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Dynamical correlations in the spin-half two-channel Kondo model A. I. Tóth and G. Zarańd Theoretical Physics Department, Institute of Physics, Budapest University of Technology and Economics, H-1521 Budapest, Hungary (Dated: February 4, 2008) Dynamical correlations of variouslocal operators are studied in thespin-half two-channelKondo (2CK) model in the presence of channel anisotropy or external magnetic field. A conformal field theory-basedscalingapproachisusedtopredicttheanalyticpropertiesofvariousspectralfunctions in the vicinity of the two-channel Kondo fixed point. These analytical results compare well with highly accurate density matrix numerical renormalization group results. The universal cross-over 8 functions interpolating between channel-anisotropy or magnetic field-induced Fermi liquid regimes 0 0 andthetwo-channelKondo,non-Fermiliquid regimes are determinednumerically. Theboundaries 2 of the real 2CK scaling regime are found to be rather restricted, and to depend both on the type of the perturbation and on the specific operator whose correlation function is studied. In a small n magnetic field, a universal resonance is observed in the local fermion’s spectral function. The a dominant superconducting instability appears in thecomposite superconductingchannel. J 8 PACSnumbers: 71.10.Hf,71.10.Pm,71.27.+a,72.15.Qm,73.43.Nq,75.20.Hr 2 ] I. INTRODUCTION the impurity spin. If the coupling of the spin to one of l e the channels is stronger than to the other then electrons - r Deviations from Fermi liquid-like behavior observed in the more strongly coupled channel screen the spin, t while the other channel becomes decoupled. However, s e.g. in the metallic state of high-temperature cuprate . for equal exchange couplings, the competition between t superconductors,1,2 or in heavy fermion systems3,4 a the two channels leads to overscreening and results in a m prompted physicists to look for new non-Fermi liquid non-Fermiliquidbehavior: Amongothers,itischaracter- (NFL) compounds. So far a large number of such ex- - ized by a non-trivial zero temperature residual entropy, d otic compounds has been found and investigated. In a square root-like temperature dependence of the differ- n these systems electrons remain incoherent down to very ential conductance, a logarithmic divergence of the spin o low temperatures and the usual Fermi liquid descrip- c tion breaks down. To our current understanding, NFL susceptibility and the linear specific heat coefficient at [ lowtemperatures.5Thisunusualandfragilegroundstate physics may arise in many different ways: it can occur cannot be described within the framework of Nozières’ 1 due to some local dynamical quantum fluctuations often Fermi liquid theory.25 v described by quantum impurity models,5,6,7 it can also 2 be attributed to the presence of the quantum fluctua- Being a prototypical example of non-Fermi liquid 7 tions of an order parameter or some collective modes, models, the two-channel Kondo model (2CKM) has al- 2 as is the case in the vicinity of many quantum phase ready been investigated with a number of methods. 4 . transitions,6,8 or for the prototypical example of a Lut- Theseincludenon-perturbativetechniquesliketheBethe 1 tinger liquid,9,10,11,12,13 where electrons are totally dis- Ansatz, which gives full account of the thermodynamic 0 integrated into collective excitations of the electron gas. properties,26,27 boundaryconformalfieldtheory,28 which 8 NFLphysicscanalsoappearasaconsequenceofdisorder describes the vicinity of the fixed points, and numeri- 0 : like e.g. in disordered Kondo alloys.14,15 cal renormalization group (NRG) methods.29 Further- v In this paper we study a variant of the overscreened more other less powerful approximate methods such as i X multi-channel Kondo model: the spin-1, two-channel theYuval-Andersonapproach,31 Abelianbosonization,32 r Kondo (2CK) model, which is the simp2lest prototypi- large-f expansion,33,34andnon-crossingapproximation35 a calexample ofnon-Fermiliquidquantumimpurity mod- have also been used to study the 2CKM successfully. els. This model has first been introduced by Nozières Rather surprisingly, despite this extensive work, very and Blandin,16 and since has been proposed to de- littleisknownaboutdynamicalcorrelationfunctionssuch scribeavarietyofsystemsincludingdiluteheavyfermion asthespinsusceptibility,localchargeandsuperconduct- compounds,5tunneling impurities in disordered metals ing susceptibilities. Even the detailed properties of the and doped semiconductors.17,18,19 More recently, the T-matrix, essential to understand elastic and inelastic 2CK state has been observed in a very controlled way scatteringinthisnon-Fermiliquidcase,36 haveonlybeen in a double dot system originally proposed by Oreg and computed earlier using conformal field theory (which is Goldhaber-Gordon.20,2149 rather limited in energy range) and by the non-crossing The two-channel Kondo model consists of a spin-1, approximation (which is not well-controlled and is un- 2 local moment which is coupled through antiferromag- able to describe the Fermi liquid cross-over).37,38 It was netic exchange interactions to two channels of conduc- also possible to compute