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Weakly first-order transition in unconventional superconductors
Volume 145, number 5
PHYSICSLETTERSA
16 April 1990
WEAKLY FIRST-ORDER T R A N S I T I O N IN U N C O N V E N T I O N A L SUPERCONDUCTORS Yonko T. MILLEV Department of Physics, Higher Institute of Mechanical and Electrical Engineering, 1156 Sofia, Bulgaria
and Dimo I. UZUNOV G. Nadjakov Institute of Solid State Physics, Bulgarian Academy of Sciences, 1784 Sofia, Bulgaria
Received 5 July 1989; revised manuscript received 27 February 1990; acceptedfor publication 27 February 1990 Communicatedby D. Bloch
The influence of the magnetic-fieldfluctuations on the critical behaviour of unconventional superconductors is studied within the frameworkof generalized Ginzburg-Landau models. It is demonstrated, by means of the renormalization group, that a fluctuation-induced weaklyfirst-order phase transition takes place. The results are related to the description of heavy-fermionand high-Tosuperconductors.
In usual superconductors the phase transition to the superconducting state in the absence of an external magnetic field is a fluctuation-induced first-order transition (the Halperin-Lubensky-Ma effect [ 1 ] ). It has been demonstrated with the help of mean-field and renormalization-group ( R G ) arguments [ 1 ] that this phenomenon is brought about by the gauge-invariant coupling of the order parameter V to the vector potential ,4 in the Ginzburg-Landau free energy. It has been shown that the R G equations for a generalised m-component complex order parameter have no fixed points (FPs) for any m < 183. This has been interpreted as an indication of a first-order transition. Further studies have revealed more details about the vector-potential influence on the order of the phase transitions in pure [2-4 ] and impure [ 5-7 ] superconductors. The same effect has been predicted for liquid crystals [ 8,9 ] and, very recently, it has been experimentally detected [ 10 ]. In this Letter we demonstrate that the effect may occur in unconventional superconductors, i.e., in heavyfermion and high-To superconductors. Unconventional types of pairings are expected to take place in these systems. One of the consequences is that ¥ might be a one-, two- or three-component complex vector field (m = 1, 2, 3). The phenomenological description of such fields was introduced in the study of heavy-fermion systems [ 11,12 ]. Later it was applied to the analysis of experimental data in high-Tc oxides such as YBa2Cu307_,~ [ 13 ]. The fluctuation phenomena near the critical point of high-To oxides can be used to derive information about the symmetry of the ordered superconducting states and, in particular, about the number m of the orderparameter components. Such a programme, however, implies a precise knowledge of the order of the phase transition and of the scaling behaviour in the critical region [ 14 ]. In most cases, the magnetic-field fluctuations cannot be ruled out and their contribution to the fluctuation phenomena has to be considered. We use the R G analysis to show that the weakly first-order transition exists in unconventional superconductors and that it is essentially modified by the underlying pairing anisotropy. We neglect the anisotropy of the effective mass m* of the carrier pairs. This latter anisotropy plays an essential role in studies relying on the mean-field Ginzburg-Landau ( G L ) equations [ 11 ]; however, it is not connected with the basic mechanism of the fluctuation-induced first-order transition [ 9 ]. 287
