Exactly solvable model of strongly correlated d -wave superconductivity
Malte Harland, Sergey Brener, Mikhail I. Katsnelson, Alexander I. Lichtenstein
EExactly solvable model of strongly correlated d -wave superconductivity Malte Harland, Sergey Brener, Mikhail I. Katsnelson, and Alexander I. Lichtenstein Institute of Theoretical Physics, University of Hamburg, Jungiusstraße 9, 20355 Hamburg, Germany Institute for Molecules and Materials, Radboud University, 6525AJ, Nijmegen, the Netherlands (Dated: May 30, 2019)We present an infinite-dimensional lattice of two-by-two plaquettes, the quadruple Bethe lattice,with Hubbard interaction and solve it exactly by means of the cluster dynamical mean-field theory.It exhibits a d -wave superconducting phase that is related to a highly degenerate point in the phasediagram of the isolated plaquette at that the groundstates of the particle number sectors N = 2 , , X/Y -momenta are the main two-particle correlations of the superconducting phase. The suppressionof the superconductivity in the overdoped regime is caused by the diminishing of pair hoppingcorrelations and in the underdoped regime by charge blocking. The optimal doping is ∼ .
15 atwhich the underlying normal state shows a Lifshitz transition. The model allows for different intra-and inter-plaquette hoppings that we use to disentangle superconductivity from antiferromagnetismas the latter requires larger inter-plaquette hoppings.
I. INTRODUCTION
High-temperature superconductivity in cuprates canpersist at temperatures up to T ∼ , small superfluid density , and competing orders .Cuprates share a common quasi two-dimensional struc-ture of layered copper-oxide compounds that are insu-lating and become superconducting upon doping withcharge carriers . The different compounds of that fam-ily share a d -wave character of superconducting gap andantiferromagnetic order in the undoped insulating state.Furthermore, at larger hole doping and temperaturesabove the critical temperature they exhibit a very in-coherent metal behavior characterized in particular by alinear temperature dependence of the resistivity and bya pseudogap formation .The Cu atoms of the stacked Cu-planes form a squarelattice. On their bonds are oxygen atoms whose p -orbitals mediate electronic transitions from one Cu d -orbital to its nearest neighbor’s d -orbital. This processis modeled by effective d − d hopping with the amplitude t that competes with the local screened Coulomb repul-sion U . Further hoppings also exist, but they are smallerin their amplitude. The bandwidth of the d -orbitals iscomparable to the interaction energy U . The broadlyaccepted minimal model to account for these competingelectronic effects is the Hubbard model . Despite itssimple appearance that model in two and three dimen-sions can be solved by approximations only, contrary tothe one-dimensional case that is exactly solvable by BetheAnsatz .A simple but powerful approximation is the dynam-ical mean-field theory (DMFT) which accounts only for the local correlations by including only the local self-energy from an effective impurity model. The DMFTprovides an exact solution in the formal limit of infi-nite dimensions but is questionable for two dimensions(2D). Phenomena such as the Mott transition and itin-erant antiferromagnetism (Slater physics) are capturedby the infinite-dimensional DMFT, but are severely over-estimated in low dimensions.The DMFT can be extended by restricting the self-energy not to a single site, but to a cluster of severalsites. Hence, this extension is called cluster DMFT(CDMFT) . The generalization to clusters is notunique and still debated . Regardless of the particu-lar choice of CDMFT-”flavor” it was found that intersitecorrelations within the cluster are sufficient to obtain asymmetry-broken d-wave superconducting (dSC) state.The minimal cluster is the two-by-two cluster (plaque-tte), since dSC order is defined on the bonds accord-ing to d x − y -wave symmetry . In 2D CDMFT is anapproximation and long-range correlations beyond thecluster can be important for a correct description of thedSC state in cuprates . Therefore in the dSC stateCDMFT aims to describe only the local formation ofCooper pairs. For example, CDMFT gives coexistingantiferromagnetic (AFM) and dSC orders , whereas incuprates these phases do not coexist. The reason is thatCDMFT does not distinguish between long-range andshort-range AFM order if the corresponding correlationlength is much larger than the lattice constant. Thereby,it also neglects the stripe order phase of cuprates whichhas been found to be suppressed by the next-nearestneighbor hopping within the Hubbard model .In this work we present a detailed analysis of the in-finite dimensional quadruple Bethe lattice model withinthe CDMFT. Similar to the well-known Mott transitionfound in the simple Bethe lattice and the cor-related Peierls insulator transition in the double Bethelattice , we find the dSC transition in the strong-coupling quadruple Bethe lattice . This choice of setup a r X i v : . [ c ond - m a t . s t r- e l ] M a y is complementary to prior studies in the sense that weinvestigate a less accurate model of infinite dimensional-ity but in return obtain an exact solution. Compared tothe simple Bethe lattice the local Hilbert-space size is in-creased, from 4 of the Hubbard site to 256 of the Hubbardplaquette. This opens up new degrees of freedom thatcan interact with the mean-field environment. In partic-ular, we focus on those plaquette eigenstates , thatdefine the dSC and cross at a quantum critical point (QCP) of the plaquette. This point is particularly in-teresting as quantum critical behavior has beenfound for the square lattice by CDMFT studies of thepseudogap phenomenon that has been suggested to orig-inate from negative interference of hybridizing plaquettestates .In this paper we start with a presentation of thequadruple Bethe lattice and the single-particle basis weuse in Sec. II. In Sec. III we provide an overview of theisolated cluster’s Hilbert space, that is the auxiliary sys-tem of our CDMFT mapping. The opposite limit ofnon-interacting Bethe lattices is presented in Sec. IV.In Sec. V we analyse the dependence of the dSC orderparameter on the screened Coulomb repulsion and thechemical potential for small Bethe lattice hoppings, i.e.plaquette hybridizations, for that the dSC order is dom-inant and other orders are less pronounced. In Sec. VIwe show how different components of the two-particleinteraction promote or interfere with the dSC order. Dy-namical properties, such as quasiparticle characteriza-tion and spectral functions are presented in Sec. VII.Finally, larger Bethe lattice hoppings yield more pro-nounced antiferromagnetic order, see Sec. VIII, and anextended Bethe lattice hopping allows us to tune the non-interacting density of states more similar to a van-Hovesingularity, that is presented in Sec. IX. II. MODEL & METHOD
