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CuO ν 1 9( 09 ≈ < x < y 4 0 . ≈ 5 where 75, aolvA Kharkov A. Yaroslav h C ssmeared is QCP the 0 . 0 47). . T 5 e Refs. see 15, c 2 8–10 superconduc- unu rtclpoint critical quantum Cu r / 3 emosi w-iesoa antcsse hti clos is that system magnetic two-dimensional a in fermions 2 ssprto ewe h emos h ehns fthe of mechanism The fermions. the between separation is O 7 tlarger At 6+ tdop- at y 6 the 1–3 4 , 1 n lgP Sushkov P. Oleg and hneo natfroantc(F utain(param- fluctuation ex- agnon). (AF) via antiferromagnetic Fermi interact an (large electrons of change approach picture this liquid Fermi Within normal surface). of frame the in odcigpiig hsie a enrcnl consid- Chubukov recently and been Wang has by idea ered This pairing. conducting fermions. of pairing d-wave the Goldstone in result the approaches both of exchange this via In magnon. interact/pair surface. holes Fermi case small implies necessarily instead, hresprto tteQCP. the at separation charge the in incommensurability mag- aside cuprates. put commensurate we only so cuprates. has ordering, netic multi-layer here presented and model single-layer equally The However, the are conclusions the to our calculation. for applicable conceptually controlled model that a believe bilayer we the performing inter- consider of We the sake antiferro- by bilayer driven coupling. the fluctuations layer use magnetic we with host dramatic. magnet insulator most Mott the on the is criticality As coupling fermions magnetic two the strong between of coupling influence therefore the case and this In surface limit. 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Fermi ref- previous large works the earlier all some edge Ref. also in are erenced There cuprates. doped h ffc fsi-hresprto onsott the familiar to not are out We points problem. pairing separation cloud. the magnon spin-charge of divergent nontriviality of by dressing effect to The due spin hole of antcciiaiycnsgicnl nunesuper- influence significantly can criticality Magnetic h oe ne osdrto eosrtsspin- demonstrates consideration under model The 2D the in injected holes two consider we work this In nst ut-annecag processes exchange multi-magnon to onds nbtentefrin,wt potential with fermions, the between on 11 12 h ihl oe FMt nuao approach, insulator Mott AF doped lightly The cnt fteQP edemonstrate we QCP, the of icinity ne h uecnutn oeat dome superconducting the under u otesrn nst ubr repulsion Hubbard on-site strong the to Due olmi rgnlymtvtdby motivated originally is roblem ia nacmn fpiigin pairing of enhancement tical 14 oee,t h eto u knowl- our of best the to However, . 1 geial ree and ordered agnetically ,Australia 2, cnt famagnetic a of icinity 13 16 nacneto electron of context a in tmasdelocalization means It 15 e u tl in still but , 2 with any models that consider pairing of two fermions, that incorporate physics of spin-charge separation. In order to probe the interaction between two fermions we consider spin fluctuations in the system, keeping the fermions to be immobile and spatially localized, just as magnetic impurities.17 Because of the immobility, Fermi statistics of the impurities is not actually relevant for our results, however for shortness we will call the impurities as ”fermions” hereafter. Our calculations show that the single magnon exchange is getting irrelevant close to the QCP. Instead, we obtain strong inter-fermion attraction in singlet and triplet spin channel due to Casimir effect.18 Figure 1: Bilayer J J⊥ antiferromagnet model. Two black dotes on the top layer− represent holes. Each of injected fermions (holes) builds up a ”bag” of the quantum magnetic fluctuations. The fermions at- tract to each other, sharing common bag and reducing AF reads energy of the magnetic fluctuations inside of the bag. So-called ”spin-bag” mechanism of attraction in the con- H = J (S(1) S(1) + S(2) S(2))+ J,J⊥ i · j i · j text of high-Tc superconductivity in cuprates is famil- hi,ji iar from the works of J. Schrieffer et al.19, that however X J S(1) S(2), (1) did not account for the magnetic criticality, neither small ⊥ i · i i Fermi surface. The spin bag model has also conceptual X similarity to QCD bag models for nucleon binding such 20 21 The superscripts (1), (2) in Eq. (1) indicate the layers, as MIT and chiral bags that have being extensively i, j denotes summation over nearest neighbour sites. studied from 1970’s up to now. h i (1) 1 † Here Si = 2 ciµ,1σµν ciν,1 is spin of an electron at site i The structure of the paper is following. In the Section † on the top plane and ciσ,1/ciσ,1 is creation/annihilation II we introduce bilayer J J⊥ antiferromagnet, which − operator of an electron with spin σ = , at site i, σµν are is simple but instructive model and contains all essen- Pauli matrices. The Hamiltonian describes↑ ↓ the antiferro- tial physics of magnetic criticality. In the Section II A magnetic coupling in the each layer as well as between the we characterize magnetic quantum critical point driven two layers. It is known that without holes (half-filling) by interlayer coupling J⊥/J and describe magnon exci- the model has an O(3) magnetic QCP at J⊥/J = 2.525 tations for undoped AF in disordered phase in the frame (see Refs.22–25) separating the AF ordered and the mag- of spin-bond mean field theory. Next, in the Section II B netically disordered phase of spin dimers. Note that since we move to the hole-doped J J model and show how − ⊥ we consider zero temperature case, the magnetic ordering holes interact with magnons. In the Section III, which is in the AF phase is consistent with the Mermin-Wagner the main content of the paper, we consider hole-hole pair- theorem. We dope the first layer with two holes. For ing problem at the QCP and show that pairing can not simplicity we set hopping integrals equal to zero, there- be described in terms of one-magnon exchange. In Sec- fore the holes are immobile. The holes interact with each tion III A we develop effective theory for Casimir interac- other via magnetic fluctuations of the spins, i.e. exchang- tion of the fermions, considering double-fermion ”atom” ing by magnons. which can be either in singlet or triplet state. In Section In the subsections A and B of the current section we III B we present results of solution to Dyson’s equations will briefly present formalism, which describes magnon for singlet and triplet Green’s functions and finally show excitations and hole-magnon interaction on the basis of how binding energy in both spin channels depends on the bilayer model. For more detailed explanations see16; inter-fermion distance r. Finally, we draw our conclu- a reader which is not interested in these technical details sions in the Section IV and provide supplementary ma- can go directly to the Section III. terial in the Appendix.
