Fluctuation conductivity due to the preformed local pairs

We investigated the properties of a system where the itinerant electrons coexist and interact with the preformed local pairs. Using the nonperturbative continuous unitary transformation technique we show that Andreev-type scattering between these charge carriers gives rise to the enhanced diamagneti...

Full description

Saved in:

Bibliographic Details
Date:	2016
Main Authors:	Domański, T., Barańska, M., Solovjov, A.L.
Format:	Article
Language:	English
Published:	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України 2016
Series:	Физика низких температур
Subjects:	К 30-летию открытия высокотемпературной сверхпроводимости
Online Access:	https://nasplib.isofts.kiev.ua/handle/123456789/129314
Tags:	Add Tag No Tags, Be the first to tag this record!
Journal Title:	Digital Library of Periodicals of National Academy of Sciences of Ukraine
Cite this:	Fluctuation conductivity due to the preformed local pairs / T. Domański, M. Barańska, A.L. Solovjov // Физика низких температур. — 2016. — Т. 42, № 10. — С. 1177-1183. — Бібліогр.: 47 назв. — англ.

Institution

Digital Library of Periodicals of National Academy of Sciences of Ukraine

id	nasplib_isofts_kiev_ua-123456789-129314
record_format	dspace
spelling	nasplib_isofts_kiev_ua-123456789-1293142025-02-09T20:14:49Z Fluctuation conductivity due to the preformed local pairs Domański, T. Barańska, M. Solovjov, A.L. К 30-летию открытия высокотемпературной сверхпроводимости We investigated the properties of a system where the itinerant electrons coexist and interact with the preformed local pairs. Using the nonperturbative continuous unitary transformation technique we show that Andreev-type scattering between these charge carriers gives rise to the enhanced diamagnetic response and is accompanied by appearance of the Drude peak inside the pseudogap regime ω ≤ 2Δpg . Both effects are caused by the short-range superconducting correlations above the transition temperature Tc. In fact, the residual diamagnetism has been detected by the torque magnetometry in the lanthanum and bismuth cuprate superconductors at temperatures up to ~1.5 T c. In this work we show how the superconducting correlations can be observed in the ac and dc conductivity. Remove selected T.D. acknowledges discussions with J. Ranninger and R. Micnas. This work is supported by the National Science Centre in Poland through the project DEC2014/13/B/ST3/04451 (TD). 2016 Article Fluctuation conductivity due to the preformed local pairs / T. Domański, M. Barańska, A.L. Solovjov // Физика низких температур. — 2016. — Т. 42, № 10. — С. 1177-1183. — Бібліогр.: 47 назв. — англ. 0132-6414 PACS: 74.25.N–, 72.10.–d, 05.10.Cc, 71.10.Li https://nasplib.isofts.kiev.ua/handle/123456789/129314 en Физика низких температур application/pdf Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
topic	К 30-летию открытия высокотемпературной сверхпроводимости К 30-летию открытия высокотемпературной сверхпроводимости
spellingShingle	К 30-летию открытия высокотемпературной сверхпроводимости К 30-летию открытия высокотемпературной сверхпроводимости Domański, T. Barańska, M. Solovjov, A.L. Fluctuation conductivity due to the preformed local pairs Физика низких температур
description	We investigated the properties of a system where the itinerant electrons coexist and interact with the preformed local pairs. Using the nonperturbative continuous unitary transformation technique we show that Andreev-type scattering between these charge carriers gives rise to the enhanced diamagnetic response and is accompanied by appearance of the Drude peak inside the pseudogap regime ω ≤ 2Δpg . Both effects are caused by the short-range superconducting correlations above the transition temperature Tc. In fact, the residual diamagnetism has been detected by the torque magnetometry in the lanthanum and bismuth cuprate superconductors at temperatures up to ~1.5 T c. In this work we show how the superconducting correlations can be observed in the ac and dc conductivity. Remove selected
format	Article
author	Domański, T. Barańska, M. Solovjov, A.L.
author_facet	Domański, T. Barańska, M. Solovjov, A.L.
author_sort	Domański, T.
title	Fluctuation conductivity due to the preformed local pairs
title_short	Fluctuation conductivity due to the preformed local pairs
title_full	Fluctuation conductivity due to the preformed local pairs
title_fullStr	Fluctuation conductivity due to the preformed local pairs
title_full_unstemmed	Fluctuation conductivity due to the preformed local pairs
title_sort	fluctuation conductivity due to the preformed local pairs
publisher	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
publishDate	2016
topic_facet	К 30-летию открытия высокотемпературной сверхпроводимости
url	https://nasplib.isofts.kiev.ua/handle/123456789/129314
citation_txt	Fluctuation conductivity due to the preformed local pairs / T. Domański, M. Barańska, A.L. Solovjov // Физика низких температур. — 2016. — Т. 42, № 10. — С. 1177-1183. — Бібліогр.: 47 назв. — англ.
