Marginal Fermi Liquid in a lattice of three-body bound-states. A. F. Ho1,2 and P. Coleman1,2 1Serin Laboratory, Rutgers University, P.O. Box 849, Piscataway, New Jersey 08855-0849 2Department of Physics, University of Oxford, 1 Keble Road, Oxford OX1 3NP, UK We study a lattice model for Marginal Fermi liquid behavior, involving a gas of electrons coupled to a dense lattice of three-body bound-states. The presence of the bound-states changes the phase space for electron-electron scattering and induces a marginal self-energy amongst the electron gas. 8 9 When the three-body bound-states are weakly coupled to the electron gas, there is a substantial 9 window for marginal Fermi liquid behavior and in this regime, the model displays the presence 1 of two relaxation times, one linear, one quadratic in the temperature. At low temperatures the bound-states develop coherence leading to a cross-over to conventional Fermi liquid behavior. At n strong-coupling, marginal Fermi liquid behavior is pre-empted by a pairing or magnetic instability, a J and it is not possible to produce a linear scattering rate comparable with the temperature. We discuss the low temperature instabilities of this model and compare it to the Hubbard model at 2 half-filling. ] l e - 72.15.Nj, 71.30+h, 71.45.-d r t s . t a m I. INTRODUCTION In this paper, we return to the original proposal, ask- ing whether a marginalFermi liquid can form in a dense - d The conceptofamarginalFermiliquid(MFL) wasin- electronicsystem. Wepursueanearlyspeculation,dueto n vented as a phenomenological description of the asymp- Ruckenstein and Varma9, that the marginal self-energy o totic properties of of high temperature superconductors might derive from the scattering of conduction electrons c in their normal state1. These include a linear electrical off a dispersionless localized bound-state Φ at the Fermi [ resistivity ρ T over at least 2 decades in tempera- energy, giving rise to the interaction: 1 ture and a qu∼asi-particle scattering rate which may be 4v proportional to frequency up to energies as high as 750 Hint =λ(c+i↑ci↑c+i↓Φ+h.c.). (3) 0 meV.2,3 Even in underdoped cuprate superconductors, ThepresenceofsuchstatespreciselyattheFermisurface 0 optical data indicates that marginal behavior develops would mean that the three-particle phase space grows 1 at scales above the spin gap.4 We shall take a marginal linearly with energy. Inelastic scattering into the local- 0 Fermi liquid to be a system of fermions with an inelastic ized bound-state then leads to a marginal self-energy in 8 scattering rate of the form leading order perturbation theory, as illustrated in Fig. 9 1. (The localised state is represented by the dash line.) / Γ max[ ω ,T], (1) t ∝ | | Since the hypothetical object at the Fermi surface scat- a m wherethereisnosignificantmomentumdependenceofΓ. ters electrons in triplets, Ruckenstein and Varma identi- Inthecuprates,theconstantofproportionalityisoforder fied it as a three-body bound state. - d unity. Analyticitythenensuresthattheappropriateself- n energy takes the form o c Σ(ω) ωlnmax[ ω ,T]. (2) : ∼ | | v i Many proposals have been made to explain the origin of X this unusual behavior. One prevalent idea, is that the r marginal Fermi liquid behavior derives from scattering a off a soft bosonic mode. This idea underpins the Van Hove scenario5, gaugetheory models6 and the quantum- critical scattering7 picture of the cuprates. By contrast, Anderson8 proposes that the cuprate metal is a fully FIG. 1. Marginal self-energy diagram for the band elec- developed Luttinger liquid, with power-law self-energies trons. Notation: thick lines denote band electron propaga- which have been mis-identified as a logarithm. Since the tors, dashed lines denote thelocalised state propagator. soft-modetheoriesfurnishanelectronself-energywhichis