Effective pairing theory for strongly correlated d-wave superconductors
aa r X i v : . [ c ond - m a t . s up r- c on ] A ug Effective pairing theory for strongly correlated d-wave superconductors
Debmalya Chakraborty,
Nitin Kaushal,
1, 3 and Amit Ghosal Indian Institute of Science Education and Research-Kolkata, Mohanpur, India-741246 Institut de Physique Th´eorique, CEA, Universit´e Paris-Saclay, Saclay, France ∗ Department of Physics and Astronomy, The University of Tennessee, Knoxville, TN 37996, USA
Motivated by recent proposals of correlation induced insensitivity of d-wave superconductors to impurities,we develop a simple pairing theory for these systems for up to a moderate strength of disorder. Our descrip-tion implements the key ideas of Anderson, originally proposed for disordered s-wave superconductors, but inaddition takes care of the inherent strong electronic repulsion in these compounds, as well as disorder inducedinhomogeneities. We first obtain the self-consistent one-particle states, that capture the effects of disorder ex-actly, and strong correlations using Gutzwiller approximation. These ‘normal states’, representing the interplayof strong correlations and disorder, when coupled through pairing attractions following the path of Bardeen-Cooper-Schrieffer (BCS), produce results nearly identical to those from a more sophisticated Gutzwiller aug-mented Bogoliubov-de Gennes analysis.
I. INTRODUCTION
One of the outstanding puzzles of the disordered supercon-ductors is the insensitivity of the high temperature cupratesuperconductors to weak and moderate disorder.
In con-trast, the conventional wisdom developed along the lines ofAbrikosov-Gorkov (AG) theory predicts an extreme sensitiv-ity of these materials to impurities. The original idea, based onperturbative expansions, had been refined subsequently, lead-ing to self-consistent T-matrix calculations, but the broadsensitivity of these materials to disorder survived.The effects of dopant disorder, however, on cuprates haveremained rather benign. The inhomogeneities in local dop-ing of the charge carrier induce local variations in the gapmap seen from the scanning tunneling microscopy measure-ments. Surprisingly, these nanoscale inhomogeneities donot affect the low energy density of states – as if, the d-wavenodes are “quantum protected”. The superfluid density and T c undergo only a modest reductions in spite of the d-wave nature of the anisotropic order parameter. Otherunconventional superconductors, e.g. organics and pnic-tides, which belong to the intermediate coupling category,also feature anomalies. On the other hand, addition of strongsubstitutional impurities in these materials weakens super-conducting correlations significantly.A number of non-BCS features of high T c cuprate super-conductors make them deviate from a favorable play-ground of AG-type theories. These include the presenceof strong repulsive correlations between the charge carriers,short coherence lengths, ξ , non-monotonic dependence of T c on the doping level, small superfluid density etc. In addition,neglect of the spatial fluctuations in the pairing amplitude ina disordered environment in AG formalism calls for a carefulmicroscopic relook into the role of impurities on these sys-tems.Inclusion of the spatial inhomogeneities of the pairing am-plitude for short-coherence length d-wave BCS superconduc-tors (dSC) within a Bogoliubov-de Gennes (BdG) formal-ism indeed enhances the robustness of dSC to impurities. Recent advances of incorporating the effects of strong elec-tronic repulsions on top of the inhomogeneous background resulted in a Gutzwiller-renormalized theory (referred toas GIMT). These analyses make these superconductors amaz-ingly immune to disorder, up to its strength as large as thebandwidth ! Such remarkable robustness of the supercon-ducting correlations naturally implies a similar robust-ness of T c , at least within the mean-field description of therenormalized theory. This raises an intrinsic question: DoesAnderson’s theorem, or an equivalent, apply even for thesestrongly correlated d-wave superconductors?We address this question by exploring the fate of a simple-minded pairing theory following Anderson’s original idea of‘pairing of exact eigenstates’. But we upgrade it now to in-clude the inherent strong correlations in these systems, as wellas the exact treatment of disorder induced inhomogeneitiesin our numerical calculations. It is well established that the‘pairing of exact eigenstates’ leads to Anderson’s theorem fors-wave superconductors (sSC) for weak disorder. However,the same ideas had been successfully extended to incorporatedetails of inhomogeneities and localization effects in its nu-merical implementation (for sSC). Here, we expand it fur-ther by implementing similar concepts for strongly correlatedsuperconductors with an anisotropic order parameter.At the outset, we emphasize that our developments per-tain to dSC with impurities up to a moderate strength andexclude strong substitutional scatterers. Studies of dSC withstrong substitutional impurities, in the limit of unitary scat-terers are also available.
There are subtleties in han-dling strong correlations and also strong impurities in a mean-field formalism, and the results depend crucially on their rel-ative strengths.In this article, we demonstrate that the complexity ofstrongly correlated disordered superconductors, such ascuprates, can be understood in terms of a simple pairing the-ory. However, the true potential of our developments lies inidentifying the underlying effective one-particle states, whichwe termed as ‘normal states’ (NS). It is these states whichparticipate in Cooper-pairing in these materials following thestandard BCS path. We posit that the properties of the truenormal state dictate the response of anisotropic superconduc-tors to impurities, providing a deeper insight to the physics ofstrongly correlated unconventional superconductors.
II. MODEL AND METHODSA. Anderson’s prescription
The original proposition of the pairing of exact eigenstates,that leads to Anderson’s theorem, relies on two important con-ceptual ideas: (a) The problem of non-interacting electrons indisorder potential is solved at the first stage to generate its‘exact eigenstates’. BCS type attractive pairing interactionsthen couple specific pairs of these states producing Cooper-pairs at the second stage and phase coherence of these pairsproduce superconductivity in the disordered background. Weemphasize that such decoupling of these two stages in theabove mechanism necessarily demands that the pairing inter-actions has no role in determining the exact eigenstates. (b)The specific states participating in Cooper-pairing (at the sec-ond stage) are the time reversed exact eigenstates derived inthe first stage. This is simply motivated by the BCS theory,which Anderson’s pairing method must reduce to, in the cleanlimit.Each of these two points are important for establishing An-derson’s theorem for disordered sSC. Can they work for thestrongly correlated d-wave superconductors as well? In orderto explore this question we first set up the formalism below.