series	Физика низких температур
work_keys_str_mv	AT domanskit fluctuationconductivityduetothepreformedlocalpairs AT baranskam fluctuationconductivityduetothepreformedlocalpairs AT solovjoval fluctuationconductivityduetothepreformedlocalpairs
first_indexed	2025-11-30T10:09:09Z
last_indexed	2025-11-30T10:09:09Z
_version_	1850209573798936576
fulltext	Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 10, pp. 1177–1183 Fluctuation conductivity due to the preformed local pairs T. Domański Institute of Physics, M. Curie Skłodowska University, 20–031 Lublin, Poland E-mail: doman@kft.umcs.lublin.pl M. Barańska Institute of Physics, Polish Academy of Sciences, 02–668 Warsaw, Poland A.L. Solovjov B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine 47 Pr. Nauky, Kharkiv 61103, Ukraine Received May 21, 2016, published online August 29, 2016 We investigated the properties of a system where the itinerant electrons coexist and interact with the preformed local pairs. Using the nonperturbative continuous unitary transformation technique we show that Andreev-type scattering between these charge carriers gives rise to the enhanced diamagnetic response and is accompanied by appearance of the Drude peak inside the pseudogap regime ω ≤ 2∆pg. Both effects are caused by the short-range superconducting correlations above the transition temperature Tc. In fact, the residual diamagnetism has been de- tected by the torque magnetometry in the lanthanum and bismuth cuprate superconductors at temperatures up to ~ 1.5Tc. In this work we show how the superconducting correlations can be observed in the ac and dc conductivity. PACS: 74.25.N– Response to electromagnetic fields; 72.10.–d Theory of electronic transport; scattering mechanisms; 05.10.Cc Renormalization group methods; 71.10.Li Excited states and pairing interactions in model systems. Keywords: cuprate superconductors, preformed pairs, short-range correlations, Drude peak. 1. Introduction One of important aspects concerning the role of elec- tron correlations in the cuprate oxides refers to the pseudo- gap phase (existing above the transition temperature cT ) and its relationship with the true superconducting state [1]. Origin of the entire pseudogap state is still a matter of con- troversy [2], but a number of experimental data [3–12] clear- ly indicate that the superconducting correlations emerge gradually upon approaching cT from above. Precursor sig- natures are seen, e.g., in a weak diamagnetic response above the superconducting dome (reported by the torque magnetometry [4]) or the short-scale superconducting corre- lations (detected by the early ultrafast spectroscopy [5] and the recent transient effects [13,14]). Physically these effects are driven by the preformed pairs which are correlated above cT only on some finite spatial and/or temporal scales. Consequences of the short-range correlated preformed pairs can be also probed by the finite-frequency optical conductivity. Rich experimental data on the electrodyna- mic properties [15,16] have been so far discussed in terms of the extended Drude model, determining the frequency- dependent relaxation time ( )τ ω . Interpretation of the pre- cursor effects within such framework is rather complicated because, on one hand, the depleted single-particle spectrum suppresses the subgap optical weight, and, on the other hand, appearance of the pair correlations gives rise to the zero- frequency Drude peak [17], signalling a fragile superfluid stiffness. Similar fluctuation effects have been also report- ed for the thin samples of the strongly disordered s-wave superconductors [18]. These physical processes have been studied within the diagrammatic approximation for the cur- rent-current response function, using the dressed single particle propagators [19–21], imposing the selfconsistent conserving scheme [22] or inventing other sophisticated methods for the vertex corrections [23,24]. In this paper we address qualitative changes of the con- ductivity driven by the preformed pairs, going beyond © T. Domański, M. Barańska, and A.L. Solovjov, 2016 T. Domański, M. Barańska, and A.L. Solovjov the usual perturbative framework. For this purpose we adopt phenomenological scenario describing the local (pre- formed) pairs coexisting and interacting with single (un- paired) electrons. We treat on equal footing the boson and fermion degrees of freedom, by means of the continuous unitary transformation [25,26] that is reminiscent of the numerical renormalization group techniques [27]. Such non- perturbative scheme has been used by us [28] to determine the response function beyond the BCS approximation [29]. Here we focus on its physical implications for the real part of the frequency-dependent conductance due to the pre- formed local pairs. In particular, we show that the Drude- like feature appears in the subgap (infrared) regime and it acquires more and more spectral weight upon approaching cT from above. We confront this prediction with the exper- imental data obtained for Bi2223 cuprates. 