Anomalous thermodynamic properties of quantum critical superconductors
AAnomalous thermodynamic properties of quantum critical superconductors
Maxim Khodas, Maxim Dzero, and Alex Levchenko Racah Institute of Physics, Hebrew University of Jerusalem, Jerusalem 91904, Israel Department of Physics, Kent State University, Kent, Ohio 44242, USA Department of Physics, University of Wisconsin–Madison, Madison, Wisconsin 53706, USA (Dated: August 27, 2020)Recent high-precision measurements employing different experimental techniques have unveiledan anomalous peak in the doping dependence of the London penetration depth which is accompa-nied by anomalies in the heat capacity in iron-pnictide superconductors at the optimal compositionassociated with the hidden antiferromagnetic quantum critical point. We argue that finite tempera-ture effects can be a cause of observed features. Specifically we show that quantum critical magneticfluctuations under superconducting dome can give rise to a nodal-like temperature dependence ofboth specific heat and magnetic penetration depth in a fully gaped superconductor. In the presenceof line nodes in the superconducting gap fluctuations can lead to the significant renormalization ofthe relative slope of T -linear penetration depth which is steepest at the quantum critical point. Theresults we obtain are general and can be applied beyond the model we use. Introduction . Quantum phase transitions and quan-tum criticality are among the central concepts in thephysics of correlated electrons [1, 2]. In general, quan-tum fluctuations (QF) near magnetic e.g. spin-density-wave (SDW) quantum critical point (QCP) give rise tonon-Fermi liquid behavior that manifests in singularitiesand nonanalyticity of various electronic characteristics[3–6]. This problem is further enriched in the situationswhen magnetic instability competes with superconduc-tivity (SC) [7–11], Fig. 1(a). This is the case in the con-text of cuprate- and iron-based superconductors whereinterest to the topic is constantly fueled by a multitudeof experimental activities (for the recent detailed reviewssee e.g. [12, 13] and references herein). (a) (b)
FIG. 1. Phase diagram of the (a) magnetic SDW quantumcriticality without SC, and (b) with SC coexistence computednumerically from the so-called disorder model presented be-low. In the superconducting case the fan region extendingaway from the QCP represents an anomalous part of the phasediagram where nodal-like behavior of gaped fermions emerges.
The key signatures of QCP behavior in SCs includecorrelated anomalies in both transport coefficients andthermodynamic properties, which emerge in differenttemperature regimes of the phase diagram when systemis tuned by an external control parameter (e.g. doping x ) to an optimal composition x c . Indeed, some of theseanomalies persist in the normal state such as linear-in- T resistivity observed in various materials at x c [14–18]. It is typically accompanied by anomalies in the trans-verse Hall and thermoelectric responses [19–21]. Whenthe system is brought to the proximity of the phasetransition, then thermally activated fluctuations of mag-netic and superconducting orders start to play a dom-inant role. This translates into the nonmonotonic dis-continuity of the specific heat jump which also peaks at x c [22–24]. When the system is cooled into the super-conducting state, quantum fluctuations proliferate andtheir effect becomes most pronounced near the transi-tion line that separates pure superconducting and mixedphase coexisting with magnetism that ultimately termi-nates at the QCP. Near that region one often detectsan enhanced critical supercurrents [25], and observes theapparent sharp peak in the magnetic penetration depth[26–30].In part motivated by these results, the interplay of pos-sible magnetic and structural quantum phase transitionsshielded by the superconductivity was a subject of an im-mediate scrutiny [31]. In a parallel vein of studies, variousmodels of Planckian resistivity were proposed [32–35],thermal and electrical transport properties across antifer-romagnetic quantum transition were considered [36], andfurther extensions to anomalous Hall phenomena weredeveloped [37, 38]. Thermodynamic signatures of QCPwere analyzed theoretically in