A. Magnons at QCP
Magnetic excitations in the magnetically disordered II. MODEL phase are magnons, which are also in literature called triplons. In the present paper we will use terms magnons and triplons as synonyms. To describe the magnons Our model system is J J⊥ square lattice bilayer we employ the spin-bond operator mean field technique. Heisenberg antiferromagnet− at zero temperature, where This approach has been previously applied to quantum magnetic fluctuations are driven by interlayer coupling disordered systems such as bilayer antiferromagnets, spin 26–31 J⊥ (see Fig. 1). The Hamiltonian of the undoped host chains, spin ladders, Kondo insulators etc. . It is 3 known27,28 that this simple technique gives the position by introducing infinite on-site repulsion of triplons; how- of the QCP at (J /J) 2.31, which is close to the ex- ever, this technique is quite involved. Another, more ⊥ c ≈ act value (J⊥/J)c =2.525 known from Quantum Monte simple, way is to employ mean-field approach, account- Carlo calculations22,23, series expansions24, and involved ing for the constraint (2) via a Lagrange multiplier µ in analytical calculations with the use of the Brueckner the Hamiltonian technique25. The spin-bond technique being much sim- † † pler than the Brueckner technique has sufficient accuracy HJ,J⊥ HJ,J⊥ µ (si si + tiαtiα 1). (5) → − − i for our purposes. X The bond-operator representation describes the system Further analysis is straightforward. We replace sin- in a base of pairs of coupled spins on a rung, which can glet operators by numbers, s† = s =s ¯ (Bose- either be in a singlet or triplet (triplon) state. So, we h i i h ii † † † † Einstein condensation of spin singlets); and diagonalize define singlet si and triplet (tix,tiy,tiz) operators that the quadratic in t Hamiltonian by performing the usual create a state at site i with spin zero and spin one, which Fourier and Bogoliubov transformations is polarized along one of the axises (x,y,z). The four types of bosons obey the bosonic commutation relations. To restrict the physical states to either singlet or triplet, 1 t = eiqri u b + v b† . (6) the above operators are subjected to the constraint iα N q qα q −qα r q X † † Here N is the number of spin dimers in the lattice; the s s + t t =1. (2) i i iα iα diagonalized Hamiltonian reads α X † In terms of these bosons, the spin operators in each layer Hm(µ, s¯) = E0(µ, s¯)+ ωqbqαbqα, (7) (1) (2) q Si and Si can be expressed as X where ω = A2 4B2 and coefficients A = J⊥ µ + q q − q q 4 − (1,2) 1 † † † 2 2 S = ( s tiα t si iǫαβγt tiγ ), (3) 2Js¯ γ , B =qJs¯ γ . Here we define iα 2 ± i ± iα − iβ q q q 1 see Ref.32. Substituting the bond-operator representa- γ = (cos(q ) + cos(q )) . (8) q 2 x y tion of spins defined in Eq. (3) into the HJ,J⊥ in Eq. (1) we obtain The lattice spacing is set to unity. The ground state energy
2 HJ,J⊥ = H1 + H2 + H3 + H4, 3J s 3 E (µ, s¯)= N ⊥ µs¯2 +µ + (ω A ) (9) 3 1 0 − 4 − 2 q − q H = J s†s + t† t , q 1 ⊥ −4 i i 4 iα iα X i X just shifts energy scale, and therefore is irrelevant for J † † † † H2 = (si sjtiαtjα + si sj tiαtjα + h.c.), our purposes. The Bogoliubov coefficients uq and vq are 2 given by hXi,ji J † † H3 = iǫαβγ(t t tiγ sj + h.c.), A 1 A 1 2 jα iβ q q hi,ji uq = + , vq = sign(Bq) . (10) X s2ωq 2 − s2ωq − 2 J H = (t† t† t t t† t† t t ). 4 2 iα jβ iβ jα − iα jα iβ jβ The parameters µ ands ¯ are determined by the sad- hi,ji X dle point equations: ∂E0(µ, s¯)/∂µ = ∂E0(µ, s¯)/∂s¯ = 0. (4) Solution to these equations gives position of the QCP at J⊥/J = 2.31 and values of ”chemical potential” The Hamiltonian (4) contains quadratic, cubic and quar- µ = 2.706 and singlet densitys ¯ = 0.906. We see that tic terms in magnon operators t. The most important even− at the QCPs ¯ is close to unity, which again justi- for us are the quadratic terms, because they provide fies smallness of the nonlinear terms H3 and H4 in the quantum criticality. The only effect due to the nonlinear Hamiltonian. terms H3 and H4 is renormalization of parameters near The dispersion of magnons is the QCP, such as position of the QCP, magnon velocity, 2 2 2 magnon gap and etc. This does not affect physics at the ωk = c (k Q) + ∆ , Q = (π, π), (11) QCP, and therefore we will neglect these terms in further − considerations. in the vicinity ofp the wave-vector Q, here ∆ is the magnon The next step for treating the Hamiltonian (4) is to ac- gap and c is the velocity of magnons c = 2Js¯2 = 1.64J, count for the hard-core constraint (2). It could be done where the more precise value is c = 1.9J, see Ref.24. In 4 the AF ordered phase the magnons are Goldstone bosons and thus necessarily gapless. On the contrary, in the = + + disordered phase the gap opens up and the spin-bond approach gives ∆ (J J ) which is not far from ∝ ⊥ − ⊥,c the prediction for O(3) universality class systems ∆ + . . . (J J )η with critical index η = 0.71 (see Ref.33∝). ⊥ − ⊥,c So, the spin-bond method provides a reasonably accurate Figure 2: Dyson’s equation for single hole Green’s function in description of the QCP. Self-Consistent Born Approximation. Solid and waivy lines correspond to hole and magnon Green’s functions.
B. Hole-magnon interaction that the infrared divergence of the fermion-magnon cou- We dope our system with two immobile holes, by re- pling at the QCP results in strong power-law attraction moving two electrons from the upper plane of the bilayer between the fermions. antiferromagnet. Hence we define the hole creation op- Importance of spin-charge separation for single- † fermion problem at the QCP could be seen from analysis erator a with spin projection σ = , by its action on iσ of analytical structure of the hole Green’s function. Stan- the spin singlet bond s ↑ ↓ | i dard approach in order to calculate one-fermion Green’s function is to use 1/ expansion for the O( ) group, N N a† s = c† 0 , a† s = c† 0 , (12) where = 3 is the number of magnon components. i↑| i i↑,2| i i↓| i i↓,2| i SummationN of leading terms in the expansion arises in where 0 is vacuum. The electron creation/annihilation Self-Consistent Born Approximation (SCBA), see Fig. 2. operator| i in the upper plane can be expressed in terms of † 29 hole creation/annihilation operators aiσ/aiσ (see Ref. ), and after substitution in (1) it gives following part of the Hamiltonian which describes hole-magnon interaction } ) ǫ Js¯ ( † † G
H = (t + t )σ + (t + t )σ { hm − 2 j j i i i j − hXi,ji n o Im /π
J † † 1
i(σ [t t ]+ σ [t t ]). (13) − 2 i j × j j i × i hXi,ji − − − − − − Here σ = a† σ a . The first line in the Hamilto- i iµ µν iν ǫ/J nian (13) corresponds to hole-magnon interaction ver- tex. The terms, describing hole-double-magnon vertex, Figure 3: Spectral function 1/π Im G(ǫ) of a single immo- − 34 { } which come from the second line of (13) will be neglected bile hole obtained in SCBA (see ). The green dashed curve below, because they are irrelevant in the infrared limit. corresponds to the QCP (∆ = 0), and the black solid line corresponds to the magnon gap ∆ = 0.1J. Note that at the Performing again standard Fourier and Bogoliubov trans- QCP the quasiparticle pole disappears. formations (6) the Hamiltonian (13) can be rewritten as
qr qr Calculations of the hole Green’s function in the disor- H σα g (b ei i + b† e−i i ). (14) hm ≈ i q qα qα dered phase in SCBA have been performed in Refs.17,34. i q X X The results show that away from the QCP, in the disor- The hole-magnon vertex is equal to dered magnetic phase, quasiparticle pole in the fermion Green’s function is separated by ∆ from incoherent part Js¯ of the Green’s function. But when approaching to the gq = γq(uq + vq). (15) −√N QCP the Green’s function instead of normal pole has just branch cut singularity. This is a consequence of the Note, that at the QCP the vertex diverges at q Q = → infrared singularity of the coupling constant gq. (π, π), because of the singularity of Bogoliubov coeffi- Spectral density of the fermion Green’s function (see cients uq, vq 1/√ωq . ∝ → ∞ Fig. 3) has inverse square root behaviour G(ǫ) The divergence of gq is crucial for the physics of 1/√ǫ ǫ in the vicinity of the singularity point ǫ and∝ 0 − 0 fermion-magnon coupling at the QCP. In fact, it re- quasiparticle residue is approaching to zero Z √∆ at sults in the phenomenon of spin-charge separation for the QCP. Here ∝ the single-fermion problem.16 The spin of the hole is dis- tributed in the power-law divergent cloud of magnons and ǫ0 0.97J (16) in this sense is separated from the charge of the hole, lo- ≈− calized on the hole’s site. Later in the paper we will show is the position of the branching point of the Green’s func- 5 tion and has a meaning of fermion energy shift due to interaction with magnons, where we set bare energy of √Z λΓω=0 √Z the hole to zero.
