dk is related to the correlation length m n ¼ hc=2dk which diverges as n ðl lc Þ ). At finite temperatures, the deviations from T ¼ 0 behavior are distinguished by the scale set by the thermal coherence length nT T 1=z (where z is the dynamic critical exponent). The QC region is then defined by the inequality nT < n. The magnetic scattering cross-section for neutrons which is proportional to the dynamic structure factor SðQ; xÞ is directly related to the spin–spin correlation function hS0 ð0Þ Sj ðtÞi [8]: Z þ1 X 1 dteixt eiQrj hS0 ð0Þ Sj ðtÞi SðQ; xÞ ¼ 2p 1 rj 1 ¼ ½1 þ nðxÞv00 ðQ; xÞ p
67
R
ð6Þ
where v00 ðQ; xÞ is the imaginary part of the dynamic susceptibility and nðxÞ ¼ ½1 þ expðx= 1 kB T Þ is the Bose function. An integration of the spectral function v00 ðQ; xÞ over wave vectors Q yields the integrated (local) spin susceptibility
ð7Þ
The averages over the product of angular momentum operators can be conveniently performed using functional differentiation by adding to the Lagrangian (2) sources which couple linearly to Llm . As a result we obtain in the momentum-frequency representation for the propagator (7): 4u X GðQ; x‘ Þ ¼ Ga ðq; mn Þ½1 þ uðx‘ 2mn Þ2 bN q;mn
Ga ðQ q; x‘ mn Þ:
ð8Þ
To proceed, we perform now the analytic continuation of GðQ; x‘ Þ to real frequencies ixn ! x þ i0þ (where 0þ stands for the positive infinitesimal). After integration over external momenta we obtain finally the imaginary part of the local dynamic susceptibility v00 ðxÞ in the scaling form: x l D v00 ðxÞ ¼ v00 ; ; kB T kB T kB T 2 3 x D 1 l k k T T 5 Uv 4 B ; B ; Xc kB T kB T Xc kBlT Xc kBlT ð9Þ where
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 4y X cosh ð2y‘Þ K1 ð2y‘Þ; XðyÞ ¼ 1 þ 2 p ‘¼1 ‘ " # Z þ1 3 2 C1 ðyÞ ¼ djqðjÞ þ : 3=2 3=2 ð3 jÞ ðy þ 3 jÞ 1 ð10Þ Here, K1 ðxÞ stands for the modified Bessel function [7] and qðjÞ is the density of states for 3D s.c. lattice:
68
T.K. Kopec, T.A. Zaleski / Physica C 387 (2003) 65–68
1 qðjÞ ¼ 3 p
Z
minð1;2jÞ
dw pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 w2 maxð1;2jÞ "rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi#
j þ w 2
K 1 Hð3 jjjÞ 2
ð11Þ
where KðxÞ and HðxÞ is the elliptic integral of the first kind [7] and the unit step function, respec1=2 1=4 1=2 tively. Subsequently, pffiffiffiffiffiffiffiffi Xc ðyÞ ¼ p 2 C1 ðgc =J Þ XðyÞ and D ¼ J =u. Finally, Uv ðy1 ; y2 ; y3 Þ Z þ1 p y1 dfqðfÞq f y1 ¼ y3 y22 1 y22 h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi y1 þ 2 2 þ ð3 fÞy22 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 þ ð3 fÞy22 2 þ ð3 fÞy22 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi y3 2 þ ð3 fÞy22
coth 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi y3 y1 y3 2 2 þ ð3 fÞy2 þ ð12Þ coth 2 2
is the universal scaling function. The appearance of the temperature as the relevant energy scale for spin dynamics may be very significant because of its possible relation to the normal state characteristics of superconducting cuprates. 4. Final remarks In conclusion, we have performed a quantitative study of the quantum critical point scenario within the concept of the SO(5) group based the-
ory of superconductivity and antiferromagnetism for high-Tc materials. The physical properties governed by the proximity to the AF/SC QCP account for the universal scaling for the finite temperature behavior. The general prerequisite of a given QCP scenario is that it must involve concept of a broken symmetry. Several questions related to this issue arise in applying other alternative QCP based theories (like charge instability near optimum doping, see e.g., [9]) to cuprates: what are the relevant phases with order and associated broken symmetries? Some possibilities that address these problems are being currently debated [10], however further theoretical work on quantum critical dynamics in cuprate materials is clearly called for.
