New fluctuation-driven phase transitions and critical phenomena in unconventional superconductors
aa r X i v : . [ c ond - m a t . s up r- c on ] A p r New fluctuation-driven phase transitions and criticalphenomena in unconventional superconductors
Dimo I. Uzunov
CP Laboratory, Institute of Solid State Physics, Bulgarian Academy of Sciences,BG–1784, Sofia, Bulgaria.
Abstract
Using the renormalization group method, new type of fluctuation-driven firstorder phase transitions and critical phenomena are predicted for certain classes offerromagnetic superconductors and superfluids with unconventional (spin-triplet)Cooper pairing. The problem for the quantum phase transitions at extremely lowand zero temperatures is also discussed. The results can be applied to a wide classof ferromagnetic superconductive and superfluid systems, in particular, to itinerantferromagnets as UGe and URhGe. Pacs:
Keywords: superconductivity, ferromagnetism, fluctuations, quantum phase transition, criticalpoint, order, symmetry.
1. Introduction
In this paper an entirely new critical behavior in unconventional ferromagnetic superconductorsand superfluids is established and described. This phenomenon corresponds to an isotropicferromagnetic order in real systems but does not belong to any known universality class [1]and, hence, it might be of considerable experimental and theoretical interest. Due to crystaland magnetic anisotropy a new type of fluctuation-driven first order phase transitions occur,as shown in the present investigation. These novel fluctuation effects can be observed nearfinite and zero temperature (“quantum”) phase transitions [1, 2] in a wide class of ferromagneticsystems with unconventional (spin-triplet) superconductivity or superfluidity.The present investigation has been performed on the concrete example of intermetallic com-pounds UGe and URhGe, where the remarkable phenomenon of coexistence of itinerant fer-romagnetism and unconventional spin-triplet superconductivity [3] has been observed [4]. Forexample, in UGe , the coexistence phase occurs [4] at temperatures 0 ≤ T < < P < P ∼ . P, T ) phase diagrams of itinerant ferromagneticcompounds [4] is sketched in Fig. 1, where the lines T F ( P ) and T c ( P ) of the paramagnetic(P) -to-ferromagnetic(F) and ferromagnetic-to-coexistence phase(C) transitions are very close to eachother and intersect at very low temperature or terminate at the absolute zero ( P , M ( r ) = { M j ( r ); j = 1 , ..., m } and the complex order pa-rameter vector of the spin-triplet Cooper pairing [3], ψ ( r ) = { ψ α ( r ) = ( ψ ′ α + iψ ′′ α ); α = 1 , ....n/ } ( n = 6) cannot be neglected [1] and, as shown here, this interaction produces new fluctuationphenomena. F T c T FC P PT
Figure 1: ( P, T ) diagram with a zero-temperature multicritical point ( P , T f ( P ) and T c ( P ) of P-F and F-C phase transitions,respectively. Both thermal fluctuations at finite temperatures (
T >
0) and quantum fluctuations (correla-tions) near the P –driven quantum phase transition at T = 0 should be considered but at afirst stage the quantum effects [2] can be neglected as irrelevant to finite temperature phasetransitions ( T F ∼ T c > m = 3) exhibit a quite particularmulti-critical behavior for any T >
0, whereas the magnetic anisotropy ( m = 1 ,
2) generatesfluctuation-driven first order transitions [1]. Thus the phase transition properties of spin-tripletferromagnetic superconductors are completely different from those predicted by mean field theo-ries [5, 6]. The results can be used in the interpretation of experimental data for phase transitionsin itinerant ferromagnetic compounds [7].The study presents for the first time an example of complex quantum criticality characterizedby a double-rate quantum critical dynamics. In the quantum limit ( T →
0) the fields M and ψ have different dynamical exponents, z M and z ψ , and this leads to two different upper criticaldimensions: d MU = 6 − z M and d Uψ = 6 − z ψ . The complete consideration of the quantumfluctuations of both fields M and ψ requires a new RG approach in which one should eitherconsider the difference ( z M − z ψ ) as an auxiliary small parameter or create a completely newtheoretical paradigm of description. The considered problem is quite general and presents achallenge to the theory of quantum phase transitions [2]. The results can be applied to anynatural system within the same class of symmetry although this report is based on the exampleof itinerant ferromagnetic compounds.