some of the dynamical corre- tion electrons. Electrons in both channels try to screen lation functions in case of extreme spin anisotropy us- 2 ing Abelian bosonization results,32 though these calcu- the 2CKM we connect it to a dimensionless approxima- lations reproduce only partly the generic features of the tion of it suited to our DM-NRG calculations. We also spin-isotropic model.39 Local correlations in the Ander- provide the symmetry generators used in the conformal son model around the non-Fermi liquid fixed point have field theoretical and DM-NRG calculations. In Section alreadybeeninvestigatedwiththeuseofNRG,although IIIweuseboundaryconformalfieldtheorytoclassifythe intheabsenceofchannelanisotropyandmagneticfield.40 boundary highest-weight fields of the electron-hole sym- However, a thorough and careful NRG analysis of the metrical 2CKM by their quantum numbers and identify T =0 temperatureT-matrixofthe 2CKMhas been car- the relevant perturbations around the 2CK fixed point. ried out only very recently,24,36 and the T = 0 analysis Based on this classification the fields are then expanded 6 still needs to be done. inleadingorderinterms ofthe operatorsofthe freethe- The main purpose of this paper is to fill this gap by ory. In Section IV we describe the technical details of giving a comprehensive analysis of the local correlation ourDM-NRGcalculations. InSectionsV,VIandVIIwe functionsatzerotemperatureusingthe numericalrenor- study the real and the imaginary parts of the retarded malization group approach. However, in the vicinity of Green’sfunctionsofthelocalfermions,theimpurityspin the rather delicate two-channel Kondo fixed point, the andthelocalsuperconductingorderparameters. Ineach conventionalNRGmethodfailsanditsfurtherdeveloped of these sections we first discuss the analytic forms of version, the density matrix NRG (DM-NRG)41 needs to the susceptibilities in the asymptotic regions of the two- be applied. Furthermore, a rather large number of mul- channel and single channel Kondo scaling regimes, as tiplets must be kept to achieve good accuracy. We have they follow from scaling arguments. Then we confirm therefore implemented a modified versionof the recently our predictions by demonstrating how the expected cor- developed spectral sum conserving DM-NRG method, rectionsduetotherelevantperturbationsandtheleading where we use non-Abelian symmetries in a flexible way irrelevant operator present themselves in the DM-NRG to compute the real and the imaginary parts of various data. Furthermore we determine the boundaries of the local correlation functions.42 2CK scaling regimes and derive universal scaling curves connecting the FL and NFL fixed points for each oper- ToidentifytherelevantperturbationsaroundtheNFL ator under study. Finally, our conclusions are drawn in fixed point we apply the machinery of boundary confor- Section VIII. mal field theory. Then we systematically study how the vicinity of fixed points and the introduction of relevant perturbations such as a finite channel anisotropy or a fi- II. HAMILTONIAN AND SYMMETRIES nite magnetic field influence the form of the dynamical response functions at zero temperature. We mainly fo- cus onthe strongcoupling regimeofthe 2CKmodel and The two-channelKondo model consistsof animpurity theuniversalcross-overfunctionsintheproximityofthis with a magnetic moment S = 1 embedded into a Fermi 2 region induced by an external magnetic field or channel liquid(FL)oftwotypesofelectrons(labeledbytheflavor anisotropy. We remark that these cross-over functions, or channel indices α = 1,2), and interacting with them describingthe cross-overfromthe non-Fermiliquidfixed through a simple exchange interaction, point to a Fermi liquid fixed point, as well as the response functions can currently be computed reliably at DF H = dk kc† (k)c (k) (1) all energy scales only with NRG. However, we shall be α,µ α,µ α,µZ−DF abletousetheresultsofboundaryconformalfieldtheory, X more precisely, the knowledge of the operator content of + Jα DF dk DF dk′ S~ c† (k)~σ c (k′). the two-channel Kondo fixed point and the scaling di- 2 αµ µν αν α µ,ν Z−DF Z−DF mensionsofthevariousperturbationsaroundit,tomake XX verygeneralstatements onthe analytic propertiesofthe Here c† (k) creates an electron of flavor α in the l = 0 various cross-overand spectral functions. α,µ angular momentum channel with spin µ and radial mo- We shall devote special attention to superconducting mentum k measured from the Fermi momentum. In the fluctuations. It has been proposed that unusual super- Hamiltonian above we allowed for a channel anisotropy conducting states observed in some incoherent heavy of the couplings, J = J , and denoted the Pauli ma- 1 2 fermion compounds could also emerge as a result of trices by ~σ. In the 6first, kinetic term, we assumed a local superconducting correlations associated with two- spherical Fermi surface and linearized the spectrum of channel Kondo physics.5,43,44 Here we investigate some the conduction electrons, ξ(k) v k = k, but these as- F possible superconducting order parameters consistent sumptionsarenotcrucial: Apar≈tfromirrelevanttermsin with the conformalfield theoreticalpredictions, andfind the Hamiltonian, our considerations below carry over to