Thermal fluctuations in superconducting phases with chiral d+id and s symmetry on a triangular lattice
aa r X i v : . [ c ond - m a t . s t r- e l ] S e p Thermal fluctuations in superconducting phaseswith chiral d + id and s symmetry on a triangularlattice A G Groshev and A K Arzhnikov
Udmurt Federal Research Center of the Ural Branch of the Russian Academy ofSciences, T. Baramzinoy st. 34, Izhevsk 426067, RussiaE-mail: groshev − [email protected] E-mail: [email protected]
Abstract.
The behavior of thermal fluctuations of a superconducting orderparameter with extended s and chiral d + id symmetry is investigated. Thestudy is carried out on a triangular lattice within the framework of the quasi-two-dimensional single-band model with attraction between electrons at neighboringsites. The method of consistent consideration of the order parameter fluctuationsand the charge carrier scattering by fluctuations of coupled electron pairs, basedon the theory of functional integration is used. The distribution functions ofthe phase fluctuation probabilities depending on temperature and charge carrierconcentration are obtained. The temperature dependences of the amplitudes ofthe averaged superconducting order parameter are calculated. A phase diagram ofsuperconducting states is constructed for the entire range of variation in the chargecarrier concentration 0 < n <
2. Near the boundaries of this range, topologicallytrivial superconducting states with extended s symmetry are realized, while asuperconducting state with topologically nontrivial chiral d + id symmetry isrealized between them. The calculated anomalous self-energies are compared withthe experimental ones obtained using machine learning techniques. Keywords : high-temperature superconductivity, triangular-lattice superconductors,layered compounds, chiral d + id superconducting phase, thermal fluctuations, theoryof functional integration.
Submitted to:
J. Phys.: Condens. Matter hermal fluctuations in superconducting phases with chiral d + id
1. Introduction
Recently, there has been an increased interestin studying superconductivity in layered materi-als with a triangular lattice, such as: sodiumcobaltites
N a x CoO · yH O intercalated with wa-ter, organic dielectrics k − ( bis ( ethylenedithio ) − tetrathiaf ulvalene ) Cu ( CN ) and k − ( BEDT − T T F ) Cu ( CN ) [1], In Cu V O [2], SrP tAs [3],molybdenum disulfide
M oS [4], bilayer graphene sam-ples [5], and (111) bilayer perovskite transition metaloxides N a IrO ( Li IrO ) [6]. One of the reasonsfor this interest is a possibility to advance in solvingthe complicated long-standing problem of the natureof high-temperature superconductivity (HTSC). In ad-dition, the triangular lattice is frustrated with respectto antiferromagnetic ordering, and superconductivityhas a topologically nontrivial chiral d x − y + id xy sym-metry, which is of interest per se. The presence of atopological phase opens the potential for a topologi-cal quantum transition under changes in the chargecarrier concentration [7, 8] and for the appearance ofMajorana states [9, 10, 11]. Furthemore, a number ofmaterials are promising in practical application, see,for example, [12]. Since the compounds under studyare highly anisotropic systems with effectively reduced(quasi-2D) dimension, it is important to take accountof the increasing role of the order parameter (OP) fluc-tuations [13] when describing their properties. The al-lowance for these fluctuations substantially (by severaltimes [14]) reduces the temperature of superconductingtransition T c and, in some cases, leads to a change inthe phase transition type [15, 16]. Usually, when con-sidering superconducting properties, either these fluc-tuations are neglected, or only the amplitude or phasefluctuations of the OP ∆ = | ∆ | exp ( iφ ) are taken intoaccount. Sometimes, such approximations are justi-fied [17]. However, in systems with a reduced dimen-sion, it is important to take into account the ampli-tude and phase fluctuations of the OP [16, 14] simul-taneously. This is due to the fact that the phase andamplitude fluctuations turn out to be effectively re-lated. Such a relationship was considered in [15, 18] inthe framework of the variational approximation and in[14] based on the self-consistent equations of the the-ory of continual integration in the coherent potentialapproximation. As indicated in [14], the approxima-tions used in [15, 18] have some limitations. In par-ticular, they do not take into account renormalizationof the single-particle states resulting from the scatter-ing of charge carriers on fluctuations of coupled elec-tron pairs. Renormalization of the spectrum of single-particle states as a result of such scattering is describedby the self-energy of the single-particle Green function,which defines the spectral density, so its consistent cal-culation seems necessary in explaining both the super- conducting transition and the experimental data of an-gular resolution photoelectron spectroscopy (ARPES).Given the increased interest in superconducting statesin layered materials with a triangular lattice, we believeit would be important to study the effect of a consis-tent consideration of the superconducting OP thermalfluctuations and the renormalization of single-particlestates.