References [1] S.L. Sondhi, S.M. Girvin, J.P. Carini, D. Shahar, Rev. Mod. Phys. 69 (1979) 315. [2] G. Aeppli, T.E. Mason, S.M. Hayden, H.A. Mook, J. Kulda, Science 287 (1997) 1432. [3] S. Sachdev, J. Ye, Phys. Rev. Lett. 69 (1992) 2411. [4] S.C. Zhang, Science 275 (1997) 1089. [5] T. Zaleski, T.K. Kopec, Phys. Rev. B 62 (2000) 9059. [6] S. Chakraverty, B.I. Halperin, D.R. Nelson, Phys. Rev. B 39 (1989) 2344. [7] M. Abramovitz, I. Stegun, Handbook of Mathematical Functions, Dover, New York, 1970. [8] W. Marshall, R.D. Lowde, Rep. Prog. Phys. 31 (1968) 705. [9] C. di Castro, L. Benefatto, S. Caprara, C. Castellani, M. Grilli, Physica C 341 (2000) 1715. [10] C.M. Varma, Phys. Rev. Lett. 83 (1999) 3538.
Quantum criticality in the SO(5) theory of antiferromagnetism and superconductivity T.K. Kopec *, T.A. Zaleski Institute of Low Temperature and Structure Research of the Polish Academy of Sciences, 50-950 Wrocław, Poland
Abstract The quantum critical point scenario is studied within the unified theory of superconductivity and antiferromagentism based on the SO(5) symmetry. The quantum-critical scaling function for the dynamic spin susceptibility is obtained from the lattice SO(5) quantum nonlinear r-model in three dimensions, revealing that in the quantum-critical region the frequency scale for spin excitations is simply set by the absolute temperature. Implications for the non-Fermi liquid behavior of the normal-state resistivity due to spin fluctuations in the quantum-critical region are also noted. Ó 2003 Elsevier Science B.V. All rights reserved. Keywords: Quantum criticality; SO(5) group; Antiferromagnetism; High-Tc theory
1. Introduction The principal idea behind the universality at phase transitions––that the physics at the transition point is ruled only by interaction at macroscopic length scales––leads to critical behavior, where classical thermal fluctuations govern the phase transition scenario. However, there is an ample class of systems for which quantum effects dominate the phase transition, resulting in a critical point accessible by tuning the relevant variable other than the temperature [1]. Recently, there have been a number of detailed dynamic scattering measurements on superconducting cuprates [2] showing that the overall frequency x scale of the spin excitation spectrum is given simply by the
*
Corresponding author. E-mail address: [email protected] (T.K. Kopec´).
absolute temperature kB T . Interestingly, this x=kB T scaling is consistent with an existence of the nearby quantum critical point (QCP). The possibility of a quantum critical behavior governing the physical properties of the superconducting cuprates has attracted great attention in recent years motivated by the fact that quantum criticality affords a controlled study of the non-Fermi liquid behavior [3]. The proximity of magnetism and superconductivity (and competition between these phases) is one of the most important hallmarks in the physics of high-Tc superconductors. In a recent theoretical framework antiferromagnetic (AF) and superconducting (SC) order parameters form a five-dimensional supervector n and the SO(5) rotation turns AF into SC states and vice versa [4]. With increasing quantum fluctuations ordering in some phases can be suppressed opening the possibility to a QCP where the AF state goes into SC
0921-4534/03/$ - see front matter Ó 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0921-4534(03)00642-7
66
T.K. Kopec, T.A. Zaleski / Physica C 387 (2003) 65–68
state through a direct second-order phase transition [4]. In this paper we characterize the quantum critical behavior which develops in the vicinity of this special point.
