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Quantum effects on the competition between antiferromagnetism and superconductivity in heavy-fermion systems
Solid State Communications 130 (2004) 321–325 www.elsevier.com/locate/ssc
Quantum effects on the competition between antiferromagnetism and superconductivity in heavy-fermion systems Andre´ S. Ferreiraa, Mucio A. Continentinoa,*, Eduardo C. Marinob a
Instituto de Fı´sica, Universidade Federal Fluminense, Campus da praia Vermelha, Av. Litoraˆnea s/n, Nitero´i 24210-340 RJ, Brazil b Instituto de Fı´sica, Universidade Federal do Rio de Janeiro, Rio de Janeiro RJ 21945-970, Brazil Received 6 November 2003; received in revised form 23 January 2004; accepted 9 February 2004 by C. Lacroix
Abstract We study a Ginzburg – Landau theory of two coupled fields describing superconductivity and antiferromagnetism in a metal. A coupling between the two-components superconductor and the antiferromagnetic (AF) field is included in the classical action. The classical results are improved calculating the quantum corrections to one-loop order with the method of the effective potential near the AF phase, but in the paramagnetic side. We discuss the influence of these corrections, including the possibility of fluctuation induced first order transitions. A scaling approach is used to obtain the critical and shift exponents at a quantum bicritical point. q 2004 Elsevier Ltd. All rights reserved. PACS: 74.20.De; 74.70.Tx Keywords: D. Superconductivity; D. Antiferromagnetism
One of the intriguing mysteries about superconductivity concerns its interface with magnetic states. In strongly correlated electronic systems, as high-Tc superconductors [1,2] and heavy-fermions [3– 7], antiferromagnetism (AF) is in close proximity or even coexists with superconducting states. This phenomenon has been recently subject of intense study [8,9] and debate. In this communication we address the quantum aspects of this problem and in particular the nature of the quantum critical points (QCPs) separating these phases for the specific case of heavy fermion metals. In general the QCP arises in these systems due to the competition between long range order and the Kondo effect. It is becoming clear however that superconductivity is an essential ingredient, which must be taken into account. We consider a Ginzburg – Landau functional which contains both magnetic and superconductor order parameters and restrict the model to zero temperature and to the paramagnetic side of the magnetic QCP. Differently * Corresponding author. Tel.: þ55-212-629-5816; fax: þ 55-212629-5887. E-mail address: [email protected] (M.A. Continentino). 0038-1098/$ - see front matter q 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ssc.2004.02.017
from previous approaches, to take into account the metallic nature of the nearly AF system we have to use a dissipative propagator to describe the dynamics of the spin fluctuations in the metal [10]. We include quantum effects through the effective potential method in one loop [11,12], using the known propagator solutions to the quadratic functionals associated with the uncoupled fields. The method is generalized to take into account the dissipative dynamics of the paramagnons in the nearly AF metal. In d ¼ 3; we find that a possible scenario is that of a quantum bicritical point, which universality class we identify, separating a metallic antiferromagnet from a superconducting phase. The model also describes the case AF and SC transitions occur at distinct QCPs. We show that the proximity to the AF instability drives the SC transition into a first order one and enlarges the region of the phase diagram where superconductivity is found. Although the calculations are done for zero temperature our results allow to make predictions for T – 0; as the shape of the critical lines close to the quantum bicritical point. The Ginzburg– Landau model contains three real fields. One corresponds to the AF order parameter, the sub-lattice