2. Preformed pairs scenario Effects of the preformed pairs (of whatever origin) can be studied using the boson-fermion Hamiltonian † † , ˆ ˆˆ ˆ ˆ=H c c E b bσσ σ ξ + +∑ ∑k k q q qk k q ( )† † † , , 1 ˆ ˆˆ ˆ ˆ ˆ .g b c c b c c N ++ ↓ ↑ ↑ ↓ + +∑ k p k pk p k p k p k p (1) This model describes the itinerant electrons (fermion oper- ators (†)ĉ σk ) coexisting with the tightly bound pairs (boson operators (†)b̂q ), where ξk measures the energy with respect to chemical potential µ and Eq is the energy of preformed pairs measured with respect to 2µ . For treating the Bose– Einstein (BE) condensed pairs (i.e., =q 0 mode) one can simplify (1) to the standard BCS Hamiltonian with = , ˆ = b g N −∆ − q 0 k k k . It is the purpose of our study here to address the role of finite momentum pairs b̂ ≠q 0. Specific argumentation in favor for the boson-fermion scenario (1) has been discussed by various groups [30–37]. This Hamiltonian can be derived from the plaquettized Hubbard model using the contractor method [33]. Such mo- del has been shown [34] to capture the Anderson’s idea of the resonating valence bond picture. The Hamiltonian (1) has been also deduced on phenomenological grounds [35–37] as realistic prototype for the correlated electrons (holes) in CuO2 planes. It also accounts for the resonant Feshbach interaction operating in the ultracold fermion atoms such as 6Li or 40K [38–40]. 3. Single particle vs collective features For studying influence of the preformed pairs on the single-particle electron spectrum (and vice versa) we con- struct the unitary transformation ˆ ( )U l , diagonalizing the Ha- miltonian †( ) = ( ) ( )H l U l HU l in a continuous manner. The transformed Hamiltonian ( )H l evolves with respect to a formal parameter l via the flow equation [25,26] ˆ ( ) ˆˆ= [ ( ), ( )]dH l l H l dl η (2) with the generating operator 1( )ˆ ( ) ( )dU ll U l dl −η ≡ . Hamiltonians 0 1 ˆ ˆ ˆ( ) = ( ) ( )H l H l H l+ (where 0 ˆ ( )H l describes the diagonal part and 1 ˆ ( )H l is the off-diagonal term) can be asymptotically diagonalized 1 ˆ ( ) = 0lim l H l →∞ (3) applying the following generating operator [25] 0 1 ˆ ˆˆ ( ) = ( ), ( ) .l H l H l η   (4) During the unitary transformation all the model parameters are continuously renormalized to their asymptotic (fixed point) values [26]. Adopting this algorithm (4) we have constructed [41,42] the continuous unitary transformation for the model (1), choosing † † 0 , ˆ ˆˆ ˆ ˆ( ) = ( ) ( )H l l c c E l b bσσ σ ξ +∑ ∑k k q q qk k q and 1 0 ˆ ˆ ˆ( ) = ( ) ( )H l H l H l− . The generating operator (4) is then given by ( )† † , , 1 ˆˆ ˆ ˆ( ) = ( ) h.c. ,l l b c c N + ↑ ↓ η α −∑ k p k p k p k p (5) where , ,( ) = ( )[ ( ) ( ) ( )]l g l l l E l+α ξ + ξ −k p k p k p k p . Substitut- ing (5) to the flow equation (1) one obtains [41] 2 ,ln ( ) = ( ) ( ) ( ) .d g l l l E l dl + − ξ + ξ − k p k p k p (6) This Eq. (6) implies an exponential decay of the boson-fer- mion coupling , ( )g lk p in the limit l →∞ . Simultaneously, the fermion and boson energies are renormalized according to the flow equations [41] , , 2( ) = ( ) ( ) Bd l l g l n dl N − −ξ α∑k k q k k q k q q , (7) , , , , 2( ) = ( ) ( ) 1 ,F Fd E l l g l n n dl N − − − ↑ ↓  − α − −  ∑q k k q k q k k q k k (8) where , Fn σk ( Bnq ) denotes the fermion (boson) occupancy. We have selfconsistently solved the Eqs. (6)–(8) for the fix- ed charge concentration 1178 Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 10 Fluctuation conductivity due to the preformed local pairs tot , , = 2F Bn n nσ σ +∑ ∑k q k q . The (asymptotic) dispersions ( )liml l→∞ξ ≡ ξk k  and ( )limlE E l→∞≡q q revealed that [41,42]: a) the fermionic spectrum is gaped around µ with the Bo- golubov-type quasiparticle branches existing below and above cT (see Fig. 1), b) the low-energy bosonic spectrum is characterized above cT by the parabolic function 2( ) / 2 Bmq with tem- perature-dependent effective mass Bm (Fig. 2) which evolves at