2. The model We begin by writing down the SO(5) Hamiltonian for a simple cubic (s.c.) three-dimensional (3D) lattice: H¼
X X 1 X 2 Li V ðnÞ 2l L15 Jij ni nj i 2u i i i
where i ¼ 1; . . . ; N (N is the number of lattice sites). The locally defined five-component superspin vector ni ¼ ðn1 ; n2 ; n3 ; n4 ; n5 Þi describes AF ðn2 ; n3 ; n4 Þi and SC ðn1 ; n5 Þi degrees of freedom. In the ZhangÕs formulation [4] these are treated as mutually commuting coordinates and their dynamics is given by their conjugate momenta: pli ¼ io=onli and ½nl ; pm ¼ idlm . Here, Llm i ¼ nli pmi nmi pli are the generators of SO(5) algebra and L15 i is a charge operator coupled to the chemical-potential l (measured from the half-filling). The parameter u in turn regulate the quantum fluctuation effects by controlling the kinetic energy of rotors (u is simply the moment of inertia of the rigid rotor). Furthermore, Jij ( J for nearest neighbors and 0 otherwise) corresponds to the stiffness in the charge and spin channel. In the presence of SO(5) symmetry breaking, P a quadratic term of the form V ðni Þ ¼ ðg=2Þ i ðn22i þ n23i þ n24i Þ is also allowed. The anisotropy constant g selects either the ‘‘easy plane’’ in the SC space, or an ‘‘easy sphere’’ in the AF space, depending on the sign of g. Given H , one can simply perform a Legendre transformation to obtain the corresponding Euclidean Lagrangian in the Matsubara ‘‘imaginary time’’ s formulation (0 6 s 6 1=kB T b): L½p; n ¼ ipðsÞ
d nðsÞ þ H ðn; pÞ: ds
ð2Þ
The partition function Z ¼ TreH =kB T of the system is then given by
Z¼
Z Y
½Dni
Z Y Dp i
2p i i Rb dsL½p;n :
dð1 n2i Þdðni pi Þe 0
ð3Þ
Due to the rigid nature of the quantum rotors one must be cautious and integrate only over the momenta transverse to the supervector (pi ni ¼ 0) with fixed length (n2i ¼ 1). To this end, we cast the problem in terms of the exactly soluble quantum spherical model [5] by taking the superspin components as continuous variables, i.e., 1 < ni ðsÞ < 1, but constrained (on average) to have unit length. The spherical closure amounts Q relation 2 then to the replacement: dð1 n Þ ! dðN i i P 2 n Þ. i i The convenient way to enforce the spherical constraint is to use the functional analog of the R þ1 Dirac-d––function representation dðxÞ ¼ 1 ðdk= 2pÞeikx which introduces the Lagrange multiplier kðsÞ [5]. In the thermodynamic limit N ! 1 the method of steepest descents becomes exact and at the saddle point kðsÞ k0 will satisfy 1 X 1¼ ½2Gs ðk; x‘ Þ þ 3Ga ðk; x‘ Þ ð4Þ bN k;x ‘
where Ga ðk; x‘ Þ ¼ 1=½2k0 J ðkÞ þ ux2‘ g Gs ðk; x‘ Þ ¼ 1=½2k0 J ðkÞ þ uðx‘ þ 2ilÞ2
ð5Þ
are the (order parameter) susceptibilities in the AF and SC sector, respectively. The summation in Eq. (4) is performed over all wave vector components k (pp=Na 6 ka 6 pp=Na, where a ¼ x; y; z; p are integers 0 6 p 6 N 1 and a is the lattice spacing). Furthermore, x‘ ¼ 2p‘=b ð‘ ¼ 0; 1; 2; . . .Þ are the (Bose) Matsubara frequencies arising from the restriction to periodic functions along the imaginary time direction. Finally, J ðkÞ ¼ J eðkÞ where eðkÞ ¼ cosðakx Þ þ cosðaky Þ þ cosðakz Þ is the structure factor for the sc lattice in 3D. At the critical lines for the AF and SC phases the Lagrange multiplier k assumes values at which corresponding uniform and static order parameter susceptibilities diverge: G1 a ðk ¼ 0; x‘ ¼ 0Þ ¼ 0 and G1 ðk ¼ 0; x ¼ 0Þ ¼ 0, which yields ka0 ¼ ð3J þ ‘ s s gÞ=2 for AF and k0 ¼ ð3J þ 4l2 uÞ=2 for SC state,
T.K. Kopec, T.A. Zaleski / Physica C 387 (2003) 65–68
respectively. On examining of the saddle point equation (4) one finds that g > 0 favors the AF state whereas the l > 0 term prefers p theffiffiffiffiffiffiffiffiffiffi SC state. For ka0 ¼ ks0 (which happens for lc ¼ g=4u) both AF and SC transition lines merge at the bicritical point at (Tbc ; lc ). However, for sufficiently small parameter u quantum fluctuations will drive Tbc to zero giving rise to a quantum critical point.