322
A.S. Ferreira et al. / Solid State Communications 130 (2004) 321–325
magnetization in one direction [13]. This is a single component real field as most heavy fermions antiferromagnets are strongly anisotropic, Ising-like systems [14] as evidenced, for example, from their large spin-wave gaps. The other two correspond to the two components of the superconductor order parameter (the ground state wave function, for example). The propagator associated with the free action of the superconductor is given by, G0 ðkÞ ¼ G0 ðq; vÞ ¼
i k 2 2 m2
ð1Þ
where k2 ¼ v2 2 q2 : The magnetic part is described by a quadratic functional which takes into account the dissipative nature of the paramagnons near the magnetic phase transition. Its propagator is [10] D0 ðq; vÞ ¼
i ilvlt 2 q2 2 m2p
ð2Þ
which shows the damping of the paramagnon excitations. The classical potential is given by,
vertex matrix ½Mlm ¼ 2iK0l
V ð1Þ ½fc ¼
f43 þ uðf21 þ f22 Þf23
ð3Þ
where f1 and f2 are the superconducting fields and f3 is the field associated with the single component AF order parameter. We have two mass terms, which measure the distance to the corresponding QCPs where the superconducting ðm ¼ 0Þ or the magnetic instability ðmp ¼ 0Þ occur. The other terms represent the self-interaction of the superconductor fields and of the paramagnon field; the last term is the minimum interaction between the relevant fields. Notice that for u . 0; which is the case considered here, superconductivity and antiferromagnetism are in competition. The first quantum correction to the effective potential can be obtained by the summation of all one loop diagrams (Fig. 1). We apply the general method proposed by Coleman [15] with minimum modification due to the unusual propagator of Eq. (2). In this case the free propagator (K0l ¼ G0 or D0 ) must be included in the definition of a
i ð 4 " d kln det½1 2 MðkÞ: 2
! 1 ð d4 k ðl=6Þf2c 1 ð d4 k þ ln 1 þ 4 2 2 2 ð2pÞ 2 ð2pÞ4 k þm ! ðl=2Þf2c 1 ð d4 k þ ln 1 þ 2 2 2 ð2pÞ4 k þm ! 2uf2c ð6Þ ln 1 þ lvlt þ q2 þ m2p
where we have rotated to Euclidean space so that k2 ¼ v2 þ q2 and used " ¼ 1 units. The first two integrations depend only on the modulus of the four-dimensional vector k and time enters as an extra dimension, as we are dealing with a Lorentz invariant case. This arises since the QCP of the superconductor transition (SQCP) has an associated dynamic exponent z ¼ 1: Therefore a cut-off regularization can be done as usual. However, in the last integration we have anisotropy between time and space since the dynamic exponent which characterizes the scaling of time takes the value z ¼ 2: Hence, if we use a cut-off L to the momentum, the correspondent frequency cut-off must be Lz [10]. The integrations in Eq. (5) can be easily performed and yield a regularized solution for the renormalized effective potential given by, Vef ðfc ; f3c ¼ 0Þ ¼ Vcl ðfc ; f3c ¼ 0Þ þ V ð1Þ ðfc Þ
Fig. 1. One loop diagrams. The superconductor fields are represented by a or b ¼ 1; 2. We have chosen the outgoing propagator of each vertex to include in the associated matrix element of Eq. (4) (that is why we use arrows). The dotted line represents the unusual propagator of Eq. (2).
ð5Þ
We simplify the 3 £ 3 matrix M if we choose the classical minimum of the superconductor fields imposing f2c ¼ 0 (this can be done because the minimum depends only on the modulus f21c þ f22c ). Since the effective potential should be valid only in the paramagnetic side we also have f3c ¼ 0 and the quantum correction can be finally written ðf1c ; fc Þ as, V ð1Þ ðfc Þ ¼
1 2 2 1 l g m ðf1 þ f22 Þ þ m2p f23 þ ðf21 þ f22 Þ2 þ 2 2 4! 4!