temperatures < cT T into the sound-wave Gold- stone dispersion \| \|sv q . We would like to emphasize that the Bogolubov quasi- particle branches surviving above cT have been later on confirmed experimentally by the angle resolved photo- emission spectroscopy for the bismuth [9] and lanthanum compounds [10]. Similar effect has been also reported by the k-resolved radiofrequency spectroscopy for the ultra- cold potassium atoms [44]. This typical superconducting feature has been observed in the normal state even in ab- sence of the long-range pair coherence. The high- cT cuprate oxides are nearly two-dimensional materials where the superconductivity is driven in CuO2 planes. For this reason we can interpret the temperature dependent mass Bm of the preformed pairs as a quantity related with the residual Meissner effect in the reduced dimensions [46]. This aspect has been recently emphasized by the ETH group [47] within the quantum Monte Carlo studies of the present model (1). Following the same rou- tine we show in Fig. 3 the diamagnetic magnetization ( )dM T obtained from the continuous unitary transfor- mation for the two-dimensional case with tot = 2n . We can notice that the increasing mobility of the preformed pairs substantially enhances the magnetization. This behavior can be independently explained by the direct calculation of the current-current response function (discussed in the next section). 4. Effect of the preformed pairs on the response function The residual Meissner effect and the conductivity can be obtained from the response function , ( , )α βΠ τ ≡q , , ˆ ˆ ˆ( )T j jτ α − β≡ − 〈 τ 〉q q (where ,α β denote the Cartesian , ,x y z coordinates) with the current operator defined as † ,, = ,2 ˆ ˆ ˆ= c c + σσ+ σ ↑ ↓ ∑ ∑q q k qkkk j v (9) and velocity 1−= ∇ εk k kv  . Within the continuous unitary transformation it is convenient to compute the current-cur- rent response function , ( , )iα βΠ νq using the statistical av- erages with respect to the diagonalized Hamiltonian ˆ ( )H ∞ . This, however, requires that the current operator (9) has to be analyzed in the same transformation routine as the Ha- miltonian. Some technical details concerning derivation of , ( , )iα βΠ νq are outlined in the Appendix. In the asymptotic Fig. 2. (Color online) Enhancement of the preformed pairs’ mo- bility 1 / Bm with the decreasing temperature obtained from selfconsistent solution of the flow equations (7), (8) for the con- stant charge concentration tot = 2n . The bandwidth D is used as a unit for energies. Inset shows the bosonic dispersion Eq for a few representative temperatures. Fig. 3. (Color online) Residual diamagnetism induced above cT by the preformed electron pairs. Magnetization has been comput- ed using ( ) 1 / ( )B dM T m T∝ suitable for the two-dimensional hard-core boson gas [46] in analogy to quantum Monte Carlo (QMC) studies [47] of the present model. Fig. 1. (Color online) Schematic view of the gaped fermion spec- trum with the Bogolubov-type quasiparticle branches surviving above cT . Results are obtained for the boson-fermion model (1) using the procedure discussed in Ref. 43. Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 10 1179 T. Domański, M. Barańska, and A.L. Solovjov limit l →∞ we obtain two contributions to the response function, from: (a) the BE condensed pairs and (b) the fi- nite momentum preformed pairs (see Fig. 1 in Ref. 28). Explicit form of the response function is given by [28] ___________________________________________________ , ,, , 2 2 1 1( , ) = ( ) ( )FD FDi f f i iα β + + α + β + +     Π ν ξ − ξ − +    ν + ξ −ξ ν − ξ + ξ    ∑ q q k q k q kk k k q k k q kk q       v v  , , 1 ( ) ( )1 ( ) ( ) ( ) FD FD BE BE f f f E f i EN ′+ ′ ′ ′− + ′ ′+ −′  − ξ − ξ  + − ξ + ξ −  ν − ξ + ξ − ∑ k q k k k q k k k q k k q k k kk         1 ( ) ( ) ( ) ( ) , ( ) FD FD BE BE f f f E f i E ′+ ′ ′− + ′ ′+ − − ξ − ξ   − − ξ + ξ   ν+ ξ + ξ −  k q k k k k q k k q k k k        (10) ______________________________________________ with the Fermi–Dirac [ ] 1( ) = exp ( / ) 1FD Bf k T −ω ω + and Bose–Einstein [ ] 1( ) = exp ( / ) 1BE Bf k T −ω ω − functions, respectively. The coefficients , , , , ,+ − − −≡ + +k q k q k q q k q k q     , , , ,− − + −+ +k q k q k q k q q    , (11) ( ), , , ( ), , ( ),′ ′ ′+ − + − + − + −≡ − ×k k q k q k q q k q k q q   ( ), , , ,′ ′− −× −k k q k k q  (12) denote the asymptotic values of parameters introduced in the l-dependent current operator (A.1). This response func- tion (10) generalizes