B. Normal states: the equivalent of “exact eigenstates” forstrongly correlated dSC
In the limit when the electron-electron repulsion is strong, itis believed that the phases of the strongly correlated cupratescan be well described by the “ t − J ” model : H t − J = X ijσ t ij (˜ c † iσ ˜ c jσ + h . c . ) + X ij J ij (cid:16) ˜ S i . ˜ S j − ˜ n i ˜ n j (cid:17) (1)The first term indicates hopping of electrons on a D squarelattice of N sites. Here, J is the exchange interaction, as-sumed to arise from a Hubbard-type onsite repulsion U viaSchrieffer-Wolff transformation, yielding J ij = 4 t ij /U .We take t ij = − t , when i and j are nearest neighbors, de-noted as h ij i , and t ij = t ′ , when i and j are next-nearestneighbors, with the notation of hh ij ii . We choose t ij = 0 forall other pairs of i and j . Correspondingly, we have J ij = J for h ij i , J ij = J ′ for hh ij ii . Here, ˜ c iσ = c iσ (1 − n i ¯ σ ) is the electron annihilation operator in the ‘projected Hilbertspace’ that prohibits double-occupancy at any site i , and sim-ilarly for the electron creation operator. We introduce disor-der by redefining H t − J to H t − J + P iσ ( V i − µ ) n iσ , where µ is the chemical potential that fixes the average density ofelectrons, ρ = N − P iσ h n iσ i , in the system to a desiredvalue. Such a simple re-definition of the Hamiltonian uponinclusion of disorder, however, would not work for strongdisorder ( V ≥ t ) and a modified treatment of Schrieffer-Wolff transformation is necessary. Here, we use themodel of Box-disorder , where V i ’s on all sites i of the lat-tice are drawn from a uniform ‘box’ distribution, such that, V i ∈ [ − V / , V / uniformly, thus defining V as the strengthof disorder.We studied the Hamiltonian H t − J at zero temperature ( T =0 ), upon including disorder, over a wide range of parameters.Here we present results for U = 12 t and t ′ = t/ , andwe express all energies in the units of t . We choose the av-erage density of electrons, ρ = 0 . , which coincides with theoptimal doping. It is the optimal doping where dSC is thestrongest in a typical phase diagram of cuprates, in addition tobeing reasonably free from the complex effects of other com-peting orders. While the phenomenology of competingorders attract interesting and active research in the underdopedregime, our goal here is to focus only on the interplay ofimpurities and strongly correlated dSC, and hence we choosethe optimal doping for our study. We carry out our numericalsimulations typically on a × lattice, and we collect statis-tics on our results for each disorder strength V from − independent realizations of disorder.The Hilbert space restriction, that prohibits any double oc-cupancy in the limit of strong correlations, are reflected inthe transformation: c iσ → ˜ c iσ , and makes it difficult to han-dle these creation and annihilation operators in the projectedspace. To make progress, we use Gutzwiller approximation(GA) to implement the phase space restrictions. GAamounts to renormalizing the parameters t and J of H t − J locally by density-dependent factors, such that, they mimicthe projection due to strong repulsions. For example, the re-stricted hopping reduces t ij due to double-occupancy prohibi-tion, whereas, the effective J ij increases because of enhancedoverall single-occupancy. The real advantage of GA lies in thefact that it turns the problem into an effective weak-couplingone redefined in the unprojected Hilbert space, which is nowamenable to simple mean field treatments. It has been shownthat GA is capable of describing non-BCS and non-trivial fea-tures of cuprate superconductors in the clean limit.Upon carrying out the inhomogeneous Hartree-Fock meanfield decoupling of the Gutzwiller renormalized H t − J suchthat no symmetry of H t − J is broken , we arrive at the following‘normal state’ Hamiltonian: H NS = X i,δ,σ { t iδ g ti,i + δ − W FS iδ } c † iσ c i + δσ + X i, ˜ δ,σ { t i ˜ δ g ti,i +˜ δ } c † iσ c i +˜ δσ + X i,σ ( V i − µ + µ HS i ) n iσ (2)Here, we have written the Hamiltonian on bonds connectingsites i and j , where j = i + δ , with δ = ± x or ± y , ˜ δ = ± ( x ± y ) . As the name suggests, we refer to the eigenstatesof H NS in Eq. (2) as the normal states, NS GIMT (here
GIMT in the subscript of normal states, NS, stands for Gutzwiller-augmented inhomogeneous Hartree-Fock mean-field theory).It is crucial to include the effect of strong correlations in H NS following the above construction, even though the final one-particle Hamiltonian without broken symmetries is similar instructure to the disordered tight binding model (or Andersonmodel of disorder). Yet, these normal states distinguish them-selves from the ‘exact eigenstates’ (eigenstates of the Ander-son model of disorder) in accounting for the strong correlationeffects through Gutzwiller factors, as well as the Hartree- andFock-shifts. These considerations naturally make the solutionof H NS already a self-consistent problem. We also empha-size that these normal states are defined at T = 0 , and arenot to be confused with the common notion of the high tem-perature normal