that the dominant instability emerges in the so-called essentially any local density of states with electron-hole composite superconducting channel, as it was proposed symmetry. The fields c† (k) are normalized to satisfy by Coleman et al.44 the anticommutation relαa,tµions The paper is organized as follows. In Section II start- ing from the one-dimensional, continuum formulation of c† (k),c (k′) =δ δ δ(k k′) , (2) α,µ β,ν α,β µ,ν − (cid:8) (cid:9) 3 andthereforethecouplingsJ arejustthedimensionless SU (2) SU (2) in the charge and spin sectors for ar- α C2 S × couplings, usually defined in the literature. Since we are bitrary couplings, J . α interested in the low-energyproperties of the system, an ToperformNRGcalculations,weusethefollowingap- energy cut-off D is introduced for the kinetic and the proximation of the dimensionless Hamiltonian,29 F interactionenergies. Inheavyfermionsystems,thislarge eEnFe,rgwyhislceafloeriqsuianntthuemrdaontgse,iotfisthoeftFheeromrideernoerfgtyh,eDsiFng∼le D (12H+Λ−1) ≈ J˜2α S~f0†,α,µ~σµνf0,α,µ F particlelevelspacingofthedot,δǫoritschargingenergy, α µ,ν XX E , whichever is smaller. ∞ CThe Hamiltonian above possesses various symmetries. + tn(fn†,α,µfn+1,α,µ+h.c.), (11) Toseeit,itisworthtointroducetheleft-movingfermion n=0α,µ,ν X X fields, with Λ a discretization parameter and J˜ = 4J /(1+ α α ψ (x) DF dk e−ikxc (k), (3) Λ−1). The operator, f0 creates an electron right at the α,µ ≡ α,µ impurity site, and can be expressed as Z−DF and to rewrite the Hamiltonian as 1 DF f = dk c (k). (12) 0,α,µ α,µ √2D dx FZ−DF H = ψ† (x)i∂ ψ (x) 2π α,µ x α,µ TheHamiltonianEq.(11)isalsocalledtheWilsonchain: α,µZ X it describes electrons hopping alonga semi-infinite chain + JαS~ψ†(0)~σψ(0). (4) with a hopping amplitude tn Λ−n/2, and interacting 2 with the impurity only at site 0∼. In the NRG procedure, α X thisHamiltonianisdiagonalizediteratively,anditsspec- Then the total spin operators Ji defined as trum is used to compute the spectral functions of the various operators.29 dx Ji Si+ Ji(x), (5) We remark that the Wilson Hamiltonian is not iden- ≡ 2π tical to H, since some terms are neglected along its Z Ji(x) 1 :ψ†(x)σiψ (x): (6) derivation.29 Nevertheless, similar to H, the Wilson ≡ 2 α α Hamiltonian also possesses the symmetry SUC1(2) Xα SU (2) SU (2) for arbitrary J and J couplings.4×5 C2 S 1 2 × commute with the Hamiltonian and satisfy the standard The corresponding symmetry generators have been enu- SU(2) algebra, meratedinTableI. Wecanthenusethesesymmetriesto label every multiplet in the Hilbert space and every op- Ji,Jj =iǫijkJk . (7) erator multiplet by the eigenvalues J~2 = j(j+1) and Inthe previouseq(cid:2)uationsw(cid:3)esuppressedspinindices and C~α2 = cα(cα +1). Throughout this paper, we shall use these quantum numbers to classify states and operators. introduced the normal ordering :...: with respect to the Inthe presenceofa magneticfield, i.e.,whena term50 non-interactingFermisea. Inasimilarwaywecandefine the“chargespin”densityoperators,forthechannelsα= H = gµ B Sz (13) 1,2 as magn − B 1 isaddedtoH,thesymmetryofthesystembreaksdown Cz(x) :ψ†(x)ψ (x): toSU (2) SU (2) U (1),withthesymmetryU (1) α ≡ 2 α α C1 × C2 × S S correspondingto the conservationof the z-componentof Cα−(x) ≡ ψα↑(x)ψα↓(x), Cα+(x)≡ψα†↓(x)ψα†↑(x), the spin, Jz (see Table I). In the rest of the papers we Cα±(x) ≡ Cαx(x)±iCαy(x), (8) shall use units where we set gµB ≡1. and the corresponding symmetry generators III. THE NON-FERMI LIQUID FIXED POINT Ci dx Ci(x) (i=x,y,z). (9) AND ITS OPERATOR CONTENT α ≡ 2π α Z ForJ =J =J andintheabsenceofanexternalmag- ThegeneratorsCi,whicharerelatedtotheelectron-hole 1 2 α netic field, the Hamiltonian, H possesses a dynamically symmetry,45 satisfythesameSU(2)algebraastheJi-s, generatedenergyscale,theso-calledKondotemperature, Cαi,Cβj = iδαβ ǫijkCβk , (10) TK DF e−1/J. ≈ h i and they also commute with the Hamiltonian, Eq. (4). ThedefinitionofT issomewhatarbitrary. Inthispaper, K Thus the Hamiltonian H has a symmetry SUC1(2) TK shallbe definedas the energyω atwhichfor J1 =J2 × 4 Symmetrygroup Generators ∞ ∞ SU (2) C+ = ( 1)nf† f† , Cz = 1 f† f 1 , C− =C+† Cα α − n,α,↑ n,α,↓ α 2 n,α,µ n,α,µ− α α nX=0 nX=0 X∞µ “ ” SU (2) ~J=S~+ 1 f† ~σ f S 2 n,α,µ µν n,α,ν n=0α,µ,ν XX TABLE I: Generators of the used symmetries for the two-channel Kondo model computations. Sites along the Wilson chain are labeled byn whereas α and µ,ν are thechannel and spin indices, respectively. the spectral function of the composite fermion drops to struction to write the kinetic part of the Hamiltonian as half of its value assumed at ω = 0 (for further details see the end of this Section and Fig. 1). For B = 0 and H0 = HC1+HC2+HS +HI , (15) bJy1 =theJ2s,o-bceallolwedtthwiso-ecnhearngnyeslcKaloentdhoefipxheydsipcosiinst.governed HCα = 1 dx :C~α(x)C~α(x):, 3 2π Z The physics of the two-channelKondo fixedpoint and 1 dx its vicinity can be captured using conformal field the- HS = :J~(x)J~(x): . 