2. Model and method of accounting forthermal fluctuations
In this paper, we use the method proposed in [14]which allows one to consistently take into accountthe renormalization of single-particle states and theeffective relationship between the amplitude andphase fluctuations. This section presents only themain statements and features connected with thetriangular lattice symmetry. We consider the one-bandHamiltonian of the t − V model with attraction betweenelectrons at the nearest sites of a triangular lattice:ˆ H = X i,j,s t ij ˆ c + is ˆ c js − X j µ ˆ n j − V X j,δ ˆ n j ↑ ˆ n j + δ ↓ , (1)where t ij = − t are the matrix elements of electronjumps to the nearest sites; ˆ c + js (ˆ c js ) are the operatorsof creation (annihilation) of an electron at site j with spin projection s ; n js = ˆ c + js · ˆ c js is theoperator of the number of electrons at site j with spinprojection s ; n j is the operator of the total numberof electrons at site j ; µ is the chemical potential; V is the parameter of inter-site attraction betweenelectrons. The choice of such a Hamiltonian is justifiedby the presumed mechanisms of the appearance ofsuperconductivity in HTSC. In this work, we do notspecify the nature of the attraction between electronsat neighboring sites, assuming it to be due to eitherantiferromagnetic spin fluctuations [12], or the stateof resonating valence bonds [19], or other mechanisms(for example, the polaron one) which, in the simplestapproximation, provide such an effective attraction[20, 21, 22, 23]. Moreover, for simplicity, it isassumed that in the considered temperature rangethis attraction is weakly temperature-dependent, see[20]. The quasi-two-dimensional character of thecompounds under study is taken into account by anapproximation in which there are no fluctuations thatdestroy superconductivity in systems with dimension D ≤
2. According to the Mermin-Wagner-Hohenbergtheorem [24, 25, 26] in a strictly two-dimensionaldegenerate system, the long-range order is absent atany nonzero temperature and superconducting statescan manifest themselves only in phase transitions of theBerezinskii-Kosterlitz-Thouless type [27]. The use ofcommutation transformations reduces the Hamiltonian(1) to the Hamiltonian of interacting electron pairs hermal fluctuations in superconducting phases with chiral d + id O + j,δ = ˆ c + j ↑ ˆ c + j + δ ↓ and annihilation ˆ O j,δ = ˆ c j + δ ↓ ˆ c j ↑ of an electron pairat site j and its nearest neighbor j + δ . Thisproblem is solved within the method of continualintegration, which proved effective earlier in studyingthe effect of temperature [28] and atomic disorder[29] on the magnetic phase separation and parametersof spiral magnetic structures in the framework ofthe quasi-two-dimensional one-band t − t ′ Hubbardmodel in the coherent potential approximation. Inthis paper, the problem under consideration is solvedin the approximation of the average t -matrix, whichreproduces quite well the results of the coherentpotential approximation used in [14], but allowsone to significantly reduce the computational cost.The Hubbard-Stratonovich transformation in theconsidered method allows us to reduce the problemof calculating the partition function of interactingelectron pairs to that of calculating the partitionfunction of independent electron pairs in the spaceof time-independent (in the static approximation)auxiliary fluctuating fields. The static approximationdoes not take into account the quantum fluctuationsof the superconducting OP, which, in our opinion, areimportant only at sufficiently low temperatures. Thisis also evidenced