Volume 145, number 5
PHYSICS LETTERS A
16 April 1990
We consider the G L fluctuation model for an m-component complex order parameter in d dimensions of space:
F(~) =
f
ddx [71 ( V - i q o A ) ~ 1 2 + ~ l ( V × A ) 2 + U ( ~ ) ] ,
(1)
where
U(~u)=rl¢lZ+½ulq/]4+½fi[~t212+½v ~ I~u~]4,
(2)
r~c T - To; Tc is the critical temperatures; qo = 2e/hc; 7= 1/2m*, and # is the magnetic permeability. The vector potential A obeys the Coulomb gauge: divA = 0 . The v term comes from the cubic anisotropy and the a term describes orthorhombicity [ 11 ]. One o f these anisotropy terms can be omitted in several systems [ 11,12 ]. These cases are covered as particular cases o f our treatment. The derivation of the R G equations to first order in t = 4 - d follows known rules [ 1,14]. The R G equations for the parameters ~,/z, and qo are explained in a most detailed fashion for the standard (isotropic) case in ref. [ 7 ]. Here, the important point is that, just as in the standard case, there are no interaction contributions to these R G equations so that they are the same as for the standard case (see, for instance, ref. [7] with their random-field parameter ho set to zero). The fact of the absence o f v and g insertions in these equations has been checked most carefully, since there are diagrammatic modifications due to the anisotropic interaction terms [ 15,16 ]. N o w then, the same equations yield the same anomalous dimensionalities [ 1 ] ~/~,and qA for the fluctuating fields ~u and A, respectively. The Fisher exponent is q~= - 9 ~ / m , while qA= - ~ . The equations for r, u, a, and v are dr
3
d l - ( 2-qv')r+ K4[ ( m+ 1)u+ 2a+ 2v] (1 - r ) + ~ r,
du
d--~= ( ~ - 2 ~ / ~ ) u - K 4 [ (m+4)u2+4ua+4uv+4a2+ 12n2z 2 ] ,
da d l - (e_2tb,)a_K4(ma2+6ua+2av),
dv ~ = ( ~ - 2 q ~ , ) v - K 4(5v 2 + 6uv+ 8av)
(3)
where l is the R G flow parameter, r=q~l~, and K4= 1/8n 2. Eqs. (3) have two types of FPs. (1) "Uncharged" fixed points (r*=O). There are eight FPs of this type and they have already been studied up to second order in ~ [ 17 ]. All o f them are unstable towards qo-perturbations, i.e., towards the influence o f the vector potential. (2) "Charged"fxed points (z*# 0). The nonzero coordinate of r is r* = 6ne/m. We have found four couples of FPs: (A)
u+-
4~2~ m(m+4)
( m + 1 8 + D , l/z ),
z/± = v ± = 0 ,
(4)
where Da=m z - 1 8 0 m - 5 4 0 ~ > 0 for m>~ma= 183. 4=2~
(B)
u± = m3+4m=_24m+144 [(l+18/m)(mZ-4m+48)+D~,/2] , 8=2~
6
m
m
a±=--(l+18/m)---u+,
-
v+=0 '
where Db=m4--204m 3 - 1356mz--864m-- 155521>0 for m>~mb=211. 288
(5)
Volume 145, number 5
(C)
u+-
4~2~ m(5m-4)
PHYSICS LETTERS A
(m+18+-D~/2)'
~+ = 0 ,
16 April 1990
v+ = 8 n 2 e ( l + 1 8 / m ) - 6 u + ,
(6)
where D~ = m 2_ 5364m + 4644 1>0 for m >t rn~ = 5364.
(D)
47C2~ u+ = 5m2(5m3_36m2 + 120m+80) [re(m+ 18)(5m2+4m+368)+D~/2] 3
t 2 + - 15 -m- - - - 6 [ 8 n 2 E ( l + 1 8 / m ) - 6 u + ] '
m-8
v + - 51 m - ~-
,
[8n2~(l+18/m)-6u+]'
(7)
where Dd = m 2 ( 2 5 m 6 - 137660m5+ 1 745 556m 4 - 1 1 0 1 3 9 2 0 m 3 + 3 5 1 9 8 2 7 2 m 2 - - 4 3 591 6 8 0 m + 3 0 9 6 5 7 6 ) >10 for m>_-md=5494. The FPs A + are known from ref. [ 1 ]. Here they are additionally unstable towards a and v. All new FPs B +, C +-, and D + do not exist for r n = 1, 2 or 3. We conclude that a weakly first-order transition takes place for the physically relevant values o f m in systems described by the model ( 1 ), (2). The critical values above which the FPs exist obey the inequalities md > mc > mb > ma. From eqs. ( 4 ) - ( 7 ) one can observe that this sequence strongly correlates with the symmetry of the