As stated above, the correlated d -electrons of thecopper-oxide planes are described by the Hubbard model H = (cid:88) ijσ t ij c † iσ c jσ + U (cid:88) i c † i ↑ c i ↑ c † i ↓ c i ↓ , (1)with fermionic creation/annihilation operators c † / c , sites i, j and spins σ . It contains a hopping term t i,j thatfor lattice structures becomes diagonal in k -space andtherefore promotes delocalization of the charge. Albeit,the quadruple Bethe lattice is only a pseudolattice in thatregard since it does not exhibit translational invariance.But still, its sites are equivalent due to its self-similarstructure. The screened local Coulomb repulsion U isdiagonal in site-space and promotes charge localization.The chemical potential µ can be written explicitly, or itcan be absorbed into the diagonal, local part of t ij .The quadruple Bethe lattice is constructed from fourBethe lattices, that are plaquette-wise connected, see FIG. 1. Quadruple Bethe lattice, four Bethe lattices (dottedlines) interconnected via plaquettes (solid lines). The coordi-nation number for each Bethe lattice of this figure is set to z = 3, and six sites of each Bethe lattice are depicted. Anentire Bethe lattice exhibits an infinite number of sites withself-similar structure. Next-nearest neighbor hoppings of theplaquette are omitted for convenience. Fig. 1, i.e. equivalent sites of the four Bethe lattices forma two-by-two plaquette. The coordination number of theBethe-lattices is set to z = ∞ corresponding to infinitedimensions. We introduce three types of hopping. Thefirst hopping t connects sites of the Bethe-lattice withequivalent points of two neighboring Bethe-lattices, i.e.within plaquettes. We use t = − t (cid:48) connects with equivalent points of the one remainingBethe-lattice and thus occurs on the next-nearest neigh-bor bond of the plaquette. The third hopping t b connectssites within the Bethe-lattices, i.e. between plaquettes.We write the plaquette hopping matrix in site basis as t p = − µ t t t (cid:48) t − µ t (cid:48) tt t (cid:48) − µ tt (cid:48) t t − µ . (2)Then, we can decompose the kinetic energy H t , the firstterm of Eq. (1), into hopping within plaquettes and be-tween plaquettes, i.e. within Bethe lattices H t = t b (cid:88)
FIG. 2. Illustration of the plaquette orbitals/momentaΓ , X, M ( Y omitted), i.e. the basis that diagonalizes the hop-ping in plaquette-site space (0 , , , take four possible values, Γ , M, X, Y . The transforma-tion applied to plaquette-site space reads T = 12 e i Γ R . . . e i Γ R e iMR . . . e iMR e iXR . . . e iXR e iY R . . . e iY R = 12 − − − −
11 1 − − (4)with ( R , ..., R ) = (cid:18) (cid:19) a, (Γ , M, X, Y ) = (cid:18) (cid:19) πa (5)with a unit length a . Due to the symmetry of the site-space, we can diagonalize the quadratic parts of theHamiltonian H in plaquette-momentum space. At thispoint the quadruple Bethe lattice can be regarded as amultiorbital simple Bethe lattice, see Fig. 2.The interaction has to be transformed to K basis aswell. From the fact that it is local in site basis, one canalready expect many terms in the plaquette-momentumbasis. We apply the rank-2 tensor transformation U (cid:55)→ T T U T † T † also using plaquette-momentum conservationand obtain H U = (cid:88) rK ...K U K ...K c † rK ↑ c rK ↑ c † rK ↓ c rK ↓ (6)with the Hubbard interaction tensor U K ...K = U δ (2 π/a ) K + K ,K + K /
4, where δ (2 π/a ) K ,K = 1, when K − K = 2 πn/a with integer n and = 0 otherwise. Depend-ing on the relative values of the four plaquette mo-menta K . . . K we can classify the terms of U K ...K into intra-orbital ( ×
4, e.g. U XXXX ), inter-orbital ( × U XXY Y ), pair-hopping ( ×
12, e.g. U XY XY ), spin-flip( ×
12, e.g. U XY Y X ), and correlated hopping ( ×
24 withall four momenta pairwise distinct).Regarding the superconducting order we use theNambu spinor basis for its description˜ c † r = (cid:16) c † r Γ ↑ c r Γ ↓ c † rM ↑ c rM ↓ c † rX ↑ c rX ↓ c † rY ↑ c rY ↓ (cid:17) . (7)It can be constructed efficiently by particle-hole trans-forming the spin- ↓ part of the conventional spinor rep-resentation. This is sufficient if no other than S z = 0spin-structures are considered for the pairing. Next, weaddress the Hubbard-Hamiltonian in Nambu basis. Weapply the Nambu spinor construction and examine how t ij , µ and U transform. This is done by using the anti-commutation rules. We obtain˜ t pσ = t p ( δ σ ↑ − δ σ ↓ ) , ˜ µ σ = ( µ + U ) δ σ ↑ − µδ σ ↓ , ˜ U = − U, (8)where ↑ and ↓ denote the indices of the Nambu spinorentries, i.e. spin- ↑ particles and spin- ↓ holes. The Bethehopping t b transforms under the Nambu spinor construc-tion in the same way as the plaquette hopping t p .We use the CDMFT to map the lattice prob-lem to the Anderson impurity model of an impurity, i.e.the cluster, with (quartic) interaction and a bath of non-interacting, but potentially renormalized, particles. Theenvironment is defined by the dynamical mean-field (bathGreen function) G ( iω n ). In particular for the quadrupleBethe lattice with infinite coordination of the Bethe lat-tices, the CDMFT becomes exact as the self-energyΣ( iω n ) = G − ( iω n ) − G − ( iω n ) (9)exists only within the plaquettes and not between them.Eq. (9) is the Dyson equation and relates Σ to the lo-cal Green function G and the bath Green function G .They depend on Matsubara frequencies ω n = π (2 n +1) /β with inverse temperature β . In this study we use for thequadruple Bethe lattice calculations β = 100 through-out (though for calculations on the isolated plaquette wealso use β = 30). The self-consistency condition to treatAFM and dSC order reads G − AKσ,AK (cid:48) σ (cid:48) ( iω n ) = iω n δ KK (cid:48) δ σσ (cid:48) + ( µδ KK (cid:48) − t pKK (cid:48) ) σ zσσ (cid:48) − t b (cid:88) ττ (cid:48) σ zστ G BKτ,BK (cid:48) τ (cid:48) ( iω n ) σ zτ (cid:48) σ (cid:48) − Σ AKσ,K (cid:48) jσ (cid:48) ( iω n ) . (10) K, K (cid:48) are labels for the plaquette momenta. σ, σ (cid:48) , τ, τ (cid:48) label the Nambu-space, i.e. spin- ↑ electrons or spin- ↓ holes. The Nambu representation also requires a trans-formation of the single-particle energies (Eq. (8)), thechemical potential µ , the matrix of plaquette-hoppings t p and the scalar Bethe hopping t b . For that reason thethird Pauli matrix σ z appears in Eq. (10).The fact that the Bethe lattice is bipartite allows usto additionally consider the possibilty of AFM symmetrybreaking. We can divide the lattice into two sublattices ofwhich we know how to transform their local Green func-tions into each other analytically. We describe the AFMof the sublattices A and B with Nambu-Green functionsas G BKσ,BK (cid:48) σ (cid:48) ( iω n ) = − (cid:88) ττ (cid:48) R στ G ∗ AKτ,AK (cid:48) τ (cid:48) ( iω n ) R † τ (cid:48) σ (cid:48) (11)with the rotation matrix R = e iπσ y / . (12)Eq. (11) describes a spin-flip accompanied by a particle-hole transformation due to the Nambu spinor formalism.For the diagonal entries of the Green function there isno difference in using the first ( σ x ) or the second ( σ y )Pauli matrix for the rotation of Eq. (12). In contrast, off-diagonal (anomalous) entries obtain an additional minussign from σ y . We use this Berry phase in order not tochange the dSC order for the Bethe sublattices A and B . σ x would change the dSC according to an X / Y -flip.Thus, Eq. (11) defines staggered spin, but homogenousdSC order.The spin order within the plaquette can still be diversefor different solutions. A/B sublattices not only supportAFM order, but also a spin order that is ferromagneticwithin the plaquette and antiferromagnetic with respectto the Bethe sublattices