strongly momentum dependent, none of these proposals actuallygivesrisetoamarginalFermiliquidasoriginally Agreatdifficulty with this picture is that itcannotbe envisaged. made self-consistent. At the same level of perturbation 1 Ψ(1)(0)Ψ(2)(0)Ψ(3)(0)=AΦ(0), (6) where A is the amplitude for forming this pole Φ. The residualinteractionwiththebulkspindegreesoffreedom in the low energy world then gives rise to a vertex of the form λΨ(1)Ψ(2)Ψ(3)Φ. The challenge here is to see if suchamechanismcouldbegeneralisedtoamorerealistic lattice model. The work described below is an attempt to make a first step in this direction. FIG.2. Self-energy for thelocalised state. II. CONSTRUCTION OF MODEL We now use these ideas to motivate a simple lattice model of a marginal Fermi Liquid. First note that the theory that furnishes a marginal self-energy, the three- Hubbardmodelathalf-fillingcanberewritteninaMajo- body state must scatter off the electrons to produce a ranafermionrepresentation11,bythefollowingtwosteps. self-energy term of the form shown in Fig 2. This self- The Hubbard model is: energy correction inevitably moves the resonance of the three-body bound-state away from the Fermi energy, in- HHubbard =t (c†i+a,σci,σ+H.c.) troducing an unwanted scale into the problem and caus- i,a,σ ing the singular scattering to disappear. X An unexpected resolution of this problem recently ap- +U (c†i↑ci↑−1/2)(c†i↓ci↓−1/2). (7) peared in the context of the single impurity two-channel i X Kondo model10. Marginal Fermi liquid behavior does Inthefirststep,weassumethatthelatticehasabipartite develop in this model, and the mechanism by which it structure,anddoagaugetransformationonelectronop- occurs is remarkably close to the original three-body eratorsbelongingtoonesublatticeofthebipartitelattice: bound-state proposal,with one critical difference: in the c ic . Also let the hopping term connect a sub- iσ iσ → − two-channel Kondo model, the three-body bound-state latticeAsiteonlytoasublatticeBsite,andnottoother formed at the impurity site carries no internal quantum sublattice A sites. Then the interaction is unchanged, numbers (spin and charge). The associated bound-state but the kinetic energy becomes: fermion is represented by a Hermitian operator: HK.E. =it (c†i+a,σci,σ +H.c.). (8) Φ=Φ†. (4) i,a,σ X ThistypeoffermionisknownasaMajoranafermion14 . Next,rewritetheelectronoperatorsusingc = 1 (Ψ(1) ∗ ↑ √2 − The effective action for such a field must have the form: iΨ(2)),c = 1 (Ψ(3)+iΨ(0))wherethe Ψ(a)’sareMa- i↓ −√2 jorana fermions, and we get: S = dωΦ( ω)(ω Σ(ω))Φ(ω). (5) − − 3 Z H =it Ψ(a)Ψ(a) Since Φ( ω)Φ(ω) = Φ(ω)Φ( ω) due to the Grassma- Hubbard i+c i − − − i,c a=0 nian nature of Φ, Σ(ω) will be an odd function of fre- XX quency, so that Σ(0) = 0. In other words, the particle- U Ψ(0)Ψ(1)Ψ(2)Ψ(3). (9) − i i i i hole symmetry of the Φ field guarantees that its energy i X is pinned to the Fermi surface. (Note the sign change for the interaction.) In Majorana In the two-channel Kondo model (after dropping representation,weseeexplicitlytheSO(4)symmetry1112 the charge degrees of freedom, thanks to spin-charge of the Hubbard model at half-filling. decoupling16), the total spin S~ of the two conduction The crucialgeneralisationofthis paperis to breakthe channels at the impurity site can be written in the form: SO(4) symmetry down to O(3), setting: S~ = iΨ~(0) Ψ~(0) where Ψ~ = (Ψ(1),Ψ(2),Ψ(3)) is a −2 × t : a=1,2,3 triplet of Majorana fermions. These three high energy t t = . (10) degrees of freedom bind at the impurity site to form the → a t0 : a=0. (cid:26) localisedthree-bodybound-stateΦ(0),representedasthe With Φ Ψ(0), contraction of the three fermions: i ≡ i 3 H =it Φ Φ +it Ψ(a)Ψ(a) 0 i+c i i+c i i,c i,c a=1 X XX ∗Such an object can always be constructed as a linear