4 2π ory. The two-channelKondo finite size spectrum and its Z operator content has first been obtained using boundary InH0,thefirsttwotermsdescribethechargesectors,and conformalfieldtheorybyAffleckandLudwig.28However, have central charge c = 1, while H describes the spin S instead of charge SU(2) symmetries, Affleck and Ludwig sector, and has central charge c = 3/2. The last term used flavor SU(2) and charge U(1) symmetries to obtain corresponds to the coset space, and must have central thefixedpointspectrum.28TheuseofchargeSU(2)sym- charge c =1/2, since the free fermion model has central metries, however, has a clear advantage over the flavor charge c = 4, corresponding to the four combinations of symmetry when it comes to performing NRG calcula- spin and channel quantum numbers. This term can thus tions: While the channel anisotropy violates the flavor be identified as the Ising model, having primary fields symmetry, it does not violate the charge SU(2) symme- 1l,σ,ǫ with scaling dimensions 0,1/16,1/2, respectively. tries. Therefore,eveninthe channelanisotropiccase,we We can then carry out the conformal embedding in the have three commuting SU(2) symmetries. If we switch usual way, by comparing the finite size spectrum of the on a local magnetic field, only the spin SU(2) symmetry free Hamiltonian with that of Eq. (15), and identifying is reduced to its U(1) subgroup. Using charge symme- the allowed primary fields in the product space. The tries allows thus for much more precise calculations, and the fusion rules obtained this way are listed on the left in fact, using them is absolutely necessary to obtain sat- side of Table II. The finite size spectrum at the two- isfactorily accurate spectral functions, especially in the channelKondofixedpoint canbe derivedby fusing with presence of a magnetic field. theimpurityspin(whichcouplestothespinsectoronly), To understand the fixed point spectrum and the oper- following the operator product expansion of the Wess– ator content of the 2CKM, let us outline the boundary Zumino–Novikov–Witten model, 1/2 0 1/2, 1/2 ⊗ → ⊗ conformalfieldtheoryinthisSU (2) SU (2) SU (2) 1/2 0 1,1/2 1 1/2(seeRHSofTableII). Finally, language. First, we remark thaCt1the×spinCd2ens×ity oSper- the o→per⊕ator con⊗ten→t of the fixed point can be found by ators, Ji(x) satisfy the SU(2) Kac-Moody algebra of performing a second fusion with the spin. The results of k=2 level k =2, thisdoublefusionarepresentedinTableIII. InTableIII the leading irrelevant operator, ~ φ~ , is also included. −1 s Although it is not a primary fieJld,28 close to the 2CK k Ji(x),Jj(x′) = δij δ′(x x′) fixed point, this operator will also have impact on the 2 − form the correlation functions. (cid:2) (cid:3) + i2πδ(x x′)ǫijkJk(x), (14) What remains is to identify the scaling operators in − terms of the operators of the non-interacting theory. In while the charge density operators, Ci(x) defined in the general, an operator of the non-interacting theory can α previous section satisfy the Kac-Moody algebra of level be written as an infinite series in terms of the scaling k =1: operators and their descendants. Apart from the Ising sector, which is hard to identify, we can tell by looking atthevariousquantumnumbersofanoperatoractingon k Ci(x),Cj(x′) = δijδ δ′(x x′) the Wilson chain, which primary fields could be present α β 2 αβ − in it. In this way, we can identify, e.g. φ~ as the spin h i + i2π δαβδ(x−x′)ǫijkCαk(x). operator S~. Thus the spin operator can besexpressed as We can use these current densities and the coset con- S~ =A φ~ +... (16) s s 5 c01 c02 0j 1Il Ef0ree c011 c002 0j12 1σIl E2C01KM sinAσgσln′eotǫt,σhσea′rΓnτ1dσγch2aτa′nsσd′.icdhaatregTehiwsspoiunolpsderca1tbo=er icst2hae=lsoo1p/ae2r.alotcoaIrtl, 2 8 cPontains the following component of the composite 1 0 1 σ 1 0 1 0 σ 1 2 2 2 2 8 superconducting order parameter 0 1 1 σ 1 1 1 1 1l 1 2 2 2 2 2 2 2 1 1 1 1l 1 1 0 1 σ 5 2 2 2 8 f† S~~σ iσ f† . (18) 1 1 0 ǫ 1 0 1 1 σ 5 OSCC ≡ 0,1 y 0,2 2 2 2 8 1 1 1 ǫ 1 2 2 2 From their transformation properties it is not obvious, which one of the above superconducting order parame- TABLE II: Left: Primary fields and the corresponding finite ters gives the leading singularity. However, NRG gives size energies at the free fermion fixed point for anti-periodic a very solid answer and tells us that, while the suscepti- boundary conditions. States are classified according to the groupSUC1(2) SUC2(2) SUS(2)andtheIsingmodel. The bility of the traditional operator does not diverge as the excitationenerg×iesEfreear×egiveninunitsof2π/L,withLthe temperature or frequency goes to zero, that of the com- sizeofthechiralfermionsystem. Right: Finitesizespectrum posite order parameter does. It is thus this latter oper- at thetwo-channel Kondofixed point. ator that can be identified as φττ′. Note that, in case ∆ of electron-hole symmetry, the composite hybridization operator where the dots stand for all the less relevant operators that are present in the expansion of S~, and some high- f† S~~σ f (19) frequency portions which are not properly captured in Omix ≡ 0,1 0,2 the expansion above. The weight, A , can be deter- s has the same singular susceptibility as since they mined from matching the decay of the spin-spin corre- OSCC are both components of the same tensor operator. This lationfunctionatshortandlongtimes. Thiswayweend is, however, not true any more away from electron-hole up with A 1/√T . s ∼ K symmetry. Furthermore, superconducting correlations We remark that there are infinitely many operators are usually more