respective FPs: as the symmetry is lowered, the critical value of m increases. We have, therefore, proven that the simultaneous action Of the underlying anisotropies and the vector-potential fluctuations brings about, at least theoretically, well-established first-order transitions. Since there are no terms of the type za and rv in the R G equations (3), one should not hesitate to conclude that the vectorpotential fluctuations and the anisotropies act independently rather than interactively enhancing each other. The A + FPs are unstable towards both ~ and v which means that, even if m > ma, the usual second-order transition does not occur. The B -+ and C + FPs are unstable towards v and a, respectively. The stability properties of the D + FPs cannot be resolved analytically. We have tackled the problem numerically and have found that the D + FP is a stable one. Besides, we have explored the large-m limit. For m - - , ~ , u, ~, and v are yu= - ~ / 5 , ya= - 3 ~ / 5 , and yv= - ~ , respectively. These exponents coincide with the exponents for the uncharged case (qo = 0) [ 15,16 ]. The result is not surprising as z* vanishes when m goes to infinity. The main scaling behaviour governed by the D + FP is completely determined by q~,, q/A, and the correlationlength exponent u = ½+
1 [ (2m 2 - 1 9 m - 3 6 ) ~ + m ( S m 2 - 2 3 m + 4 8 ) K 4 u + ] , 4 m ( S m - 16)
(8)
where u+ is given by eq. (10). This critical behaviour corresponds to symmetry indices rn> 5494. At present, we are not aware o f physical systems where it may have the chance to occur. In summary, we have shown that the R G equations of the model (1), (2) have no FPs for m = 2 or m = 3; hence, the models o f unconventional superconductors lead to a weakly first-order phase transition. The simultaneous action of the magnetic-field fluctuations and the possible pairing anisotropy leaves no theoretical ground for second-order transitions in pure systems. The results might be o f interest in discussions of experimental data for fluctuation effects near the critical points of heavy-fermion or high-To superconductors.
References [ 1] B.I. Halperin, T.C. Lubensky and S. Ma, Phys. Rev. Lett. 32 (1974) 292. [ 2 ] J.-H. Chen, T.C. Lubensky and D.R. Nelson, Phys. Rev. B 17 ( 1978) 4275. 289
Volume 145, number 5
PHYSICS LETTERS A
16 April 1990
[3] I.D. Lawrie, Nucl. Phys. B 200 (1982) 1. [4] N.S. Tonchev and D.I. Uzunov, J. Phys. A 14 ( 1981 ) 521. [5] D. Boyanovsky and J.L. Cardy, Phys. Rev. B 25 (1982) 7058. [ 6 ] D.I. Uzunov, E.R. Korutcheva and Y.T. Millev, J. Phys. A 17 (1984) 247. [7] G. Busiello, L. De Cesare and D.I. Uzunov, Phys. Rev. B 34 (1986) 4932. [8] B.I. Halperin and T.C. Lubensky, Solid State Commun. 14 (1974) 947. [ 9 ] T.C. Lubensky and J.-H. Chen, Phys. Rev. B 17 (1978) 366. [ 10] M. Anisimov, V.P. Voronov, E.E. Gorodetskii, V.E. Pondek and F. Kholmorodov, Pis'ma Zh. Eksp. Teor. Fiz. 45 (1987) 336 [ JETP Lett. 45 ( 1987 ) 425 ]. [ 11 ] G.E. Volovik and UP. Gor'kov, Zh. Eksp. Teor. Fiz. 88 (1985) 1412 [Sov. Phys. JETP 61 (1985) 8431 ]. [ 12] K. Ueda and T.M. Rice, Phys. Rev. B 31 (1985) 7114. [ 13 ] J. Annett, M. Randeria and S. Renn, Phys. Rev. B 38 ( 1988 ) 4600. [ 14] S. Ma, Modern theory of critical phenomena (Benjamin, New York, 1976). [ 15 ] Y.T. Millev and D.I. Uzunov, Phys. Lett. A 138 (1989) 523. [ 16] E.J. Blagoeva, G. Busiello, L. De Cesare, Y.T. MiUev, I. Rabuffo and D.I. Uzunov, Phys. Rev. B 40 (1989) 7357. [ 17] E.J. Blagoeva, G. Busiello, L. De Cesare, Y.T. Millev, I. Rabuffo and D.I. Uzunov, preprint, Salerno University (1989).