com- U Φ(0)Ψ(1)Ψ(2)Ψ(3). (11) bination of two charged fermions Φ= √12(a+a†). − Xi i i i i 2 When t = 0, this Hamiltonian describes a lattice of lo- conductivity has the classic marginal Fermi liquid be- 0 calised three-body bound-states Φ coupled to the con- haviour. i tinuum. Within the marginal Fermi liquid regime, the Ψ(a) re- This toy model provides a simple system to study the tainsits marginalself-energy. TheΦ mode,while having properties of these pre-formed bound-states at each site. no self-energy corrections at T = 0 and ω = 0, acquires A microscopic model would provide an explanation of a Fermi liquid on-site self-energy because of the virtual the originof this symmetry breaking field leading to for- fluctuations into the three Ψ(a) (with the same diagram mation of these bound-states. In this work, we do not as Fig.3, except that site i is the same as site j). In address this issue, but instead, ask whether the single this regime then, our model has 2 distinct quasiparticle impurity marginal Fermi Liquid mechanism survives in relaxation times: the lattice, in this model of reduced symmetry. We shallshow that the main physicaleffect of the lat- ΓΦ = ImΣΦ(ω) ω2+π2T2 (14) − ∝ tice (in the absence of a brokensymmetry phase) is that Γ = ImΣ (ω) max[ ω ,T] (15) Ψ Ψ the previously localised mode Φ can now move from − ∝ | | i site to site via virtual fluctuations into the Fermi Sea: This suggests an intriguing link to the two relaxation see Fig.3, thereby providing the lattice coherence energy time phenomenology observed in the electrical and Hall scale t U2/t, below which this marginal Fermi Liq- conductivitiesofthecuprates13,whereelectricalconduc- 0 ∼ uid revertsto a FermiLiquid. Unfortunately,this means tivity is dominated by the slower relaxation rate, while that the marginal Fermi liquid phenomenology does not theHallconductivityisproportionaltotheproductofthe persist to large U, which makes the application of our two relaxation times. Unfortunately, in our model, the model to the cuprates rather problematic. conservedcurrentjadoesnotincludetheΦfermion,thus transport quantities constructed from this O(3) current donotrevealtheslow,quadraticrelaxation. Butanother conservedquantity in our model is the totalenergy. Fol- Φ Φ lowing the same strategy as for the O(3) current, it can i j be shown that the conserved thermal current is just a sum of currents due to each of the Φ and Ψ(a). Now the propagators are diagonal in the Φ or Ψ operators, thus the thermal conductivity proportional to the ther- mal current-current correlator will just be a sum of the relaxation times: κ/T 3t2 + t20 . Also, any mixed FIG.3. Leadingorderdiagram thatgeneratesadispersion ∝ ΓΨ ΓΦ correlators jaQ0 will be identically zero, where Q0 is for the Φi fermion. Note that there are no arrows on the h i the thermalcurrentdue to Φ. Insummary,it isunfortu- propagatorlines,asallfermionsarerepresentedbyHermitian natelyimpossibletoseethetworelaxationtimesentering operators. multiplicatively inany transportquantities ofourmodel: the various conductivities derived from the O(3) current To probe the marginal fermi liquid phenomenology, will only depend on Ψ(a), whereas the thermal conduc- we have calculated the “optical conductivity”. This tivity will be dominated by the largest relaxation time: cannot be the ordinary electrical current, because the that of the Φ fermion. Hamiltonian of Eqn.15 does not conserve particle num- Allofourpreviousconsiderationsassumethatthesys- ber unless t = t (this is most easily seen with orig- tem does not develop into a broken symmetry state at 0 inal electron operators c,c+). The model does have low temperatures. In fact, as our Hamiltonian is a gen- the O(3) symmetry, so there is the conserved quantity eralisationofthe Hubbardmodelathalf-filling,it is per- Sa = 1 3 Ψ(α)Ta Ψ(β), where Ta are the haps not surprising that it displays similar magnetic or −2 j α,β=1 j αβ j charge ordering instabilities, due to Fermi surface nest- three generators of O(3). In the representation where P P ing. The main qualitative difference is the presence of a Ta =iǫ , αβ aαβ large marginal Fermi Liquid regime in the T U phase − diagram. i Sa =−2 ǫabcΨ(jb)Ψ(jc). (12) The plan of the paper is as follows: in Section III, we j set out the formalism of Dynamical Mean Field Theory X forsolvingthelatticemodelintheweakcouplingregime. This leads to the conserved (Noether) current: Section IV presents the results and Section V discusses the lattice coherence scale, and the relationship to the jia+xˆ =itǫabcΨ(i+b)xˆΨ(ic). (13) two-channelKondo lattice. We alsodiscuss the low tem- perature phase of the lattice when the Fermi surface has (See Methods for its derivation.) One can then define a strong nesting instability. Finally, we touch on the diffi- “conductivity” which is the linear response of this O(3) culties this model faces in modelling the cuprates. currentto anapplied field. We shallshow thatthis O(3) 3 III. METHOD the lattice, and is not the originallattice non-interacting localGreen’s function. Itplays the role analogousto the We study the Hamiltonian: Weiss mean field for conventionalmean field theory, and has to be determined self-consistently: see below. 3 The effective single-site problemdressedGreen’s func- H =it˜ Φ Φ +it˜ Ψ(a) Ψ(a) 0 j+c j j+c j tion G is related to via: j,c j,c a=1 G X XX −U ΦjΨ(j1)Ψ(j2)Ψ(j3). (16) Σa(iωn)=Ga−1(iωn)−G−a1(iωn), a=0,1,2,3, (19) j X where the self-energies Σ are calculated from S . To a eff where each fermion is a canonical Majorana fermion: lowest order in U, they are given by the following dia- Ψ(a),Ψ(b) =δ δ , Φ ,Φ =δ ,and Ψ(a),Φ =0. grams in Fig.4. { i j } ab ij { i j} ij { i j} To gain some insight into the properties of the model in the weak coupling limit, we use dynamical mean field theory(DMFT)18,amethodsuitedtosystemswherethe ψ ψ dominant interaction is on-site and spatial fluctuations i i are unimportant, and when the on-site temporal fluctu- ations at all energy scales are to be taken into account. The d limit (where DMFT is exact), requires the usual s→cali∞ng t˜ = t /√d , with t ,U O(d0). For a a a a lattice of localised bound states Φ , t˜ =∼0. j 0 ThecrucialsimplificationinDMFT isthatthe selfen- Φ Φ ergies are k-independent as any intersite diagram (such i i as in Fig.3) is at most of order 1/√d relative to on-site diagrams. Now the diagram of Fig.3 is precisely that which causes the Φ fermion to propagate: omitting it means that t remains at zero, and Φ stays localised, 0 i strictly in infinite spatial dimension. In finite d, we will need to incorporateits effect, since propagatingΦ fields FIG.4. On-siteself-energies. i will lead to the destruction of the marginal scattering mechanism. A rigorous 1/d expansion appears to be formidable. Nevertheless, the essential effects of finite The effective single site dressedGreen’s function must dimensionsmaybeincludedbyintroducingafinitevalue then be related back to the k average of the original − for t˜0, and treat it as a fixed parameter of the model. lattice dressed Green’s function Ga(k,iω), via the mean Defining t˜ = t /√d, we estimate t by calculating the field self-consistency equations18: 0 0 0 diagram (Fig.3) in finite d at T =0. In Appendix A, we sahnodwctihsaat st0m/atll=cocn(Ust/atn)t2./dF,inwaitllhy,tade=fintefaorzearo=th1,c2o,m3-, Ga(iωn)= (2dπdk)dGa(k,iωn), Z ponent Ψ(0) Φ to get: j ≡ j where since in d limit, self-energies have no → ∞ k dependence, G (k,iω )=1/(iω ǫ (k) Σ (iω )). 