2. Renormalization-group investigation
The relevant part of the fluctuation Hamiltonian of unconventional ferromagnetic superconduc- ors [5, 6] can be written in the form H = X k (cid:20)(cid:0) r + k (cid:1) | ψ ( k ) | + 12 (cid:0) t + k (cid:1) | M ( k ) | (cid:21) + ig √ V X k , k M ( k ) . [ ψ ( k ) × ψ ∗ ( k + k )](1)where V ∼ L d is the volume of the d − dimensional system, the length unit is chosen so thatthe wave vector k is confined below unity (0 ≤ k = | k | ≤ g ≥ M and the vector product ( ψ × ψ ∗ ) for symmetryindices m = ( n/
2) = 3, and the parameters t ∼ ( T − T f ) and r ∼ ( T − T s ) are expressed by thecritical temperatures of the generic ( g ≡
0) ferromagnetic and superconducting transitions. Asmean field studies indicate [5, 6], T s ( P ) is much lower than T c ( T ) and T F ( P ) = T f ( P ).The fourth order terms ( M , | ψ | , M | ψ | ) in the total free energy ( effective Hamiltonian) [5, 6]have not been included in Eq. (1) as they are irrelevant to the present investigation. The simpledimensional analysis shows that the g − term in Eq. (1) corresponds to a scaling factor b − d/ and,hence, becomes relevant below the upper borderline dimension d U = 6, while fourth order termsare scaled by a factor b − d as in the usual φ − theory and are relevant below d < b > d =6 − ǫ where the g –term in Eq. (1) describes the only relevant fluctuation interaction. Moreover,the total fluctuation Hamiltonian [5, 6] contains off-diagonal terms of the form k i k j ψ α ψ ∗ β ; i = j and/or α = β . Using a convenient loop expansion these terms can be completely integrated outfrom the partition function to show that they modify the parameters ( r, t, g ) of the theory butthey do not affect the structure of the model (1). So, such terms change auxiliary quantities,for example, the coordinates of the RG fixed points (FPs) but they do not affect the main RGresults for the stability of the FPs and the values of the critical exponents. Here we ignore theseoff-diagonal terms.One may consider several cases: (i) uniaxial magnetic symmetry, M = (0 , , M ), (ii) tetragonalcrystal symmetry when ψ = ( ψ , ψ , XY magnetic order ( M , M , m = 3) when all components of thethree dimensional vectors M and ψ may have nonzero equilibrium and fluctuation components.The latter case is of major interest to real systems where fluctuations of all components ofthe fields are possible despite the presence of spatial crystal and magnetic anisotropy thatnullifies some of the equilibrium field components. In one-loop approximation, the RG analysisreveals different pictures for anisotropic (i)-(iii) and isotropic (iv) systems. As usual, a Gaussian(“trivial”) FP ( g ∗ = 0) exists for all d > d > d <
6, where the critical behavior is usually governed by nontrivial FPs ( g ∗ = 0). Inthe cases (i)-(iii) only negative (“unphysical” [9]) FP values of g have been obtained for d < g takes the form g ′ = b − d/ − η g (cid:0) g K d ln b (cid:1) , (2)where g ′ is the renormalized value of g , η = ( K d − / g is the anomalous dimension (Fisher’sexponent) [1] of the field M ; K d = 2 − d π − d/ / Γ( d/ g ) ∗ = − π ǫ . For d < d > g as one may ee from the positive value y g = − ǫ/ > y g definedby δg ′ = b y g δg . Therefore, a change of the order of the phase transition from second order inmean-field (“fluctuation free”) approximation to a fluctuation-driven first order transition whenthe fluctuation g –interaction is taken into account takes place. This conclusion is supported bygeneral concepts of RG theory [1] and by the particular property of these systems to exhibitfirst order phase transitions [6] in mean field approximation for broad variations of T and P .In the case (iv) of isotropic systems the RG equation for g is degenerate and the ǫ -expansionbreaks down. A similar situation is known from the theory of disordered systems [9] but herethe physical mechanism and details of description are different. Namely for this degenerationone should consider the RG equations up to the two-loop order. The derivation of the two-loopterms in the