the context of the specificheat [39–41] and Josephson effect [42, 43]. However, de-spite much of the efforts [44–49] there is no consensuson the explanation of the observed peak in the Londonpenetration depth.In this work we show that finite-temperature effects ofquantum spin-density-wave fluctuations yield anomalousthermodynamic properties of gaped fermions with pro-nounced power-law dependencies in both specific heatand London penetration depth which is reminiscent ofthat of nodal superconductors. We also demonstrate gen-erality of these results, in particular robustness to effectsof disorder. a r X i v : . [ c ond - m a t . s up r- c on ] A ug Disorder model of SC-SDW coexistence . Weadopt the two-band model which is defined by theHamiltonian [39, 48]: H = H + H sdw + H sc + H dis . Thefirst term describes non-interacting fermions occupyingtwo (one electron- and one hole-like) bands: H = (cid:80) k ξ k Ψ † k τ ρ σ Ψ k , where we take simple parabolicband dispersion ξ k = k / m − µ , defined relative tothe chemical potential, µ , and all momenta are countedrelative to the center of the corresponding pocket.Ψ † k = (ˆ c † k ↑ , ˆ c † k ↓ , -ˆ c − k ↓ , ˆ c − k ↑ , ˆ f † k ↑ , ˆ f † k ↓ , - ˆ f − k ↓ , ˆ f − k ↑ ) iscomposed of electron- c and hole- f creation/annihilationoperators. Three sets of Pauli matrices ( τ, ρ, σ ) operatein the band, Nambu and spin spaces, respectively. Thesecond term describes magnetic inter-pocket interactionbetween fermions H sdw = − ( g sdw / (cid:80) Q S Q S − Q ,where the magnetization fluctuation at momentum Q is S Q = (1 / (cid:80) k Ψ † k + Q Ξ Ψ k , Ξ = τ ρ σ . Thethird term captures pairing interaction and in themodel of s ± order parameter changing sign be-tween the hole and electron pockets takes the form, H sc = − ( g sc / (cid:80) kk (cid:48) B k B k (cid:48) , where the fermion bi-linearis defined as B k = Ψ † k ( τ ρ σ )Ψ k . In this low-energydescription we impose high-energy cutoff Λ, and consideronly angle-independent interactions in the SDW channeland in the s ± SC channel with the couplings g sdw and g sc . With the last term we introduced a disorderpotential into the problem. We account for two typesof scattering processes: the intra-band disorder withpotential U , which scatters quasiparticles within thesame band, and inter-band scattering between theFermi pockets mediated by the potential U π . In thebasis of spinors Ψ k the disorder term reads H dis = (cid:80) kk (cid:48) R j Ψ † k [ U ( τ ρ σ ) + U π ( τ ρ σ )] Ψ k (cid:48) e i ( k − k (cid:48) ) R j ,where summation goes over the random locations R j of individual impurities. When performing disorderaveraging within the self-consistent Born approximationwe assume that concentration of impurities is n imp .This naturally introduces two scattering rates into themodel Γ ,π = πν F n imp | U ,π | /
4, where ν F is the totalquasiparticle density of states at the Fermi energy [50].The mean-field (MF) analysis of this model proceedsin a standard way by decoupling interaction terms viaHubbard-Stratonovich transformation with magnetic M and superconducting ∆ order parameters, and integrat-ing out fermions [39, 48]. In this treatment, the pureSC transition temperature T c is suppressed only by theinter-band scattering as described in accordance with theAbrikosov-Gor’kov scenario ln (cid:16) T c T c (cid:17) = ψ (cid:16) + Γ π πT c (cid:17) − ψ (cid:0) (cid:1) , where T c (cid:39) Λ e − /ν F g sc and ψ ( x ) is the di-gammafunction. This is similar to the equation for T c in con-ventional single-band s -wave superconductors with mag-netic impurities, and in the unconventional d -wave super-conductors with potential impurities. In contrast, pureSDW transition temperature T s is suppressed by the to- tal scattering rate, ln (cid:16) T s T s (cid:17) = ψ (cid:16) + Γ +Γ π πT s (cid:17) − ψ (cid:0) (cid:1) ,where T s (cid:39) Λ e − /ν F g sdw . As a result of different sensi-tivity to disorder, there exists a finite parameter range inΓ ,π where both orders M and ∆ can coexists simulta-neously. The magnetic QCP is defined by the condition T s (∆) = 0, which corresponds to M = 0 for certain val-ues of Γ ,π , see Fig. 1b for an example. We note that thisphase-diagram was calculated numerically for the choiceof parameters when Γ π / Γ = 0 .