3 a n n a a n H =i ta Ψ(a) Ψ(a) U Ψ(0)Ψ(1)Ψ(2)Ψ(3), D−oing the integration then gives18: − − √d j+c j − j j j j j,c a=0 j XX X 1(iω )=iω +it sgn(ω ), a=0,1,2,3 (20) t : a=1,2,3 Ga− n n a n t = (17) a t : a=0. (cid:26) 0 for the Lorentzian density of state (DOS): Da(ǫ) = We shall use both Φ and Ψ(0) interchangeably. ta/(π(ǫ2+t2a))(correspondingtoinfiniterangehopping); j j or alternatively: Following standard procedures of DMFT18, we map the lattice problem to an effective single-site problem 1(iω )=iω t2G (iω ), a=0,1,2,3 (21) with the effective action: Ga− n n− a a n β 3 for the semi-circular DOS Da(ǫ) = π1ta 1−(ǫ/2ta)2 S = dτdτ Ψ(a)(τ) 1(τ τ )Ψ(a)(τ ) (correspondingto nearest-neighbourhopping ona Bethe eff ′{ Ga− − ′ ′ p Z0 a=0 lattice). Equations17-19and20or21,andFig.4together X β define our DMFT. The Lorentzian DOS is tractable an- U dτ(Ψ(0)Ψ(1)Ψ(2)Ψ(3))(τ), (18) alyticallyasthe self-consistencyequationsaredecoupled − Z0 from G; however this also means that the effect of the where (τ)isthedynamicalmeanfieldatthesinglesite lattice enters rather trivially as just a renormalisationof a G which includes time dependent influence of the rest of the bandwidth. To check these results we also use the 4 semi-circular DOS where the self-consistency equations IV. RESULTS are solved computationally using iterated perturbation theory19. In the d limit (ie. t =0), is the same as the 0 0 One quantity of particular interest in the context of single impu→rit∞y model, by Eqn 19 Gor 20, so the bound- the marginalFermi liquid is the opticalconductivity. As states described by Ψ(0) are localised, with a self-energy mentioned inSection II, the Hamiltonianof Eqn.17does i that has a Fermi liquid form: not conserve particle number unless t = t, thus the or- 0 dinaryelectricalcurrentproportionaltotheparticlecur- 4ω πN Σ (ω+)= (UN )2 +i 0(ω2+(πT)2) , (25) rentisnotuseful. Wecanhowevergeneralisetheconcept Φ − 0 π 2 of the optical conductivity to this model, using the fact (cid:26) (cid:27) that the total isospin20 Sa = i −21Ψ(iα)TαaβΨ(iβ) is con- where ω+ = ω +i0+ and N0 is the DOS at the Fermi served,whereTa =iǫ aretheO(3)generators. Com- surface. We give a brief derivation of this result in Ap- biningthecontiαnβuityeqauαβation:Pi∂τSib+ aˆ(jib+aˆ−jib)=0 peffeencdtiixveBs.inFgolreΨsi(itae),paro=bl1e,m2,i3s,othfethmeesaanmfieelfdorGmaaosf tthhee (aˆ are the unit lattice vectors), and the equation of mo- tion ∂ Sb =[H,Sb] leads to the conservPed current: single impurity model bare Green’s function, using the τ i i Lorentzian DOS for the d lattice. This is due to → ∞ jb =itǫ Ψ(α)Ψ(β). (22) the DOS in the effective problembeing smooth at Fermi i+xˆ bαβ i+xˆ i energy, and for T t, equaling a constant N , just as 0 ≪ in the single impurity model10 We obtain the marginal This is the Noether current associated with the O(3) self-energy: symmetry of our Hamiltonian. We can then introduce adivmeecntosiropnoatlesnptaiacel,ficeoldupA~le(da)(t~xo)t=he(eAle(1ac)tAro(2na)s·a·s·Afo(dlalo))wisn: d ΣΨ(ω+)= (UN0)2ω lnΛ Ψ 1 iω + πT , (26) − T − − 2πT iω (cid:20) (cid:18) (cid:19) (cid:21) H =it Ψ(α) exp i d~l A~(a)Ta Ψ(β), (23) where Λ is a cut-offproportionalto t, Ψ is the Digamma A i+c · i function, γ 0.6 is the Euler constant (Appendix B). i,c (cid:26) Z (cid:27)αβ ∼ X This has the following limiting behavior where the line integral goes from site i to i+c, and a ( ωln2πΛ iπ ω ), (ω T) summationisimpliedoverdummyindices. Sincethereis − ω − 2| | ≫ Σ (ω+)=(UN )2 | | itsoottrhoeptyhfeorA~a(1)=c1o,m2p,3o,newnetnjex(1e)d(ωo)nl=y stuydσyxyth(ωe)rAes(y1p)o(nωs)e, Ψ 0 × (−ωlnΛTeγ −iπT), (ω ≪T) which can be described by a Kubo forPmula21:  (27) 1 At finite d, the lattice coherence energy scale t gen- 0 σ (iω )= Π(~q,iω ) , xx n ω n erated from the diagram of Fig.3 becomes finite, with Π(~q,iωn)=−− nβdτeiωnτh(cid:12)(cid:12)(cid:12)(cid:12)Tq~τ=j0x†(~q,τ)jx(~q,0)i. stt0ist∼<sUot2lto/hnte.gMtahsarertge0-in<baolTdFy≪erbmotui.nlidqAu-tsitdlaotbweeeshrabtveeigmoiunpretwroaitlplurnroeopswaTgpae≪tre- Z0 