A/B . We will refer to the latteras plaquette antiferromagnetism (PAFM).The dynamical mean-field is constructed as G − AKσ,AK (cid:48) σ (cid:48) ( iω n ) = iω n δ KK (cid:48) δ σσ (cid:48) + ( µδ KK (cid:48) − t pKK (cid:48) ) σ zσσ (cid:48) − t b (cid:88) ττ (cid:48) σ zστ G BKτ,BK (cid:48) τ (cid:48) ( iω n ) σ zτ (cid:48) σ (cid:48) . (13)Together with the local interaction, G defines the impu-rity setup. Eq. (13) shows that the mean-field of sublat-tice A is constructed from the local properties of sublat-tice B . In the following we drop the Bethe lattice index r = A/B for convenience. The numerical solution of theimpurity Green function is obtained by the hybridiza-tion expansion continuous time quantum Monte-Carlomethod (CTHYB). The self-consistency is closedwith the Dyson equation and by demanding that the locallattice Green function equals the impurity Green functionwhich is inserted into the right-hand side of Eq. (10) untilconvergence is reached. In our implementation Eq. (10)is also used to iteratively find µ in the case of a certainfilling is set as a parameter rather than µ directly. The numerics can be implemented efficiently usingsymmetries and blockstructure of the Green function.For our setup the Matsubara-Green function has thestructure Γ M X YG = γ a − γ ∗ a ∗ a m a ∗ − m ∗ x − d ˜ a π − d − x ∗ − π ˜ a ∗ ˜ a − π y dπ ˜ a ∗ d − y ∗ . (14)It contains the two-by-two Nambu blocks of spin- ↑ parti-cles and spin- ↓ holes and additionally four-by-four blocksin plaquette momentum basis, the Γ M - and the XY -blocks. d and a /˜ a stand for dSC and AFM orders, respec-tively. AFM breaks the plaquette point-group symmetryin such a way, that Γ, M and X , Y are pairwise coupled.In the plaquette momentum basis AFM order is reflectedby non-zero a /˜ a off-diagonals. Furthermore, dSC orderbreaks the plaquette symmetries so that the X - Y degen-eracy is lifted, but off-diagonals are introduced only inNambu-space, d and − d . The diagonal-part of the X / Y -block is not affected by the dSC symmetry breaking, andthus y = x . The entries of π describe spin-triplet su-perconductivity πSC which we study in Sec. VIII. Theanomalous part of the Green function has non-zero ele-ments only in the XY -block. It can be written as F = (cid:32) − d π − π d (cid:33) , (15)for that the entries of π show the symmetry F ↑↓ XY = − F ↑↓ Y X = F ↓↑ XY and hence also the spin-triplet pairing.In contrast d = F ↑↓ XX = − F ↓↑ XX which is a spin-singletstructure. Note, that in the present study non-zero en-tries for π occur only simultaneously with the coexistenceof dSC and AFM.DMFT calculations of broken symmetries can be doneefficiently by introducing seeds with the proper symmetryfor the first DMFT-iteration and subsequently runningadditional loops until convergence. For example regard-ing dSC, we initialize the anomalous Green function with d init ( iω n ) = d β δ n, − + δ n, ) (16)for some small d . This function transforms into a cosinein imaginary time that is symmetric and real. III. TWO-BY-TWO PLAQUETTE
The low-energy many-body states of the Hubbard two-by-two plaquette, around (cid:104) N (cid:105) = 3 filling, have been con- U c U − . µ c . . µ N = 4 N = 3 N = 2 χ p a i r XX ( ω = ) FIG. 3. Retarded pairing susceptibility χ pair ( ω ) of pairs withplaquette momentum X in the isolated plaquette dependenton the screened Coulomb repulsion U and chemical potential µ . The groundstate sectors N = 2 , , U c = 2 .
78 and µ c = 0 . χ pair ( ω ) (black circle) lies at the N = 2 , U c
3. Calculations for t (cid:48) = 0 showqualitatively similar results although the QCP is shiftedto larger values of µ and U .The instability towards dSC order can be observed al-ready in the isolated plaquette using exact diagonaliza-tion. The pairing susceptibility χ pair XX ( τ ) = (cid:68) T τ c X ↑ ( τ ) c X ↓ ( τ ) c † X ↓ (0) c † X ↑ (0) (cid:69) , (17)with imaginary time ( τ ) ordering operator T τ can be cal-culated using the Lehmann representation. The retardedpairing susceptibility at Fermi level χ pair ( ω = 0) showslarge values in the µ / U -phase diagram at the boundaryof N = 2 ,
4, see Fig. 3. In the N = 4 sector, where | , , Γ (cid:105) is the groundstate, | , , Γ (cid:105) describes a bosonictwo-hole excitation . χ pair XX has its maximum close to U = 2. Moreover, in this phase diagram the quantumcritical point where 6 many-body states of the sectors N = 2 , , U c = 2 .
78 and µ c = 0 . µ for constant U = U c . Addition- . . . . . . . µ − . − . . . . . . . E − E n | , , Γ > | , , XY > | , , Γ > | , , M > | , , Γ > | , , M > | , , XY > | , , Γ > normal dSCsSCπSC FIG. 4. Plaquette eigenenergies E as functions of the chemi-cal potential µ around µ = 0 . , U = 2 .
78 at that N = 2 , , E n . The kets label the normalstates particle number N , spin number S and plaquette mo-mentum K . The superconducting ( SC ) fields are ∆ x = 0 . x : d -wave dSC , s -wave sSC and spin-triplet πSC . ally, we add different N -symmetry breaking fields h dSC = ∆ dSC ( c ↑ X c ↓ X − c ↑ Y c ↓ Y ) + h . c .,h sSC = ∆ sSC ( c ↑ X c ↓ X + c ↑ Y c ↓ Y ) + h . c .,h πSC = ∆ πSC ( c ↑ X c ↓ Y + c ↓ X c ↑ Y ) + h . c . (18)of spin-singlet s -wave (sSC), spin-singlet d -wave (dSC)and spin-triplet ( π SC) symmetries. The groundstate en-ergy lowering by the dSC order is the largest at the crit-ical point µ c ∼ .
24, see Fig. 4. Different absolute valuesof the slopes in Fig. 4 correspond to different particlenumber sectors of the normal state. The small- µ andlarge- µ part have | , , Γ (cid:105) and | , , Γ (cid:105) as groundstates,respectively. The dSC -groundstate is a superposition ofmainly these two and there crossing is avoided by thesymmetry breaking. A contribution of N = 3 to the dSC groundstate is excluded since Cooper-pairs containtwo electrons and therefore the groundstate has even par-ity, i.e. it is a superposition of particle number sectors ofeven particle numbers.Regarding ∆ sSC , only the | , , Γ (cid:105) is lowered in en-ergy, but not due to mixing with the low-energy | , , Γ (cid:105) of the normal state as this one is unaffected. For the πSC field, the degeneracy of the spin-triplet | , , M (cid:105) is liftedas only the S z = 0-state mixes with | , , Γ (cid:105) . The split-ting of the two is visible, whereas one state is lowered inenergy, the other is increased relative to the correspond-ing normal state. Without field, i.e. in the normal state, | , , Γ (cid:105) and | , , M (cid:105) cross around µ = 0 .
35, this crossingis avoided in the πSC state. Among the considered sym-metry breakings, the energy splitting of low-energy stateswith dSC -field is the largest. It is noticeable, that themain instability in the many-body physics of the Hub-bard two-by-two plaquette is towards dSC order as itlowers the energy the most. − − − − ω A K ( ω ) | , , XY > ← | , , Γ > & | , , Γ > ←| , , XY > | , , XY > →| , , M > µ = 0 . µ = 0 . µ = 0 . µ = 1 . | , , Γ > ←| , , Γ > | , , XY > ←| , , Γ > K Γ XM FIG. 5. Plaquette momentum K resolved Spectral function A K ( ω ) of the isolated plaquette for different chemical poten-tials µ . The peaks are identified with single-particle transi-tions of the plaquette eigenstates. U = 2 . t (cid:48) = 0 . β = 30and Lorentzian broadening (cid:15) = π/β . In addition to the transition of pairs we investigate alsosingle-particle transitions of the isolated plaquette, seeFig. 5. At half-filling ( µ = 1 .