ð4Þ
where the set of fields {f} assumes the classical values {fc }: The sum of diagrams with the correct Wick factors is formally done and yields in momentum space,
Vcl ðf1 ; f2 ; f3 Þ ¼
›2 V ›fl ›fm {f}¼{fc }
ð7Þ
where Vcl is given by Eq. (3). The minima of this full effective potential occur close to those of the classical potential, at fc < 0; as we can show by numerical inspection of this potential. Close to the transition we can expand the effective potential in powers of the field
A.S. Ferreira et al. / Solid State Communications 130 (2004) 321–325
to obtain Vef ðfc Þ <
1 2 2 l p2 m fc þ f4c þ 4! 2 ð2pÞ4 8 8 ð2uÞ5=2 f5c þ 2 ð2uÞ5=2 kflf4c £ 15 3 4 8 £ þ ð2uÞ3=2 m2p f3c 2 m3p uf2c : 3 3
ð8Þ
The term in brackets is the first quantum correction of order ": As usual, the effective potential depends on the value of its minimum kfl which determines if the system is in the normal or in the broken symmetry superconducting phase. Since l is a small quantity, terms in l2 and ml arising from the quantum corrections have been neglected when compared to the classical terms of the same order. In general we have to consider three possible phase diagrams [16] as shown in Fig. 2 [6]. In this communication we study only cases (A) and (B), at T ¼ 0; since the diffusive paramagnon propagator, Eq. (2), is appropriate to describe the paramagnetic, nearly AF region. We start with the case (A) where both magnetic and superconducting critical points coincide. This requires substituting m2 ¼ 2m2p in Eq. (6), such that, m2 , 0 and the system is in the superconducting phase. It is clear from the quantum potential that the classical mass term is corrected by a new quadratic term that renormalizes the superconducting mass as um3p ump 2 m2 ! m2 2 ¼ m 1 þ : ð9Þ 3p2 3p2 The symmetry is broken for m2 , 0 and due to the multiplicative form of the mass renormalization in the equation above, the location of the QCP remains unchanged ðm2 ¼ 0Þ: However, the character of the mass renormalization changes the properties of the superconductor itself as, for example, the constant k which determines whether the system is a type I or type II superconductor will be affected by this renormalization. The analysis of the extrema of the potential can be carried out with no further difficulty and we find, varying the mass, a second order phase transition between a state with kfl – 0 to a symmetric phase exactly on the quantum bicritical point (QBP) at m2 ¼ 0: This point has SO(3) symmetry [8] and the associated dynamic exponent is z ¼ 2 being determined by the slow relaxation dynamics of the magnetic component. Scaling close to the
Fig. 2. Possible phase diagrams for the system.
323
QBP allows to obtain the finite temperature properties of the system. For d ¼ 3; assuming that a unique correlation length diverges as T is reduced at the QBP, we obtain that both the antiferromagnetic ðTN Þ and superconducting ðTS Þ critical lines of finite temperature phase transitions rise with the distance d to the QBP, as TN;S / ldlc ; i.e. with the same shift exponent c: Furthermore, we find c ¼ nz; where n and z are the correlation length and the dynamic exponents associated with the QBP. Since the effective dimension deff ¼ d þ z ¼ 5 is larger than the upper critical dimension, dc ¼ 4; for the bicritical point, the correlation length exponent assumes the mean-field value n ¼ 1=2 and the quantum bicritical crossover exponent nz ¼ 1 [17]. The correlation length, at the QBP (pdffiffi¼ 0), will diverge with decreasing temperature as, j / 1= T and the specific pffiffi heat along this line has a nonFermi liquid, CP ðTÞ / T ; temperature dependence [17] (see Fig. 3). The results above are valid in the close vicinity of the QBP. Next we consider case (B) of Fig. 2. For convenience, in Eq. (6) we introduce the separation D2 ¼ m2 þ m2p . 0 between the magnetic and superconductor QCPs and study the phase diagram as D2 is reduced. This is also taken as a small quantity besides m2 and m2p : We find that the quantum corrections induce a symmetry breaking extending the region where superconductivity is found in the phase diagram. The shift of the superconductor QCP occurs towards the antiferromagnetic quantum critical point (AF-QCP), but in the non-magnetic side and consequently it is correctly described by the present method. Neglecting higher order kfl5 terms, the extrema of the effective potential are given by kfl ¼ 0
ð10Þ "
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# 3 2l 2 kfl ¼ M 2g ðu; mp Þ ^ ½g ðu; mp Þ2 2 l 3 R
ð11Þ
where
gðu; mp Þ ¼
ð2uÞ3=2 m2p 4p2
ð12Þ
and MR2 is the superconductor renormalized mass MR2 ¼ m2 2
m3p u m3p u 2 2 ¼ D 2 m 2 : p 3p2 3p2
ð13Þ