q,ω = 0 III. HOLE-HOLE INTERACTION, MEDIATED BY MAGNONS
λΓω=0 Now we are ready to move to the actual problem of √Z √Z magnon mediated pairing of fermions and demonstrate new results. Adding up magnon Hamiltonian Hm, Eq. Figure 4: One-magnon exchange diagram that provides (1) (7) and hole-magnon interaction Hamiltonian Hhm, Eq. fermion-fermion interaction potential Vint (q). Note that (14), we arrive to effective Hamiltonian for two interact- renormalization factor √Z should be referred to each fermion ing holes, located at the sites with coordinates r and line. Fermion-magnon vertices also come renormalized λ 1 → r2, λΓω=0.
† Heff = ωqbqαbqα + λΓω q X = + + . . . α iqri † −iqri σi gq(bqαe + bqαe ). (17) i=1,2 q q,ω X X The Hamiltonian (17) is applicable only if distance be- Figure 5: The fermion-magnon vertex. tween holes r = r1 r2 > 1, as long as we put aside direct exchange| interaction− | between two neighbouring λ λΓ . Here ω is the frequency of the exchange holes. → ω=0 The effective model (17) can be formulated in the lan- magnon, which is equal to zero. In the coordinate repre- guage of a field theory. In fact, it is equivalent to the sentation the potential reads problem of two spin 1/2 fermions coupled to a vector λ2 r∆ φ r (1) 2 2 Qr σ σ field ( ), described by O(3)-symmetric theory with La- Vint (r)= 2 Z Γω=0 cos( ) 1 2 K0 , grangian −2πc h i c (20) 2 2 1 2 c 2 ∆ 2 where K0 is the Macdonald function of zero’s order. The = (∂ φ) (∇φ) φ (1) t (18) potential energy V (r) ln(r) is logarithmic at small L 2 − 2 − 2 − int ∝ λ(φ(r1)σ1 + φ(r2)σ2), distances r
Creation operator for singlet state is 1 Ψ† = (a† a† a† a† ) (22) S √2 1↑ 2↓ − 1↓ 2↑ and for triplet state 1 Ψ† = − (a† a† a† a† ), T,x √2 1↑ 2↑ − 1↓ 2↓ Figure 6: Dependence of spin channel which provides attrac- † i † † † † tion between holes on a mutual positioning of the holes in ΨT,y = (a1↑a2↑ + a1↓a2↓), (23) the lattice. Two holes with spins up symbolicaly represent √2 triplet channel which provides negative interaction energy for † 1 † † † † rx+ry ΨT,z = (a1↑a2↓ + a1↓a2↑). Pr = ( 1) = +1, two holes with opposite spins repre- √ − 2 sent singlet channel which results in attraction when Pr = 1 − . According to the selection rules for interaction of the ”atom” with magnon, there are three types of tran- sitions S Tα, Tα Tβ and Tα S, where An approach to evaluate Casimir interaction energy → → → S means singlet state and Tα denotes triplet state between two spins can be drawn from analogy with with polarization α. The only one invariant kine- calculation of Van der Vaals force between two atoms. matic structure that provides coupling between S and The most elegant way to do so was first developed by T, α states with emission (absorption) of one magnon is Dzyaloshinsky. The Van der Vaals potential is given by g (q)Ψ† Ψ (b + b† )+ h.c. . In similar way, tran- ”box diagram” with two-photon exchange between the ST T,α S qα qα atoms.37 nsition of the type Tα Tβ iso governed by the term † → † √Z √Z igT T (q)εαβγ ΨT,αΨT,β(bqγ + bqγ). The coefficients gST (q) λΓω λΓ−ω and gT T (q) are coupling constants for these transitions. Therefore, the interaction of two-fermion system with magnon field in the singlet-triplet representation reads q,ω q, ω − † † = δαβΨT,αΨS gST (q)(bqβ + bqβ)+ h.c. + λΓω λΓ−ω H ( q ) √Z √Z X † † iεαβγΨT,αΨT,β gT T (q)(bqγ + bqγ). (24) Figure 7: The box diagram for two-magnon exchange between q fermions. Note that the renormalization factor √Z should be X referred to each external fermion line. The effective vertices can be calculated by evaluating matrix elements of the Hamiltonian (17) between states We can try to follow this approach to calculate Casimir (22), (23) : pairing energy between two fermions in the vicinity of the qr g (q)= g∗ (q)=2ig sin , QCP. The diagram is shown in the Fig. 7. However, like ST TS q 2 in one magnon exchange diagram, we have to account for qr (25) g (q)=2g cos . renormalization of external fermion lines, that results in T T q 2 additional factor Z2 0. In such logic we again obtain zero for pairing energy→ at the QCP, and therefore we need Let us define retarded Green’s function for the singlet another technique to solve the problem. and triplet state