325 and T s /T c = 1 . SDW fluctuation propagator in SC state . Ex-tending theory beyond the mean field we consider thecritical fluctuations that mediate an effective interactionin the spin channel ( S z ) represented by the propaga-tor L Q, Ω m = (cid:0) g − − Π zQ, Ω m (cid:1) − . The polarization op-erator, Π zQ, Ω m = Tr[Ξ z G Q + k ,ω n +Ω m Ξ z G k ,ω n ] is definedvia the disorder averaged Green functions, [ G k ,ω n ] αβ = − (cid:82) T − dτ e iω n τ (cid:104) Ψ k α ( τ )Ψ † k β (cid:105) . The trace includes convo-lution over all the indices and the summation over theMatsubara frequency ω n = πT (2 n + 1) with n ∈ Z andmomenta Tr = (cid:82) k T (cid:80) ωαβ .The critical paramagnon described by the spin correla-tion function in L Q, Ω m with Ω m = 2 πmT softens towardsthe QCP, g − − Π zQ, Ω m = πν F (cid:0) γ + Q /Q c + Ω m / Ω c (cid:1) reached at Γ = Γ c such that, γ (Γ) ≈ γ (cid:48)± | Γ − Γ c | , γ (cid:48)± = | dγ/d Γ | taken at Γ = Γ c ± + . We find in this modelrather generally that the QCP is located at Γ c = 2 πqT c where precise numerical value of q depends the choice oftwo ratios between scattering rates and bare interactionparameters. Furthermore, while the ratio γ (cid:48) + /γ (cid:48)− can bearbitrary, the low-energy expansion coefficients Q c andΩ c may be computed right at the QCP. Further detailsand generalities of calculation of L Q, Ω m are relegated toRef. [55]. Specific heat near QCP . We now focus on theimpact of quantum SDW fluctuations on the low-temperature behavior of the specific heat inside the domeof s ± superconductivity. Recall that at the level of themean-field analysis, the low- T asymptotic behavior ofthe specific heat in a fully gaped SC state is exponen-tial C MF ∝ (∆ /T ) / e − ∆ /T for T (cid:28) ∆. Our intent isto investigate the fate of this result as the system ap-proaches a QCP by accounting for the extra contributionof the spin fluctuations. Following the standard proce-dure, we integrate out these soft magnetic modes fromthe action. At the Gaussian level we thus get a renor-malized free energy of a superconductor per unit layerarea F = F SC (∆ , M ) + δF QF that can be expressed interms of the SDW propagator, δF QF T = N (cid:104) L − Q, Ω m (cid:105) , (1)where N counts the number of soft modes. In our model N = 3 at x > x c and N = 1 at x < x c as only the lon-gitudinal mode has a mass changing with Γ. The factor1 / δC QF = − T ∂ T δF QF . We thus find in a broad regimeof temperatures ∆ QCP < T < ∆, δC QF = 9 ζ (3) π (cid:18) v F Q c Ω c (cid:19) (cid:18) Tv F (cid:19) , (2)where we introduced gap to QCP as ∆ QCP (Γ) = (cid:112) γ (Γ)Ω c . The most striking feature of this result is thatproliferation of quantum fluctuations to finite tempera-tures gives a power-law instead of exponential behavioreven in the presence of a full SC gap. As is known, apower-law in the specific heat occurs only in the uncon-ventional superconductors having nodal structure of thegap. In particular ∝ T is a characteristic of a gap struc-ture with first-order nodes at isolated points.We note that the details of the microscopic model enterEq. (2) only via the ratio v F Q c / Ω c so that T depen-dence is a model independent result. Furthermore, asΓ c / πT c (cid:28) v F Q c = ∆ √ π and Ω c = ∆ (cid:112) π/ δC QF = ζ (3) π (cid:16) Tv F (cid:17) . Ulti-mately, at the lowest temperatures, T < ∆ QCP , specificheat crosses over to exponential dependence, δC QF ∝ (∆ QCP /T ) e − ∆ QCP /T . We note that the same conclusionhas been reached independently