09) we observe a four-peakstructure of the spectral function. Since we are interestedin hole-doping, we focus on the transitions below Fermi-level. The lowest has plaquette momentum Γ and is atransition of the groundstate, | , , Γ (cid:105) → (cid:12)(cid:12) , , Γ (cid:11) . Thesecond lowest and closest to Fermi level has plaquette mo-mentum X/Y and corresponds to | , , Γ (cid:105) → (cid:12)(cid:12) , , XY (cid:11) .Thus, (cid:12)(cid:12) , , XY (cid:11) describes the low-energy, one-particle ex-citation of plaquette momentum X/Y of the N = 4 sys-tem with groundstate | , , Γ (cid:105) .Upon reducing µ the system gets hole-doped and thelower peak of A X ( ω ) crosses Fermi-level. At µ = 0 . | , , Γ (cid:105) → (cid:12)(cid:12) , , XY (cid:11) and (cid:12)(cid:12) , , XY (cid:11) → | , , Γ (cid:105) occur onthe same ω . Furthermore, the peak has a pronouncedshoulder from the transition (cid:12)(cid:12) , , XY (cid:11) → | , , M (cid:105) . Thusin total around the QCP three prominent one-particletransitions exist close to Fermi level. IV. NON-INTERACTING QUADRUPLE BETHELATTICE
For the non-interacting case ( U = 0, Σ = 0) the Greenfunction G ( iω n ) of Eq. (10) becomes the bare Green func-tion G ( iω n ). Thus, we can solve Eq. (10) analyticallyand obtain G K ( iω n ) = 2 σ z ξ K − (cid:112) ξ K − t b ,ξ K = iω n σ z + ( µ − t pK ) . (19)The third Pauli matrix σ z stems from the particle holetransformation of the hoppings and acts on Nambu space.The derivation of the analytical solutions depends on the t + 2 t − t t − t ω + µ . . . . . A ( ω ) Γ X/Y M Γ X/Y M total W i = 4 t b t b . . FIG. 6. Semicircular densities of states of the non-interacting( U = 0) quadruple Bethe lattice for the different or-bitals/momenta and bethe-lattice hoppings t b ( t (cid:48) = 0 . W i for scalar t b . fact, that all quantities can be diagonalized in spinor and K -space by a unitary transformation.The spectral function corresponding to G ( iω n ) isshown in Fig. 6. It consists of four semicirculars of thattwo are degenerate corresponding to X and Y . The semi-circulars have a bandwidth of W = 4 t b each. The posi-tions of the semicirculars are defined by the eigenvalues ofthe hopping within the plaquette. Therefore, we have thelowest momentum/orbital Γ at ω Γ = t (cid:48) + 2 t − µ , the high-est M at ω M = t (cid:48) − t − µ and X/Y at ω X/Y = − t (cid:48) − µ .The model is particle-hole symmetric for t (cid:48) = 0 and largevalues of t (cid:48) or t b can make the orbitals overlap.The dependence of the filling on the chemical potentialand the Bethe-hopping are shown in Fig. 7. In orderto relate states of the isolated plaquette to solutions ofthe quadruple Bethe lattice, it can be useful to knowthe effect of t b . From the non-interacting case we canlearn how t b and µ change the filling. For small t b and0 . < (cid:104) n (cid:105) < t b reduces the particle occupation. Thereare mainly two effects that define this dependence. First,the semicircular at Fermi-level broadens, depending onwhether its maximum is above or below Fermi-level itincreases or decreases the filling. Second, an additionalsemicircular can broaden enough to also touch the Fermi-level and thereby change the filling. V. µ - U PHASE DIAGRAM
In order to get an overview of the phases of the quadru-ple Bethe lattice and their relation to the states of theisolated plaquette, Fig. 8 presents several phase diagramsin the µ - U -plane for different plaquette next-nearest-neighbor hopppings t (cid:48) and Bethe hoppings t b . The ordersdescribed by the selfconsistency condition of Eq. (10),that exist for small t b , are dSC, AFM and PAFM. Their . . . . . t b − . − . − . − . − . − . . µ . . . . < n > FIG. 7. Filling (cid:104) n (cid:105) dependence on the chemical potential µ and the Bethe hopping t b for the non-interacting ( U = 0)quadruple Bethe lattice ( t (cid:48) = 0 . order parameters are defined asΨ dSC = 14 (cid:88) RR (cid:48) (cos [ X ( R − R (cid:48) )] − cos [ Y ( R − R (cid:48) )]) × (cid:104) c R ↑ c R (cid:48) ↓ − c R ↓ c R (cid:48) ↑ (cid:105) , Ψ AF M = 14 (cid:88) R e iMR (cid:104) S zR (cid:105) , (20)Ψ P AF M = 14 (cid:88) R e i Γ R (cid:104) S zR (cid:105) , with the local spin along quantization axis S zR = ( n R ↑ − n R ↓ ) /
2. By the symmetries of Eq. (14) and Eq. (15) weobtain Ψ dSC = Tr iω n F XX ( iω n ). The order parametersare calculated broad region around the QCP, where thegroundstates | , , Γ (cid:105) , (cid:12)(cid:12) , , XY (cid:11) and | , , Γ (cid:105) cross. Westress that Fig. 8 combines information of two differentsystems, i.e. the phase boundaries of the isolated plaque-tte ( t b = 0, T = 0) and order parameters of the quadrupleBethe lattice ( t b > T = 0 . t (cid:48) and t b of Fig. 8 is the dSC. For small t b the dSC region isrelatively narrow as a function of µ . It broadens, andits maximum Ψ maxdSC decreases with increasing t b , as if itis smeared. t b increases the width of the semicirculardensity of states keeping its area constant and therebydecreases its height. Thus, t b increases the energy win-dow for fluctuations, i.e. more plaquette eigenstates fromhigher energies contribute to the solution of the quadru-ple Bethe lattice, but at the same time the amplitudes ofthe quantum fluctations can become smaller. This givesat least an intuition of t b ’s effect, the quantitative detailsare hidden in the CDMFT self-consistency.The QCP of the plaquette shifts to smaller µ andsmaller U as t (cid:48) is increased. For t (cid:48) = 0 .
3, we also findan additional crossover from the spin-doublet (cid:12)(cid:12) , , XY (cid:11) to the spin-quadruplet (cid:12)(cid:12) , , M (cid:11) , that is recognized bya kink in the phase boundaries around U ∼
6. In thequadruple Bethe lattice, at t b = 0 .
2, in that region PAFM t (cid:48) µ ( U c ) = µ c µ (1) opt U (0) opt U (1) opt .
72 0 . ± .
02 2 .
93 1 . . .
24 0 . ± .
05 1 .
82 5 . t b models for the opti-mal chemical potential µ opt and optimal Hubbard interaction U opt for different next-nearest-neighbor hoppings t (cid:48) . The off-set of µ opt , i.e. µ , is calculated in the isolated plaquette, itis the chemical potential at that | , , Γ (cid:105) and | , , Γ (cid:105) of theisolated plaquette cross. order is observed. It is spin-3 / t - J model. How-ever, it is unclear how the PAFM order found here couldbe related to those.AFM is found for t (cid:48) = 0 . , t b = 0 . µ , close tohalf-filling, with a relatively small order parameter, butin the considered parameters of Fig. 8 AFM is mostlyabsent. Heisenberg AFM is promoted by double occupa-tions of sites that occur at half-filling. The effective spinexchange J appears in the strong coupling regime of theHubbard model, i.e. for large U . Therefore the pre-dominant abscence of AFM within the phase diagramsof Fig. 8 seems reasonable as t b is small and U has inter-mediate values. The fact that it appears only at t (cid:48) = 0 . t (cid:48) can cause an effectively enhanced U . Amore detailed view on the AFM order will be providedbelow, in Sec. VIII where we discuss larger t b .In the following we locate and study the optimal pa-rameter set ( µ opt , U opt , t optb ) that corresponds to Ψ maxdSC using linear fits for fixed t (cid:48) = 0 and t (cid:48) = 0 .