Fig. 3. A quantum bicritical point separating an antiferromagnet from a superconductor. Both lines of finite temperature phase transitions rise with the same exponent c: For d ¼ 3; c ¼ nz ¼ 1:
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For MR2 , 0 but m2 . 0; the value kfl ¼ 0 is a maximum and the other values of kfl – 0 in Eq. (7) are both minima. Consequently, near the superconducting phase ðm2p < D2 Þ; the symmetry of the normal phase is broken extending the superconducting region. When we move further towards the magnetic phase (decreasing the value of mp ) MR2 changes sign and kfl ¼ 0 turns into a minimum. However, the global minimum still occurs at a finite, kfl – 0 and is given by the smallest value of Eq. (7). The point S1 ¼ MR2 ¼ 0 is an inflection point and corresponds to a spinodal of the system. This spinodal marks the limit of metastability of the paramagnetic phase inside the stable SC phase. Further decreasing mp ; the origin, i.e. kfl ¼ 0 becomes the global minimum before passing through a situation of coexistence of phases which determines the new quantum phase transition. The position of the new quantum phase transition point can be expressed in terms of the distance to the AF-QCP, ðm2p Þ: The critical value ðm2p Þc ¼ m2c is determined by the coexistence condition Vðfc ¼ kflÞ ¼ Vðfc ¼ 0Þ ¼ 0
ð14Þ
where kfl is the asymmetric minimum given by the smaller value of Eq. (7). The potential and consequently the ground state energy are analytic in the neighborhood of the transition allowing an expansion around the critical point EG ¼ EG ðmp ¼ mc Þ þ A^ lmp 2 mc l
ð15Þ
where A^ are the amplitudes of the ground state energy on different sides of the transition. We can associate the amplitude difference mc u ð2uÞ3=2 dA < mc 1 þ kfl3c ð16Þ kfl2c þ 2mc 2 2p 12p2 to a latent heat [17]. Above, kflc is the asymmetric minimum calculated on the quantum phase transition point. mc : As mp decreases beyond the first order transition, there is still a local minimum at a kfl – 0; given by Eq. (7), up to a value of MR2 ¼ S2 at which the argument of the square root in this equation vanishes. Between the two spinodals S1 and S2 there is an interval of coexistence of the superconducting phase with regions of strong AF fluctuations. We conclude pointing out that the one loop quantum effective potential is a good approximation near both magnetic and superconducting critical points in the paramagnetic side. The effective potential allowed us to study the influence of the proximity to an AF phase on superconductivity in heavy fermion metals. For noncoincident QCPs the second order nature of the quantum transition is modified as it changes from continuous to a first order transition. Hence, the spin fluctuations change the nature of the superconducting transition in the same way that the coupling with a magnetic field does [17,18]. In the case of coincident QCPs the transitions remain second order and the finite temperature behavior can be extracted from
Fig. 4. Influence of the quantum corrections in the phase diagram and transition. The spinodal points are labelled S1 and S2 (see text).
the scaling properties of the quantum bicritical point. The results for non-coincident QCPs are summarized schematically in Fig. 4. We have shown that if a superconductor QCP is located close to an AF instability, it is strongly affected by the AF spin fluctuations in the metal. The effects of the competition between these instabilities are highly non-trivial as we have shown here. Magnetic fluctuations may drive the superconducting transition first order and modify the properties of the superconducting state. They can also enhance superconductivity, increasing the range in the phase diagram where this phase exists. These modifications arise from the quantum corrections, obtained up to one-loop order, to the effective potential. The standard method to calculate these corrections had to be generalized to deal with the diffusive nature of the excitations in the nearly AF metal.
Acknowledgements Work partially supported by the Brazilian Agencies, FAPERJ and CNPq (PRONEX98/MCT-CNPq-0364.00/00).
References [1] R. Nagaragan, Chandan Mazumdar, Zakir Hossain, et al., Phys. Rev. Lett. 72 (1994) 274. [2] B.K. Cho, P.C. Canfield, D.C. Johnston, et al., Phys. Rev. Lett. 77 (1996) 163. [3] H. Hegger, C. Petrovic, E.G. Moshopoulou, M.F. Hundley, et al., Phys. Rev. Lett. 84 (2001) 4986. [4] J.D. Thompson, R. Movshovich, Z. Fisk, et al., J. Magn. Mater 226 (2001) 5. [5] C. Petrovic, R. Movshovich, M. Jaime, et al., Europhys. Lett. 53 (2001) 354. [6] N.D. Mathur, F.M. Grosche, S.R. Julian, et al., Nature 394 (1998) 39. [7] H.J. Kang, P. Dai, J.W. Lynn, et al., Nature 423 (2003) 522. [8] S.C. Zhang, Science 275 (1997) 1089.