in the considerations ofa different model [41]. Penetration depth near QCP . We turn our atten-tion to the anomalies in the magnetic penetration depth, λ ( T, x ), where numerous recent measurements [26–30] re-vealed a distinct peak in the low-temperature limit T (cid:28) ∆ concentrated around the putative QCP x → x c . Themodel we explore in this study with x = Γ is perhaps bestsuited to experiments of Ref. [30] on Ba(Fe − x Co x ) As .This material is in the disordered limit with fully gapedFermi surface as opposed to BaFe (As − x P x ) , which israther clean system that displays nodal superconductiv-ity. However, the arguments we put forward are rathergeneric, and in fact apply to both compounds.It is natural to account for soft bosonic modes in thefermionic electromagnetic response function that defines λ ( x, T ). However, the one-loop fluctuation correctionwhile giving a good approximation outside the criticalregion, is inapplicable inside this region. In the present (a) (b) FIG. 2. (a) Contour plot of the London penetration depth λ − (Γ , T ) (arb. units) calculated within the MF theory ap-proximation as a function of temperature and disorder scat-tering rate Γ assuming Γ π = 0 . . We note that already atthe MF the width of the region in which λ − has maximumvalue narrows upon an increase in temperature. (b) Normal-ized quantum fluctuation correction to the electromagneticresponse kernel [Eq. (6)] as a function of the proximity to theQCP gap, γ , showing emergent peak in a color-plot. context, this implies that as a matter of principle, thecharacter of the λ ( x ) singularity cannot be determinedon the level of a one-loop approximation. Indeed, themean field theory predicts a deep in λ ( x ) [48, 53, 54],see also Fig. 2 for the further illustration. Therefore, inorder to turn the deep into a peak the fluctuation cor-rection must exceed the mean field value. According tothe Ginzburg criterion, however, this cannot happen inthe region of validity of one-loop approximation. For thisreason, the problem has to be solved inside a critical re-gion, and is essentially non-perturbative.Such a solution valid in critical region is possible at T = 0 for the model of electrons coupled to critical bosonswith the mass term ∝ ( x − x c ) [45]. In this model thereis a universal relation between the critical scaling of λ at x above and below x c . When the bosons are viewed ascollective fermion excitations as captured by L Q, Ω m sucha universal relation is lost as the ratio of the paramagnonmasses at x = x c ± δ is a model dependent number,while in e.g. Ising boson theory it is 2. In our specificmodel this number, γ (cid:48) + /γ (cid:48)− is a non-universal function ofΓ ,π (see [55]). Despite this discrepancy with the purelybosonic approach, the x dependence of λ established inRef. [45] remains monotonic in our model as well. Thisleaves us with the puzzle of the peak in λ ( x ).Our resolution to this puzzle builds on the strong x dependence of the T -dependent part of the penetrationdepth, λ ( T ) − λ ( T = 0). In distinction with the T = 0case, at the mean field level δλ ( T ) = λ ( T ) − λ (0) ∝ e − ∆ /T is suppressed exponentially at T (cid:28) ∆. Therefore, justoutside the critical region the one-loop correction gives areliable estimate of fluctuation correction to δλ ( T ). Thiscorrection yields the peak in λ ( T ) at the temperatures T (cid:38) ∆ /E F much smaller than ∆.To quantify these statements we express the fluctua-tion correction to λ = λ + δλ QF through the correctionto