by the estimate of the contributionof quantum fluctuations to the suppression of thesuperconducting transition temperature [30]. Thus,for T = 0, the functional integration methodreduces to the Hartree-Fock (HF) approximation.When calculating the partition function, the auxiliaryfluctuating fields are considered in polar variables: themodulus ∆ j,δ and the phase φ j,δ , which determinethe fluctuating complex OP ∆ j,δ · exp( iφ j,δ ) in sitenotation. In the ground state, thermal fluctuationsof the superconducting OP are absent and φ j,δ = α j,δ ,where α j,δ is the OP phase in the HF approximation,which determines its symmetry. Modern high-precisionNMR data point to the spin-singlet Cooper pairingin the considered compounds [31, 32]. Therefore, inthis work, we restrict ourselves to the study of singletsuperconducting phases with extended s symmetry inwhich the superconducting gap depends on the wavevector according to the law ∆( k ) ∝ cos( k ) + cos( k ) +cos( k − k ), and chiral d x − y + id xy - symmetrywith dependence ∆( k ) ∝ cos( k )+exp( i π/
3) cos( k )+exp( − i π/
3) cos( k − k )[38], where k are the valuesof the wave vector along the basic reciprocal latticevectors. Such symmetry types are admissible inthe group-theoretical analysis of the singlet orderparameter on a triangular lattice. When taking intoaccount the interaction between electrons within thefirst coordination sphere, one should consider the OP inwhich only the phase depends on the nearest neighbors ∆ j,δ · exp( iα j,δ ) = ∆ · exp( iα δ ): α δ = , δ = ± a ,α, δ = ± a , − α, δ = ± ( a − a ) , (2)where a and a are the basic vectors of the triangularlattice. The phase value α = 0 corresponds tothe s -symmetry, and α = 2 π/ d + id -symmetry. To simplify the problem and reducecomputational effort when calculating the integral overthe amplitude field ∆, the saddle point approximationis used, in which the fluctuating field ∆ is replacedby its value at the saddle point ∆( φ ). Thus, inthis approximation, the most probable amplitude andphase fluctuations turn out to be related. Thisapproximation is valid when amplitude fluctuationsbecome so much faster than phase fluctuations thatthe amplitude field has time to adjust to the phasedistribution in an equilibrium manner. As shownin [14], this approximation works well over a widetemperature range up to T c . The minimum condition ∂ Ω /∂ ∆ = 0 of the thermodynamic potential Ω servesas an equation for finding the saddle point, being at thesame time a self-consistent equation for determiningthe superconducting OP amplitude ∆( φ ). At T =0 it coincides with the self-consistency equation forthe superconducting OP of the mean field theory(HF, BCS). However, at finite temperatures in themethod of continual integration, the solution ∆ = 0is lacking (see [14]), so to determine the temperatureof the superconducting transition T c , the conditionfor the averaged OP to be zero h ∆( φ ) · exp( iφ ) i =0 is used. Therefore, the transition to the normalstate in the considered approach occurs as a resultof the loss of phase coherence of the fluctuatingcomplex OP. In consequence of the approximationsmade, the partition function per one electron pair isrepresented as an integral only over the phase field φ . Together with the equations for determining thechemical potential µ and the self-energy Σ in theapproximation of the average t -matrix (see Section 3),this set of self-consistent equations is solved by theiteration method. The solution with the minimumvalue of the thermodynamic potential determinesthe superconducting properties of the model underconsideration.