290
PHYSICSLETTERSA
16 April 1990
WEAKLY FIRST-ORDER T R A N S I T I O N IN U N C O N V E N T I O N A L SUPERCONDUCTORS Yonko T. MILLEV Department of Physics, Higher Institute of Mechanical and Electrical Engineering, 1156 Sofia, Bulgaria
and Dimo I. UZUNOV G. Nadjakov Institute of Solid State Physics, Bulgarian Academy of Sciences, 1784 Sofia, Bulgaria
Received 5 July 1989; revised manuscript received 27 February 1990; acceptedfor publication 27 February 1990 Communicatedby D. Bloch
The influence of the magnetic-fieldfluctuations on the critical behaviour of unconventional superconductors is studied within the frameworkof generalized Ginzburg-Landau models. It is demonstrated, by means of the renormalization group, that a fluctuation-induced weaklyfirst-order phase transition takes place. The results are related to the description of heavy-fermionand high-Tosuperconductors.
In usual superconductors the phase transition to the superconducting state in the absence of an external magnetic field is a fluctuation-induced first-order transition (the Halperin-Lubensky-Ma effect [ 1 ] ). It has been demonstrated with the help of mean-field and renormalization-group ( R G ) arguments [ 1 ] that this phenomenon is brought about by the gauge-invariant coupling of the order parameter V to the vector potential ,4 in the Ginzburg-Landau free energy. It has been shown that the R G equations for a generalised m-component complex order parameter have no fixed points (FPs) for any m < 183. This has been interpreted as an indication of a first-order transition. Further studies have revealed more details about the vector-potential influence on the order of the phase transitions in pure [2-4 ] and impure [ 5-7 ] superconductors. The same effect has been predicted for liquid crystals [ 8,9 ] and, very recently, it has been experimentally detected [ 10 ]. In this Letter we demonstrate that the effect may occur in unconventional superconductors, i.e., in heavyfermion and high-To superconductors. Unconventional types of pairings are expected to take place in these systems. One of the consequences is that ¥ might be a one-, two- or three-component complex vector field (m = 1, 2, 3). The phenomenological description of such fields was introduced in the study of heavy-fermion systems [ 11,12 ]. Later it was applied to the analysis of experimental data in high-Tc oxides such as YBa2Cu307_,~ [ 13 ]. The fluctuation phenomena near the critical point of high-To oxides can be used to derive information about the symmetry of the ordered superconducting states and, in particular, about the number m of the orderparameter components. Such a programme, however, implies a precise knowledge of the order of the phase transition and of the scaling behaviour in the critical region [ 14 ]. In most cases, the magnetic-field fluctuations cannot be ruled out and their contribution to the fluctuation phenomena has to be considered. We use the R G analysis to show that the weakly first-order transition exists in unconventional superconductors and that it is essentially modified by the underlying pairing anisotropy. We neglect the anisotropy of the effective mass m* of the carrier pairs. This latter anisotropy plays an essential role in studies relying on the mean-field Ginzburg-Landau ( G L ) equations [ 11 ]; however, it is not connected with the basic mechanism of the fluctuation-induced first-order transition [ 9 ]. 287
Volume 145, number 5
PHYSICS LETTERS A
16 April 1990