state of the material in which thermal fluctu-ations destroy superconductivity. The Fock-shift ( W FS iδ ) andthe Hartree-shift ( µ HS i ) terms in Eq. 2 are given by, W F Siδ = J ( g xyi,i + δ − ! τ δi ) (3) µ HSi = − t X δ,σ ( ∂g ti,i + δ ∂ρ i τ δi ) + 4 t ′ X ˜ δ,σ ( ∂g ti,i +˜ δ ∂ρ i τ ˜ δi ) − J X δ,σ ∂g xyi,i + δ ∂ρ i (cid:16) τ δi (cid:17) (4)where, ρ i = P σ h n iσ i and τ ij ≡ h c † i ↓ c j ↓ i ≡ h c † i ↑ c j ↑ i .Here, hi denotes the expectation value in the unprojectedspace. The Gutzwiller factors in Eq. (3) and (4) are givenin terms of the local density: g tij = s − ρ i )(1 − ρ j )(2 − ρ i )(2 − ρ j ) , g xyij = 4(2 − ρ i )(2 − ρ j ) (5)As mentioned, the above construction of the NS GIMT ex-cludes any broken symmetry order parameters, e.g. mag-netism, charge density wave etc. However, unbroken sym-metry is not a fundamental requirement of NS GIMT . In fact,we need to include them in H NS (except, of course, any su-perconducting order through Bogoliubov channels), when westudy the effects of such additional orders competing with su-perconductivity.Considering the unitary transformation to diagonalize H NS in the { α } -basis: c iσ = N X α =1 ψ αi c ασ , (6)we obtain H NS = P α,σ ξ α c † ασ c ασ . Here, the self-consistent { ψ αi } are the eigenvectors of H NS , and they constitute our“normal states”, i.e. the NS GIMT . C. Pairing of normal States (PNS)
To study the superconducting properties of H t − J in Eq. (1),we now introduce the pairing term, H P = 12 X h ij i ∆ ij ( c † i ↑ c † j ↓ − c † i ↓ c † j ↑ ) + h . c . (7)in addition to H NS , where, ∆ ij = − J (cid:16) g xyij + 14 (cid:17) h c i ↑ c j ↓ i − h c i ↓ c j ↑ i . (8) The pairing part of the Hamiltonian in Eq. (7) can be thoughtto arise from a mean-field decoupling of the original H t − J inthe Bogoliubov channel. Note that, the form of H P ensuresthat we have chosen the singlet pairing channel on the links.Writing H P in the { α } -basis, we have, H P = 12 X αβ ∆ αβ { c † α ↑ c † β ↓ − c † α ↓ c † β ↑ } + h . c . (9)where, ∆ αβ = X h ij i ∆ ij ( ψ αi ) ∗ ( ψ βj ) ∗ , (10)leaving the total Hamiltonian as: H total = X α,σ ( ξ α − µ p ) c † ασ c ασ + 12 X αβ (cid:16) ∆ αβ { c † α ↑ c † β ↓ − c † α ↓ c † β ↑ } + h . c . (cid:17) (11)Here, we introduced µ p to fix the final average density (afterpairing) to the desired value ρ = 0 . . Note that, the µ in H NS fixes the density to the same desired value, but only in thenormal state. Pairing at the second stage (after inclusion of H P ) can deviate ρ from this value. We use µ p to tune it backto the chosen value. We also note that there is no restriction,in principle, on α , β in the definition of ∆ αβ appearing inEq. (10), though we will see in Sec. (III D) that the dominantcontribution comes from those α , β for which ξ α ≈ ξ β . D. Self-consistent pairing amplitude
Evidently, H total in Eq. (11) carries the BCS structure andwe diagonalize it using a modified Bogoliubov transforma-tion: c pσ = N X n =1 (cid:16) u p,n γ nσ − σv ∗ p,n γ † n ¯ σ (cid:17) (12)where, γ † nσ ( γ nσ ) are fermionic quasiparticle creation (anni-hilation) operators.Starting with guess values of ∆ ij on all the N bonds wefirst obtain the N numbers of ∆ αβ using the normal stateeigenfunctions ψ αi ’s in Eq. (10). The eigenvalues and eigen-vectors of H total allows us to re-calculate ∆ ij and ρ i usingEq. (8) and the self consistency conditions: h c i ↑ c j ↓ i = N X p ,p =1 ψ p i ψ p j h c p ↑ c p ↓ i (13) ρ i = 2 N X p ,p =1 ( ψ p i ) ∗ ψ p i h c † p ↓ c p ↓ i . (14)We then iteratively update the guess values of ∆ ij and ρ i for the inputs in Eq. (11) in order to achieve the final self-consistency until the inputs and corresponding outputs in −0.3 −0.1 0 0.1 0.30123 V eff P ( V e ff ) GIMTNS
V=1.75 (e) −0.1 −0.05 0 0.05 0.101345 t eff P ( t e ff ) GIMTNS(f)
V=1.75
FIG. 1: (a-d) Spatial density map of V eff and t eff from NS and GIMTshows similar spatial anticorrelation between V eff and t eff . They alsohighlight the spatial correlation in V eff . Comparison of distributions P ( V eff ) in (e) and P ( t eff ) in (f) for NS and for the full GIMT out-puts at V = 1 . . The distributions match rather well in the twocalculations validating the basis of PNS calculations. We subtractedthe homogeneous components of t eff ( t eff ( V = 0) = 0 . ) and V eff ( V eff ( V = 0) = 1 . ), arising from the Fock- and Hartree-shiftrespectively. The resulting distributions in (e) and (f) feature zeromean – this is broadly true for all V . Eq. (13) and (14) match within tolerance. For accelerating theconvergence, we used combinations of linear, Broyden andmodified Broyden schemes of mixing of the input and out-put at every iteration. III. RESULTS
We will discuss in this section our findings from the pair-ing of normal states (PNS) and compare them with GIMTfindings. Here, GIMT refers to the full BdG calculation aug-mented with Gutzwiller renormalization. However, it is trulyilluminating to focus our attention first on the distinguishingfeatures of NS GIMT that separate them from their uncorre-lated counterparts – the “exact eigenstates” of the Anderson’smodel of disorder.