RG equations is quite nontrivial because of the special symmetry properties of theinteraction g -term in Eq. (1). For example, some diagrams with opposite arrows of internallines, as the couple shown in Fig. (2), have opposite signs and compensate each other. Theterms bringing contributions to the g –vertex are shown diagrammatically in Fig. 3. The RGanalysis is carried out by a completely new ǫ / -expansion for the FP values and ǫ / -expansionfor the critical exponents; again ǫ = (6 − d ). The RG equations are quite lengthy and here onlythe equation for g is discussed. It has the form g ′ = b ( ǫ − η ψ − η M ) / g (cid:2) Ag + 3(2 B + C ) g (cid:3) , (3)where A = K d (cid:2) b + ǫ (ln b ) + (1 − b )(2 r + t ) (cid:3) , (4) B = K d − K d h b − − b − b ) i , (5) C = 3 K d − K d h ln b + 2 (ln b ) i , (6) η M and η ψ are the anomalous dimensions of the fields M and ψ , respectively. The one-loopapproximation gives correct results to order ǫ / and the two-loop approximation brings suchresults up to order ǫ . In Eq. (4), r and t are small expansion quantities with equal FP values t ∗ = r ∗ = K d g . Using the condition for invariance of the two k -terms in Eq. (1) one obtains η M = η ψ ≡ η , where η = K d − g (cid:18) − K d − g (cid:19) . (7)Eq. (3) yields a new FP g ∗ = 8 (cid:0) π (cid:1) / (2 ǫ/ / , (8)which corresponds to the critical exponent η = 2(2 ǫ/ / − ǫ/ d = 3, η ≈ − . M = [( ∂µ i /∂µ j ); ( µ , µ , µ ) = ( r, t, g )] canbe solved by the expansion of the matrix elements up to order ǫ / . When the eigenvalues λ j = A j ( b ) b y j of ˆ M are calculated dangerous large terms of type b and b (ln b ), ( b ≫
1) [8] inthe off-diagonal elements of the matrix ˆ M ensure the compensation of redundant large termsof the same type in the diagonal elements ˆ M ii . This compensation is crucial for the validity ofscaling for this type of critical behavior. Such a problem does not appear in standard cases ofRG analysis [1, 8]. As in the usual φ –theory [8] the amplitudes A j depend on the scaling factor A sum of g –diagrams equal to zero. The thick and thin lines correspond to correlationfunctions h| ψ α | i and h| M j | i , respectively; vertices ( • ) represent g –term in Eq. (1). b : A = A = 1 + (27 / b ǫ , A = 1 − (81 / ǫ (ln b ) . The critical exponents y t = y , y r = y and y g = y are b –invariant: y r = 2 + 10 r ǫ
13 + 19739 ǫ, (9) y t = y r − ǫ/ / , and y g = − ǫ > d <
6. The correlation length critical exponents ν ψ = 1 /y r and ν M = 1 /y t corresponding to the fields ψ and M are ν ψ = 12 − r ǫ
13 + 103156 ǫ, (10) ν M = 12 + 2 r ǫ − ǫ . (11)These exponents describe a quite particular multi-critical behavior which differs from the nu-merous examples known so far. For d = 3, ν ψ = 0.78 which is somewhat above the usual value ν ∼ . ÷ . ν M = 1 .
76 at the samedimension ( d = 3) is unusually large. The fact that the Fisher’s exponent [1] η is negative for d = 3 does not create troubles because such cases are known in complex systems, for example,in conventional superconductors [10]. Perhaps, a direct extrapolation of the results from thepresent ǫ -series is not completely reliable because of the fact that the series has been derivedunder the assumptions of ǫ ≪ ǫ / b < ǫ / (ln b ) ≪ b >
1. These conditions are stronger than those in the usual φ -theory [1, 8]. Using the knownrelation [1] γ = (2 − η ) ν , the susceptibility exponents for d = 3 take the values γ ψ = 2 .
06 and γ M = 4 .
65. These values exceed even those corresponding to the Hartree approximation [1]( γ = 2 ν = 2 for d = 3) and can be easily distinguished in experiments. Note, that here wefollow the interpretation of the asymptotic ǫ -series in the way given by Lawrie et al. [9]. Thispoint of view is quite comprehensive, in particular, for an avoiding artificial conclusions from Diagrams for g ′ of third and fifth order in g . The arrows of the thick lines have been omitted. the RG analysis of complex systems with competing effects, such as the systems described bythe Eq. (1). Notes about the quantum effects on the phase transitions.