the static, long wave length limit of the current corre-lation function K = K + δK QF , δλ QF λ = − δK QF K , K = 12 ν F e v F , λ − = 4 πw K , (3)where w is the inter-layer separation as appropriate tothe quasi-2D systems. The one-loop correction of theelectromagnetic kernel K QF contains both effective massrenormalization, captured by the density of states (DOS)type diagrams, and vertex renormalization expressed bythe Maki-Thompson (MT) type interference processes.The Aslamazov-Larkin vertex corrections cancel for thecase when the gaps on hole and electron Fermi surfacesare of equal magnitude (and opposite sign), which is im-plicit in our model. The cancellation occurs at the levelof fermionic triangular blocks as for each block there aretwo ways to arrange electron and hole Green’s functionlines and their corresponding momenta which cancel eachother. We thus have ∂ γ δK QF = N e v F Tr [ ∂ γ L Q, Ω m ] F l , F l = F DOS + F MT . (4)Apart from excluding transverse spin excitations, takingthe derivative of δK QF makes the integration over theboson energies and momenta convergent at the ultra-violet. This means that at γ (cid:28) Q and Ω are within the region of applicabilityof low-energy expansion of L Q, Ω m assumed above. Atthe same time the integrations over fermion and bo-son energies and momenta in Eq. (4) factorize, and thefermionic loop F l can be taken at zero boson energyand momentum ( Q, Ω m ) →
0. The factorization inEq. (4) is possible thanks to the energy scale separa-tion of fermions, ∆ and bosons ∆
QCP (cid:28) ∆. The indi-vidual terms are F DOS = 2 Tr (cid:2) V S Gτ G Ξ z G Ξ z Gτ (cid:3) and F MT = Tr (cid:2) V S Gτ G Ξ z Gτ G Ξ z (cid:3) , where, the spin vertexrenormalization V S can be evaluated at ( Q, Ω m ) → π = 0, V S = (1 + Γ / (cid:112) ∆ + ω n ) − ,where ω n is a frequency argument of Green functions. Inthe wide range of parameters ( T, Γ c ) (cid:28) ∆, F l (cid:39) ν F / ∆ .We further separate δK QF = δK QCP + δK SDW into zero-temperature ( δK QCP ) and finite-temperature( δK SDW ) terms. For the former we straightforwardly findin a limit, Γ c (cid:28) ∆ δK QCP K = − (cid:114) π π N
16 ∆ E F √ γ (5)up to a constant with the Fermi energy, E F = πν F v F / √ γ such that the fluctuation correction, ∂ γ δK QCP becomes comparable to the mean field value. From (5),Gi = ∆ /E F . It follows that the loop expansion is seriesin powers of Gi /γ . For instance, the two-loop contribu-tion can be estimated to give a correction to δK QCP /K of the form, Gi γ − / , see [55].We proceed to analyze the temperature dependent partof the response kernel. After the Matsubara sum we ar-rive at ∂ γ δK SDW K = − N π Ω c F l ν F (cid:90) Q f ( E Q / T ) E Q , (6)where E Q = Ω c (cid:112) γ + ( Q/Q c ) and f ( z ) = coth( z ) − z/ sinh ( z ). In the temperature range above the QCPgap, ∆ QCP < T < ∆, we find for the penetration depth δλ QF λ = N TE F ln (cid:18) γ (cid:19) . (7)so that the peak height is estimated as δλ maxQF /λ (cid:39) ( T / ∆)Gi ln(1 / Gi). At temperatures within the QCP gap,
T < ∆ QCP , we instead find an exponential dependence δλ QF /λ ∝ e − ∆ QCP /T . The linear in T result holdsin both paramagnetic and magnetically ordered phases.The only difference originates from the difference in thecoefficients γ (cid:48)± describing the paramagnon softening inthe two phases as introduced above. Discussion . To address implications of these resultsin light of experiments we stress that measurements ofRef. [30] on Ba(Fe − x Co x ) As were carried out at T ∼ .