3. Green functions in the approximation of theaverage t -matrix When solving the problem under study, it is convenientto use the representation of Nambu site matricesˆ c jδ ( τ ) = (cid:20) ˆ c j ↑ ( τ )ˆ c + j + δ ↓ ( τ ) (cid:21) , ˆ c + jδ ( τ ) = (cid:2) ˆ c + j ↑ ( τ ) ˆ c j + δ ↓ ( τ ) (cid:3) , (3) hermal fluctuations in superconducting phases with chiral d + id G jδ ( τ − τ ′ ) = − D T τ ˆ c jδ ( τ )ˆ c + jδ ( τ ′ ) E == " G ↑↑ j,j ( τ − τ ′ ) G ↑↓ j,j + δ ( τ − τ ′ ) G ↓↑ j + δ,j ( τ − τ ′ ) G ↓↓ j + δ,j + δ ( τ − τ ′ ) , ∆ ˆ U (∆ , φ, τ ) = X j,δ ˆ c + jδ ( τ )∆ U jδ ˆ c jδ ( τ ) , ∆ U jδ = " U ↑↓ j,j + δ ∆ ˆ U ↓↑ j + δ,j , ∆ ˆ U ↑↓ j,j + δ = V exp ( iα δ ) (cid:2) ∆ − ∆( φ ) exp ( iφ ) (cid:3) , ∆ ˆ U ↓↑ j + δ,j = V exp ( − iα δ ) (cid:2) ∆ − ∆( φ ) exp ( − iφ ) (cid:3) , (4)where ∆ is the average OP amplitude, G ↑↑ j,j ( τ − τ ′ ) and G ↓↓ j + δ,j + δ ( τ − τ ′ ) are the normal, and G ↓↑ j + δ,j ( τ − τ ′ )and G ↑↓ j,j + δ ( τ − τ ′ ) the anomalous Matsubara Greenfunctions defined by standard relations G ↑↑ j,j ′ ( τ − τ ′ ) = − D T τ ˆ c j ↑ ( τ )ˆ c + j ′ ↑ ( τ ′ ) E ,G ↓↓ j,j ′ ( τ − τ ′ ) = − D T τ ˆ c + j ↓ ( τ )ˆ c j ′ ↓ ( τ ′ ) E ,G ↓↑ j,j ′ ( τ − τ ′ ) = − D T τ ˆ c + j ↓ ( τ )ˆ c + j ′ ↑ ( τ ′ ) E ,G ↑↓ j,j ′ ( τ − τ ′ ) = − (cid:10) T τ ˆ c j ↑ ( τ )ˆ c j ′ ↓ ( τ ′ ) (cid:11) . (5)The fluctuating potential ∆ ˆ U introduced in (4) toaccount for thermal OP fluctuations determines thescattering matrix ( t -matrix) in the Dyson site equation G jδ ( iω n ) == G AVjδ ( iω n ) + G AVjδ ( iω n )∆ U jδ G jδ ( iω n ) == G AVjδ ( iω n ) + G AVjδ ( iω n ) T jδ ( iω n ) G AVjδ ( iω n ) , (6)where G AVjδ ( iω n ) is the Fourier transform of theMatsubara Green function with averaged OP; T jδ ( iω n )is the Fourier transform of the electron pair scatteringmatrix; ω n = (2 n +1) πT are the Matsubara frequenciesfor Fermi particles. Thus, when fluctuations aretaken into account in the considered model (1), theproblem of off-diagonal disorder arises in disorderedsystems. In this paper, this problem is solvedwithin the framework of the two-site approximationof the average t -matrix, which significantly reducesthe computational cost in comparison with the self-consistent approximation of the coherent potential.In the average t -matrix approximation, the effectivemedium that preserves the full symmetry of the systemunder consideration and consists of electron pairs witheffective parameters is determined by the self-energyin the Dyson site equation for the effective average F jδ ( iω n ) = h G jδ ( iω n ) i Matsubara Green function F jδ ( iω