G (t t )= i 0 Ψ (t )Ψ† (t ) 0 θ(t t ), T,αβ 2 − 1 − h | T,β 2 T,α 1 | i 2 − 1 A. The ”Lamb shift” technique for calculation of G (t t )= i 0 Ψ (t )Ψ† (t ) 0 θ(t t ), (26) Casimir interaction S 2 − 1 − h | S 2 S 1 | i 2 − 1 where 0 is a ground state of the system and theta- In this section we introduce a new technique to treat function| i is Casimir pairing energy. To incorporate ”Casimir effect” physics, we consider composite two-fermion ”atom”, 1,t> 0; θ(t)= (27) which has total spin either zero (singlet state) or one 0,t< 0. (triplet state). Next, we calculate ”Lamb shift” in energy of this composite ”atom” due to radiation of magnons as Due to the O(3) rotational invariance the triplet Green’s a function of separation between fermions. function should be of the form GT,αβ(t) = δαβGT (t). Let’s consider effective theory for the composite object. Note that our definition of the Green’s functions G (t S 2 − 7
or introduce effective momentum cutoff and integrate an- ΣS (ǫ)= alytically over angle in momentum space and then inte- grate over radial component of the momentum q′ = | | q Q Λq 1 numerically. We have checked that there| − is| a ≤ good≈ agreement between these two methods. ΣT (ǫ)= + However, the effective momentum cutoff method is much more efficient for numerics and provides better precision Figure 8: Diagrams for singlet ΣS (ǫ) and triplet ΣT (ǫ) self- of the computations, therefore we mostly used the later energies in SCBA. Double dashed line and dashed line rep- approach. resent double-fermion Green’s function in singlet and triplet The limit r of infinite separation between the channels correspondingly. Analytical expressions for the dia- fermions corresponds→ ∞ to the case, when the vertices in grams are given in Eq. (30). the equations (30) are subsituted to the averaged ones 2 2 2 over q oscillations gST (q) , gT T (q) 2 gq . The position of singularity| of triplet| | and singlet| → Green’s| | func- t ), G (t t ) assumes that the fermions, which consti- 1 T 2 − 1 tions gives us energy E∞ of the two-fermion system sep- tute the composite ”atom”, are both created at the same arated by infinite distance. It is clear that in such limit moment of time t1 and then both annihilated at the mo- the Green’s functions in both spin channels should coin- ment t . Fourier transform of Eq. (26) gives the Green’s 2 cide GS(ǫ) = GT (ǫ) which guarantees the same value functions in the frequency representation for the asymptotic energy E∞ in singlet and triplet ∞ states. So, we refer interaction energy as a difference i(ǫ+iη)t Vint(r)= E(r) E∞. GS,T (ǫ)= dte GS,T (t), (28) − (a) (b) Z0 where η = +0. Dyson’s equations for singlet and triplet state Green’s functions read 1 GS,T (ǫ)= . (29) ǫ ΣS,T (ǫ)+ iη − Figure 9: Diagrams contributing to E∞. Top and bottom We use SCBA to evaluate singlet and triplet self-energy solid line correspond to single hole Green’s function. Diagram (a) is accounted in SCBA (30), diagram (b) is not included 2 in (30) and corresponds to 1/ correction to SCBA. ΣS(ǫ) =3 gST (q) GT (ǫ ωq), N q | | − X Σ (ǫ)= g (q) 2G (ǫ ω )+ T ST S q (30) We found that the value E∞ 1.55J (at the QPC) q | | − ≈ − X is about 20% smaller compared to doubled energy of an 2 isolated single hole 2ǫ 1.94J, see Eq. (16). This 2 gT T (q) GT (ǫ ωq). 0 ≈ − q | | − difference is due to the fact that certain diagrams, which X are presented in SCBA for single hole Green’s function, The diagrams for the singlet and triplet self-energies are are not included in SCBA for the two hole Green’s func- presented in Fig. 8. The combinator factors here come tion GS,T (see Fig. 9). This deviation shows precision of from contraction of the corresponding tensor structures our method, which can be improved by calculating 1/ N of the coupling vertices in (24) and have a meaning of corrections to SCBA (30). the number of the polarizations of intermediate state. As it’s seen from the structure of effective vertices ′ g (q) 2 =2g2(1 P cos q r), | ST | q − r 2 2 ′ (31) B. Solution to Dyson’s equations for singlet and gT T (q) =2g (1 + Pr cos q r), | | q triplet states and hole-hole interaction energy the system’s behaviour greatly depends on the ”parity” rx+ry Pr = ( 1) of the inter-fermion distance. The holes In order to find interaction energy of two fermions, prefer− to form singlet (triplet) spin state for negative we numerically