We consider the G L fluctuation model for an m-component complex order parameter in d dimensions of space:
F(~) =
f
ddx [71 ( V - i q o A ) ~ 1 2 + ~ l ( V × A ) 2 + U ( ~ ) ] ,
(1)
where
U(~u)=rl¢lZ+½ulq/]4+½fi[~t212+½v ~ I~u~]4,
(2)
r~c T - To; Tc is the critical temperatures; qo = 2e/hc; 7= 1/2m*, and # is the magnetic permeability. The vector potential A obeys the Coulomb gauge: divA = 0 . The v term comes from the cubic anisotropy and the a term describes orthorhombicity [ 11 ]. One o f these anisotropy terms can be omitted in several systems [ 11,12 ]. These cases are covered as particular cases o f our treatment. The derivation of the R G equations to first order in t = 4 - d follows known rules [ 1,14]. The R G equations for the parameters ~,/z, and qo are explained in a most detailed fashion for the standard (isotropic) case in ref. [ 7 ]. Here, the important point is that, just as in the standard case, there are no interaction contributions to these R G equations so that they are the same as for the standard case (see, for instance, ref. [7] with their random-field parameter ho set to zero). The fact of the absence o f v and g insertions in these equations has been checked most carefully, since there are diagrammatic modifications due to the anisotropic interaction terms [ 15,16 ]. N o w then, the same equations yield the same anomalous dimensionalities [ 1 ] ~/~,and qA for the fluctuating fields ~u and A, respectively. The Fisher exponent is q~= - 9 ~ / m , while qA= - ~ . The equations for r, u, a, and v are dr
3
d l - ( 2-qv')r+ K4[ ( m+ 1)u+ 2a+ 2v] (1 - r ) + ~ r,
du
d--~= ( ~ - 2 ~ / ~ ) u - K 4 [ (m+4)u2+4ua+4uv+4a2+ 12n2z 2 ] ,
da d l - (e_2tb,)a_K4(ma2+6ua+2av),
dv ~ = ( ~ - 2 q ~ , ) v - K 4(5v 2 + 6uv+ 8av)
(3)
where l is the R G flow parameter, r=q~l~, and K4= 1/8n 2. Eqs. (3) have two types of FPs. (1) "Uncharged" fixed points (r*=O). There are eight FPs of this type and they have already been studied up to second order in ~ [ 17 ]. All o f them are unstable towards qo-perturbations, i.e., towards the influence o f the vector potential. (2) "Charged"fxed points (z*# 0). The nonzero coordinate of r is r* = 6ne/m. We have found four couples of FPs: (A)
u+-
4~2~ m(m+4)
( m + 1 8 + D , l/z ),
z/± = v ± = 0 ,
(4)
where Da=m z - 1 8 0 m - 5 4 0 ~ > 0 for m>~ma= 183. 4=2~
(B)
u± = m3+4m=_24m+144 [(l+18/m)(mZ-4m+48)+D~,/2] , 8=2~
6
m
m
a±=--(l+18/m)---u+,
-
v+=0 '
where Db=m4--204m 3 - 1356mz--864m-- 155521>0 for m>~mb=211. 288
(5)
Volume 145, number 5
(C)
u+-
4~2~ m(5m-4)
PHYSICS LETTERS A
(m+18+-D~/2)'
~+ = 0 ,
16 April 1990
v+ = 8 n 2 e ( l + 1 8 / m ) - 6 u + ,
(6)
where D~ = m 2_ 5364m + 4644 1>0 for m >t rn~ = 5364.
(D)
47C2~ u+ = 5m2(5m3_36m2 + 120m+80) [re(m+ 18)(5m2+4m+368)+D~/2] 3
t 2 + - 15 -m- - - - 6 [ 8 n 2 E ( l + 1 8 / m ) - 6 u + ] '
m-8
v + - 51 m - ~-
,
[8n2~(l+18/m)-6u+]'
(7)
where Dd = m 2 ( 2 5 m 6 - 137660m5+ 1 745 556m 4 - 1 1 0 1 3 9 2 0 m 3 + 3 5 1 9 8 2 7 2 m 2 - - 4 3 591 6 8 0 m + 3 0 9 6 5 7 6 ) >10 for m>_-md=5494. The FPs A + are known from ref. [ 1 ]. Here they are additionally unstable towards a and v. All new FPs B +, C +-, and D + do not exist for r n = 1, 2 or 3. We conclude that a weakly first-order transition takes place for the physically relevant values o f m in systems described by the model ( 1 ), (2). The critical values above which the FPs exist obey the inequalities md > mc > mb > ma. From eqs. ( 4 ) - ( 7 ) one can observe that this sequence strongly correlates with the symmetry of the respective FPs: as the symmetry is lowered, the