n ) == G AVjδ ( iω n ) + G AVjδ ( iω n )Σ jδ ( iω n ) F jδ ( iω n ) , (7)where F jδ ( iω n ) is the Fourier transform of the effectiveMatsubara Green function; Σ jδ ( iω n ) is the Fourier transform of the self-energy. Then from the matrixequations (6) and (7) the self-energy Σ jδ ( iω n ) isexpressed in terms of the average t -matrixΣ jδ ( iω n ) == (cid:2) h T jδ ( iω n ) i G AVjδ ( iω n ) (cid:3) − h T jδ ( iω n ) i . (8)The transition to the quasi-momentum representationoccurs as a result of Fourier transformation of theNambu site matrices. In this representation, theHamiltonian of the system considered with averagedOP ˆ H AV has the following formˆ H AV (∆ , α ) = 1 N X k ˆ c + k H AV ( k )ˆ c k , H AV ( k ) = (cid:20) H ↑↑ AV ( k ) H ↑↓ AV ( k ) H ↓↑ AV ( k ) H ↓↓ AV ( k ) (cid:21) , H ↑↑ AV ( k ) = ε k − µ, H ↓↓ AV ( k ) = − ε k + µ H ↑↓ AV ( k ) = − V ∆ V k ( α ) , H ↓↑ AV ( k ) = (cid:16) H ↑↓ AV ( k ) (cid:17) ∗ ,ε k = − t [cos k + cos k + cos ( k − k )] ,V k ( α ) = cos k + exp( iα ) cos k ++ exp( − iα ) cos ( k − k ) , (9)where ˆ c + k , ˆ c k are the Nambu matrices in quasi-momentum representation; ε k is the dispersion lawof the electron energy on the triangular lattice withjumps within the first coordination sphere; V k ( α ) is thedispersion law of superconducting OP with symmetrygiven by the phase value α . Thus, the Hamiltonian ofthe effective medium is determined by the expressionsˆ H eff ( iω n ) = ˆ H AV + ˆΣ( iω n ) , ˆΣ( iω n ) = 1 N X k ˆ c + k Σ k ( E )ˆ c k , Σ k ( iω n ) = (cid:20) Σ ↑ ( E ) Σ ↑↓ k ( iω n ) − H ↑↓ AV ( k )Σ ↓↑ k ( iω n ) − H ↓↑ AV ( k ) Σ ↓ ( iω n ) (cid:21) , Σ ↑↓ k ( iω n ) = 2Σ ↑↓ ( iω n ) V k ( α ) , Σ ↓↑ k ( iω n ) = 2Σ ↑↓ ( iω n ) V ∗ k ( α ) , (10)where Σ k ( iω n ) is the self-energy (8) in the quasi-momentum representation. To take into account onlycontributions with the considered symmetry types tothe OP, it is necessary that the coefficient Σ ↑↓ ( iω n )in the anomalous self-energy (ASE) (10), definingthe effective OP, be a real energy function, whereasthe normal self-energies (NSE) Σ ↑ ( iω n ) and Σ ↓ ( iω n )are in general complex functions. The effectiveMatsubara Green function (7) in the quasi-momentumrepresentation is defined through the effective medium hermal fluctuations in superconducting phases with chiral d + id H eff ( iω n ) (10) F k ( iω n ) = 1 iω n − H eff ( k ) == (cid:20) F ↑ k ( iω n ) F ↑↓ k ( iω n ) F ↓↑ k ( iω n ) F ↓ k ( iω n ) (cid:21) ,F ↑ ( ↓ ) k ( iω n ) = iω n ± ε k ∓ µ − Σ ↓ ( ↑ ) ( iω n ) (cid:0) iω n − E + k )( iω n − E − k (cid:1) ,F ↑↓ ( ↓↑ ) k ( iω