solve the system of two Dyson’s equa- (positive) ”parity” Pr at given r. tions (30) in the square Brillouin zone for different inter- First, let us consider the case, when the system is away fermion separations r, measured in units of lattice spac- from the QCP, ∆ > 0. We plot spectral functions ings. Energy grid in our computation is ∆ǫ = 10−3J. (0) 1 The zero approximation Green’s function is GS,T (ǫ) = AS,T (ǫ)= Im GS,T (ǫ) (32) 1/(ǫ+iη) and for artificial broadening we take η = 10−3J. −π { } In order to perform numerical integration in equations for singlet and triplet Green’s functions at different r, (30) we can directly integrate over square Brillouin zone, see Fig.10 (a, b). We see well defined quasiparticle peak 8
r = 4 (a)r = 5 (b) 0 00 ) ) ∆ = 0.08J ǫ ǫ ( ( 0 0
S,T S,T A A 0 0
− − − − − − /J ) 0 0 0 00 r ǫ/J ǫ/J ( int
V 0 0 0 0 Figure 10: Spectral functions AS(ǫ), AT (ǫ) of double-fermion Green’s functions in singlet and triplet channels close to the 0 0 QCP (∆ = 0.08J). Panel (a) corresponds to inter-hole dis- 0 0 ∆ = 0.67J tance r = 4 and (b) corresponds to r = 5. Blue dashed lines 0 0 0 correspond to triplet state, red solid lines correspond to sin- 0 glet state. 0 0 0 r in the triplet (singlet) spin channel at r = 4 (r = 5). Figure 11: Interaction energy of two holes Vint(r) at finite However, in the opposite spin channel the peak broadens magnon gaps as a function of inter-hole distance r. Red tran- and submerges to continuum. This effect can be inter- gles and blue squares show the results of the ”Lamb shift” preted as a formation of an excited decaying state, which technique in singlet and triplet state. Red and blue solid lines coupled via magnons to the ground state. represent theoretical prediction from one-magnon exchange In Fig. 11 we plot the fermion-fermion interaction mechanism (20) for singlet and triplet channels. The main energy versus distance. The inset displays the interac- plot corresponds to small magnon gap (∆ = 0.08J), the inset tion energy when the system is away from the QCP (the corresponds to large magnon gap (∆ = 0.67J). magnon gap is large, ∆ = 0.67J). Squares and triangles show results of our ”Lamb-shift” technique calculations and solid lines represent the single magnon exchange for- mula (20). There is an excellent agreement between the (a) two approaches. The main part of Fig. 11 shows the r = 20 same quantities, but close to the QCP (the magnon gap is r = 4 small, ∆ = 0.08J). Here we observe a dramatic disagree- ) ǫ ( ment between the result of the ”Lamb-shift” technique − − − and the single magnon exchange potential (20). The sin- S,T A gle magnon exchange approximation fails in the vicinity of the QCP. Let us now consider the most interesting case of pairing − − − between fermions at the QCP (∆ = 0). As in the case ǫ/J of a single fermion at the QCP, the Green’s functions GS(ǫ) and GT (ǫ) have just power-law cuts, instead of (b) r = 21 quasiparticle peaks, with branching point E = E(r), see r = 5 Fig. 12 (a, b). The position E(r) of the branching point ) ǫ gives the ground state energy of the system. Spin channel ( − − − of the ground state is specified by the spin state in which S,T the Green’s function is singular at E(r). The state, in A which the Green’s function is not singular, corresponds to decaying state. Imaginary part of both singlet and − − − triplet Green’s functions emerges at the same branching ǫ/J point E(r) for any fixed r (see Fig. 12). This is due to transitions between the states with emission of soft Figure 12: Spectral functions AS(ǫ), AT (ǫ) of double-fermion magnons with ωq 0. In the Fig. 12 (b) we see distinct Green’s functions in singlet and triplet channels at the QCP discontinuity of both→ singlet and triplet spectral functions (∆ = 0). On the panel (a) the main plot corresponds to inter- at the same branching point E(r). In the Fig. 12 (a) the hole distance r = 4, the inset plot to r = 20. On the panel (b) the main plot corresponds to r = 5, the inset plot corresponds branching points are also coincide, but singlet spectral to r = 21. Red solid lines show singlet state, blue dashed lines function has small spectral weight in the vicinity of the show triplet state. branching point. Our results for the interaction energy Vint(r) at the 9