A.S. Ferreira et al. / Solid State Communications 130 (2004) 321–325 [9] S. Sachdev, S.C. Zhang, Science 295 (2002) 452. E. Demler, S. Sachdev, Y. Zhang, Phys. Rev. Lett. 87 (2001) 067202. [10] J.A. Hertz, Phys. Rev. B 14 (1976) 1165. [11] G. Jona-Lasinio, Nuovo Cimento 34 (1964) 1790. [12] S. Coleman, E. Weinberg, Phys. Rev. D 7 (1973) 1888. [13] H. Kohno, H. Fukuyama, M. Sigrist, J. Phys. Soc. Jpn 68 (1999) 1500. [14] J. Flouquet, Y. Haga, P. Haen, et al. Cond-mat/0306157, W. Snafo, et al. Cond-mat/0302487.
325
[15] S. Coleman, Aspects of symmetry, Cambridge University Press, Cambridge, 1985, Chapter 5. [16] The finite temperature part of the the diagrams in Fig. 2 is merely a sketch. [17] See M.A. Continentino, Quantum scaling in many-body systems, World Scientific, Singapore, 2001. [18] B.I. Halperin, T.C. Lubensky, Shang-keng Ma, Phys. Rev. Lett. 32 (1974) 292.
Quantum effects on the competition between antiferromagnetism and superconductivity in heavy-fermion systems Andre´ S. Ferreiraa, Mucio A. Continentinoa,*, Eduardo C. Marinob a
Instituto de Fı´sica, Universidade Federal Fluminense, Campus da praia Vermelha, Av. Litoraˆnea s/n, Nitero´i 24210-340 RJ, Brazil b Instituto de Fı´sica, Universidade Federal do Rio de Janeiro, Rio de Janeiro RJ 21945-970, Brazil Received 6 November 2003; received in revised form 23 January 2004; accepted 9 February 2004 by C. Lacroix
Abstract We study a Ginzburg – Landau theory of two coupled fields describing superconductivity and antiferromagnetism in a metal. A coupling between the two-components superconductor and the antiferromagnetic (AF) field is included in the classical action. The classical results are improved calculating the quantum corrections to one-loop order with the method of the effective potential near the AF phase, but in the paramagnetic side. We discuss the influence of these corrections, including the possibility of fluctuation induced first order transitions. A scaling approach is used to obtain the critical and shift exponents at a quantum bicritical point. q 2004 Elsevier Ltd. All rights reserved. PACS: 74.20.De; 74.70.Tx Keywords: D. Superconductivity; D. Antiferromagnetism
One of the intriguing mysteries about superconductivity concerns its interface with magnetic states. In strongly correlated electronic systems, as high-Tc superconductors [1,2] and heavy-fermions [3– 7], antiferromagnetism (AF) is in close proximity or even coexists with superconducting states. This phenomenon has been recently subject of intense study [8,9] and debate. In this communication we address the quantum aspects of this problem and in particular the nature of the quantum critical points (QCPs) separating these phases for the specific case of heavy fermion metals. In general the QCP arises in these systems due to the competition between long range order and the Kondo effect. It is becoming clear however that superconductivity is an essential ingredient, which must be taken into account. We consider a Ginzburg – Landau functional which contains both magnetic and superconductor order parameters and restrict the model to zero temperature and to the paramagnetic side of the magnetic QCP. Differently * Corresponding author. Tel.: þ55-212-629-5816; fax: þ 55-212629-5887. E-mail address: [email protected] (M.A. Continentino). 0038-1098/$ - see front matter q 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ssc.2004.02.017
from previous approaches, to take into account the metallic nature of the nearly AF system we have to use a dissipative propagator to describe the dynamics of the spin fluctuations in the metal [10]. We include quantum effects through the effective potential method in one loop [11,12], using the known propagator solutions to the quadratic functionals associated with the uncoupled fields. The method is generalized to take into account the dissipative dynamics of the paramagnons in the nearly AF metal. In d ¼ 3; we find that a possible scenario is that of a quantum bicritical point, which universality class we identify, separating a metallic antiferromagnet from a superconducting phase. The model also describes the case AF and SC transitions occur at distinct QCPs. We show that the proximity to the AF instability drives the SC transition into a first order one and enlarges the region of the phase diagram where superconductivity is found. Although the calculations are done for zero temperature our results allow to make predictions for T – 0; as the shape of the critical lines close to the quantum bicritical point. The Ginzburg– Landau model contains three real fields. One corresponds to the AF order parameter, the sub-lattice