dangerous, since in the Cooper chan- that contain the scaling fields in their expansion. As nel any small attraction would lead to ordering when a an example, consider the operators φτσ. Here the label ψ1 regularlattice model of two-channelKondo impurities is σ = , refers to the spin components of a j = 1/2 {↑ ↓} considered. spinor, while τ = refer to the charge spins of a charge ± The knowledge of the operator content of the two- c = 1/2 spinor. To identify the corresponding operator channel Kondo fixed point enables us to describe the on the Wilson chain, we first note that f† transforms 0,1,σ effects of small magnetic fields and small channel asaspinorunderspinrotations. Itcaneasilybeseenthat anisotropies (J = J ). For energies and temperatures the operator f˜† iσ f also transforms as a spinor. 1 6 2 0,1 ≡ y 0,1 below TK, the behavior of the model can be described We can then form a four-spinor out of these operators, by the slightly perturbed two-channelKondo fixed point aγs1 ≡a {spfi0†n,1o,σr,ufñ0†,d1,eσr}.SUItCis1(e2a)syrottoasthioonwstahsatwγe1ll,trtahnussforφmτψσ1s HBamilTtKon,itahni.sFHoarmJi1lt≈onJia2nacnadninbeaesxmpraellssmedagansetic field, ≪ could be identified as γ = f† ,f˜† . 1 { 0,1,σ 0,1,σ} However, we can construct another operator, F† = ∗ + 1 ≡ H H2CK sfp0†,i1nS~o~σr aonudt iotfstchoeumnt:erΓp1ar≡t,F˜{1†F1≡†,σi,σF˜y1†F,σ1},.anTdhfoisrmopaerfoautorr- +D01/2 κ0 φanis+D01/2~h0 φ~s+D0−1/2 λ0 J~−1φ~s+...(2.0) has the same quantum numbers as γ , and in fact, both 1 opTerhaetoorps’ereaxtpoarnφsiτoτn′ icsonoftasipnescφiaτψlσ1i.nterest, since it is rel- HisetrheeHd2∗iCmKenissiotnhlees2sCcKoufipxliendgptoointtheHachmainltnoenliaann,isaontrdopκy0 ∆ evant at the two-channel Kondo fixed point, just like field,φ ,whereasthe effective magneticfield,~h , cou- anis 0 the spin. Its susceptibility therefore diverges logarith- ples to the “spin field”, φ . Both of them are relevant s mically. Good candidates for these operators would be perturbations at the two-channel Kondo fixed point and σσ′ǫσσ′γ1τσγ2τ′σ′, since these are spin singlet operators they must vanish to end up with the two-channelKondo that behave as charge1/2 spinorsin both channels. The fixed point at ω,T 0. The third coupling, λ , couples 0 τP= τ′ = + component of this operator corresponds to to the leading irrel→evant operator (see Tab. III), which the superconducting order parameter dominates the physics when κ=h=0. The energy cut- off D in Eq. (20) is a somewhat arbitrary scale: it can 0 OSC ≡f0†,1,↑f0†,2,↓−f0†,1,↓f0†,2,↑ , (17) be though of as the energy scale below which the two- channel Kondo physics emerges, i.e. D T . Then 0 K ∼ while the + components describe simply a local opera- the dimensionless couplings κ , λ and h are approxi- tor that hyb−ridizes the channels, f† f . mately related to the coupling0s of0the orig0inal Hamilto- ∼ 0,1,σ 0,2,σ 6 c1 c2 j I x2CK scaling corresponding operators operators 0 0 1 1 1 φ~ S~ 2 s 1 0 1 σ 1 φτσ γ1 ≡ f0†,1,σ , (iσyf0,1)σ 2 2 2 ψ1 Γ1 ≡ “F0†,1,σ , (iσyF0,1)σ” 0 1 1 σ 1 φτσ “ γ2 ” 2 2 2 ψ2 Γ2 1 1 0 1l 1 φττ′ f0†,1S~~σiσyf0†,2 −f0†,1S~~σf0,2 2 2 2 ∆ −f0,1σy S~~σσyf0†,2 −f0,1iσy S~~σf0,2 ! 0 0 0 ǫ 21 φanis S~(f0†,1~σf0,1−f0†,2~σf0,2) 0 0 0 1l 23 J~−1φ~s S~(f0†,1~σf0,1+f0†,2~σf0,2) TABLE III: Highest-weight operators and their dimensions x2CK at the 2CK fixed point. Operators are classified by the symmetry group SUC1(2) SUC2(2) SUS(2) and the scaling operators of the Ising model. The constants c1 and c2 denote × × the charge spins in channels 1 and 2, respectively, while j refers to the spin, and I labels the scaling operators of the Ising model: 1l,σ,ǫ. Superscripts τ,τ′ = refer to the two components of charge spinors, while σ = , label the components of a spin- 1 spinor. ± ↑ ↓ ±2 nian, Eq. (11), as The prefactors in Eqs. (28) are somewhat arbitrary, and depend slightly on the precise definition one uses J J κ K 4 1− 2 , (21) to extract these scales. In this paper, we shall use the 0 ≈ R ≡ (J1+J2)2 spectral function of the composite fermion to define the h0 B/TK , (22) scales TK and T∗. We define TK to be the energy at ≈ which for K =0 the spectral function of the composite λ O(1) . (23) R 0 ≈ fermion takes half of its fixed point value (i.e. the value However, the arbitrary scale D in Eq. (20) can be assumed at ω = 0). Whereas T∗ is the energy at which 0 changedattheexpenseofchangingthecouplings: D0 for KR > 0 it takes 75% of its fixed point value (see → D,κ κ(D), h h(D) and λ λ(D) in such a Fig. 1). 