QCP as a function of distance r, obtained within the triplet channels. We calculate the attraction energy due ”Lamb shift” technique, are presented in Fig. 13. We see to multi-magnon exchange processes as a ”Lamb shift” of from the data that the interaction between two fermions energy of a two fermion ”atom” due to emission of mul- is attractive, when the ”parity” Pr is negative (positive), tiple magnons. The fermions interact, sharing common see Fig. 6. The binding becomes stronger at smaller ”bag” of magnetic fluctuations and reducing energy of inter-fermion distances r. The interaction energy has a fluctuations inside of the ”bag”. Therefore, the physics power-law form of inter-fermion attraction in the vicinity of the QCP is due to ”Casimir bag” mechanism. V (r)= a/rν , ν 0.75 (33) int − ≈ with prefactor a 0.3J, where ν and a are found from IV. CONCLUSIONS the least-square fit≈ of our numerical data. The values for prefactor a and power exponent ν are slightly different for singlet (a = 0.3J, ν = 0.76) and triplet (a = 0.33J, In conclusion, we considered interaction between two ν = 0.74) cases. The variations of the values of a and ν spin 1/2 fermions embedded in a two dimensional antifer- are negligible within the accuracy of our calculations. romagnetic system at the QCP, which separates ordered and disordered magnetic phases. As a model system we study bilayer antiferromagnet at T = 0 with two injected 0 00 holes, in which magnetic criticality is driven by inter- layer coupling. We have shown that in the vicinity of the 0 0 QCP the interaction between fermions can not be de- scribed by simple one-magnon exchange, unlike the case
0 0 when the system is away from the QCP. The interac-
/J tion mechanism is similar to Casimir effect and is due to ) 0 0 r
( multi-magnon exchange processes. To incorporate fea- 0
int tures of Casimir physics we developed a new approach,
V 0 which we call a ”Lamb shift” technique. We considered 0 0 composite two-fermion ”atom” and calculated it’s energy 0 shift (”Lamb shift”) provided by radiation of magnons. 0 We found strong attraction between the fermions in spin 0 0 0 0 singlet and triplet states depending on the ”parity” of the 0 0 0 0 0 inter-fermion distance r, which is positive (negative) for even (odd) r. Positive (negative) ”parity” corresponds r to attraction in triplet (singlet) channel. The attractive potential has power-law form V (r) 1/rν with the exponent ν 0.75. ∝ − ≈ Figure 13: Interaction energy Vint(r) of two holes at the QCP, We suppose that our work sheds light on the influence ∆ = 0. Red trangles and blue squares show the results of the of magnetic criticality on fermion pairing mediated by ”Lamb shift” technique for singlet and triplet state. Red and ν magnons. We also believe that our results are conceptu- blue solid lines represent power-law fits Vint(r) = a/r for ally applicable to cuprates. singlet and triplet channel. The main plot corresponds− to SCBA, in the inset we represent Vint(r), which includes first 1/ correction to the SCBA. The exponent for all curves is V. AKNOWLEDGEMENTS approximatelyN ν 0.75. ≈ We gratefully acknowledge A. Chubukov, G. Khaliullin In the inset in Fig. 13 we show Vint(r) which includes vertex corrections to SCBA (30). Leading in 1/ correc- and I. Terekhov for useful discussions. This research was N supported by Australian Research Council (Grant No. tions for the singlet and triplet self-energy δΣS and δΣT are presented in Appendix (see diagrams in Fig. 14 and DP110102123). formulas (A1), (A2)). These corrections increase binding by about 20%, leaving critical index ν almost unchanged. Thus we conclude that corrections in 1/ to SCBA do not change qualitative and quantitative picture,N given by SCBA. From our calculations we observe very strong long- range attraction between fermions in the vicinity of the QCP. We clearly see that one magnon exchange contri- bution to the interaction energy vanishes at the QCP. On the contrary, accounting for multi-magnon exchange processes we obtain significant binding in singlet and 10
Appendix A: Leading 1/ corrections to SCBA for two-fermion Green’sN function