322
A.S. Ferreira et al. / Solid State Communications 130 (2004) 321–325
magnetization in one direction [13]. This is a single component real field as most heavy fermions antiferromagnets are strongly anisotropic, Ising-like systems [14] as evidenced, for example, from their large spin-wave gaps. The other two correspond to the two components of the superconductor order parameter (the ground state wave function, for example). The propagator associated with the free action of the superconductor is given by, G0 ðkÞ ¼ G0 ðq; vÞ ¼
i k 2 2 m2
ð1Þ
where k2 ¼ v2 2 q2 : The magnetic part is described by a quadratic functional which takes into account the dissipative nature of the paramagnons near the magnetic phase transition. Its propagator is [10] D0 ðq; vÞ ¼
i ilvlt 2 q2 2 m2p
ð2Þ
which shows the damping of the paramagnon excitations. The classical potential is given by,
vertex matrix ½Mlm ¼ 2iK0l
V ð1Þ ½fc ¼
f43 þ uðf21 þ f22 Þf23
ð3Þ
where f1 and f2 are the superconducting fields and f3 is the field associated with the single component AF order parameter. We have two mass terms, which measure the distance to the corresponding QCPs where the superconducting ðm ¼ 0Þ or the magnetic instability ðmp ¼ 0Þ occur. The other terms represent the self-interaction of the superconductor fields and of the paramagnon field; the last term is the minimum interaction between the relevant fields. Notice that for u . 0; which is the case considered here, superconductivity and antiferromagnetism are in competition. The first quantum correction to the effective potential can be obtained by the summation of all one loop diagrams (Fig. 1). We apply the general method proposed by Coleman [15] with minimum modification due to the unusual propagator of Eq. (2). In this case the free propagator (K0l ¼ G0 or D0 ) must be included in the definition of a
i ð 4 " d kln det½1 2 MðkÞ: 2
! 1 ð d4 k ðl=6Þf2c 1 ð d4 k þ ln 1 þ 4 2 2 2 ð2pÞ 2 ð2pÞ4 k þm ! ðl=2Þf2c 1 ð d4 k þ ln 1 þ 2 2 2 ð2pÞ4 k þm ! 2uf2c ð6Þ ln 1 þ lvlt þ q2 þ m2p
where we have rotated to Euclidean space so that k2 ¼ v2 þ q2 and used " ¼ 1 units. The first two integrations depend only on the modulus of the four-dimensional vector k and time enters as an extra dimension, as we are dealing with a Lorentz invariant case. This arises since the QCP of the superconductor transition (SQCP) has an associated dynamic exponent z ¼ 1: Therefore a cut-off regularization can be done as usual. However, in the last integration we have anisotropy between time and space since the dynamic exponent which characterizes the scaling of time takes the value z ¼ 2: Hence, if we use a cut-off L to the momentum, the correspondent frequency cut-off must be Lz [10]. The integrations in Eq. (5) can be easily performed and yield a regularized solution for the renormalized effective potential given by, Vef ðfc ; f3c ¼ 0Þ ¼ Vcl ðfc ; f3c ¼ 0Þ þ V ð1Þ ðfc Þ
Fig. 1. One loop diagrams. The superconductor fields are represented by a or b ¼ 1; 2. We have chosen the outgoing propagator of each vertex to include in the associated matrix element of Eq. (4) (that is why we use arrows). The dotted line represents the unusual propagator of Eq. (2).
ð5Þ
We simplify the 3 £ 3 matrix M if we choose the classical minimum of the superconductor fields imposing f2c ¼ 0 (this can be done because the minimum depends only on the modulus f21c þ f22c ). Since the effective potential should be valid only in the paramagnetic side we also have f3c ¼ 0 and the quantum correction can be finally written ðf1c ; fc Þ as, V ð1Þ ðfc Þ ¼
1 2 2 1 l g m ðf1 þ f22 Þ þ m2p f23 þ ðf21 þ f22 Þ2 þ 2 2 4! 4!