3. The op-timal chemical potential µ opt , that corresponds to Ψ maxdSC as a function of U is found on a line in µ - U -plane thatis parallel to the line µ of the plaquette’s | , , Γ (cid:105) - | , , Γ (cid:105) -crossing, even if these two are not the ground-states. Fig. 8 shows this for small 0 . ≤ t b ≤ .
3. InFig. 7 we see a linear t b -dependence of µ at constant fill-ing. We write the linear model to fit the optimal chemicalpotential for a constant t (cid:48) µ opt ( U, t b ) (cid:39) µ ( U ) + µ (1) opt t b . (21) µ ( U ) is calculated on the isolated plaquette and thecoefficient of the linear shift by t b , namely µ (1) opt , is fit-ted to the numerical results of the quadruple Bethe lat-tice, see Tab. I for the coefficients. The maxima in thedoping-dependence of the dSC order parameter Ψ dSC ( δ )have been calculated via quadratic fits to the largest val-ues. The data is presented in Fig. 9 (insets). Fig. 9shows, that the t b -dependence of the Ψ maxdSC is indeed lin-ear. Furthermore, the extrapolation to t b = 0 points to µ of the isolated plaquette, that for U = 4 . t (cid:48) = 0 U . . . . . µ t b = 0 . , t = 0 . µ opt ( U, . U t b = 0 . , t = 0 . t b = . . . . . . µ t b = 0 . , t = 0 . µ opt ( U, . t b = 0 . , t = 0 . t b = . . . . . . µ t b = 0 . , t = 0 . t = 0 . µ opt ( U, . µ opt ( U opt ( t b ) , t b )) µ opt ( U opt (0 . , . ,U opt (0 . dSCAF MP AF M t b = 0 . , t = 0 . t = 0 . t b = . FIG. 8. Phase diagrams of the quadruple Bethe lattice dependent on the chemical potential µ and Hubbard interaction U for several next-nearest-neighbor hoppings t (cid:48) and Bethe hoppings t b . Considered spontaneously broken symmetries are d -wavesuperconductivity (dSC), antiferromagnetism (AFM) and plaquette antiferromagnetism (PAFM). The black lines denote thegroundstate crossovers of the regions N = 2 , , N = 2 , µ opt ( U, t b ) and U opt ( t b ) fits corresponding to Ψ maxdSC , respectively. and U = 2 . t (cid:48) = 0 . maxdSC at t b = 0 . t (cid:48) = 0 and t (cid:48) = 0 .
3. For very small t b the quadruple Bethe lattice turns into isolated plaquettesand dSC vanishes.So far, we have focused on a description in terms ofenergies and thus on µ rather than the observable holedoping δ . In Fig. 9 (insets) we present Ψ dSC dependingon the doping. For small t b t (cid:48) = 0 and t (cid:48) = 0 . δ ∼ .
15, that is the optimal dopingof cuprates . In particular, for the data of t (cid:48) = 0 .
3, atthat we have also calculated solutions of t b = 0 .
5, Ψ maxdSC shifts towards half-filling. It is remarkable, that the max-imum at δ ∼ .
15 is such a stable feature for different t (cid:48) and U at small t b ∼ .
1, i.e. weakly hybridized plaque-ttes. Larger t b make the dSC dome results similar to 2DCDMFT studies at larger temperatures, where the dSCdome is closer to half-filling. In the 2D approximationof CDMFT the hybridization is solely determined by theself-consistency condition and there is no analogue to t b .The present context can raise the question whether long-range correlations that are neglected by 2D CDMFT caneffectively turn the system into more weakly hybridizedplaquettes.With the fit of µ opt ( U, t b ) we can predict optimal dop-ing, next we fit a linear model for optimal Hubbard in- teraction U opt ( t b ) (cid:39) U (0) opt + U (1) opt t b , (22)to find the optimal U opt ( t b ) that maximizes Ψ dSC alongthe line described by µ opt ( U, t b ) in the µ − U phase dia-gram. But contrary to µ opt ( U, t b ), we need to fit the slope U (1) opt and the offset U (0) opt . Furthermore, there is no mo-tiviation from the non-interacting case as in the µ opt -fit.We use it only to estimate the position of the maximumΨ maxdSC within the µ - U phase diagram, also for different t b .Fig. 10 (top) shows the linear fit of U opt , though only fewpoints are taken into account. The fitted models predictthe position, ( µ opt , U opt ), of Ψ maxdSC dependent on t b in the µ - U plane, see Fig. 8.Along the line of t b -dependent ( U opt , µ opt ) the dSCorder parameter exhibits a maximum at t b = 0 .
1, seeFig. 10 (bottom), that is an order of magnitude largerthan the temperature T = 0 .
01 and smaller than theplaquette hopping | t | = 1. The steep slope of Ψ dSC inFig. 10 (bottom) at small t b is difficult to resolve accu-rately since the filling is very sensitive and small errorsin the µ opt -estimate can cause strong noise. The steepslope is caused by the transition of the quadruple Bethelattice into disconnected plaquettes. The t b dependenceof U opt ( t b ) is stronger for t (cid:48) = 0 . t (cid:48) = 0 (Tab. I).In order to sum up the numerical calculations shown . . . . . . t b . . . µ o p t ( U c , t b ) QCP t = 0 . . . . µ o p t ( U c , t b ) QCP t = 0 . . . . δ . . Ψ dSC t b . . . . . . . δ . . Ψ dSC FIG. 9. Chemical potential µ opt , that maximizes the d -wavesuperconducting order parameter Ψ dSC as a function of theBethe hopping t b , U = U c . Linear fits (dashed) are per-formed for the next-nearest neighbor hoppings t (cid:48) = 0 (top)and t (cid:48) = 0 . t b = 0. The quantum critical point (QCP) of the isolatedplaquette is shown, too. Ψ dSC is also shown as a function ofthe hole doping δ (insets) for different t b (color-coded). . . . . . . t b . . . . . Ψ d S C ( µ o p t , U o p t , t b ) U opt ( t optb ) = 3 . U opt ( t optb ) = 2 . . . . . . . U o p t ( t b ) t . . FIG. 10. Top: Linear fit (dashed) of the optimal Hub-bard interaction U opt as a function of the Bethe hopping t b atoptimal doping ( µ opt ). The fit is performed for different next-nearest-neighbor hoppings t (cid:48) separately. Bottom: d -wave su-perconducting order parameter along the optimal µ - U -line asa function of t b . in this section we present an overview of the fitted mod-els of the quadruple Bethe lattice’s µ opt and U opt in thecontext of the isolated plaquette groundstate phase dia-gram, see Fig. 11. At small U the plaquette exhibits atransition from | , , Γ (cid:105) to | , , Γ (cid:105) at µ . For U > U c this crossover is not a groundstate crossover. However, / / U/W plaquette . . . µ µ µ µ µ opt ( U, t b ) t b = . t b = . t b = . Ψ maxdSC U opt ( t b ) | , , Γ > | , , XY > | , , Γ > | , , M > QCP χ pairmax ( ω = 0) PlaquetteQuadruple BetheFIG. 11. Chemical potential µ , Hubbard interaction U -phase diagram of the isolated plaquette (black, solid) withthe groundstates | N, S, K (cid:105) ( t (cid:48) = 0 . W plaquette is the en-ergy range of the plaquette-hopping. The crossover of | , , Γ (cid:105) and | , , Γ (cid:105) is also shown for U > U
QCP , where it is not agroundstate crossover (black, dash-dotted). The highly de-generate quantum critical point (QCP) and the maximum ofthe retarded pairing susceptibility χ pairmax ( ω = 0) of the pla-quette are marked. Linear fits of µ opt ( U, t b ) (red, dashed)and U opt ( t b ) (red, solid) of the quadruple Bethe lattice areshown. The maximum Ψ maxdSC corresponding to the parameterset ( µ opt , U opt , t optb ) is marked by +. µ opt of the quadruple Bethe lattice is parallel to it, in-dicating that the optimal plaquette state superpositionfor dSC requires a certain, t b -proportional, gapsize be-tween | , , Γ (cid:105) and | , , Γ (cid:105) . Upon varying t b , Ψ maxdSC ofthe quadruple Bethe lattice stays in the µ − U diagramcloser to the QCP than to the maximum of the pairingsusceptibilty of the isolated plaquette.The quadruple Bethe lattice effectively provides an en-vironment for the states of the isolated plaquette. Nei-ther of the two distinct points, QCP and χ maxdSC , in µ - U -diagram of the isolated plaquette is the optimal param-eter set for the maximum of the dSC order parameterof the quadruple Bethe lattice Ψ maxdSC . This is due to ef-fective environment shifting dependent on t b the crucialproperties of the QCP, in particular the spectral densitypeak (Fig. 5), to different values of µ and U . The peakat the Fermi level is due to the N = 2 , , maxdSC in Sec. VII. The qualitative behavior around theQCP for different t (cid:48) are very similar despite the QCP be-ing at different ( µ , U ). Thus, at least for small t b ∼ . t b > . | , , Γ (cid:105) , (cid:12)(cid:12) , , XY (cid:11) and | , , Γ (cid:105) become active.Those will also change the optimal doping as shown inFig. 9.In Fig. 11 we choose to present U with respect to theenergy range of the plaquette hopping W plaquette = 4 | t | .This ratio is interesting in a sense that the square lat-tice, that is more accurately applied as a description for0 − . − . . . . . . ∆ U ijkl /U ijkl . . . . . . . . Ψ d S C − . ± . . ± . . ± . − . ± . ijkl XXXXXXY YXY XYXY Y X
X Y ↑↓↑ ↓
FIG. 12. d -wave superconducting order parameter Ψ dSC asa function of the change in different entries of the Hubbard-interaction ∆ U ijkl normalized by its initial value U ijkl ( t (cid:48) =0 . t b = 0 . δ = 0 .