The critical behavior discussed sofar may occur in a close vicinity of finite temperature multi-critical points ( T c = T f >
0) insystems possessing the symmetry of the model (1). In certain systems, as shown in Fig. 1, thismulti-critical points may occur at T = 0. In the quantum limit ( T → T ≪ µ ; µ ≡ ( t, r ); k B = 1] the thermal wavelengths of the fields M and ψ exceed the inter-particle interaction radius and the quantum correlations fluctuationsbecome essential for the critical behavior [2, 11]. The quantum effects can be considered by RGanalysis of a comprehensively generalized version of the model (1), namely, the action S of thereferent quantum system. The generalized action is constructed with the help of the substitution( −H /T ) → S [ M ( q ) , ψ ( q )]. Now the description is given in terms of the (Bose) quantum fields M ( q ) and ψ ( q ) which depend on the ( d + 1)-dimensional vector q = ( ω l , k ); ω l = 2 πlT is theMatsubara frequency ( ~ = 1; l = 0 , ± , . . . ). The k -sums in Eq. (1) should be substituted byrespective q -sums and the inverse bare correlation functions ( r + k ) and ( t + k ) in Eq. (1)contain additional ω l − dependent terms, for example[2, 11] h| ψ α ( q ) | i − = | ω l | + k + r. (12)The bare correlation function h| M j ( q ) |i contains a term of type | ω l | /k θ , where θ = 1 and θ = 2for clean and dirty itinerant ferromagnets, respectively [11]. The quantum dynamics of the field ψ is described by the bare value z = 2 of the dynamical critical exponent z = z ψ whereas thequantum dynamics of the magnetization corresponds to z M = 3 (for θ = 1), or, to z M = 4(for θ = 2). This means that the classical-to-quantum dimensional crossover at T → d → ( d + 2) and, hence, the system exhibits a simple mean field behavior for d ≥
4. Justbelow the upper quantum critical dimension d (0) U = 4 the relevant quantum effects at T = 0are represented by the field ψ whereas the quantum ( ω l –) fluctuations of the magnetization arerelevant for d < d < onfirmed by the analysis of singularities of the relevant perturbation integrals. Therefore thequantum fluctuations of the field ψ have a dominating role below spatial dimensions d <
4, andfor dimensions 3 < d < , d < ψ and completely neglecting the ω l –dependence of the magnetization M , ǫ = (4 − d )–analysis of the generalized action S has beenperformed within the one-loop approximation (order ǫ ). In the classical limit ( r/T ≪
1) onere-derives the results already reported above together with an essentially new result, namely,the value of the dynamical exponent z ψ = 2 − (2 ǫ/ / which describes the quantum dynamicsof the field ψ . In the quantum limit ( r/T ≫ T →
0) the static phase transition propertiesare affected by the quantum fluctuations, in particular, in isotropic systems ( n/ m = 3).For this case, the one-loop RG equations corresponding to T = 0 are not degenerate and givedefinite results. The RG equation for g , g ′ = b ǫ / g (cid:18) g π ln b (cid:19) , (13)yields two FPs: ( a ) a Gaussian FP ( g ∗ = 0), which is unstable for d <
4, and ( b ) a FP( g ) ∗ = − π ǫ which is unphysical [( g ) ∗ <
0] for d < d ≥
4. Thus thenew stable critical behavior corresponding to
T > d < T →
0. At the absolute zero and any dimension d > P − driven phase transition (Fig. 1)is of first order. This can be explained as a mere result of the limit T →
0. The only role of thequantum effects is the creation of the new unphysical FP ( b ). In fact, the referent classical systemdescribed by H from Eq. (1) also looses its stable FP (8) in the zero-temperature ( classical ) limit T → g -term in the equation for g ′ ; see Eq. (13). At T = 0 the classical system has a purely mean field behavior [2] which ischaracterized by a Gaussian FP ( g ∗ = 0) and is unstable towards T –perturbations for 0 < d < φ − theory this picture holds for d <
4. One may suppose that the quantumfluctuations of the field ψ are not enough to ensure a stable quantum multi-critical behavior at T c = T F = 0 and that the lack of such behavior in result of neglecting the quantum fluctuationsof M . One may try to take into account these quantum fluctuations by the special approachesfrom the theory of disordered systems, where additional expansion parameters are used to ensurethe marginality of the fluctuating modes at the same borderline dimension d U (see, e.g., Ref. [6]).It may be conjectured that the techniques known from the theory of disordered systems withextended impurities cannot be straightforwardly applied to the present problem and, perhaps,a completely new supposition should be introduced.
3. Final remarks
The present results may be of use in interpretations of recent experiments [7] in UGe , wherethe magnetic order is uniaxial (Ising symmetry) and the experimental data, in accord with thepresent consideration, indicate that the C-P phase transition is of first order. Systems withisotropic magnetic order are needed for an experimental test of the new multi-critical behavior.Besides, the present investigation exhibits several new essential problems which are a challengeto the theory of quantum phase transitions. cknowledgements. The author thanks the hospitality of JINR-Dubna where a part of thiswork has been written. Financial support by grants No.P1507 (NFSR, Sofia) and No.G5RT-CT-2002-05077 (EC, SCENET-2, Parma) is acknowledged.
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