FIG. 2: A schematic evolution of the inhomogeneity in space thatleads to the renormalization of V eff ( i ) and spatial anti-correlationbetween V eff ( i ) and t eff ( i ) , upon including electronic repulsionsthrough Gutzwiller approximation (GA). Consider in (a) the site i having a high hill of disorder potential (also assumed that V i ± δ = 0 ),that would normally yield a low ρ i compared to ρ i ± δ ≈ ρ , as shownin (b), rarely populating the site i . However, GA insures that t eff onbonds connecting i to its neighbors is enhanced, according to Eq. (5),increasing charge flow to this site. This in turn reduces the dip in ρ i as seen in (c), so that the corresponding V eff ( i ) , that would have nor-mally produced the ρ i in (c), is far weaker than its bare value, shownin (a). Exactly similar arguments would yield a similar weakening ofdeep potential well by strong correlations. A. Structure of the normal states
For the convenience of our discussions below, it is useful tocast the normal state Hamiltonian H NS in the following form: H NS = − X i,δ,σ t eff ( i, δ ) c † i,σ c i + δ,σ + X i,σ V eff ( i ) n i,σ , (15)to emphasize H NS as a tight binding model, but with effec-tive disorder both on the links ( t eff ), as well as on the sites( V eff ). However, these disorder terms now contain order pa-rameters, as seen from Eq. (2), (3), and (4), and hence, mustbe evaluated self-consistently, as mentioned already. We findthem to develop spatially correlated structures, and are illus-trated in Fig. (1 a-d). For a justified comparison between thespatial structures of V eff and t eff , we transformed the bondvariable t eff ( i, δ ) to a site variable using relation: t eff ( i ) = P δ t eff ( i, δ ) . Spatial associations are found, firstly, in theprofile of V eff ( i ) itself, showing conglomeration of regionswith large and small V eff , but more importantly, through theexplicit anti-correlation of regions of V eff and t eff in space.We also compare the distributions P ( V eff ) and P ( t eff ) for V = 1 . from the NS GIMT and GIMT results in Fig. (1e,f), using statistics over realizations of disorder. Such afavorable comparison of NS GIMT outputs of t eff and V eff withthose from GIMT validates the conceptual basis of the PNS −0.200.20.40.6 −1.5 −1 0 1 1.5−0.200.2 V(i) V e ff ( i ) (b)(a) Fit Equation:Fit Equation:y=0.195x+0.04y=0.189x+0.1 V=1.0V=2.5
FIG. 3: Scatter plots of V eff ( i ) against the bare potential V ( i ) for: (a) V = 1 . , and (b) V = 2 . . The red lines are the best fit to the data.The slope of the solid line in both panels are close to the averagedoping ( δ = 0 . ). For V = 2 . , the data tend to deviate from the fitfor larger | V ( i ) | , signalling higher order effects. formalism. The role of strong electronic repulsions on thedisordered normal states has a simple and intuitive rationale,as we describe below. Random impurity potential tends togenerate charge inhomogeneities in space, whereas, repulsiveinteractions smear out such heterogeneities, trying to restoreits homogeneous distribution. The key ingredient of NS GIMT ,that distinguishes it from the ‘exact eigenstates’, lies in its im-purity renormalization – a footprint of electronic repulsion in NS GIMT . This is ascribed to the modification the hoppingamplitudes based on local density, which smear out chargeaccumulation near deep potential wells, and also partly popu-lating potential hills, as explained in Fig. (2). As a schematicdescription, we consider in Fig. (2), a site i having a high hillof local potential, and hence it ordinarily supports little den-sity of electrons there, compared to the average density on itsneighbors, assumed to have no disorder. This local chargeimbalance leads to an interesting feedback loop through g tij ,absent in the uncorrelated systems. The low electronic den-sity at i enhances g tij according to Eq. (5), which in turnenhances the charge fluctuations across site i , leading to anlarger effective ρ i than what would be its value in the absenceof the Gutzwiller factors. This leads to a much weaker effec-tive disorder to account for the enhanced ρ i . In addi-tion to impurity renormalization, the above argument shedslight on the spatial anti-correlations of V eff and t eff . Boththese features make NS GIMT distinct from the plain ‘exacteigenstates’. However, in the limit U → , the NS GIMT and ‘exact eigenstates’ would be identical. How strong issuch renormalization of disorder? In order to get a quantita-tive estimate of the impurity renormalization, we present thescatter plot of V eff against bare V in Fig (3) from our self- −0.1 −0.05 0 0.05 0.102040 R ∆ d P ( R ∆ d ) −0.1 −0.05 0 0.05 0.10204060 R ρ P ( R ρ ) V=0.5V=1.0V=2.0V=2.5 (a)(b)
FIG. 4: The distribution P ( R ∆ d ) in (a) and of P ( R ρ ) in (b) areshown for various disorder strengths. Sharply peaked nature of thesedistributions (with small variance) validates the PNS formalism. consistent NS-calculations (statistics collected over 10 real-izations of disorder). Our results show a simple linear trend: V eff ≈ δV for low V , with weak corrections for stronger V .Here, δ = (1 − ρ ) is the average doping. This low- V linearityis consistent with earlier findings from a single-impurity cal-culation. This is easily comprehended: Since t → g t t ∼ δt ,we must rescale V by the same factor for a justified compari-son, yielding V eff ∼ δV . For the cuprate superconductors, wetypically have δ ≤ . . The above considerations then implythat the Fermi’s golden rule estimate of the inverse scatteringtime of the electrons in the underlying NS GIMT is an order ofmagnitude smaller compared to the ‘usual’ exact eigenstates: τ − ∼ ˜ g (0) V ∼ δτ − , where ˜ g (0) is the density of statesat Fermi energy of NS GIMT . A similar dependence of τ − has also been been predicted recently from the T-matrix es-timation. We focus next on Cooper-pairing between thesestrongly correlated NS GIMT states.