0 coherently,causingacross-overtoFermiliquidbehavior. In the d limit, the absence of vertex corrections to Thisisborneoutbyanalyticalcalculations;hereweillus- the cond→uct∞ivity bubble18 permits us to write: tratewithcomputationalresults(using the semi-circular DOS) in Fig.5 showing the effective quasiparticle scat- σ(νm)= T (vkx)2Ga(k,iωn+iνm)Ga(k,iωn). tering rate Γ(ω) = ωReσ(ω)/Imσ(ω) ImΣa(ω+), ∼ − X~k Xωn where σ(ω) is the optical conductivity defined in Meth- ods. In Fig.6 we plot the the resistivity ρ(T) showing As usual, at temperatures much lower than the band- the largelinearT regimeatweakcoupling,andthe inset width, doing the Matsubara sum leads to a function shows the crossoverto the T2 Fermi Liquid regime. peaking largely near k , and we replace (v )2 by F ~k kx n/m dǫ. Doingtheenergyintegralandanalyticallycon- P tinuing to real frequencies: V. DISCUSSION AND CONCLUSION R n +∞ f(ω ) f(ω+) σ(ν+iδ)= dω − − In this paper we have shown how a lattice of three- m iν Z−∞ (cid:20) − (cid:21) body bound-states induces marginalFermiliquid behav- [ν (ΣR(ω+) ΣR(ω ))+i(Γ(ω+)+Γ(ω ))]−1 (24) ior above a lattice coherence temperature to where a × − − − − Fermi liquid forms. Since t U2/t, a substantial win- 0 where Σ(ω i0+)=ΣR(ω) iΓ(ω) and ω =ω ν/2. dowintemperatureformargin∼alFermiLiquidbehaviour ± ∓ ± ± existsonlyforsmallU. The emergenceofthis lattice co- herenceenergyisexpectedtobequitegeneral: whenever 5 a localised mode is allowed to interact with conduction electrons,itwillbedifficulttopreventhybridizationsbe- tween fields at different sites. These effects however, are missed in a strict d calculation. 1.2 →∞ This brings us onto the question of the relationship U = 0.2 t 03 t between our lattice model and the two-channel Kondo 1.0 t = 0.08 t 0. -2 0 T = lattice model. The single-impurity version of the Hamil- 0 tonian of Eqn.11 (with U < 0) was originally derived10 * 1 0.8 = 0.01 t imnotdheel2c4o.nTtehxattodfetrhiveastiinognlet-oimokpuadrivtaynttwagoe-chofanspnienl-Kchoanrdgoe / t 0.6 T separation16 to throw away uncoupled (charge) degrees ω) of freedom, and it has been shown via bosonization to Γ( 0.4 be exactly equivalent to the original model17. Unfortu- 0.2 T = 0.0 2 t nately there is no such relation between the two-channel Kondo lattice and the Majorana lattice considered here. In the above calculations, we have assumed that the 0 marginalFermiLiquidstateisunstableonlytotheFermi 0.1 0.2 0.3 0.4 0.5 Liquid state at low T. In fact, just as in the Hubbard ω /t model,ameanfieldcalculationindicatesthatforaFermi surface with a strong nesting instability (for example, nearestneighbourhoppinginahypercubiclattice),there is a phase transitionto antiferromagneticorder(for U > FIG. 5. Plot of quasi-particle scattering rate 0). TheorderparameterisavectorthatreflectstheO(3) Γ(ω)=ωReσ/Imσ. symmetry of the model: Va(~xj)=eiQ·xjhc†ασαaβcβi 1 a =ieiQ·xj Ψ(0)Ψ(a) Ψ~ Ψ~ , (28) 4.0 − 2 × (cid:28) (cid:16) (cid:17) (cid:29) 0.06 where Q=(π,...,π) is the nesting vector. From the di- 3.5 vergence in the susceptibility, we find that at weak cou- pling, T <t , except when t =0: c 0 0 -20 3.0 0.03 1 T lnt /t (lnt /t)2 1+t /t * 2.5 c =exp 0 0 + 0 , (29) Λ " 4 −s 4 2(UN0)2# ) 0 1.0 2.0 n m/ 2.0 (T/t)2*10-5 where Λ is a cut-off (Λ < t). (Note that this reduces (t to the Hubbard model value when t0 = t.) At t0 = 0, / 1.5 T is identical to the Hubbard case. For 0 < t < t, T c 0 c ) T is enhanced relative to the Hubbard case. Hence a re- ρ( 1.0 U = 0.05 t gionofFermiliquidphaseseparatesthelowtemperature antiferromagneticphase from the marginalFermi Liquid t = 0.02 t 0.5 0 regime: Fig.7. There are further similarities to the Hubbard model 0 at half-filling. Both of the SO(4) and O(3) models are invariant under U U and Ψ(0) Ψ(0). The lat- 