15 and U = 2 . the cuprates, has a bandwidth of W d = 8 | t | and thisestimated factor of 2 = W d /W plaquette puts our resultin a context with U -induced correlations of Mott physicsstudied before with (C)DMFT. With this normalizationthe QCP lies at U/W plaquette ≈ .
75 and the maximumdSC order parameter at
U/W plaquette ≈ .
5, which canbe regarded as intermediate coupling strengths . VI. COMPONENT ANALYSIS OF THEHUBBARD INTERACTION
In Eq. (6) we transform the local interaction U into theplaquette-momentum/orbital basis and observe the exis-tence of many two-particle couplings between the pla-quette momenta , that we classify into intra-orbitalrepulsion, inter-orbital repulsion, spin-flip, pair-hop andcorrelated hopping terms. In this section we investigatethe effect of those on the dSC order, but we restrict thediscussion to the X/Y -orbitals, that are close to Fermilevel and describe the dSC order parameter.Regarding the notation we introduce the tensor U ijkl for convenience. Initially all of its values are either “0”or “ U/ U ijkl by 20%(∆ U ijkl /U ijkl = ± .
2) and observe its effect on the dSCorder parameter. Throughout, we change all terms fallinginto the same class, e.g. a reduction of U XXY Y meansalso a reduction of U Y Y XX . The terms of U ijkl , shownin Fig. 12, have the same degeneracy. Also, we adjust µ so that δ = 0 .
15. Changing certain parts of U ijkl ,we can decrease as well as increase Ψ dSC . Whereas pairhoppings ( U XY XY ) and inter-orbital repulsion ( U XXY Y )promote the dSC, spin flips ( U XY Y X ) and intra-orbitalrepulsion ( U XXXX ) diminish it. By the magnitude ofthe change in Ψ dSC , we can identify two competitions in − − − ω A ( ω ) H S S H µ . . FIG. 13. Spectral function A ( ω ) for different chemical poten-tials µ at approximately half-filling δ ≈
0. For µ = 0 . t (cid:48) = 0 . t b = 0 . U = 2 . . the two-particle processes. First, the pair hopping hasthe same slope as the negative slope of the intra-orbitalrepulsion ( U ∆Ψ dSC / ∆ U ∼ .
23) and second, the spinflip has the same slope as the negative slope of the inter-orbital repulsion ( U ∆Ψ dSC / ∆ U ∼ . δ = 0 .
15 the fluctuations arecharacterized by pair hopping and intra-orbital repul-sion rather than spin flips and inter-orbital repulsion.Both competitions occur between a density-density anda fluctuation term. The dominant contribution to thedSC stems from the pair hoppings that compete withthe intra-orbital repulsion. The two-particle interac-tion terms in the plaquette orbital basis reminds of theKanamori interaction of a multi-orbital atom with pecu-liar values of the Hund’s exchange coupling. Indeed, asupersite formed by only the next-nearest neighbors ofthe plaquette has been proposed for a unified descriptionof the superconductivity in cuprates and pnictides. VII. SPECTRAL PROPERTIES & DOPINGDEPENDENCE
The cuprates become superconducting upon dopingwhereas at half-filling they are insulating. The insulat-ing state is of interest as it can exhibit crucial corre-lations, but without free charge carriers. The theoret-ical concepts of the quantum spin liquid and the res-onating valence bond state originate from this insulatingbehavior . At low temperatures this insulator is hid-den behind antiferromagnetic ordering. Antiferromag-netic correlations and insulating behavior at half-fillingcan be explained by the Mott insulator and the DMFT .The Mott insulator is characterized by a divergence in themass renormalization of the quasiparticles and has alsobeen suspected to affect the dSC .The value of the Hubbard interaction U to model the1cuprates is known only approximately , and it is de-batable whether dSC is a weak- or strong-coupling phe-nomenon. In Fig. 13 we present the density of statesof the quadruple Bethe lattice at δ ≈
0. It is obtainedby the stochastic optimization analytic continuation of the (impurity) Green function. At µ = 1 . µ , but still within the gap,so that δ ≈
0, an asymmetry develops. The hole exci-tation peak becomes sharper and shifts towards Fermilevel. A structure similar to this four-peak structureof two Slater peaks within the Hubbard gap has beenfound in a prior study for t (cid:48) = 0, and is character-istic of Slater physics that include short-range singletcorrelations . Correlated singlets also appear in thedouble Bethe lattice , and define the low-energy ex-citations at intermediate coupling strengths.The hole-doped copper-oxide superconductors have apeculiar phase of the pseudogap at underdoping and tem-peratures above T c . CDMFT studies have shown that itsopening can be related to a topological Lifshitz transitionat that the Fermi surface turns from electron- to hole-like . It can be defined as the point at that the renor-malized quasiparticle energy of the K = X/Y points˜ (cid:15) K = Z K ( t pK + Re Σ K (0)) (23)cross the Fermi level. Z K is the quasiparticle residue.The importance of a particle-hole symmetry has alsobeen pointed out in the dSC state . Particularly forthe Bethe lattice model we can also define a renormal-ized band model for the semicircular density of states ˜ W K = Z K t b . (24)We compare the plaquette momenta of the quadrupleBethe lattice to the high-symmetry points of the Brillouinzone of the square lattice, and thus the Lifshitz transitionis defined by ˜ (cid:15) X/Y .Fig. 14 shows the evolution of the quasiparticle residue Z and the renormalized quasiparticle bands ( ˜ W , ˜ (cid:15) ) withdecreasing µ . We use it to continuously tune the insulatorinto the hole-doped regime. The approximate half-fillingregion δ ≈ . > µ > .