n ) = Σ ↑↓ ( ↓↑ ) k ( iω n ) (cid:0) iω n − E + k )( iω n − E − k (cid:1) ,E ± k = (cid:2) Σ ↑ ( iω n ) + Σ ↓ ( iω n ) (cid:3) / ±± h(cid:0) ε k − µ + (cid:2) Σ ↑ ( iω n ) + Σ ↓ ( iω n ) (cid:3) / (cid:1) ++Σ ↑ ( iω n )Σ ↓ ( iω n )+ | Σ ↑↓ k ( iω n ) | i / . (11)To determine the self-energy from the Dyson equa-tions (6) and (7), the matrix elements of the Mat-subara Green functions are required in the represen-tation of the Nambu site matrices. Their explicit ex-pressions can be obtained using the symmetry proper-ties of the considered system under Fourier transfor-mation. First of all, this is the property of the in-version symmetry of the dispersion laws of the elec-tron energy and superconducting OP (9) F − k ( iω n ) = F k ( iω n ). In addition, in the triangular lattice with s - and d + id -symmetry, the off-diagonal matrix ele-ments of the effective Matsubara Green function (11)in the quasi-momentum representation have the sym-metry property F ↑↓ ( ↓↑ ) k ,k ( iω n ) = exp ( ± iα ) F ↓↑ ( ↑↓ ) k ,k ( iω n ), F ↑↓ ( ↓↑ ) k ,k ( iω n ) = exp ( ± iα ) F ↓↑ ( ↑↓ ) k − k ,k ( iω n ) and F ↑↓ ( ↓↑ ) k ,k ( iω n ) =exp ( ∓ iα ) F ↓↑ ( ↑↓ ) k ,k − k ( iω n ). As a result, the matrix el-ements in the site representation obey the relations F ↑↓ j,j ± δ ( iω n ) = exp ( iα δ ) F ↑↓ j,j ± a ( iω n ), F ↓↑ j ± δ,j ( iω n ) =exp ( − iα δ ) F ↓↑ j ± a ,j ( iω n ). It is easy to verify thatthis is also true for the matrix elements of theMatsubara Green functions of the system consid-ered with averaged OP and off-diagonal matrix el-ements: ∆ U ↑↓ j,j ± δ = exp ( iα δ )∆ U ↑↓ j,j ± a , ∆ U ↓↑ j ± δ,j =exp ( − iα δ )∆ U ↓↑ j ± a ,j and Σ ↑↓ j,j ± δ = exp ( iα δ )Σ ↑↓ j,j ± a ,Σ ↓↑ j ± δ,j = exp ( − iα δ )Σ ↓↑ j ± a ,j . Therefore, the explicit ex-pressions for the matrix elements of the Green functionin the representation of Nambu site matrices G j,δ ( iω n ),determined from the Dyson site equation (6), also havethis property G ↑↓ j,j ± δ ( iω n ) = exp ( iα δ ) G ↑↓ j,j ± a ( iω n ), G ↓↑ j ± δ,j ( iω n ) = exp ( − iα δ ) G ↓↑ j ± a ,j ( iω n ). This propertyallows the use of one self-consistent equation to de-termine the superconducting OP amplitude ∆( φ ), be-cause it is not any more dependent on the nearest neighbors:∆ ( φ ) − ∆( φ ) K ( φ )2 − βV = 0 ,K ( φ ) = 1 β X n exp ( iφ ) G ↓↑ j + a ,j ( iω n )++ 1 β X n exp ( − iφ ) G ↑↓ j,j + a ( iω n ) . (12)In the general case, for example, with an anisotropichopping integral or interaction, to determine thesuperconducting OP amplitudes, it is necessary tosolve a system of self-consistent equations.