0 0.02 0.04 0.06 0.08 0.1 → − → − termapcorrespondstoaparticle-holetransformationfor T / t the down spin only: c c+. This implies that in going fromthepositiveU mo↓d↔elto↓ thenegativeU model,mag- netic ordering turns into charge ordering25 . It can also be shownthat neither model mixes chargeand magnetic FIG. 6. Plot of resistivity vs.T. Inset shows the very low ordering. Further, both models reduce to the Heisen- temperature crossover to T2 behaviour: the y-axis is in the berg antiferromagnet as U . Thus our model has same units as thebig plot. → ∞ very similar properties to the half-filled Hubbard model, exceptforthe marginalFermiLiquidphaseatweakcou- pling. Whatinsightdoesourmodelbringtowardstheunder- standing of the marginal Fermi liquid behaviour in the 6 cuprates? While our model does provide a simple lattice realisation of a marginal Fermi liquid, it unfortunately suffers from a number of defects: It has a wide window of marginal Fermi liquid • behaviour only for small coupling. The cuprates are believed to be in the strong coupling regime22, but at strong coupling, our system has charge or magnetic instabilities. Related to this is the fact that in the cuprates, the inelastic scattering rate Γ=Γ max[ω,T] has Γ /t a constant of order one, 0 0 whereas our model has Γ /t proportional to the 0 coupling squared. incoherent The modelneedstobe athalffilling: upondoping, • a chemical potential term µ(Ψ(0)Ψ(3) +Ψ(1)Ψ(2)) T~U leads to a width ∆ µ2 for Φ, with Fermi liquid ∝ properties when T < ∆. This seems to require t fine tuning, contradicting the rather robust linear- / T T resistivity observed even in underdoped systems MFL (above the “spin gap” scale). PI 1 Despite the presence of two relaxation rates in the • system,transportquantitieswillnotinvolveamul- tiplicative combination of the Φ and Ψ relaxation rates, as is postulated in the two-relaxation-times AF T=T ~1/U T=t ~tg2 FL c phenomenology for the cuprates. (See Section II.) 0 In conclusion we have demonstrated the persistence T=T ~ e-1/g c of marginal Fermi Liquid behaviour at weak coupling in a toy model of a marginal Fermi Liquid in an infinite 0 1 g=U N 8 0 dimensional lattice. For finite d, the lattice coherence energycuts offmarginalFermiLiquidbehaviourandthe system reverts to a Fermi Liquid at low temperatures. Since this cut-off grows with the coupling, there will be FIG. 7. Schematic phase diagram of the Majorana lat- no marginal Fermi Liquid regime at strong coupling. It tice model. The dimensionless coupling is g ≡ UN where 0 remains to be seen if a strong coupling marginal Fermi N ∝ 1/t is the density of states at the Fermi surface. 0 t = ctg2/d. MFL is the marginal Fermi liquid phase, FL Liquid exists in any finite dimensions. 0 istheFermiliquidphase. AFistheantiferromagnetic phase, Acknowledgement We acknowledge useful discussions PIistheparamagneticinsulatingphase. Atweakcoupling,Tc with Andrew Schofield, Gunnar Pa´lsson and Revaz Ra- goes as exp(−1/g), while at strong coupling, it goes as t2/U. mazashvili. This work was supported by NSF grant WehavenocomputationforstrongcouplingforT >Tc,thus DMR-96-14999. Part of the work was done while the wedonotknowthecontinuationoftheT =t line. However, 0 authors were in the Non-Fermi Liquid Workshop at the wemightexpectthatwhenU isofordert,Tc andt0 willalso InstituteofTheoreticalPhysics(ITP),UCSB,fundedun- beof ordert. Onescenario is thentheT =t linemeets and 0 der NSF grant PHYS94-07194 and DMR-92-23217. We ends at the T =Tc line around t0. Also, at strong coupling, thank the staff at ITP for their hospitality. the paramagnetic insulator crosses over to a metal at T ∼U (inanalogytotheHubbardmodel),andintheweakcoupling regime, there will be a similar high temperature cross-over VI. APPENDIX A: CALCULATION OF THE from marginal Fermi Liquid to an