Let us consider 1/ corrections to the self-energies δΣS (ǫ)= + N ΣS(ǫ) and ΣT (ǫ), calculated in Self-Consistent Born Ap- δΣT (ǫ)= + + proximation (see Eqs. (30)). In order to do this we ac- count for vertex corrections δΣS(ǫ) and δΣT (ǫ) to self- energies obtained in SCBA, corresponding diagrams are shown in Fig. 14. The vertex correction to the singlet self-energy reads Figure 14: Diagrams for the leading 1/ corrections δΣS (ǫ) N and δΣT (ǫ) (a) to singlet and (b) to triplet self-energies. Dou- δΣ (ǫ)=3 g (q) 2 g (k) 2G G G ble dashed line and dashed line represent two-fermion Green’s S | ST | | ST | T,q T,k S,qk− q k function in singlet and triplet channels correspondingly. X, ∗ ∗ 6 gST (q)gST (k)gT T (k)gT T (q)GT,q GT,kGT,qk. q k Here we are using shorten notations Gn,q = Gn(ǫ ωq), X, − (A1) Gn,k = Gn(ǫ ωk) and Gn,qk = Gn(ǫ ωq ωk) for singlet and triplet Green’s− functions (n = S,T− ).− One can check The combinator factors come from contractions of the that in the limit r the correction δΣS will be sup- → ∞ (2 loop) corresponding tensor structures of the effective vertices pressed by the factor 1/ =1/3 with respect to ΣS,T in Eq. (24). In the similar way the vertex correction to calculated in SCBA withinN two loop approximation. the triplet self-energy is given by The relative shift of binding energy, calculated with and without vertex correction, does not exceed 20%. It δΣ (ǫ)= g (q) 2 g (k) 2G G G can be considered as a confirmation of applicability of T | ST | | ST | S,q S,k T,qk− q,k 1/ expansion for the effective ”Lamb-shift” theory de- X N ∗ ∗ scribed by the Hamiltonian (24). 2 gT T (q)gT T (k)gST (q)gST (k)GT,q GT,kGS,qk+ (A2) q k X, 2 g (q) 2 g (k) 2G G G . | T T | | T T | T,q T,k T,qk q k X, 1 M. A. Hossain it et. al., Nat. Phys. 4, 527 (2008). 17 M. Vojta, C. Buragohain, and S. Sachdev, Phys. Rev. B 2 R-H He et. al., New J. Phys. 13, 013031 (2011). 61, 15152 (2000). 3 H.-B. Yang et. al., Phys. Rev. Lett. 107, 047003 (2011). 18 H.B.C. Casimir, D. Polder , Phys. Rev. 73, 360 (1948). 4 N. Doiron-Leyraud, C. Proust, D. LeBoeuf, J. Levallois, J. 19 J. R. Schrieffer, X. G. Wen, and S. C. Zhang, Phys. Rev. B. Bonnemaison, R. Liang, D. A. Bonn, W. N. Hardy, and Lett. 60, 944 (1988). L. Taillefer, Nature 447, 565 (2007). 20 A. Chodos, R. L. Jaffe, K. Johnson, and C. B. Thorn, Phys. 5 B. Vignolle, A. Carrington, R. A. Cooper, M. M. J. French, Rev. B 10, 8 (1974). A. P. Mackenzie, C. Jaudet, D. Vignolles, C. Proust and 21 G.E. Brown, M. Rho, Phys. Lett. B 82, 177-180 (1979). N. E. Hussey, Nature, 455, 952 (2008). 22 A. W. Sandvik and D. J. Scalapino, Phys. Rev. Lett. 72, 6 Z. Liu and E. Manousakis, Phys. Rev. B 45, 2425 (1992). 2777 (1994). 7 A. I. Milstein and O. P. Sushkov, Phys. Rev. B 78, 014501 23 A. W. Sandvik, A. V. Chubukov, and S. Sachdev, Phys. (2008). Rev. B 51, 16483 (1995). 8 C. Stock, et al, Phys. Rev. B 77, 104513 (2008). 24 ZhengWeihong, Phys. Rev. B 55, 12267 (1997). 9 V. Hinkov, et al, Science 319, 597 (2008). 25 V. N. Kotov, O. Sushkov, ZhengWeihong , and J. Oitmaa, 10 D. Haug, et al, New J. Phys., 12, 105006 (2010). Phys. Rev. Lett. 80, 5790 (1998). 11 D. J. Scalapino, Phys. Rep. 250, 329 (1995); P. Monthoux 26 M. Vojta and K. W. Becker, Phys. Rev. B 60, 15201 and D. Pines, Phys. Rev. B 47, 6069 (1993); A. Abanov, (1999). A. V. Chubukov, and J. Schmalian, Adv. Phys. 52, 119 27 Y. Matsushita, M. P. Gelfand, and C. Ishii, J. Phys. Soc. (2003). Jpn. 68, 247 (1999). 12 M. Yu. Kuchiev and O. P. Sushkov, Physica C 218, 197 28 D. K. Yu, Q. Gu, H. T. Wang, and J. L. Shen, Phys. Rev. (1993); V. V. Flambaum, M. Yu. Kuchiev, O. P. Sushkov, B 59, 111 (1999). Physica C 227, 267 (1994). 29 C. Jurecka and W. Brenig, Phys. Rev. B 63, 094409 (2001). 13 Y. Wang and A. V. Chubukov, Phys. Rev. B 88, 024516 30 R. Eder, Phys. Rev. B 57, 12832 (1998). (2013). 31 Y. Saito, A. Koga, and N. Kawakami, J. Phys. Soc. Jpn 14 P. Krotkov and A. V. Chubukov, Phys. Rev. Lett., 96, 72, 1208 (2003). 107002 (2006). 32 A. V. Chubukov, JETP Lett. 47, 129 (1989); S. Sachdev 15 E. G. Moon and S. Sachdev, Phys. Rev. B 80, 035117 and R. Bhatt, Phys. Rev. B 41, 9323 (1990). (2009). 33 J.Zinn-Justin, Quantum field theory and critical phenom- 16 M. Holt, J. Oitmaa, W. Chen, and O. P. Sushkov, Phys. ena (3rd ed.), Oxford University Press, (1996). Rev. Lett. 109, 037001 (2012); Phys. Rev. B 87, 075109. 34 O. P. Sushkov, Phys. Rev. B 62, 12135 (2000). 11
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