B. Self-consistent order parameters
Inducing pairing through BCS-type attraction as describedin Sec, (II C), we find that the self-consistent PNS outputs ofthe spatial profiles of the pairing amplitude ∆ ij , local den-sity ρ i , or τ ij are nearly indistinguishable from the results ofGIMT calculations. In order to quantify the strength of PNSformalism, we find it easier to define the relative difference in ∆ O P / ∆ O P ( ) GIMTPNS (from NS
GIMT )IMTPNS (from NS
IMT ) FIG. 5: Evolution of ∆ OP is presented against V . The V -dependences of both the PNS and GIMT results show nearly identi-cal behaviour. The inset shows an expanded region of the main panelestablishing that the PNS findings match excellently with those fromGIMT within the error bars. The results for ∆ OP from IMT calcu-lations, shown by the magenta curve (forcing all Gutzwiller factorsto unity, and thereby neglecting strong electronic repulsions), deviatesignificantly from the PNS or GIMT results. However, it still com-plements the plain BdG results (red dashed line) exceedingly well. the PNS order parameters with respect to those from GIMT,in the following manner: R OP ( i ) = OP GIMT ( i ) − OP PNS ( i ) h OP i GIMT (16)where, OP represents either of ρ i , ∆ d ( i ) or τ ij . Here, hi denotes average over all sites and over configurations. Wedefine the d-wave superconducting order parameter on a siteas: ∆ d ( i ) = (∆ + xi − ∆ + yi + ∆ − xi − ∆ − yi ) . We emphasizehere that the PNS self-consistency produces for us the solu-tion of link variable ∆ ij among other things. This by itself isno confirmation of a d-wave anisotropy of the pairing ampli-tude. However, our choice of parameters in the Hamiltonian H t − J ensures that we have exclusively the d-wave ( d x − y )pairing amplitude in the clean limit. Introduction of disorderdoes generate other possibilities of bond pairing amplitude,e.g., ∆ xs , ∆ s xy , ∆ d xy . But their strengths remain negligi-bly small compared to the ∆ d component. GIMT calculationsalso confirm the same qualitative picture in this regard.We plot the normalized distribution of R ∆ d and R ρ fordifferent V in Fig. (4). These distributions, always peakedat zero, show only a weak broadening with V . Further,such smearing is essentially independent of V in the range . ≤ V ≤ . . The difference between PNS and GIMT re-mains only at about for all order parameters at V = 2 . ,emphasizing the accuracy of the proposed PNS method to de-scribe the strongly correlated dSC. C. Off-diagonal long range order
In order to illustrate the accuracy of the PNS results forphysical observables, we study the V -dependence of the su-perconducting off-diagonal long range order (ODLRO), de-fined as: ∆ = lim | i − j |→∞ F δ,δ ′ ( i − j ) (17)where, the pair-pair correlation function, F δ,δ ′ ( i − j ) = h B † iδ B jδ ′ i , and, B † iδ = ( c † i ↑ c † i + δ ↓ + c † i + δ ↑ c † i ↓ ) is the sin-glet Cooper-pair creation operator on the links connecting theneighboring sites at i and i + δ . Since F δ,δ ′ ( i − j ) can beinterpreted as simultaneous hopping of a singlet cooper-pairon a link, the Gutzwiller factor corresponding to this processbecomes g ti,j g ti + δ,j + δ ′ . We calculate F δ,δ ′ ( i − j ) using thetransformations Eq. (6) and (12). The evolution of ODLRO(normalized by its value ∆ (0)OP at V = 0 ) with V , as evaluatedfrom the PNS and GIMT calculations, are shown in Fig. (5).The main panel shows that the PNS results are nearly iden-tical with the GIMT findings (see the inset for an expandedview), ascertaining that PNS formalism serves as good a pur-pose as the GIMT method for handling the physics of strongcorrelations.An independent test for the effectiveness of the PNS for-malism comes from its comparison with a full BdG calcula-tion, when both neglects strong correlations (and will be re-ferred to as IMT, henceforth). Suppression of strong corre-lations, though unphysical for cuprates, can easily be imple-mented by setting all Gutzwiller factors to unity. In Fig. (5)we also compared ∆ OP ( V ) as obtained from pairing between NS IMT with those from corresponding plain BdG outcomes.The excellent match of the two formalisms even in the uncor-related domain strengthens PNS method as a natural descrip-tion of disordered superconductors. Note that the results differsignificantly by including and excluding Gutzwiller factors –irrespective of PNS or BdG methods (See also Sec. (III E)).
D. Pairing of limited states with close by energies
As discussed in Sec. (II C), the PNS method amounts topairing between all the eigenstates of H NS , making its numer-ical implementation computationally as demanding as that ofGIMT. However, technical gain can be insured by having topair only a limited number of normal states α and β that arenot too far from the Fermi energy, such that, ξ α ≈ ξ β . Suchan expectation is, of course, motivated by the structure of theBCS gap equation.In search of this simplification, we plot in Fig. (6 a,b), thefully self-consistent and disorder averaged profiles of | ∆ αβ | in the eigen-space of α and β . The near diagonal structuresof | ∆ αβ | implies that the states α , β which are far in ener-gies, have negligible contributions in ∆ αβ . Such a diagonalcharacter of ∆ αβ is well maintained for V ≤ .The diagonal nature is preserved when the same | ∆ αβ | isplotted against ξ α and ξ β (shown for V = 2 . in Fig. (6c)). FIG. 6: Intensity plot of | ∆ αβ | in the normal state eigen basis α - β for (a) V = 1 . , and (b) V = 2 . . We show | ∆ αβ | in a limited rangeof α, β (only the central part) for a better resolution. The presentedvalues of | ∆ αβ | are scaled by their maximum values for clarity ( . for V = 1 . and . for V = 2 . ). The near-diagonal nature ofthe pairing is evident for both V . The color scales are identical tothat in Fig. (1). (c) Density-plot of | ∆ αβ | against ξ α and ξ β acrossthe full (renormalized) bandwidth for V = 2 . . While the diagonalcharacter of | ∆ αβ | is evident, only negligible contribution to | ∆ αα | comes from the states near band edges. (d) Accuracy of PNS (withrespect to GIMT) is shown along y -axis, against the percentage ofstates paired (along x -axis). This accuracy, already impressive withabout 10% NS GIMT participating in pairing, becomes better as morestates included in Cooper-pairing.