the standard BCS result [29,45] tak- ing into account the finite momentum preformed pairs Bn ≠q 0. They enter the response function through the terms proportional to , ,′k k q and their influence leads to ap- pearance of the Drude peak in the subgap optical conduc- tivity (Fig. 4). Let us remark, that in the superconducting state the electrodynamic response is dominated by the BE-condens- ed ( =q 0) pairs. In such situation the coefficients (11), (12) simplify to the usual BCS coherence factors 2 , = ( )u u+ ++k q k q k k q kv v and , , = 1 ( ) =BEf E Nk k q q 0 2= ( )q qu u+ +−k k k kv v [28]. Since the preformed pairs are concentrated in the low-momentum (long-wavelength) states (see Fig. 4), therefore some of these BCS features can be preserved also in the pseudogap state above cT . 5. Fluctuation conductivity above Tc We now analyze how the preformed pairs show up in the ac (dynamic) conductivity defined by [45] , , 1( , ) = Im ( , ) .α β α β σ ω − Π ω ω q q (13) For specific considerations we focus on two-dimensional lattice version of the boson-fermion model (1), character- ized by the tight-binding dispersion = 2 [cos ( )xt k aξ − +k cos ( )]yk a+ −µ . In this expression t is the hopping inte- gral, and the bandwidth D ≡ 8t is used as a unit for the en- ergies. We assume that (initially) the preformed pairs are dispersionless (localized) = 2BE ∆ − µq but they acquire some itineracy due to boson-fermion coupling , .gk p We have constructed the numerical codes using the following set of parameters = 0B∆ , , = 0.08g D′k k . We have deter- mined the chemical potential ( )Tµ keeping the fixed charge concentration tot = 2n . We solved the differential equations (6)–(8) along with the flow equations (A.2), (A.5) for the parametrized cur- rent operator (A.1). We have covered the Brillouin zone by a mesh of 500×500 equidistant points and solved the cou- pled differential equations using the Runge–Kutta algo- rithm. The flow parameter l l l→ +δ has been changed with the flexible increment lδ adjusted in order to control the ongoing renormalizations. In the initial stage of trans- formation we used 2= 0.0001l D−δ , and later on we in- creased its value as discussed by us in Refs. 37, 42. To avoid summations of the sharp delta functions we have imposed a small imaginary part in the analytical con- tinuation 1 , ,( , ) ( , )i i − α β α βΠ ν →Π ω+ τq q . Roughly speak- ing τ can be regarded as some phenomenological scatter- ing time, which we assume to be constant for the discussed Fig. 4. (Color online) Gradual accumulation of the preformed pairs at low momenta, leading to appearance of the Drude peak in ac conductivity. 1180 Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 10 Fluctuation conductivity due to the preformed local pairs temperature regime. In the normal state the dynamic con- ductivity is characterized by the Drude model behavior 2 2( ) = / (1 )Nσ ω σ +ω τ with the dc conductivity 2= / F N ne mσ τ . It obeys the important f-sum rule 2( ) = / Fd ne m ∞ −∞ ωσ ω π∫ . Figure 5 shows the ac conductivity obtained for the dirty limit , 1( ) ( , )x xN σ ω ≡ σ ω∑ q q . Upon lowering temperature we observe that: (i) depletion of the single-particle (fermion) states near the Fermi level induces [through the terms (11)] the optical gap over ener- gy regime 2 ;2pg pgω∈〈− ∆ ∆ 〉, (ii) accumulation of the low-momentum preformed pairs (bosons) contributes [via the terms (12)] more and more spectral weight to the Drude peak. Transfer of this spectral weight goes hand in hand with a gradual emergence of the diamagnetism (Fig. 3) in very much the same way as it does in the sym- metry-broken superconducting state [29]. The ongoing transfer of the optical weight has the indi- rect consequence on temperature variation of the dc con- ductivity (0)σ . We observe that dc conductivity is substan- tially enhanced with decreasing temperature in the pseudogap regime. This “fluctuation enhanced conductivi- ty” is well known experimentally. As an example we show in Fig. 6 the temperature-dependent resistivity = 1/ (0)ρ σ of the bismuth cuprate superconductors. Subtracting the normal state value nρ we can notice that the reduced resis- tivity (enhanced conductivity) starts well above the transi- tion temperature, already at * 2.2 cT T . As concerns the optical gap the ac conductivity this effect has been reported for various families of the cuprate superconductors [16] in the temperature and doping regime corresponding to the residual Meissner effect [17]. Similar fluctuation effects have been observed also in the strongly disordered thin classical superconductors [18]. 