v00 ðxÞ ¼ d3 Qv00 ðQ; xÞ––an important quantity for verification of the x=kB T scaling in the QC region. To establish the connection with our SO(5) calculations we consider ‘‘imaginary-time’’ spin–spin correlation function Gij ðs s0 Þ ¼ hSi ðsÞ Sj ðs0 Þi where h i denotes the ensemble averaging with the Hamiltonian (1). Utilizing the relation between spin operators and SO(5) Lie algebra generators [4] we can write: 43 0 42 42 0 Gij ðs s0 Þ ¼ hL43 i ðsÞLj ðs Þi þ hLi ðsÞLj ðs Þi
3. QC region
32 0 þ hL32 i ðsÞLj ðs Þi:
An especially interesting regime appears in the vicinity of the QCP where kB T is significantly larger than any energy scale which measures deviations of the coupling constants from their zero-temperature critical values. In this so-called quantum critical (QC) region [6] the system feels the finite value of the temperature before becoming sensitive to the deviations of l from lc (i.e. the leading behavior of the system will be described at all scales by the T ¼ 0 critical point and its universal response to a finite temperature). To quantify this observation we introduce the deviation of k from the QCP value dk ¼ k k0 (which plays the role of the critical ‘‘mass’’ vanishing at QCP; note that dk is related to the correlation length m n ¼ hc=2dk which diverges as n ðl lc Þ ). At finite temperatures, the deviations from T ¼ 0 behavior are distinguished by the scale set by the thermal coherence length nT T 1=z (where z is the dynamic critical exponent). The QC region is then defined by the inequality nT < n. The magnetic scattering cross-section for neutrons which is proportional to the dynamic structure factor SðQ; xÞ is directly related to the spin–spin correlation function hS0 ð0Þ Sj ðtÞi [8]: Z þ1 X 1 dteixt eiQrj hS0 ð0Þ Sj ðtÞi SðQ; xÞ ¼ 2p 1 rj 1 ¼ ½1 þ nðxÞv00 ðQ; xÞ p
67
R
ð6Þ
where v00 ðQ; xÞ is the imaginary part of the dynamic susceptibility and nðxÞ ¼ ½1 þ expðx= 1 kB T Þ is the Bose function. An integration of the spectral function v00 ðQ; xÞ over wave vectors Q yields the integrated (local) spin susceptibility
ð7Þ
The averages over the product of angular momentum operators can be conveniently performed using functional differentiation by adding to the Lagrangian (2) sources which couple linearly to Llm . As a result we obtain in the momentum-frequency representation for the propagator (7): 4u X GðQ; x‘ Þ ¼ Ga ðq; mn Þ½1 þ uðx‘ 2mn Þ2 bN q;mn
Ga ðQ q; x‘ mn Þ:
ð8Þ
To proceed, we perform now the analytic continuation of GðQ; x‘ Þ to real frequencies ixn ! x þ i0þ (where 0þ stands for the positive infinitesimal). After integration over external momenta we obtain finally the imaginary part of the local dynamic susceptibility v00 ðxÞ in the scaling form: x l D v00 ðxÞ ¼ v00 ; ; kB T kB T kB T 2 3 x D 1 l k k T T 5 Uv 4 B ; B ; Xc kB T kB T Xc kBlT Xc kBlT ð9Þ where