4. Results
The calculations have been performed for the intersiteinteraction parameter V = t . This is due to thefact that the value V ≃ t was used to analyze thetopological properties of superconducting cobaltites[33, 7]. In addition, among other values, V = t was chosen when calculating the phase diagramsof superconducting states for the considered modelon a square lattice without taking into account theOP fluctuations [34]. The results of calculating thedependence of the amplitude of the averaged OP onthe charge carrier concentration at temperature T =0.0004t are shown in Fig. 1. It is seen that thelargest part of the phase diagram is occupied by asuperconducting region with chiral d + id symmetry,on both sides of which superconducting states withextended s symmetry are realized. The presenceof contiguous superconducting regions with differentsymmetry suggests a first-order phase transition andphase separation between them, which contradicts theproperties of superconducting states. For the modelon a square lattice, this contradiction can be avoidedby including in consideration an intermediate phasewith s + id -symmetry which provides a minimumof free energy and second-order phase transitions[34]. For the model on a triangular lattice, thesuperconducting states substantially differ from thosein the model on a square lattice. The point is thatthe chiral d + id superconducting phase in the modelon triangular lattice is topologically nontrivial (hasa finite value of the topological parameter Q , see[7]). Then, according to the results of this work, atsufficiently low temperatures, when there is no normal-state region between the superconducting regions with s - and d + id -symmetry, a change in the charge carrierconcentration should result in the transition from atopologically trivial state with extended s-symmetry toa superconducting state with a topologically nontrivialchiral d + id -symmetry. Besides, it is known [8, 7]that under changes in the charge carrier concentrationor other model parameters, the superconducting statewith a chiral d + id symmetry type admits a quantum hermal fluctuations in superconducting phases with chiral d + id T = 0 is impossible, sinceany arbitrarily small attractive interaction makes thesystem unstable with respect to the Cooper pairing[35]. Therefore, the appearance in Fig. 1. of a wideregion in which superconducting states are destroyedby thermal fluctuations even at such a low temperature T = 0 . t points to a small phase stiffness ofsuch states in this concentration range. The resultsof calculating the temperature dependences of theaveraged OP amplitude h ∆ i in superconducting regionswith d + id - (n ≃ s -symmetry (n ≃ ≃ s -symmetry are muchmore sensitive to thermal fluctuations than those with d + id symmetry. Taking into account fluctuations for s d+id s < ∆ > /t nV=t, T=0.0004t, Figure 1.
Dependence of the averaged OP amplitude on thecharge carrier concentration at temperature T = 0.0004t. the states with s -symmetry reduces the temperatureof transition to the superconducting state with respectto the transition temperature calculated in the HFapproximation by 11.2 times for n ≃ ≃ d + id -symmetryonly by 1.7 times. Note also a significant variationwith temperature of the behavior of the averaged OPamplitude. Another important parameter, which isoften measured experimentally, is the ratio of thegap value at low temperatures to the superconductingtransition temperature. Calculations of the ratioof the energy gap to the superconducting transitiontemperature in the HF approximation (without takingaccount of fluctuations) give the values ∆ /T c ≃ . < ∆ > /t T/td+id, V=t, n ≈ HF Figure 2.
Temperature dependences of the amplitude of theaveraged OP with d + id symmetry in different approximations. < ∆ > /t T/tS, V=t, n ≈ ≈ HFHF
Figure 3.
Temperature dependence of the amplitude of theaveraged OP with extended s-symmetry. d + id - , and ∆ /T c ≃ . ≃ /T c ≃ . ≃ /T c ≃ . d + id - , and ∆ /T c ≃ . ≃ /T c ≃ . ≃ = 2 V h ∆ i ). A larger valueof the ratio ∆ /T c as compared to the BCS theory(by 5 times and more), is observed experimentallyin layered chloronitride compounds with transitionmetals M N aCl ( M = Zr, Hf ) [36] which are highlyanisotropic effectively quasi-two-dimensional systemswith a triangular lattice in each of the backbone planes.It should be emphasized that such a large ratio cannotbe a consequence of the frustration of the triangularlattice, since the frustration effects diminish also thegap magnitude at low temperatures. The ∆ /T c ratio increases by about ≃ d + id -symmetry if the self-consistent method of coherentpotential is used in the solution instead of the t -matrix hermal fluctuations in superconducting phases with chiral d + id a P ( φ ) φ/π - 2/3 V=t, µ =2t (n ≈ T=0.0004t T=0.04t T=0.08t 0 0.05 0.1 0.15 0.2 0.25 0.3-1 -0.5 0 0.5 1 b ∆ ( φ ) /t φ/π - 2/3 V=t, µ =2t (n ≈ T=0.0004t T=0.04t T=0.08t
Figure 4. (a) The probabilities of the distribution of phasefluctuations φ of the superconducting OP with d + id symmetry,and (b) the dependence of the amplitude of the OP with d + id symmetry on φ . n ( E ) Ε/ tV=t, µ =2t, (n ≈ T=0.0004t T=0.04t T=0.08t
Figure 5.