0 0 0 → → → waythatthe physicsbelowD0 remainsunchanged. This It is much harder to relate Th to a physically measur- freedom translates to scaling equations, whose leading ablequantity. We defineditsimplythroughtherelation, terms follow from the conformalfield theory results, and B2 read T C , (29) h h ≡ T K dκ(D) 1 = κ(D)+... , (24) dx 2 where the constant was chosen to be Ch 60. This way ≈ dh(D) 1 Th corresponds roughly to the energy at which the NFL = h(D)+... , (25) finite size spectrumcrossesoverto the low-frequencyFL dx 2 spectrum. dλ(D) 1 = λ(D)+... , (26) dx −2 withx= logD. Solvingtheseequationswiththeinitial IV. NRG CALCULATIONS − conditions, D = D T and h = h , κ = κ , λ = λ , 0 K 0 0 0 ∼ we can read out the energy scales at which the rescaled Priortodiscussingtheanalyticandnumericalfeatures couplings become of the order of one, oftheresponsefunctions,letusdevotethissectiontothe shortdescriptionofthe NRGprocedureused. Allresults (J J )2 T∗ T κ2 T 1− 2 , presented in this paper refer to zero temperature. The ∝ K 0 ∼ K(J1+J2)4 NRG calculations were performed with a discretization (27) parameter Λ = 2. The sum of the dimensionless cou- T T h2 B2/T . (28) plingswasJ˜1+J˜2 =0.4foreachrun. TheNRGdatawere h ∝ K 0 ∼ K computed with a so-called flexible DM-NRG program,42 At these scales the couplings of the relevant operators whichpermits the use ofanarbitrarynumberofAbelian aresolargethattheycannolongerbetreatedaspertur- and non-Abelian symmetries (see Tab. I), and incorpo- bations. Below T∗ the single channel Kondo behavior is rates the spectral-sum conserving density matrix NRG recoveredinthemorestronglycoupledchannel,whileT (DM-NRG)algorithm.41TheDM-NRGmethodmakesit h can be interpreted as the scale where the impurity spin possible to generate spectral functions that satisfy spec- dynamics is frozen by the external field. tral sum rules with machine precision at T = 0 tem- 7 V. LOCAL FERMIONS’ SPECTRAL FUNCTIONS AND SUSCEPTIBILITIES 2 K R Let us first analyse the Green’s function of the local 0.02 ω) 00..00007255 ~fΓerm)ioGn,refe0†n,σ’,sαf↔un~γctαi.onThheascoamlrpeoasdiytebfeeremniloono’kse(dF0†i,nσt,αo↔in ( 0 0,α ρF1 detail in an earlier study of ours.24 We shall therefore not discuss its analytic properties here but use it merely as a reference to define the various energy scales in the NRGcalculations(seeFig.1). Letusnote,however,that T * T in the large bandwidth limit, ω,T D , the spectral K K ≪ F function of the composite fermion and that of the local 0 fermion are simply related, 10-9 10-6 10-3 100 ω 1 π ̺ (ω)= J2̺ (ω). (30) f F 2D − 4 FIG. 1: (color online) Spectral function ̺ of the composite F F fermion operator,F0,1,↑ asafunctionofω,andthedefinition Thus, apart from a trivial constant shift and a minus ofthescalesTK andT∗. TK isdefinedbytherelation̺F(ω= sign, the spectral function of the local fermion is that zT0e,KrKo,TK)=R0t3,h̺KeR(sωc=a=le0)0T,≡T∗ 21i=s̺F0d(,eωfiKn=ed0).,tThr=ou0g,hK̺RF=(ω0)=. FTo∗r,Tnon=- roeffltehcetecdominp̺ofsi.te fermion, and all features of ̺F are also R ≡ 4 F | R| BeforewediscusstheNRGresults,letusexaminewhat predictions we have for the retarded Green’s function of the operatorf† fromconformalfieldtheory. By look- 0,σ,α ing at its quantumnumbers, this operatorcanbe identified with the operator φ+σ (see Tab. III), i.e. ψα perature. For calculations with non-zero magnetic field the use of the DM-NRG method represents a great ad- f† =A φ+σ+... , (31) 0,σ,α f ψα vantage over conventional NRG methods,46 which loose spectralweights andviolate spectralsumrules. Conven- withtheprefactorAf 1/√DF. NotethatAf isacom- ∝ tional methods also lead to smaller or bigger jumps in plexnumber,itdoesnotneedtobereal. Thedotsinthe the spectral functions at ω =0 which hinder the compu- equation above indicate the series of other, less relevant tation of the universal scaling functions provided by the operators and their descendants, which give subleading scale Th.24 The DM-NRG method solves all these prob- correctionsto the correlationfunctionoff0†,σ,α. Further- lems if a sufficient number of multiplets is kept. On an more,theexpansionaboveholdsforthelong time behav- ordinary desk-top computer, however, we need to use as ior. The “short time part” of the correlationfunction of many symmetries as possible to keep the computation f† is not captured by Eq. (31), and gives a constant 0,σ,α time within reasonable limits. to (ω) of the order of 1/D . Thus, apart from a f F preGfactor A2, a constant s∼hift and subleading terms, the f In the present paper, where we study the electron- Green’s function of f† is that of the field φ+σ. As 0,σ,α ψα holesymmetricalcase,itispossibleto usethe symmetry we discuss it shortly in Appendix A, the Fourier trans- groupSUC1(2) SUC2(2) SUS(2)evenincaseofchannel formoftheGreen’sfunctionofanyoperatorofdimension × × anisotropy. At these calculations the maximum number x=1/2isscaleinvariantaroundthe two-channelKondo of kept multiplets was 750 in each iteration. This cor- fixed point. Since φ+σ and thus f† have a scaling di- responds to the diagonalization of 85 matrices with ψα 0,σ,α ≈ mension 1/2 at the 2CK fixed point, it follows that the matrix sizes ranging up to 630, acting on the vector spaceof≈9000multiplets c≈onsistingof≈106000states. adlismoesncsaiolenilnesvsarrieatnatr,d51ed Green’s function, DF Gf(ω), is In the presence of magnetic field we used the symmetry group SU (2) SU (2) U (1), and retained a ω T maximum of 1C3150 m×ultipCle2ts in×eacSh iteration, that cor- DF Gf(ω,T) ≡ gˆf D,D,κ(D),h(D),λ(D),... , (cid:18) (cid:19) responds to the diagonalization of 150 matrices with dgˆ ≈ f matrix sizes ranging up to 800 acting on the vector D = 0. (32) ≈ dD spaceof 18000multiplets consistingof 73000states. ≈ ≈ FromEq.