We also note that not all normal states contribute to ‘diago-nal’ pairing, particularly those states lying close to band edgescontribute only negligibly to | ∆ αα | . Such contribution wouldhave been limited only to a narrow energy window, ± ~ ω D , insimple BCS theory ( ω D being the Debye frequency). In thepresent case of strongly correlated anisotropic superconduc-tors in the presence of disorder, the energy range of contri-bution is wider. Further, the inclusion of next-nearest neigh-bor hopping, t ′ , makes the NS GIMT energy-band (and hence | ∆ αα | ) asymmetric about the Fermi energy ( ξ = 0 ). The finalprofile of | ∆ αβ | , as seen from Fig. (6c), hints that the summa-tions in Eqs. (13) and (14) can be further restricted to a limitedset (leaving out the states close to band edges) to achieve a de-sired accuracy.Motivated by these findings, we simplify the PNS calcula-tions by limiting progressively smaller number of total statescontributing to pairing. The corresponding output of ∆ OP ,as its percentage deviation from the GIMT value, is shownin Fig. (6d) against the fraction of normal states participatedin pairing. To illustrate our choice of restricted states for V = 2 . , we show the bounding box BH in Fig. (6c) by athin dotted line that includes about 19% of the normal statesfor pairing, and results into more than 99% accuracy in ∆ OP (the last data point along x -axis in Fig. (6d)). It is apparent that our bounding box encloses states that subscribe to | ∆ αβ | of significance . Evidently, PNS results achieve perfectionwhen increasing fraction of states are included. Yet, we seethat only about of NS GIMT ensures accuracy of theresults, even for disorder as large as V = 2 . ! E. Pairing theory with ‘uncorrelated’ NS and with a differentmodel of disorder
We discuss below the prospects of our PNS proposal interms of ‘uncorrelated’ normal states, in which all Gutzwillerfactors are set to unity. The impressive match of ∆ OP fromsuch pairing theory using NS IMT , in comparison with theplain BdG results has already been analysed in Sec. (III C).In fact, we found that the IMT-normal states are very closeto the original ‘exact eigenstates’, except, of course, for theHartree- and Fock-shifts, which adds only weak correctionsin the absence of Gutzwiler renormalization. While the suc-cess of PNS formalism is evident, there are practical concernsfor the applicability of such implementation. The NS IMT arenaturally incapable of accounting for the strong correlationeffects, crucial for the qualitative physics of the strongly cor-related superconductors. In addition, the pairing of NS IMT misses the near-diagonal nature of | ∆ αβ | as found in Fig. (6)for NS GIMT , making the NS IMT less useful, for deriving tech-nical advantages over IMT calculations.We also verified that the results and conclusions of PNSformalism remain valid even with a model of ‘concentrationimpurity’, in which n imp fraction of the (random) lattice sitescontain a fixed disorder potential V , provided we use V ≤ .Stronger V brings in subtle effects even in GIMT implemen-tation. IV. DISCUSSIONS
The impressive match between the PNS and GIMT resultsis inspiring from the perspective of developing simple under-standing on the complex physics of disordered and stronglycorrelated superconductors. However, we believe that it is theconceptual advances offered by PNS technique, as describedin the previous sections, which have far reaching values. Wewill discuss below a crucial notional gains from the PNS pro-posal.
A. Insensitivity of inhomogeneity in pairing
Our results make it evident that inhomogeneities are lessrelevant for pairing in case of strongly correlated dSC. Thishas already been illustrated in Ref. 40, by matching the spec-tral density of states evaluated in GIMT for V ≤ t , with itsd-wave BCS form convoluted with the near-Gaussian GIMTdistribution of ∆ ij . Here, we argue for a more direct evidenceto this assertion by noting that the spatial inhomogeneities inthe Gutzwiller factor g xyij , arising from the spatial fluctuationsin the local density, has little role in the final self-consistent FIG. 7: ∆ ij on each bond for a section of the lattice for: (a) V = 2 . ,and (b) V = 4 . for a specific realization of disorder. The π/ phasedifference between ∆ i,i +ˆ x and ∆ i,i +ˆ y survives over the entire lat-tice as seen in (a). The larger disorder strength of panel (b) still sup-ports the d-wave anisotropy in most parts (highlighted by the squareboundary). It also features regions of strong potential fluctuations(marked by circular boundary), where ∆ i,i +ˆ x and ∆ i,i +ˆ y are closerin magnitude, but only when both are vanishingly small! output of ∆ ij on the bonds. This is, however, only true, pro-vided that the correct NS GIMT is obtained by taking care ofall inhomogeneities in their construction. For concreteness,we can consider three independent implementations of g xyij ,with a progressive degree of approximations of the inhomo-geneities: (a) A full self-consistency in local density ρ i in thedefinition of g xyij is achieved during the iterative update of ∆ ij during the pairing stage following Eq. (8). (b) We fix the inho-mogeneous density profile to its form as obtained in NS GIMT ,without any update during the pairing self-consistency. (c) Inthe extreme approximation, we set g xyij = (1 − . ρ ) − for thepurpose of pairing self-consistency. Obviously, each degreeof approximation is associated with significant computationalgains. We find that even with the most drastic approximation,the resulting order parameters are in good agreement (within10%) with the GIMT findings. On the other hand, we foundthat an approximate handling of heterogeneities in the normalstate leads to significant deviation of the final results. B. What makes d-wave anisotropy of pairing so robust?