5K (with maximal T c ∼ (As − x P x ) were done at T = 1 .
2K (with maximal T c ∼ λ QF from Eq. (7) dominates over suppressed mean field be-havior δλ MF ∝ e − ∆ /T . This is exemplified in Fig. 1(b)and further in Fig. 2(b). In addition, due to renormaliza-tion of fluctuations by finite M in the phase of coexistencethe structure of the peak should be non-symmetric fromboth sides of QCP. This is supported by our model anal-ysis and is in qualitative agreement with observations.In contrast, in the P-doped case a quasi linear- T depen-dence of λ was seen and attributed to the nodal structureof the gap. However, it is crucial to point out that theslope of this linear behavior was changing with doping at-taining a maximum at QCP (see Fig. 3 of Ref. [26]). Weattribute this enhancement to SDW fluctuations whichalso result in linear-in- T penetration depth as we showin Eq. (7).A signature of the peak was also detected in(Ba − x K x )Fe As [28] concomitant with non-monotonicdoping dependence and change in δλ ∝ T n power-law[56]. While SDW fluctuations certainly play an impor-tant role near QCP, interpretation of the data in thewhole range is difficult as K-doped system displays aseries of Lifshitz topological phase transitions resultingin gaped-to-nodal change of the pairing gap. Additionalcomplications come from apparent narrow dome of s + is (cid:48) superconductivity separating gaped and nodal regions[24] capturing which is beyond our two-band model. Summary and outlook . In this work we studied theinterplay of magnetism and superconductivity in the con-text of iron-pnictides. Our principal results are Eqs. (2)and (7) for the temperature dependence of the specificheat and the London penetration depth, respectively, dueto physics associated with the QCP. These results are sig-nificant as power-law T -dependence of thermodynamicproperties of SCs is used as a hallmark diagnostic fortheir unconventional character, namely determination ofthe types of the nodes of superconducting order parame-ter. Yet we demonstrate that even in the presence of thefull gap such behavior can be promoted by the quantumcriticality under the dome of superconductivity.We further comment that there remain some unre-solved issues that warrant additional studies. In partic-ular, a double-peak structure was detected in the pene-tration depth measurements in NaFe − x Co x As [29]. Thisremarkable feature was attributed to the second putativeQCP of nematic origin. However, the mere statement ofmultiple possible QCPs under the SC dome is at oddswith the present state of the theory [31] that predictsthat magnetic and nematic transitions merge togetherinto the weak first-order quantum critical line that thusterminates at a single QCP.In closing, we mention that our results open inter-esting perspectives for the studies of transport proper-ties due to QCP, specifically for the optical conductivityand thermoelectric effects, where one may hope to obtainanomalous frequency and temperature dependencies dueto quantum fluctuations. It is also of special interestto investigate the QCP behavior due to the interplay ofcharge/pair-density-order and superconductivity, whichis highly relevant to cuprates.
Acknowledgments . We thank E. Berg, V. S. de Car-valho, A. Chubukov, R. Fernandes, S. Gazit, Y. Mat-suda, D. Orgad, R. Prozorov, S. Sachdev, J. Schmalian,and T. Shibauchi for important discussions that shapedthis study. This work was supported in part by BSFGrant No. 2016317, ISF Grant No. 2665/20 (M.K.),NSF-DMR-BSF-2002795 (M.D. and M.K.) and U.S. De-partment of Energy (DOE), Office of Science, Basic En-ergy Sciences, under Awards No. DE-SC0020313 (A.L.)and DE-SC0016481 (M.D.). This work was performed inpart at the Landau Institute for Theoretical Physics, MaxPlanck Institute for the Physics of Complex Systems, andat the Aspen Center for Physics, which is supported byNational Science Foundation grant PHY-1607611. [1] Subir Sachdev,
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