critical value of m increases. We have, therefore, proven that the simultaneous action Of the underlying anisotropies and the vector-potential fluctuations brings about, at least theoretically, well-established first-order transitions. Since there are no terms of the type za and rv in the R G equations (3), one should not hesitate to conclude that the vectorpotential fluctuations and the anisotropies act independently rather than interactively enhancing each other. The A + FPs are unstable towards both ~ and v which means that, even if m > ma, the usual second-order transition does not occur. The B -+ and C + FPs are unstable towards v and a, respectively. The stability properties of the D + FPs cannot be resolved analytically. We have tackled the problem numerically and have found that the D + FP is a stable one. Besides, we have explored the large-m limit. For m - - , ~ , u, ~, and v are yu= - ~ / 5 , ya= - 3 ~ / 5 , and yv= - ~ , respectively. These exponents coincide with the exponents for the uncharged case (qo = 0) [ 15,16 ]. The result is not surprising as z* vanishes when m goes to infinity. The main scaling behaviour governed by the D + FP is completely determined by q~,, q/A, and the correlationlength exponent u = ½+
1 [ (2m 2 - 1 9 m - 3 6 ) ~ + m ( S m 2 - 2 3 m + 4 8 ) K 4 u + ] , 4 m ( S m - 16)
(8)
where u+ is given by eq. (10). This critical behaviour corresponds to symmetry indices rn> 5494. At present, we are not aware o f physical systems where it may have the chance to occur. In summary, we have shown that the R G equations of the model (1), (2) have no FPs for m = 2 or m = 3; hence, the models o f unconventional superconductors lead to a weakly first-order phase transition. The simultaneous action of the magnetic-field fluctuations and the possible pairing anisotropy leaves no theoretical ground for second-order transitions in pure systems. The results might be o f interest in discussions of experimental data for fluctuation effects near the critical points of heavy-fermion or high-To superconductors.
References [ 1] B.I. Halperin, T.C. Lubensky and S. Ma, Phys. Rev. Lett. 32 (1974) 292. [ 2 ] J.-H. Chen, T.C. Lubensky and D.R. Nelson, Phys. Rev. B 17 ( 1978) 4275. 289
Volume 145, number 5
PHYSICS LETTERS A
16 April 1990
[3] I.D. Lawrie, Nucl. Phys. B 200 (1982) 1. [4] N.S. Tonchev and D.I. Uzunov, J. Phys. A 14 ( 1981 ) 521. [5] D. Boyanovsky and J.L. Cardy, Phys. Rev. B 25 (1982) 7058. [ 6 ] D.I. Uzunov, E.R. Korutcheva and Y.T. Millev, J. Phys. A 17 (1984) 247. [7] G. Busiello, L. De Cesare and D.I. Uzunov, Phys. Rev. B 34 (1986) 4932. [8] B.I. Halperin and T.C. Lubensky, Solid State Commun. 14 (1974) 947. [ 9 ] T.C. Lubensky and J.-H. Chen, Phys. Rev. B 17 (1978) 366. [ 10] M. Anisimov, V.P. Voronov, E.E. Gorodetskii, V.E. Pondek and F. Kholmorodov, Pis'ma Zh. Eksp. Teor. Fiz. 45 (1987) 336 [ JETP Lett. 45 ( 1987 ) 425 ]. [ 11 ] G.E. Volovik and UP. Gor'kov, Zh. Eksp. Teor. Fiz. 88 (1985) 1412 [Sov. Phys. JETP 61 (1985) 8431 ]. [ 12] K. Ueda and T.M. Rice, Phys. Rev. B 31 (1985) 7114. [ 13 ] J. Annett, M. Randeria and S. Renn, Phys. Rev. B 38 ( 1988 ) 4600. [ 14] S. Ma, Modern theory of critical phenomena (Benjamin, New York, 1976). [ 15 ] Y.T. Millev and D.I. Uzunov, Phys. Lett. A 138 (1989) 523. [ 16] E.J. Blagoeva, G. Busiello, L. De Cesare, Y.T. MiUev, I. Rabuffo and D.I. Uzunov, Phys. Rev. B 40 (1989) 7357. [ 17] E.J. Blagoeva, G. Busiello, L. De Cesare, Y.T. Millev, I. Rabuffo and D.I. Uzunov, preprint, Salerno University (1989).
290