6. Summary We have studied influence of the preformed local pairs on the diamagnetic response and the conductivity in the pseudogap region above cT . For specific considerations we have used the boson-fermion model, describing the itiner- ant electrons interacting via the Andreev-type scattering with the preformed local pairs. We have shown that a gradu- al suppression of the single particle states near the Fermi energy is accompanied by an increasing mobility of the preformed pairs (Fig. 2). The latter effect leads in turn to some fragile diamagnetic response of the system (Fig. 3). We have further supported this result by analysis of the preformed pairs contribution to the current-current re- sponse function, that has been determined within the flow equation procedure beyond the perturbative scheme. We have also investigated the dynamic conductivity and found that the suppressed fermionic spectrum induces the optical gap in the infrared regime \| \| 2 pgω ≤ ∆ while the accumulation of the low-momentum preformed pairs gives rise to the Drude-like peak. Upon lowering the temperature there is more and more spectral weight transferred to the Drude peak at expense of deepening the optical gap. This processes driven by the low-momentum preformed pairs does amplify (via f-sum rule) the dc conductivity. Finally, we have confronted such fluctuation conductivity with the experimental data obtained for the Bi2223 cuprate super- conductors. Fig. 5. (Color online) The dynamic conductivity ( )σ ω revealing the Drude peak caused by the low-momentum preformed pairs and the optical gap \| \| pgω ≤ ∆ due to the depleted single particle states (i.e., pseudogap). Energy ω is expressed in units of the pseudogap pg∆ at low temperature = 0.02T D . Fig. 6. (Color online) Temperature dependence of the dc resistivi- ty of Bi2223 cuprate superconductors. Notice that the fluctuation conductivity occurs below * 2.2 cT T≈ . Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 10 1181 T. Domański, M. Barańska, and A.L. Solovjov Acknowledgments T.D. acknowledges discussions with J. Ranninger and R. Micnas. This work is supported by the National Science Centre in Poland through the project DEC- 2014/13/B/ST3/04451 (TD). Appendix A: Transformation of the current operator We briefly outline here the continuous transformation for the current operator (9). We individually study both spin contributions † ,, 2 ˆ ˆ ˆ= c cσ + σσ+ ∑q q k qkk k j v because their evolution with respect to l is slightly differ- ent. From the initial ( = 0l ) derivative ˆ ˆˆ( ) = [ ( ), ( )]d l l l dl σ σηq qj j we conclude the following (l-dependent) parametrization † , ,, 2 ˆ ˆ ˆ( ) = ( )l l c c↑ + ↑↑+  +  ∑q q k q k qkkk j v  † † † , , ,, ( ), , , ˆˆ ˆ ˆ ˆ( ) ( )l c c l b c c+− ↓ − + ↓ ↑ − ↓ + + +∑k q k p q k pk k q k p q p   † , , , , ˆ ˆ ˆ( ) .F l b c c+ ↓ + ↑  +   ∑ k p q k p p k q p (A.1) The other spin contribution ˆ ( )l↓ qj has the coefficient , , ( )lk p q interchanged with , , ( )l− k p q and vice versa. The new parameters are subject to the boundary conditions , (0) = 1k q and , , , , ,(0) (0) = (0) = 0=k q k p q k p q   . Let us remark here, that restricting only to the BE condensed pairs (†) (†) , ˆ ˆ=b b −+ δp kk p 0 the constraint (A.1) exactly repro- duces the standard BCS solution [28]. For arbitrary case we can derive from the operator equation ˆ ˆˆ( ) / = [ ( ), ( )]d l dl l lσ σηq qj j the following set of flow equa- tions ( ), , , , , ( ) = ( ) ( ) Fd l l l n n dl + − − σ + α + +∑k q k q p q k p q p q k p p   ( ), , , ,( ) ( ) Fl l n nσ + + α + k p k p q p k p  , (A.2) ( ), , , , , ( ) = ( ) ( ) Fd l l l n n dl − − σ + − α + +∑k q k p p k q p k p p   ( ), , , ,( ) ( ) Fl l n n+ − − − − σ + +α + k q p q p k q p q k p  , (A.3) , , , , , , ( ) = ( ) ( ) ( ) ( ), d l l l l l dl + − −−α +αk p q k q p q k q k p p q    (A.4) , , , , , , ( ) = ( ) ( ) ( ) ( ). d l l l l l dl + − −−α +αk p q k p k q k q p q p q    (A.5) These complex equations can be either solved numeri- cally or (with some compromise) analytically. The lowest order estimation of the coefficients −  is feasible for instance if we neglect renormalizations of the fermion and boson energies on the right hand side of the Eqs. (A.2)–(A.5). Substituting the exponential scaling 2( ) , ,( ) e E l g l g − ξ +ξ − +k p k p k p k p these flow equations (A.2)–(A.5) can be solved iteratively, starting from the initial (boundary) conditions. Thus esti- mated coefficients (A.4), (A.5) are given by [28] ( ) 2 , , 2 \| \|11 2 ( ) F Bn n g E + +  + − + ξ + ξ − ∑ p k p k p k q p k p k p   ( ) 2 , 2 \| \| ( ) F Bn n g E − + + − + − + + + ξ + ξ −  p q k p k q p q k q p q k p , (A.6) , , , , g E + − + − + − ξ + ξ − k q p q k p q k q p q k p   (A.7) , , , g E + − ξ + ξ − k p k p q k p k p   , (A.8) , , ,g g + − ×∑k q k p k q p q p   , 2 2 , , , , 1F Bn n X X X X X + + − + −   +   × − +   +  p k p k p k q p q k p k p k q p q , 2 2 , , , , 1F Bn n X X X X X − + + − + − + −  +  −  +  p q k p k q p q k p k q p q k p k q p q , (A.9) where ,X E +≡ ξ + ξ −k p k p k p . 