5. The role of µ is here reminiscentof a field effect transistor experiment in that the spectralproperties of the hole excitations change due to the gatevoltage.The Mott phase is found near µ = 1 .
1, in the cen-ter of the gap, where the quasiparticle residue vanishes Z X ≈
0. The system restores coherence in the plaque-tte orbital X with decreasing µ . The renormalized bandmodel assumes that the self-energy makes only small con-tributions and renormalizes the quasiparticles of the non-interacting system. For the Mott insulator this assump-tion is not fulfilled. But for µ (cid:46) . . . . . . µ − − ˜ (cid:15) . . . Z δ ≈ δ > Γ X/YM
FIG. 14. Quasiparticle residue Z (top) and renormalizedquasiparticle energy ˜ (cid:15) and bandwidth ˜ W (bottom) of the nor-mal state as functions of the chemical potential µ , that at acertain value (dashed vertical line) hole dopes δ the system.The K -differentiation of Z is absent in the Fermi liquid (FL)( t (cid:48) = 0 . t b = 0 . U = 2 . level. According to the renormalized band model, thespectral properties change and the Mott insulator devel-ops a correlated Slater peak.In the hole-doped regime δ >
0, we have performedcalculations of the normal state for that dSC order issuppressed (Fig. 14). Thereby we can investigate quasi-particles and their contribution to the dSC mechanism. Z X has a local minimum at the Lifshitz transition ,at that ˜ (cid:15) X = 0. It is related to a strong scattering rateand suggests an avoided criticality mechanism of dSC.In the overdoped region the Fermi surface is electron-like and for large hole dopings the plaquette momentumdifferentiation in Z K is lifted. In this case, a DMFTdescription of a (site-)local self-energy can be sufficientfor a description, and the system enters the Fermi liquidregime.In Fig. 15 we show the plaquette-momentum resolvedspectral function. It is obtained by analytic continua-tion of the (local lattice) Green function and shown alsofor the symmetry-broken dSC state. The Slater peaksdescribe excitations with momentum X/Y . The split-ting of upper peaks and lower peaks is of the order of U . Decreasing the chemical potential shifts the lowerSlater peak to Fermi level, and the dSC order originatesfrom the lower Slater peak, i.e. the dSC gap appearswith the doping of the Slater peak, see Fig. 15. Thespectral function of Fig. 15 looks very similar to Fig. 5,so that it is possible to relate the plaquette transitions | , , Γ (cid:105) → (cid:12)(cid:12) , , XY (cid:11) and | , , Γ (cid:105) → (cid:12)(cid:12) , , Γ (cid:11) to the lowerSlater and Hubbard peaks, respectively. It points out thecrucial part of (cid:12)(cid:12) , , XY (cid:11) , that provides low-energy transi-tions for the electrons that will form the dSC pairs. Fur-ther does (cid:12)(cid:12) , , XY (cid:11) provide a single-particle transition to | , , Γ (cid:105) and the pairs are formed by the latter and | , , Γ (cid:105) (Fig. 3).Fig. 16 (a) shows the doping dependence of the dSC2 − − − − ω A K ( ω ) dSC µ = 0 . δ = 0 . dSC µ = 0 . δ = 0 . µ = 0 . δ = 0 . H S S H U µ = 1 . K Γ XM FIG. 15. Momentum resolved spectral function A K ( ω ) fordifferent dopings δ (corresponding to chemical potentials µ )showing a four-peak structure at half-filling ( δ = 0) of Hub-bard (H) and Slater (S) peaks and for hole-dopings δ the d -wave superconducting gap ( t (cid:48) = 0 . t b = 0 . U = 2 . . order for t b = 0 .
2. We characterize the maximum andthe endpoints by features in the correlation functions ofthe normal state with suppressed dSC order (Fig. 16 (b)-(d)). At half-filling we find the two different solutionsof insulators as discussed in Fig. 14. In the underdopedregime the single-particle excitations are hole-like andthe quasiparticle bandwidth W X is strongly renormal-ized. The renormalization is particularly strong at theLifshitz transition at that the Fermi surface changes fromhole-like to particle-like. At this point is also the max-imum of the dSC dome. At overdoping the quasiparti-cle of the X -orbital becomes more Fermi liquid-like andthe quasiparticle energy shifts away from Fermi level.The renormalized bandwidth broadens and quasiparti-cle states remain at Fermi level at the overdoping end ofthe dSC dome.In the overdoped regime the dynamics of the single-particle correlations do not show any peculiar feature.In order to understand this regime better we present thestatic two-particle correlation functions in Fig. 16 (c).In Eq. (6) we have discussed the transformation of thelocal Coulomb repulsion into plaquette momentum basis.The fluctuation terms between X and Y only, i.e. pairhopping and spinflip terms, appear symmetrically in theinteraction, but the dependence of Ψ dSC is stronger onthe pair hopping part of the interaction, see Sec. VI. Atthe overdoping end of the dSC dome pair hopping andspin flip correlations between X and Y are equally weak.Due to the small Bethe hopping (hybridization) thedSC phase is mostly governed by a few low-energy clus-ter eigenstates. Fig. 16 (d) shows that the dSC order oc-curs only where the Boltzmann weights of (cid:12)(cid:12) , , XY (cid:11) and | , , Γ (cid:105) are non-zero. The large pair hopping correlationsstem mostly from | , , Γ (cid:105) . Only a combination of both, (cid:12)(cid:12) , , XY (cid:11) which produces a peak at Fermi level and pairhopping correlations, results in the non-trivial dome-like . . . . . δ . . . . . . ρ γγ (d) γ , , Γ3 , , XY , , Γ4 , , M − . − . . < c † ↑ i c ↑ j c † ↓ k c ↓ l > pairhopspinflip (c) ijkl XY XYXY Y X − . . . ˜ (cid:15) X (b) . . . Ψ d S C (a)Mott / Slater Lifshitz pair fluctuationsFIG. 16. d -wave superconducting order parameter Ψ dSC ofthe symmetry-broken state (a), renormalized quasiparticleband(width) ˜ (cid:15) X of the normal state (b), static two-particleObservable (cid:104) ... (cid:105) of the normal state (c) and reduced densitymatrix of the normal state ρ with plaquette many-body stateindices γ (d) as functions of hole-doping δ . Points of cer-tain features in the doping dependence are marked by circles. t (cid:48) = 0 . t b = 0 . U = 2 . structure of the dSC order. At the overdoping end ofthe dSC dome the Boltzmann weight of the spin-triplet | , , M (cid:105) exceeds that of | , , Γ (cid:105) , and pair hoppings correlvanish which suppresses the dSC.Fig. 17 is a detailed view of Fig. 15 with more valuesfor δ . It shows the dSC gap of the one-particle spec-tral function. Finite hole doping and dSC order set inwith a sharp peak below Fermi level and a small peakabove. The latter grows until at optimal doping the dSCgap is approximately particle-hole symmetric. From op-timal doping to overdoping the peak of hole excitationsshifts through the Fermi level increasing spectral weightat Fermi level until the gap is closed and dSC order isabsent. In contrast to the lower edge the upper edge of3 − . . . ω . . . . . . . A ( ω ) . . . . ω δ = 0 . δ = 0 . δ = 0 . δ = 0 . δ inthe dSC state ( t (cid:48) = 0 . t b = 0 . U = 2 . . the dSC gap does not shift with doping. It suggests thattwo distinct mechanisms contribute to the formation ofthe dSC gap in the one-particle spectral function . VIII. SUPERCONDUCTIVITY &ANTIFERROMAGNETISM
Using a two-by-two plaquette as cluster, we can de-scribe AFM and dSC order on equal footing, and both arerelevant for the phase diagram of the cuprates. In Fig. 18we observe that it is largest at half-filling, at that accord-ing to experimental findings, the N´eel temperature is alsolargest. In contrast to the hole doped cuprates we findcoexistence of AFM and dSC order up to δ = 0 . and isexpected to arise from the neglecting of long-ranged cor-relations. In fact, already an eight-site cluster can sup-press dSC in proximity of half-filling .The maximum value of the AFM order parameter isΨ AF M = 0 .