(32),wecandeducevariousimportantprop- erties. Let us first consider the simplest case, T =0 and In the next sections, we shall see how the knowledge κ = h = 0. Then setting the scale D to D T we of the operator content of the two-channel Kondo fixed 0 ∼ K have pointcanhelpustounderstandtheanalyticstructureof the various dynamical correlation functions obtained by gˆκ,h,T=0(ω)=gˆ ω ,λ ,... . (33) NRG. f f D 0 (cid:18) 0 (cid:19) 8 Let us now rescale D ω , and use the fixed point ) 1 cst scaling equation (26) to→obt|ai|n λ(D), ̺ωˆ(f ≈2−log2(TωK) 1+cst ω ω ≈4 sTK gˆf = gf±1,sD| 0| λ0 . (34) ∼cst+ Tω!2   Assuming that this function is analytic in its second ar- gument we obtain for ω T K | |≪ ω gˆfκ,h,T=0(ω) = gˆf T T >0,KR=0,B=0 K (cid:16) (cid:17) ≈ g±f +g±′ f sT|ωK| +... , (35) ω) logTTK 0 logDTKF logTωK ( wHietrhegth±efsuabnsdcrgi±′ptfssomreefecrotmoptlheexceaxspeasnωsi>on0caonedffiωcie<nt0s., gReˆf− ∼vuutTωK T>0,KR=0,B=0 ± respectively. As we discussed above, the constants g±f ω depend alsoon the shorttime behaviorof (t), andare ∼T f G npoentduennitveorfseaalcihnoththisers.enTshe.eyTahreeserecloantesdtabnytsthaereconnosttirnadinet- logTωK that the Green’s function must be analytic in the upper logTTK 0 logDTKF     half-plane. Furthermore,electron-holesymmetryimplies that g =g and g′ = (g′ )∗. +f −f +f − −f Relationssimilartotheonesaboveholdforthedimen- sionless spectral function. It is defined as FIG.2: (coloronline)(top)Sketchofthedimensionlessspec- ̺ˆf(ω) 1 Imgˆf(ω), (36) tral function ̺ˆf = DF̺f of f0†,1,σ, and (bottom) the real ≡−π partofitsdimensionlessGreen’sfunction,Regˆ =D Re f F f G for T > 0 and K = 0,B = 0 as a function of log(ω/T ). andassumesthefollowingsimplerformatsmallfrequen- R K Asymptoticsindicatedforω<T werederivedthroughscal- K cies in case of electron-hole symmetry, ing arguments. The large ω-behavior is a result of perturbation theory. ω ̺ˆT,κ,h=0(ω)=r +r′ | | +... . (37) f f f sTK T must be smaller than T . The asymptotic properties K Forω TK thescalingdimensionofthelocalfermion of Θf and Θ˜f can be extracted by making use of the is xfree =≫1/2 corresponding to an ω-independent spec- facts that (i) gˆf(ω,T) must be analytic for ω T, (ii) f that Eq. (39) should reproduce the T 0 resu≪lts in the tral function. Perturbation theory in J amounts to log- → arithmic corrections of the form: 1/2 cst/log2(T /ω), limit ω T, and (iii) that by electron-hole symmetry, K ≫ − ̺ˆ must be an even function of ω. The issuing asymp- as it is sketched in the upper parts of Fig.-s 2 and 3. f totic properties together with those of the other scaling For T = 0, and κ = h = 0 using similar arguments as 6 functions defined later are summarized in Table V. The before, but now rescaling D T we find → asymptotic properties of the real part, Regˆ , can be ex- f gˆκ,h=0(ω) = gˆ ω, T ,λ tracted from those of ̺ˆf by performing a Hilbert trans- f f T T 0 form (cid:18) K (cid:19) ̺ˆ (ω˜) ω T f Regˆ (ω)= dω˜ (40) gˆ ,1, λ ,... . (38) f f 0 P ω ω˜ ≡ T rD0 ! Z − with the principal part. The obtained features are Then by expanding gˆf we obtain the following scaling sketchPed in Fig. 2 for T >0 and κ=h=0. form for the low temperature behavior of the spectral Letusnowinvestigatetheeffectofchannelanisotropy, function, i.e. κ = 0 at T = 0 temperature and no magnetic field 6 h=0. Inthiscase,wecanrescaleDtoD = ω toobtain ω T ω | | ̺ˆh,κ=0(ω) = Θ + Θ˜ +... ,(39) f f T T f T (cid:16) (cid:17) r K (cid:16) (cid:17) ̺ˆT,h=0(ω)= ± ω + |ω| ˜± ω +... , with Θ and Θ˜ universal scaling functions. Note that f Kf T∗ sTK Kf T∗ f f (cid:16) (cid:17) (cid:16) (cid:17) wemadenoassumptionontheratioω/T,butbothωand (41) 9 ̺ωˆ()f T =0,KR>0,B=0 ≈21−log2c(sTtωK) sevFeirga.l4v.a(lau)esdeopficKtsRtahseaspfuecntcrtaiolnfuonfctωio/nTKofonf0†,a1,σlogfoar- rithmic scale. The overall scaling is very similar to the 1+ ω ≈4 sTK one sketched in Fig. 3, except that the high tempera-   ture plateauis missing; this is due to the relatively large ω 2 value of T , which is only one decade smaller than the ∼ T∗! K bandwidth cut-off. Figures 4.(b e) are the numerical − confirmations of the asymptotics stated. In all these figures dashed straight lines are to demonstrate deviations from the expected behavior. In Fig. 4.(b) we show the cst+ T∗ ∼ vuutω square root-like asymptotics in the 2CK scaling regime for the channel symmetric case. This behavior is a con- logTTK∗ 21logTTK∗ 0 logDTKF logTωK sequence of the dimension of the leading irrelevant op- gωReˆ()f− ω ∼ vuutTω∗ ∼vuutTωK   T=0,KR> 0,B=0   soearfamateofiranasitysemictphthaoantsincesjluisastnsibshoeoetwrnonpdyiin,scwuthhsseeerdes.aasmIbenerlFoewgigio.tnh4e.i(mnc)cFtaihsgee. ∼T∗ 4.(d) demonstrates (1/ω)1/2-like behavior resulting from the relevant perturbation of the 2CK fixed point Hamil- logDTKF tonian with channel anisotropy. In Fig. 4.(e) the FL-like   ω2-behavior is recovered below T∗, which is typical of logTTK∗ 21logTTK∗ 0 logTωK fermionic operators in the 1CK scaling regimes. InFigs.5.(a b)we showthe universalscalingcurves, ± thatconnec−tthetwo-channelandsinglechannelfixed Kpoints at low-frequencies as a function of ω/T∗. They FIG.3: (coloronline)(top)Sketchofthedimensionlessspec- werecomputedfromrunswithnegativeandpositiveval- tral function of f† : ̺ˆ = D ̺ , and (bottom) the real 0,1,σ f F f ues of K . This universalbehavior is violated for values partofitsdimensionlessGreen’sfunction: Regˆ =D Re R f F f G for T = 0,K > 0 and B = 0 as a function of log(ω/T ). R K Asymptoticsindicatedforω<T werederivedthroughscal- K ing arguments. The large ω-behavior is a result of perturba- 0.5 K R tion theory. 