Density of electron states calculated for thesuperconducting state with d + id symmetry. approximation [14]. In addition, we did not take intoaccount the possible decrease in the effective attractionof electrons with increasing temperature [37], which a Σ ↑↓ ( Ε ) /t Ε/ t T=0.0004t 0.028 0.029 0.03 0.031 0.032 0.033 0.034 0.035 0.036 0.037 0.038 -0.4 -0.2 0 0.2 0.4 b Σ ↑↓ ( Ε ) /t Ε/ t T=0.04t
Figure 6.
Anomalous self-energy of the one-electron Greenfunction, calculated for the superconducting state with d + id symmetry for a) T=0.0004t and b) T=0.04t. also leads to an increase of this ratio. Thus, thelarge experimental values of the ratio ∆ /T c , in ouropinion, are a consequence of thermal fluctuations ofthe superconducting OP, and are hardly determinedby the details of the electron pairing mechanism. Theresults of a consistent calculation of the probabilitiesof the distribution of phase fluctuations and the mostprobable fluctuations of the amplitude, calculated inthe saddle point approximation for three differenttemperatures, are shown in Fig. 4a, and Fig. 4b.It is seen that the most probable phase value φ =2 π/ d + id -symmetry. At thesame time, considering the variation range of phasefluctuations [ − π/ , π/
3] at finite temperatures, thereis always a nonzero probability of finding the system ina superconducting state with the d − id -symmetry (i.e.,with the opposite chirality and φ = 4 π/ hermal fluctuations in superconducting phases with chiral d + id Bi Sr CaCu O and Bi Sr CuO .The obtained experimental dependence of the ASE[39] shows qualitative agreement with the theoreticallycalculated dependences presented in Fig. 6.
5. Conclusion
For a quasi-two-dimensional system with a triangularlattice, the effect of thermal fluctuations of thesuperconducting order parameter on the behaviorof the superconducting gap, one-electron density ofstates, and phase transition temperature in singletsuperconducting phases with extended s- and chiral d + id -symmetry is studied. It is shown that thephase diagram of superconducting states, constructedin the whole range of variation of the chargecarrier concentration 0 < n <
2, consists of twosuperconducting regions with a topologically trivialextended s - symmetry and a superconducting regionwith a topologically nontrivial chiral d + id - symmetrybetween them. At sufficiently low temperatures, there is no normal-state region between the superconductingregions with s - and d + id - symmetry. In this case, as aresult of variations in the charge carrier concentration,a phase transition should occur in the system froma topologically trivial superconducting state with anextended s - symmetry to a superconducting state witha topologically nontrivial chiral d + id - symmetry. Theelucidation of the type of this transition calls for furtherinvestigation. Taking into account the OP fluctuationssignificantly lowers the temperature of transition tothe superconducting state, as a consequence, the ratioof the energy gap to the superconducting transitiontemperature increases several times. Then it turnsout that superconducting states with extended s -symmetry have a lower phase stiffness and, therefore,are more sensitive to thermal fluctuations than thestates with chiral d + id symmetry. The calculatedASE values are semi-quantitatively consistent with theASE values expressed from ARPES experimental datausing machine learning techniques. Acknowledgments
This study was supported by the financing programAAAA-A16-116021010082-8.A G Groshev https://orcid.org/0000-0002-7389-5023A K Arzhnikov https://orcid.org/0000-0002-8365-1962
References [1] Yamashita M, Nakata N, Kasahara Y, Sasaki T, YoneyamaN, Kobayashi N, Fujimoto S, Shibauchi T, Matsuda Y2009 Nature Phys. No. 8 789[13] Emery V J and Kivelson S A 1995 Nature
No. 2 247[15] Curty P and Beck H 2000 Phys. Rev. Lett hermal fluctuations in superconducting phases with chiral d + id [18] Curty P and Beck H 2003 Phys. Rev. Lett (11) 8190(R)[21] Schrieffer J R, Wen G, and Zhang S 1989 Phys. Rev B (16) 11663[22] Izyumov Y A 1999 Physics-Uspekhi (3) 215[23] Scalapino D J 2012 Rev. Mod. Phys.
383 Coleman S1973 Commun. Math. Phys.41