ð4Þ
where the set of fields {f} assumes the classical values {fc }: The sum of diagrams with the correct Wick factors is formally done and yields in momentum space,
Vcl ðf1 ; f2 ; f3 Þ ¼
›2 V ›fl ›fm {f}¼{fc }
ð7Þ
where Vcl is given by Eq. (3). The minima of this full effective potential occur close to those of the classical potential, at fc < 0; as we can show by numerical inspection of this potential. Close to the transition we can expand the effective potential in powers of the field
A.S. Ferreira et al. / Solid State Communications 130 (2004) 321–325
to obtain Vef ðfc Þ <
1 2 2 l p2 m fc þ f4c þ 4! 2 ð2pÞ4 8 8 ð2uÞ5=2 f5c þ 2 ð2uÞ5=2 kflf4c £ 15 3 4 8 £ þ ð2uÞ3=2 m2p f3c 2 m3p uf2c : 3 3
ð8Þ
The term in brackets is the first quantum correction of order ": As usual, the effective potential depends on the value of its minimum kfl which determines if the system is in the normal or in the broken symmetry superconducting phase. Since l is a small quantity, terms in l2 and ml arising from the quantum corrections have been neglected when compared to the classical terms of the same order. In general we have to consider three possible phase diagrams [16] as shown in Fig. 2 [6]. In this communication we study only cases (A) and (B), at T ¼ 0; since the diffusive paramagnon propagator, Eq. (2), is appropriate to describe the paramagnetic, nearly AF region. We start with the case (A) where both magnetic and superconducting critical points coincide. This requires substituting m2 ¼ 2m2p in Eq. (6), such that, m2 , 0 and the system is in the superconducting phase. It is clear from the quantum potential that the classical mass term is corrected by a new quadratic term that renormalizes the superconducting mass as um3p ump 2 m2 ! m2 2 ¼ m 1 þ : ð9Þ 3p2 3p2 The symmetry is broken for m2 , 0 and due to the multiplicative form of the mass renormalization in the equation above, the location of the QCP remains unchanged ðm2 ¼ 0Þ: However, the character of the mass renormalization changes the properties of the superconductor itself as, for example, the constant k which determines whether the system is a type I or type II superconductor will be affected by this renormalization. The analysis of the extrema of the potential can be carried out with no further difficulty and we find, varying the mass, a second order phase transition between a state with kfl – 0 to a symmetric phase exactly on the quantum bicritical point (QBP) at m2 ¼ 0: This point has SO(3) symmetry [8] and the associated dynamic exponent is z ¼ 2 being determined by the slow relaxation dynamics of the magnetic component. Scaling close to the
Fig. 2. Possible phase diagrams for the system.
323
QBP allows to obtain the finite temperature properties of the system. For d ¼ 3; assuming that a unique correlation length diverges as T is reduced at the QBP, we obtain that both the antiferromagnetic ðTN Þ and superconducting ðTS Þ critical lines of finite temperature phase transitions rise with the distance d to the QBP, as TN;S / ldlc ; i.e. with the same shift exponent c: Furthermore, we find c ¼ nz; where n and z are the correlation length and the dynamic exponents associated with the QBP. Since the effective dimension deff ¼ d þ z ¼ 5 is larger than the upper critical dimension, dc ¼ 4; for the bicritical point, the correlation length exponent assumes the mean-field value n ¼ 1=2 and the quantum bicritical crossover exponent nz ¼ 1 [17]. The correlation length, at the QBP (pdffiffi¼ 0), will diverge with decreasing temperature as, j / 1= T and the specific pffiffi heat along this line has a nonFermi liquid, CP ðTÞ / T ; temperature dependence [17] (see Fig. 3). The results above are valid in the close vicinity of the QBP. Next we consider case (B) of Fig. 2. For convenience, in Eq. (6) we introduce the separation D2 ¼ m2 þ m2p . 0 between the magnetic and superconductor QCPs and study the phase diagram as D2 is reduced. This is also taken as a small quantity besides m2 and m2p : We find that the quantum corrections induce a symmetry breaking extending the region where superconductivity is found in the phase diagram. The shift of the superconductor QCP occurs towards the antiferromagnetic quantum critical point (AF-QCP), but in the non-magnetic side and consequently it is correctly described by the present method. Neglecting higher order kfl5 terms, the extrema of the effective potential are given by kfl ¼ 0
ð10Þ "