Why does not AG-theory capture the insensitivity ofstrongly correlated d-wave superconductors to impurities?Admittedly, such strongly coupled systems with short coher-ence length ξ , fall outside the scope of a true AG description.However, our PNS formalism offers a simple and intuitive per-spective for the distinct outcome of the GIMT findings. Suchresults (or the results from PNS, which produces essentiallyidentical results as GIMT) of the spatial profile of ∆ ij on eachbond on a square lattice is shown in Fig. (7a), for a specificrealization of disorder.We witness pairing amplitudes of opposite signs but ofnearly equal strengths on bonds along ˆ x - and ˆ y -directionsfrom each site for V = 2 . (See Fig. (7a)). Such a phasedifferences of π/ between adjacent orthogonal bonds is the hallmark of its d x − y anisotropy of pairing amplitude inthe clean systems, and remains near-perfect even at V = 2 . !With the introduction of disorder, AG theory predicts that theimpurity scattering ‘mixes-up’ such sensitive phase relations,and thereby depletes d-wave superconductivity. Instead, wefind a healthy d-wave anisotropy to survive. But, this is natu-rally expected within the PNS formalism – there is no disorderleft to scramble phases at the second stage of ‘pairing’, theyare all consumed in generating the normal states at the firststage of calculations!Do such phase relations continue to hold for stronger disor-ders? While additional considerations are necessary for push-ing the applicability of the PNS method to larger V , an ex-tension of GIMT-type calculation upon including localizationphysics for V ≥ has already been reported in Ref. 47, andthose results offer a significant pointer. By ramping up V insuch calculations, we find that for V = 4 . , the local pairingamplitude tends to zero identically on both ˆ x - and ˆ y -bonds inregions of strong fluctuation of disorder potential (marked bycircular boundary in Fig. (7b)). Yet, the d-wave anisotropyremains intact in regions possessing a healthy ∆ ij (markedby square boundary), albeit some inhomogeneity. Thus, im-purities can affect superconductivity by locally collapsing theself-consistent pairing amplitudes, which are due to the local-ization properties of the normal states, but are not because ofscrambling of the d-wave anisotropy. V. CONCLUSION
In conclusion, we presented a description of disordered andstrongly correlated d-wave superconductors by implementingsimple pairing ideas of Anderson, but extending it by includ-ing the effects of strong electronic correlations as well as dis-order induced inhomogeneities. The impressive match of theresults from the proposed PNS method and GIMT findings isencouraging. In addition to offering a deeper understandingof the GIMT findings, our formalism sheds important lighton some shortcomings of the conventional wisdom. The piv-otal advance offered by the PNS formalism lies in identifyingthe underlying effective one-particle states that participate inCooper-pairing in unconventional superconductors. This mo-tivates future survey of the properties of NS GIMT by probingthem using various means, and in particular on their temper-ature dependences. It will also be interesting to consider therobustness of the PNS formalism upon including the physicsof ‘competing orders’ in NS GIMT and their role in subsequentCooper-pairing.
Acknowledgments
We thank Indranil Paul and Kazumasa Miyake for usefuldiscussions. ∗ Present address H. Alloul, J. Bobroff, M. Gabay, and P. J. Hirschfeld, Rev. Mod.Phys. , 45 (2009). E. Dagotto, Science , 257 (2005). B. Keimer, S. A. Kivelson, M. R. Norman, S. Uchida, and J. Zaa-nen, Nature , 179 (2015). B. Nachumi, A. Keren, K. Kojima, M. Larkin, G. M. Luke, J. Mer-rin, O. Tchernysh¨ov, Y. J. Uemura, N. Ichikawa, M. Goto, et al.,Phys. Rev. Lett. , 5421 (1996). A. Garg, M. Randeria, and N. Trivedi, Nature Physics , 762(2008). A. Abrikosov and L. Gorkov, Sov. Phys. JETP , 1243 (1961). P. Hirschfeld, D. Vollhardt, and P. Wlfle, Solid State Communica-tions , 111 (1986). S. Schmitt-Rink, K. Miyake, and C. M. Varma, Phys. Rev. Lett. , 2575 (1986). R. Joynt, Journal of Low Temperature Physics , 811 (1997). N. E. Hussey, Advances in Physics , 1685 (2002). A. V. Balatsky, I. Vekhter, and J.-X. Zhu, Rev. Mod. Phys. , 373(2006). C. P´epin and P. A. Lee, Phys. Rev. B , 054502 (2001). H. Won, K. Maki, and E. Puchkaryov,