0.001 0.0075 ω) KR< 0 00..0255 ( 0.25 with T∗ the anisotropy scale defined earlier. The su- ρ< f K > 0 - 0.001 R - 0.0075 perscripts refer to the cases of positive or negative - 0.05 ± (a) - 0.25 anisotropies: the superscript “+” is used when the cou- 0 pling is larger in the channel where we measure the 0 Green’sfunctionoff† . Theasymptoticsoftheuniver- 10-6 10-3 100 103 0,α,σ ω / T salfunctions ± and ˜± canbeobtainedthroughsimilar K = 0 K K = 0.02 scaling argumKefnts asKbfefore and they differ only slightly ω) 0.32 R ω) 0.435 R from those of Θf and Θ˜f (see Table V for a summary). ρ<(f ρ<(f 0.42 The properties of ̺ˆT,h=0(ω) are summarized in Fig. 3. 0.28 (b) (c) f A remarkable feature of the spectral function is that it 0 0.3 0.6 1.6 2 contains a correction T∗/ω . This correction can (ω / T)1/2 (ω / T)1/2 ∼ | | K K be obtained by doing peprturbation theory in the small 0.07 KR = 0.02 1.5×10- 3 KR = 0.02 parameter κ(ω) at the two-channel Kondo fixed point. ω) ω) From the asymptotic forms in Table V we find that ρ<(f ρ<(f7.5×10- 4 in the local fermion’s susceptibility a new scale, T∗∗ 0.065 (d) (e) f ∼ √T∗T appears as a result of the competition be- 0 K 1.8 1.9 0 0.0002 0.0004 tween the leading irrelevant operator and the channel (ω / T* )-1/2 (ω / T* )2 anisotropy:24 It isonly inthe regimeT∗∗ <ω <T that f K the leading irrelevant operator determines the dominant FIG. 4: (color online) (a) Dimensionless spectral function of scaling behavior of the local fermion’s susceptibility, i.e., f0,1,σ: ̺ˆf(ω)=DF ̺f(ω) as a function of ω/TK for different we observe the true two-channel Kondo physics. The values of K . (b e) Numerical confirmations of the low- R − expected properties of ρˆ and the real part of its dimen- frequency asymptotics derived through scaling arguments in f sionless Green’s function gˆ in the presence of channel Sec. V. Dashed straight lines are to demonstrate deviations f asymmetryaresummarizedinFig.3. These analyticex- from theexpected √ω-like (b c), 1/√ω-like (d) and ω2-like (e) behavior. In plots (c e)−T∗/T =2.4 10−4. pectations are indeed met by our NRG results. − K × 10 AsymptoticForm Scaling Variable 2CK Scaling Function x 1, 1 x x Scaling Regime ≪ ≪ Θ (x) θ0+θ0′x2 , θ∞ f f f f ω/T T /ω Θ˜ (x) θ˜0+θ˜0′x2 , θ˜∞ x1/2 f f f f ± (x) κ± +κ±′x2 , κ± +κ±′ 1 1/2 K˜f± (x) fκ˜,±0 xf3,0/2 , f,∞ κ˜±f,∞ x ω/T∗ Tf∗∗ /ω, Tf∗∗ ∝√T∗TK Kf f,0 | | f,∞ ˛ ˛ BB˜ff,,σσ ((xx)) βfβ˜0,fσ0,σ+|βxf|0,3σ/′2x2, , βf∞,σ+βfβ∞˜,σf∞,′σ˛x1˛˛1˛/2 ω/Th Th∗∗ /ω, Th∗∗∝√ThTK Θ (x) θ0x, θ∞ sgn(x) ˛ ˛ S S S ω/T T /ω Θ˜ (x) θ˜0x, θ˜∞ sgn(x) x1/2 S S S | | K˜S ((xx)) κ˜0 sgnκ(Sx0)x,x1/2 , κ∞S sgκñ∞(xs)gn+(xκ)∞S ′′ x1 ω/T∗ Ts∗∗ /ω, Ts∗∗ ∝ T∗2TK 1/3 KS S | | S BB˜SS,,zz((xx)) β˜S0,βzS0,|zx|x1/,2 , βS∞,z+ββ˜S∞S∞,,zz′ ˛x1˛1/2 ω/Th Th∗∗ /ω, Th∗∗∝` √ThT´K ˛ ˛ TABLEIV:Asymptoticbehavioroftheuniversalcross-overfunctions. Atfinitetemperature,theboundaryofthetwo-channel Kondoscalingregimeissetbythetemperature. Atzerotemperature,thevariousboundariesofthe2CKscaling regimederive from the competition between theleading irrelevant operator and the relevant perturbation. 0.5 to a single non-universalfeature in our NRG curves. (a) Letus now turnto the effect of a finite magnetic field, ω)( 0.25 KKKRR === --- 000...00052075 tBhe6=sa0mfeorwtahyeincatsheeT2C=K0s,cKalRing=r0e.gimAes,hthaenadrgκumsceanlet ρ< f KR = - 0.0025 concerning the κ = 0 case can be repeated with minor R 6 K = - 0.001 modifications. Now,however,thespinSU (2)symmetry ΚR−f is violated, and therefore the spectral funSctions of f† 0.5 0,α,↑ KKR == 00..0052 (b) and f0†,α,↓ become different, and they are no longer even R either. Nevertheless,duetoparticle-holesymmetry,they ) K = 0.0075 ω R are still related through the relations ρ<(f0.25 KKR == 00..0000215 R Κ+ ρˆ (ω,T,κ,h,...) = ρˆ ( ω,T,κ,h,...) , f f,↑ f,↓ − ρˆ (ω,T,κ,h,...) = ρˆ (ω,T,κ, h,...) . (42) 0 f,↑ f,↓ − 10-3 100 103 106 ω / T* Wearethusfreetochoosetheorientationofthemagnetic FIG.5: (coloronline) Universalcollapse ofthedimensionless 0.2 spectral functions, ̺ˆ =D ̺ (with f in channel 1) to two f F f scaling curves, ± as a function of ω/T∗ for positive (b) and K Kf R negative (a) values of KR. 0 0.1 0.001 0.0025 ω) 0.0075 ( ofKRhigherthanthehighestonesshowninFig.5,where <gf 0 0.02 T∗ becomes comparable to T . e 0.05 K R - - 0.001 The real parts of the local fermion susceptibilities are - 0.0025 plottedin Fig.6 forseveralvaluesofKR. They wereob- -0.1 - 0.0075 tainedbyperformingtheHilberttransformationsnumer- - 0.02 ically. Theyshouldshowathree-peakstructurebasedon - 0.05 the analytic considerations (see Fig. 3). There are two 10-6 10-3 100 103 low-frequency peaks clearly visible, associated with the ω / T K cross-overs at T∗ and T . Furthermore there should be K a non-universal peak at the cut-off. For relatively large FIG.6: (color online) Realpart of thedimensionless Green’s channel anisotropies, where T∗ TK, the former two function: Regˆf = DF Re f (with f† in channel 1) as a ∼ G peaks cannotbe clearlyseparatedinFig.6. Also,due to functionofω/TK fordifferentvaluesofKR. Fromamongthe the large value of T the band cut-off, D , the peak threepeaks sketched in Fig. 3 only thetwo peaks around T∗ K F atω TK andthesme∼aredsingularityatω =DF merge and TK are shown. ∼