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# 3 2l 2 kfl ¼ M 2g ðu; mp Þ ^ ½g ðu; mp Þ2 2 l 3 R
ð11Þ
where
gðu; mp Þ ¼
ð2uÞ3=2 m2p 4p2
ð12Þ
and MR2 is the superconductor renormalized mass MR2 ¼ m2 2
m3p u m3p u 2 2 ¼ D 2 m 2 : p 3p2 3p2
ð13Þ
Fig. 3. A quantum bicritical point separating an antiferromagnet from a superconductor. Both lines of finite temperature phase transitions rise with the same exponent c: For d ¼ 3; c ¼ nz ¼ 1:
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For MR2 , 0 but m2 . 0; the value kfl ¼ 0 is a maximum and the other values of kfl – 0 in Eq. (7) are both minima. Consequently, near the superconducting phase ðm2p < D2 Þ; the symmetry of the normal phase is broken extending the superconducting region. When we move further towards the magnetic phase (decreasing the value of mp ) MR2 changes sign and kfl ¼ 0 turns into a minimum. However, the global minimum still occurs at a finite, kfl – 0 and is given by the smallest value of Eq. (7). The point S1 ¼ MR2 ¼ 0 is an inflection point and corresponds to a spinodal of the system. This spinodal marks the limit of metastability of the paramagnetic phase inside the stable SC phase. Further decreasing mp ; the origin, i.e. kfl ¼ 0 becomes the global minimum before passing through a situation of coexistence of phases which determines the new quantum phase transition. The position of the new quantum phase transition point can be expressed in terms of the distance to the AF-QCP, ðm2p Þ: The critical value ðm2p Þc ¼ m2c is determined by the coexistence condition Vðfc ¼ kflÞ ¼ Vðfc ¼ 0Þ ¼ 0
ð14Þ
where kfl is the asymmetric minimum given by the smaller value of Eq. (7). The potential and consequently the ground state energy are analytic in the neighborhood of the transition allowing an expansion around the critical point EG ¼ EG ðmp ¼ mc Þ þ A^ lmp 2 mc l
ð15Þ
where A^ are the amplitudes of the ground state energy on different sides of the transition. We can associate the amplitude difference mc u ð2uÞ3=2 dA < mc 1 þ kfl3c ð16Þ kfl2c þ 2mc 2 2p 12p2 to a latent heat [17]. Above, kflc is the asymmetric minimum calculated on the quantum phase transition point. mc : As mp decreases beyond the first order transition, there is still a local minimum at a kfl – 0; given by Eq. (7), up to a value of MR2 ¼ S2 at which the argument of the square root in this equation vanishes. Between the two spinodals S1 and S2 there is an interval of coexistence of the superconducting phase with regions of strong AF fluctuations. We conclude pointing out that the one loop quantum effective potential is a good approximation near both magnetic and superconducting critical points in the paramagnetic side. The effective potential allowed us to study the influence of the proximity to an AF phase on superconductivity in heavy fermion metals. For noncoincident QCPs the second order nature of the quantum transition is modified as it changes from continuous to a first order transition. Hence, the spin fluctuations change the nature of the superconducting transition in the same way that the coupling with a magnetic field does [17,18]. In the case of coincident QCPs the transitions remain second order and the finite temperature behavior can be extracted from
Fig. 4. Influence of the quantum corrections in the phase diagram and transition. The spinodal points are labelled S1 and S2 (see text).
the scaling properties of the quantum bicritical point. The results for non-coincident QCPs are summarized schematically in Fig. 4. We have shown that if a superconductor QCP is located close to an AF instability, it is strongly affected by the AF spin fluctuations in the metal. The effects of the competition between these instabilities are highly non-trivial as we have shown here. Magnetic fluctuations may drive the superconducting transition first order and modify the properties of the superconducting state. They can also enhance superconductivity, increasing the range in the phase diagram where this phase exists. These modifications arise from the quantum corrections, obtained up to one-loop order, to the effective potential. The standard method to calculate these corrections had to be generalized to deal with the diffusive nature of the excitations in the nearly AF metal.
Acknowledgements Work partially supported by the Brazilian Agencies, FAPERJ and CNPq (PRONEX98/MCT-CNPq-0364.00/00).
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