25, which is only half the magnitude of theplaquette’s full magnetization. This is the case, becausetwo electrons are locked in the singlet of the Γ-orbital,that is fully occupied and doesn’t touch the Fermi-level,see Fig. 6. Finite values for Ψ
AF M we find only for t b ≥ .
3, it increases sharply as function of t b and sat-urates around t b = 0 . AF M = 0 .
25. This is verydifferent from the dSC order parameter, that has its max-imum around t b = 0 . X and Y , AFM order couples also Γ and M , which aresplit and farther from the Fermi level. . . . . δ . . . . . . Ψ t b = 0 . AF MdSCπSC . . . . t b δ = 0FIG. 18. Order parameters Ψ of antiferromagnetism (AFM), d -wave superconductivity (dSC) and spin-triplet supercon-ductivity ( πSC ) dependent on hole doping δ for Bethe hop-ping t b = 0 . t b for half-filling δ = 0(right) ( U = 2 . t (cid:48) = 0 . Moreover, we observe spin-tripletsuperconductivity ( π SC) with the order parameterΨ πSC = 14 (cid:88) RR (cid:48) (cos [ X ( R − R (cid:48) )] − cos [ Y ( R − R (cid:48) )]) × e iMR (cid:48) (cid:104) c R ↑ c R (cid:48) ↓ + c R ↓ c R (cid:48) ↑ (cid:105) . (25)It is described by entries of the correlation functionsthat are off-diagonal in Nambu and plaquette-momentumspace, see Eq. (14). Further, a comparison with Eq. (20)shows also that it is a combination of AFM and dSC asit breaks the spatial symmetries of the plaquette accord-ing to AFM and dSC. Finally, the symmetry upon spin-exchange can be seen explicitly in Eq. (25) and stressesthe spin-triplet character. We find non-zero values forΨ πSC only at dopings for the the quadruple Bethe lat-tice also shows coexistence of dSC and AFM. Thus πSC is a result of the interplay between dSC and AFM. IX. EXTENDED BETHE LATTICE HOPPING
To this point the Bethe hopping exists only within oneBethe lattice and is represented by a scalar. In this sec-tion we introduce the Bethe-hopping matrix in plaquette-site space t b = t b t (cid:48) b t b t (cid:48) b t (cid:48) b t b t (cid:48) b t b (26)with the extended Bethe-lattice hopping t (cid:48) b , that appearsin entries, that in the case of the plaquette hopping ma-trix t p are occupied by the next-nearest neighbor hop-ping. It means, that for t (cid:48) b the transition between plaque-ttes is accompanied by a transition to the next-nearest4 − . − . . . . t b /t b − . . . ˜ (cid:15) X . . Ψ d S C A ( ω ) t b /t b . FIG. 19. Superconducting order parameter Ψ dSC (top) andrenormalized quasiparticle energy ˜ (cid:15) X (bottom) as functionsof the extended Bethe lattice hopping at δ = 0 .
15 ( U = 2 . t (cid:48) = 0 . t b = 0 . W . Non-Interacting semicircu-lar density of states for different next-nearest neighbor Bethelattice hoppings (inset). neighbor of the target plaquette. The effect of the non-diagonal terms in the t b -matrix is the finetuning of thewidths of the semicirculars independently. The nearestneighbor components would affect only the widths of theΓ and M bands, so we have set them to zero for simplic-ity.The CDMFT self-consistency becomes G − ( iω n ) = ( iω n + µ ) − t p − t b G ( iω n ) t b (27)with the quantities being matrices in plaquette site space(Nambu degrees of freedom are omitted for convenience),and the last term are matrix products of t b and G .The transformation Eq. (4) must be applied to t b , too.Thus it gets diagonalized. The non-interacting semicircu-lar density of states changes so that the width and heightof G , M differ from those of X , Y , but they remain semi-circulars, see Fig. 19 (inset). Depending on the sign of t (cid:48) b /t b the height increases and the width decreases or viceversa. Despite the absence of a real divergence the effectof an increased density of states of X and Y can be in-teresting in the context of the van Hove singularity in the square lattice.Fig. 19 shows that the dSC order parameter increaseswith a decreasing quasiparticle bandwidth. The dopingis set to δ = 0 .
15 that remains independent of t (cid:48) b relatedto the Lifshitz transition, i.e. the quasiparticle energy˜ (cid:15) X is almost constant. The change in Ψ dSC is small, andtherefore in Fig. 19 we also present errorbars, that arecalculated as the largest absolute deviation of eight valuesof two CDMFT-loops with local Green functions thatby symmetry have four entries of the order parameter.The quasiparticle residue of the presented calulcationsonly weekly depends on t (cid:48) b , i.e. stays Z X = 0 . ± . t (cid:48) b is almost entirely due to the renormalization ofthe non-interacting X/Y -band. The modification of the semicircular density of statesby t (cid:48) b is small and finite, but it already shows an en-hancing effect on Ψ dSC . The van Hove singularity cor-responds to an infinite density of states and can poten-tially enhance that effect much more. The role of thenext-nearest neighbor hopping t (cid:48) in the cuprates is cum-bersome. Whereas a comparison of bandstrucutre calcu-lations with experiments show a finite value for t (cid:48) as theoptimal one , calculations in the framework of strongcorrelations are not able to confirm this by includingonly local correlations. First, we observe that t (cid:48) shiftsthe quantum critical point of the plaquette, which inthe quadruple Bethe lattice is in proximity to the max-imum Ψ dSC , to smaller U . And second, we find thata hopping similar to t (cid:48) , i.e. t (cid:48) b , can have an enhancingeffect on Ψ dSC . It reduces effectively the bandwidth of X towards the optimal value t b ≈ . . X. CONCLUSION
We have formulated the CDMFT self-consistency,Eq. (10), that solves the quadruple Bethe lattice exactly,also in the d -wave superconducting state. An analysis ofthe isolated two-by-two cluster has shown that this pla-quette is even without an environment unstable towardsdSC order. The coupling to other plaquettes in the in-finite dimensional quadruple Bethe lattice allows for thespontaneous symmetry breaking. dSC order is found inproximity of a QCP of the plaquette in the µ - U diagram,where the plaquette eigenstates | , , Γ (cid:105) , (cid:12)(cid:12) , , XY (cid:11) and | , , Γ (cid:105) cross. The optimal value for the parameter thatcontrols the hybridization of plaquettes is t b = 0 . δ = 0 .
15. The latter has also beenmeasured in experiments on the cuprates.The dSC dome of the doping phase diagram lies next toa half-filling state with a vanishing quasiparticle residue,characteristic of the Mott insulator. Moreover, at half-filling the renormalized quasiparticle picture shows acrossover to an insulator with a correlated Slater peakwith decreasing µ . The hole excitations correspond-ing to the Slater peak occur around the energy of the | , , Γ (cid:105) → (cid:12)(cid:12) , , XY (cid:11) -transition of the isolated plaquette.At hole doping this hole excitation forms the supercon-ducting gap. The small density of states at Fermi level re-stricts the local pair formation in the underdoped regime,a Lifshitz transition occurs at optimal doping, and atoverdoping the superconductivity is suppressed by thevanishing of the two-particle pair-hopping correlations.For large t b = 0 . t b = 0 . π SC. Whereas AFM is staggered within eachof the four Bethe lattices of the quadruple Bethe lattice,dSC and π SC are homogenous in those.
ACKNOWLEDGMENTS
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