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 4y X cosh ð2y‘Þ K1 ð2y‘Þ; XðyÞ ¼ 1 þ 2 p ‘¼1 ‘ " # Z þ1 3 2 C1 ðyÞ ¼ djqðjÞ þ : 3=2 3=2 ð3 jÞ ðy þ 3 jÞ 1 ð10Þ Here, K1 ðxÞ stands for the modified Bessel function [7] and qðjÞ is the density of states for 3D s.c. lattice:
68
T.K. Kopec, T.A. Zaleski / Physica C 387 (2003) 65–68
1 qðjÞ ¼ 3 p
Z
minð1;2jÞ
dw pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 w2 maxð1;2jÞ "rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi#
j þ w 2
K 1 Hð3 jjjÞ 2
ð11Þ
where KðxÞ and HðxÞ is the elliptic integral of the first kind [7] and the unit step function, respec1=2 1=4 1=2 tively. Subsequently, pffiffiffiffiffiffiffiffi Xc ðyÞ ¼ p 2 C1 ðgc =J Þ XðyÞ and D ¼ J =u. Finally, Uv ðy1 ; y2 ; y3 Þ Z þ1 p y1 dfqðfÞq f y1 ¼ y3 y22 1 y22 h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi y1 þ 2 2 þ ð3 fÞy22 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 þ ð3 fÞy22 2 þ ð3 fÞy22 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi y3 2 þ ð3 fÞy22
coth 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi y3 y1 y3 2 2 þ ð3 fÞy2 þ ð12Þ coth 2 2
is the universal scaling function. The appearance of the temperature as the relevant energy scale for spin dynamics may be very significant because of its possible relation to the normal state characteristics of superconducting cuprates. 4. Final remarks In conclusion, we have performed a quantitative study of the quantum critical point scenario within the concept of the SO(5) group based the-
ory of superconductivity and antiferromagnetism for high-Tc materials. The physical properties governed by the proximity to the AF/SC QCP account for the universal scaling for the finite temperature behavior. The general prerequisite of a given QCP scenario is that it must involve concept of a broken symmetry. Several questions related to this issue arise in applying other alternative QCP based theories (like charge instability near optimum doping, see e.g., [9]) to cuprates: what are the relevant phases with order and associated broken symmetries? Some possibilities that address these problems are being currently debated [10], however further theoretical work on quantum critical dynamics in cuprate materials is clearly called for.
References [1] S.L. Sondhi, S.M. Girvin, J.P. Carini, D. Shahar, Rev. Mod. Phys. 69 (1979) 315. [2] G. Aeppli, T.E. Mason, S.M. Hayden, H.A. Mook, J. Kulda, Science 287 (1997) 1432. [3] S. Sachdev, J. Ye, Phys. Rev. Lett. 69 (1992) 2411. [4] S.C. Zhang, Science 275 (1997) 1089. [5] T. Zaleski, T.K. Kopec, Phys. Rev. B 62 (2000) 9059. [6] S. Chakraverty, B.I. Halperin, D.R. Nelson, Phys. Rev. B 39 (1989) 2344. [7] M. Abramovitz, I. Stegun, Handbook of Mathematical Functions, Dover, New York, 1970. [8] W. Marshall, R.D. Lowde, Rep. Prog. Phys. 31 (1968) 705. [9] C. di Castro, L. Benefatto, S. Caprara, C. Castellani, M. Grilli, Physica C 341 (2000) 1715. [10] C.M. Varma, Phys. Rev. Lett. 83 (1999) 3538.