Introduction to D-WaveSuperconductivity (Springer Netherlands, Dordrecht, 2001), pp.375–386. J. A. Slezak, J. Lee, M. Wang, K. McElroy, K. Fujita, B. M. An-dersen, P. J. Hirschfeld, H. Eisaki, S. Uchida, and J. C. Davis, Pro-ceedings of the National Academy of Sciences , 3203 (2008). K. McElroy, D.-H. Lee, J. E. Hoffman, K. M. Lang, J. Lee, E. W.Hudson, H. Eisaki, S. Uchida, and J. C. Davis, Phys. Rev. Lett. , 197005 (2005). S. H. Pan, J. P. O’Neal, R. L. Badzey, C. Chamon, H. Ding, J. R.Engelbrecht, Z. Wang, H. Eisaki, S. Uchida, A. K. Gupta, et al.,Nature , 282 (2001). K. M. Lang, V. Madhavan, J. E. Hoffman, E. W. Hudson,H. Eisaki, S. Uchida, and J. C. Davis, Nature , 412 (2002). P. W. Anderson, Science , 480 (2000). A. F. Kemper, D. G. S. P. Doluweera, T. A. Maier, M. Jarrell, P. J.Hirschfeld, and H.-P. Cheng, Phys. Rev. B , 104502 (2009). F. Rullier-Albenque, H. Alloul, and R. Tourbot, Phys. Rev. Lett. , 047001 (2003). S. K. Tolpygo, J.-Y. Lin, M. Gurvitch, S. Y. Hou, and J. M.Phillips, Phys. Rev. B , 12454 (1996). D. A. Wollman, D. J. Van Harlingen, J. Giapintzakis, and D. M.Ginsberg, Phys. Rev. Lett. , 797 (1995). D. A. Wollman, D. J. Van Harlingen, W. C. Lee, D. M. Ginsberg,and A. J. Leggett, Phys. Rev. Lett. , 2134 (1993). H. Ding, M. R. Norman, J. C. Campuzano, M. Randeria, A. F.Bellman, T. Yokoya, T. Takahashi, T. Mochiku, and K. Kadowaki,Phys. Rev. B , R9678 (1996). C. C. Tsuei, J. R. Kirtley, C. C. Chi, L. S. Yu-Jahnes, A. Gupta,T. Shaw, J. Z. Sun, and M. B. Ketchen, Phys. Rev. Lett. , 593(1994). J. G. Analytis, A. Ardavan, S. J. Blundell, R. L. Owen, E. F. Gar-man, C. Jeynes, and B. J. Powell, Phys. Rev. Lett. , 177002(2006). J. Li, Y. F. Guo, S. B. Zhang, J. Yuan, Y. Tsujimoto, X. Wang,C. I. Sathish, Y. Sun, S. Yu, W. Yi, et al., Phys. Rev. B , 214509(2012). M. N. Gastiasoro, F. Bernardini, and B. M. Andersen, Phys. Rev.Lett. , 257002 (2016). Y. K. Kuo, C. W. Schneider, M. J. Skove, M. V. Nevitt, G. X. Tessema, and J. J. McGee, Phys. Rev. B , 6201 (1997). E. Dagotto, Rev. Mod. Phys. , 763 (1994). P. W. Anderson, P. A. Lee, M. Randeria, T. M. Rice, N. Trivedi,and F. C. Zhang, Journal of Physics: Condensed Matter , R755(2004). P. A. Lee, N. Nagaosa, and X.-G. Wen, Rev. Mod. Phys. , 17(2006). W. A. Atkinson, P. J. Hirschfeld, and A. H. MacDonald, Phys.Rev. Lett. , 3922 (2000). A. Ghosal, M. Randeria, and N. Trivedi, Phys. Rev. B , 020505(2000). W. Atkinson, P. Hirschfeld, and A. MacDonald, Physica C: Su-perconductivity and its Applications , 1687 (2000). M. Franz, C. Kallin, A. J. Berlinsky, and M. I. Salkola, Phys. Rev.B , 7882 (1997). F. C. Zhang, C. Gros, T. M. Rice, and H. Shiba, SuperconductorScience and Technology , 36 (1988). R. B. Christensen, P. J. Hirschfeld, and B. M. Andersen, Phys.Rev. B , 184511 (2011). N. Fukushima, C.-P. Chou, and T. K. Lee, Phys. Rev. B ,184510 (2009). D. Chakraborty and A. Ghosal, New Journal of Physics ,103018 (2014). B. M. Andersen and P. J. Hirschfeld, Phys. Rev. Lett. , 257003(2008). S. Tang, V. Dobrosavljevi´c, and E. Miranda, Phys. Rev. B ,195109 (2016). P. Anderson, Journal of Physics and Chemistry of Solids , 26(1959). A. Ghosal, M. Randeria, and N. Trivedi, Phys. Rev. B , 014501(2001). A. V. Balatsky, M. I. Salkola, and A. Rosengren, Phys. Rev. B ,15547 (1995). A. Kreisel, P. Choubey, T. Berlijn, W. Ku, B. M. Andersen, andP. J. Hirschfeld, Phys. Rev. Lett. , 217002 (2015). D. Chakraborty, R. Sensarma, and A. Ghosal, Phys. Rev. B ,014516 (2017). J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Phys. Rev. , 162(1957). P. W. Anderson, Science , 1196 (1987). J. Hubbard, Proceedings of the Royal Society of London A: Math-ematical, Physical and Engineering Sciences , 238 (1963). K. A. Chao, J. Spalek, and A. M. Oles, Journal of Physics C: SolidState Physics , L271 (1977). A. Samanta and R. Sensarma, Phys. Rev. B , 224517 (2016). M. R. Norman, M. Randeria, H. Ding, and J. C. Campuzano, Phys.Rev. B , 615 (1995). V. J. Emery, S. A. Kivelson, and J. M. Tranquada, Proceedings ofthe National Academy of Sciences , 8814 (1999). E. Fradkin, S. A. Kivelson, M. J. Lawler, J. P. Eisenstein, and A. P.Mackenzie, Annual Review of Condensed Matter Physics , 153(2010). E. G. Moon and S. Sachdev, Phys. Rev. B , 035117 (2009). A. Mesaros, K. Fujita, S. D. Edkins, M. H. Hamidian, H. Eisaki,S.-i. Uchida, J. C. S. Davis, M. J. Lawler, and E.-A. Kim, Proceed-ings of the National Academy of Sciences , 12661 (2016). J. P. L. Faye and D. S´en´echal, Phys. Rev. B , 115127 (2017). B. M. Andersen, P. J. Hirschfeld, A. P. Kampf, and M. Schmid,Phys. Rev. Lett. , 147002 (2007). E. Fradkin, S. A. Kivelson, and J. M. Tranquada, Rev. Mod. Phys. , 457 (2015). X. Montiel, T. Kloss, and C. P´epin, Phys. Rev. B , 104510(2017). S. Chakravarty, R. B. Laughlin, D. K. Morr, and C. Nayak, Phys.Rev. B , 094503 (2001). C. M. Varma, Phys. Rev. B , 14554 (1997). B. Fauqu´e, Y. Sidis, V. Hinkov, S. Pailh`es, C. T. Lin, X. Chaud,and P. Bourges, Phys. Rev. Lett. , 197001 (2006). B. Edegger, V. N. Muthukumar, and C. Gros, Advances in Physics , 927 (2007). W.-H. Ko, C. P. Nave, and P. A. Lee, Phys. Rev. B , 245113(2007). A. Paramekanti, M. Randeria, and N. Trivedi, Phys. Rev. B ,054504 (2004). R. Sensarma, M. Randeria, and N. Trivedi, Phys. Rev. Lett. ,027004 (2007). D. D. Johnson, Phys. Rev. B , 12807 (1988). S. Tang, E. Miranda, and V. Dobrosavljevic, Phys. Rev. B ,020501 (2015). D. Tanaskovi, V. Dobrosavljevi´c, E. Abrahams, and G. Kotliar,Phys. Rev. Lett. , 066603 (2003). K. Seo, B. A. Bernevig, and J. Hu, Phys. Rev. Lett. , 206404(2008).73