incoherent metallic phase: these cross-overs are indicated with a dotted line. EFFECTIVE BANDWIDTH t0 We want to estimate in finite dimensions the effective kinetic energy it˜ Ψ(0)Ψ(0) from the zero-frequency 0 i,c i+c i partoftheΨ(0)self-energy,asdepictedinFig.3,tolowest P order in the coupling: β it˜ =U2 dτ[G (τ)]3 (30) 0 xˆ Z0 7 where G (τ) is the bare propagator of Ψ(a)(a = 1,2,3), Σ (τ)=U2 (τ) (τ) (τ). (35) xˆ Ψ 0 a a G G G for nearest neighbour sites, taken here to be in the xˆ dsiimrepcltiicointy., Itnakke,ωǫ~ksp=ac−e,2t˜G(~kdi,=i1ωsni)n(=ki)[,iωanp−proǫ~kp]r−ia1t.eFfoorr GpFro0o(rτpΨ)ag(=aa)t,sogrn10(,τ)s/in2cies iindetnhtiecasltrtioctthde→sing∞leliimmpitu,ritt0y=Ψ(00). thehypercubicnearestneighbourdispersion. (Weexpect i P that as T 0, the exact shape for the band does not matter.) Le→t t˜=t/√d and t˜0 =t0/√d, as required for a a(τ)=T e−iωnτ (36) properscalingofthe kinetic energyterminlargedlimit. G iωn+itasgn(ωn) Xωn Then G (τ) will be of order 1/√d, and as mentioned in xˆ the Methods section, t˜ O(d 3/2), ie., 1/d down on wherewehaveusedEqn.19for (iω )fortheLorentzian 0 − a n ∼ G the dispersion for the a=1,2,3 components. Doing the DOS. As usual, turn the Matusbara sum into a contour standard Matsubara sum leads to: integral and deform the contour onto the branch cut at the real axis to get: ddk G (τ)= f( ǫ )exp(i~k xˆ ǫ τ). (31) xˆ (2π)d − ~k · − ~k ∞ dω e−ωτ Z a(τ)= Im (1 f(ω)) . (37) G − π − ω+it Inthezerotemperaturelimit,theFermifunctionbecome Z−∞ a f(−ǫ~k)→θ(ǫkx+ǫ~′k), wherewehavesplitupthe disper- (f(ω) is the Fermi function.) As we are interested in sion into the kx part and the other d−1 part. Turning T ≪ ta, the integrand is dominated by small ω, and we the d 1 dimensional k integralinto an energy integral approximate the denominator ω +it it . Now the − − a ≈ a gives: d dimensional Lorentzian DOS at the Fermi surface is − N =1/(πt ), and thus (τ) is identical to that for the 0 a a Gxˆ(τ)= dkx eikx−ǫkxτ ∞ dǫN(ǫ)e−ǫτ, (32) single impurity calculatiGon10: (2π) Z Z−ǫkx N πT 0 (τ)= . (38) where N(ǫ) is the density of states in d 1 dimensions. Ga sin(τπT) − To make further progress, a flat density of states is used: N(ǫ) = 1/(4t) for ǫ < 2t and zero otherwise. Note that this expression is accurate for 0 τ β. Thus, defining the dimensio|n|less time variable s=2τt, Going to Matsubara frequencies ωn =(2n+1≪)πT:≪ e s s β sgn(τ) N πT 2 Gxˆ(s)=− 2−s J1(i√d), (33) ΣΨ(iωn)=U2 dτ 2 sin(0τπT) e−iωnτ Z0 (cid:18) (cid:19) πT where J1(z) = ππ 2dπxe−izsin(x)+ix is the Bessel func- =−i 2 (UN0)2In(ǫ) (39) tion of the first k−ind of the first order. Thanks to the R factor of 1/√d inside its argument, the Bessel function where asymptotically alwaysgrowsmore slowlythan the decay due to the e s factor. (For arg(z) < π and z , π−ǫ sin(2n+1) − I (ǫ)= dx . (40) J (z) ( 2 )1/2cos(z 3π/4|).) Th|us, contrib|ut|io→ns∞to n (sin(x))2 1 → πz − Zǫ theintegralinEqn.30aredominatedbytheregimewhen We have put in a cut off ǫ = πT/(2t ) 1. Integrat- theBesselfunctionisatmostoforderone,allowingusto a approximate: iJ (ix)= x/2+O(x3)inEqn.33,leading ing by parts twice, and using the tabula≪ted integral26: 1 − πdxlnsin(x)sin(2n+1)x = 2 ( 1 +ln2+γ + finally to: 0 −2n+1 2n+1 Ψ(n+1/2)), we get: R t˜0 ≃ U2t2 ∞ds e2−ss2√sd 3 I (ǫ)= 2ωn ln Λ Ψ ωn 2. (41) Z0 (cid:20) (cid:21) n πT T − 2πT − U2 (cid:18) (cid:18) (cid:19) (cid:16) (cid:17)(cid:19) = , (34) 27 3 t d3/2 Ψ(x)istheDigammafunction,Λ=t e1 γ/(πT)andγ · a − ∼ 0.6 is the Euler constant. (We have expanded in ǫ and with t˜0 of order d−3/2 as claimed. kept only the terms up to ǫ0.) Putting this all together andperforming the analytic continuationiω ω+i0+ n → gives Eqn. 26. VII. APPENDIX B: CALCULATION OF THE The on-site self-energy for Φ is: MARGINAL SELF-ENERGY Σ (τ)=U2 (τ) (τ) (τ). 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