1. K. Behnia and H. Aubin, Rep. Prog. Phys. 79, 046502 (2016). 2. V. Mishra, U. Chatterjee, J.C. Campuzano, and M.R. Norman, Nature Phys. 10, 357 (2014). 3. T. Kirzner and G. Koren, Sci. Rep. 4, 6244 (2014). 4. L. Li, Y. Wang, S. Komiya, S. Ono, Y. Ando, G.D. Gu, and N.P. Ong, Phys. Rev. B 81, 054510 (2010). 5. J. Corson, R. Mallozzi, J. Orenstein, J.N. Eckstein, and I. Bozovic, Nature 398, 221 (1999). 6. Y. Wang, L. Li, and N.P. Ong, Phys. Rev. B 73, 024510 (2006); Z.A. Xu, N.P. Ong, Y. Wang, T. Takeshita, and S. Uchida, Nature 406, 486 (2000). 7. T. Kirzhner and G. Koren, Sci. Rep. 4, 6244 (2014); N. Bergeal, J. Lesueur, M. Aprili, G. Faini, J.P. Contour, and B. Leridon, Nature Phys. 4, 608 (2008). 1182 Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 10 http://dx.doi.org/10.1088/0034-4885/79/4/046502 http://dx.doi.org/10.1038/nphys2926 http://dx.doi.org/10.1038/srep06244 http://dx.doi.org/10.1103/PhysRevB.81.054510 http://dx.doi.org/10.1038/18402 http://dx.doi.org/10.1103/PhysRevB.73.024510 http://dx.doi.org/10.1038/35020016 http://dx.doi.org/10.1038/srep06244;%2010.1038/nphys1017 Fluctuation conductivity due to the preformed local pairs 8. O. Yuli, I. Asulin, Y. Kalchaim, G. Koren, and O. Millo, Phys. Rev. Lett. 103, 197003 (2009). 9. A. Kanigel, U. Chatterjee, M. Randeria, M.R. Norman, G. Koren, K. Kadowaki, and J. Campuzano, Phys. Rev. Lett. 101, 137002 (2008). 10. M. Shi, A. Bendounan, E. Razzoli, S. Rosenkranz, M.R. Norman, J.C. Campuzano, J. Chang, M. Mansson, Y. Sassa, T. Claesson, O. Tjernberg, L. Patthey, N. Momono, M. Oda, M. Ido, S. Guerrero, C. Mudry, and J. Mesot, Eur. Phys. Lett. 88, 27008 (2009). 11. J. Lee, K. Fujita, A.R. Schmit, C.K. Kim, H. Eisaki, S. Uchida, and J.C. Davis, Science 325, 1099 (2009). 12. Y. Kohsaka, C. Taylor, P. Wahl, A. Schmidt, J. Lee, K. Fujita, J.W. Alldredge, K. McElroy, Jinho Lee, H. Eisaki, S. Uchida, D.-H. Lee, and J.C. Davis, Nature 454, 1072 (2008). 13. D. Fausti, R.I. Tobey, N. Dean, S. Kaiser, A. Dienst, M.C. Hoffmann, S. Pyon, T. Takayama, H. Takagi, A. Cavalleri, Science 331, 189 (2011); S. Kaiser, C. R. Hunt, D. Nicoletti, W. Hu, I. Gierz, H. Y. Liu, M. Le Tacon, T. Loew, D. Haug, B. Keimer, and A. Cavalleri, Phys. Rev. B 89, 184516 (2014). 14. F. Cilento, S. Dal Conte, G. Coslovich, S. Peli, N. Nembrini, S. Mor, F. Banfi, G. Ferrini, H. Eisaki, M. K. Chan, C.J. Dorow, M.J. Veit, M. Greven, D. van der Marel, R. Comin, A. Damascelli, L. Rettig, U. Bovensiepen, M. Capone, C. Giannetti, and F. Parmigiani, Nat. Commun. 5, 4353 (2014). 15. T. Timusk and B. Statt, Rep. Prog. Phys. 62, 61 (1999). 16. D.N. Basov, R.A. Averitt, D. van der Marel, M. Dressel, and K. Haule, Rev. Mod. Phys. 83, 471 (2011); D.N. Basov and T. Timusk, Rev. Mod. Phys. 77, 721 (2005). 17. A. Dubroka, M. Rössle, K.W. Kim, V.K. Malik, D. Munzar, D.N. Basov, A.A. Schafgans, S.J. Moon, C.T. Lin, D. Haug, V. Hinkov, B. Keimer, Th. Wolf, J.G. Storey, J.L. Tallon, and C. Bernhard, Phys. Rev. Lett. 106, 047006 (2011). 18. M. Mondal, Anand Kamlapure, Somesh Chandra Ganguli, John Jesudasan, Vivas Bagwe, Lara Benfatto, and Pratap Raychaudhuria, Sci. Rep. 3, 1357 (2013). 19. E. Illes, E.J. Nicol, and J.P. Carbotte, Phys. Rev. B 79, 100505(R) (2009); J.P. Carbotte, K.A.G. Fisher, J.P.F. LeBlanc, and E.J. Nicol, Phys. Rev. B 81, 014522 (2010); T.S. DeNobrega and J.P. Carbotte, Phys. Rev. B 83, 094504 (2011). 20. N. Lin, E. Gull, and A.J. Millis, Phys. Rev. B 82, 045104 (2010). 21. A. Levchenko, T. Micklitz, M.R. Norman, and I. Paul, Phys. Rev. B 82, 060502(R) (2010). 22. D. Wulin, B.M. Fregoso, H. Guo, C.-C. Chien, and K. Levin, Phys. Rev. B 84, 140509(R) (2011); H. Guo, C.-C. Chien, and K. Levin, Phys. Rev. Lett. 105, 120401 (2010). 23. L. Fanfarillo, L. Benfatto, and C. Castellani, Phys. Rev. B 85, 024507 (2012). 24. D.L. Maslov and A.V. Chubukov, Phys. Rev. B 86, 155137 (2012). 25. F. Wegner, Ann. Physik (Leipzig) 506, 77 (1994); S.D. Głazek and K.G. Wilson, Phys. Rev. D 49, 4214 (1994). 26. S. Kehrein, The Flow Equation Approach to Many-Particle Systems, Springer Tracts in Modern Physics 215, Berlin (2006). 27. W. Metzner, M. Salmhofer, C. Honnerkamp, V. Meden, and K. Schönhammer, Rev. Mod. Phys. 84, 299 (2012); R. Bulla, T.A. Costi, and Th. Pruschke, Rev. Mod. Phys. 80, 395 (2008). 28. M. Zapalska and T. Domański, Phys. Rev. B 84, 174520 (2011). 29. A.L. Fetter, J.D. Walecka, Quantum Theory of Many-Part- icle Systems, McGraw-Hill Book Company, New York (1971). 30. The original idea has been proposed by J. Ranninger and S. Robaszkiewicz, Physica B 135, 468 (1985). For a survey see the review paper R. Micnas, J. Ranninger, and S. Robaszkiewicz, Rev. Mod. Phys. 62, 113 (1990). 31. R. Friedberg and T.D. Lee, Phys. Rev. B 40, 6745 (1989). 32. R. Micnas, Phys. Rev. B 76, 184507 (2007). 33. E. Altman and A. Auerbach, Phys. Rev. B 65, 104508 (2002); A. Mihlin and A. Auerbach, Phys. Rev. B 80, 134521 (2009). 34. K. LeHur and T.M. Rice, Ann. Phys. (NY) 324, 1452 (2009). 35. V.B. Geshkenbein, L.B. Ioffe and A.I. Larkin, Phys. Rev. B 55, 3173 (1997). 36. T. Senthil and P.A. Lee, Phys. Rev. Lett. 103, 076402 (2009); Phys. Rev. B 79, 245116 (2009). 37. J. Ranninger and T. Domański, Phys. Rev. B 81, 014514 (2010). 38. M. Holland, S.J.J.M.F. Kokkelmans, M.L. Chiofalo, and R. Walser, Phys. Rev. Lett. 87, 120406 (2001). 39. Y. Ohashi and A. Griffin, Phys. Rev. Lett. 89, 130402 (2002). 40. Q. Chen, J. Stajic, S. Tan and K. Levin, Phys. Rep. 412, 1 (2005). 41. T. Domański and J. Ranninger, Phys. Rev. B 63, 134505 (2001). 42. T. Domański and J. Ranninger, Phys. Rev. Lett. 91, 255301 (2003); Phys. Rev. B 70, 184503 (2004). 43. T. Domański, Phys. Rev. A 84, 023634 (2011). 44. J.P. Gaebler, J.T. Stewart, T.E. Drake, D.S. Jin, A. Perali, P. Pieri, and G.C. Strinati, Nature Phys. 6, 569 (2010). 45. G.D. Mahan, Many-Particle Physics, Plenum Press, New- York (1990). 46. R.M. May, Phys. Rev. 115, 254 (1959). 47. K.-Y. Yang, E. Kozik, X. Wang, and M. Troyer, Phys. Rev. B 83, 214516 (2011). Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 10 1183 http://dx.doi.org/10.1103/PhysRevB.82.140502 http://dx.doi.org/10.1103/PhysRevLett.101.137002 http://dx.doi.org/10.1209/0295-5075/88/27008 http://dx.doi.org/10.1209/0295-5075/88/27008 http://dx.doi.org/10.1126/science.1176369 http://dx.doi.org/10.1038/nature07243 http://dx.doi.org/10.1126/science.1197294 http://dx.doi.org/10.1103/PhysRevB.89.184516 http://dx.doi.org/10.1038/ncomms5353 http://dx.doi.org/10.1088/0034-4885/62/1/002 http://dx.doi.org/10.1103/RevModPhys.83.471 http://dx.doi.org/10.1103/RevModPhys.77.721 http://dx.doi.org/10.1103/PhysRevB.73.024510 http://dx.doi.org/10.1038/srep01357 http://dx.doi.org/10.1103/PhysRevB.79.100505 http://dx.doi.org/10.1103/PhysRevB.81.014522 http://dx.doi.org/10.1103/PhysRevB.83.094504 http://dx.doi.org/10.1103/PhysRevB.82.045104 http://dx.doi.org/10.1103/PhysRevB.73.024510 http://dx.doi.org/10.1103/PhysRevB.73.024510 http://dx.doi.org/10.1103/PhysRevB.84.140509 http://dx.doi.org/10.1103/PhysRevLett.105.120401 http://dx.doi.org/10.1103/RevModPhys.84.299 http://dx.doi.org/10.1103/PhysRevB.84.174520 http://dx.doi.org/10.1103/PhysRevB.40.6745 http://dx.doi.org/10.1103/PhysRevB.76.184507 http://dx.doi.org/10.1103/PhysRevB.65.104508 http://dx.doi.org/10.1103/PhysRevB.80.134521 http://dx.doi.org/10.1016/j.aop.2009.02.004 http://dx.doi.org/10.1103/PhysRevB.55.3173 http://dx.doi.org/10.1103/PhysRevLett.103.076402 http://dx.doi.org/10.1103/PhysRevB.79.245116 http://dx.doi.org/10.1103/PhysRevB.81.014514 http://dx.doi.org/10.1103/PhysRevLett.87.120406 http://dx.doi.org/10.1103/PhysRevLett.89.130402 http://dx.doi.org/10.1016/j.physrep.2005.02.005 http://dx.doi.org/10.1103/PhysRevB.63.134505 http://dx.doi.org/10.1103/PhysRevLett.91.255301 http://dx.doi.org/10.1103/PhysRevB.70.184503 http://dx.doi.org/10.1103/PhysRevA.84.023634 http://dx.doi.org/10.1038/nphys1709 http://dx.doi.org/10.1103/PhysRev.115.254 http://dx.doi.org/10.1103/PhysRevB.83.214516 http://dx.doi.org/10.1103/PhysRevB.83.214516 1. Introduction 2. Preformed pairs scenario 3. Single particle vs collective features 4. Effect of the preformed pairs on the response function 5. Fluctuation conductivity above Tc 6. Summary Acknowledgments Appendix A: Transformation of the current operator

Fluctuation conductivity due to the preformed local pairs

Institution

Similar Items