PHYSICAL REVIEW B VOLUME 57, NUMBER 2 1 JANUARY 1998-II
Multichannel Kondo effect in an interacting electron system: Exact results for the low-temperature thermodynamics
Mats Granath and Henrik Johannesson Institute of Theoretical Physics, Chalmers University of Technology and Go¨teborg University, S-412 96 Go¨teborg, Sweden ͑Received 2 July 1997͒ We study the low-temperature thermodynamics of a spin-S magnetic impurity coupled to mу2 degenerate bands of interacting electrons in one dimension. By exploiting boundary-conformal-field-theory techniques, we derive exact results for the possible impurity thermal and magnetic response. The leading behavior of the impurity magnetic susceptibility is shown to be insensitive to the electron-electron interaction. In contrast, there are two types of scaling behavior of the impurity specific heat consistent with the symmetries of the problem: Either it remains the same as for the ordinary multichannel Kondo problem for noninteracting Ϫ1 electrons or it acquires a new leading term governed by a critical exponent ␣mϭ(gm Ϫ1)/m, where gmр1is a generalized ͑m channel͒ Luttinger-liquid charge parameter measuring the strength of the repulsive electron- electron interaction. We conjecture that the latter behavior is indeed realized when the impurity is exactly screened (mϭ2S). ͓S0163-1829͑98͒05002-4͔
I. INTRODUCTION We study the problem using boundary conformal field theory ͑BCFT͒,12 assuming that at low temperatures the The violation of Fermi-liquid behavior in the normal state impurity-electron interaction renormalizes onto a scale- invariant boundary condition on the bulk theory. This ap- of the high-Tc superconductors has turned the study of non- Fermi-liquid phenomena into a major theme in condensed- proach, suggested by Affleck and Ludwig for the ordinary multichannel Kondo problem,8,13 has successfully been em- matter physics.1 Additional motivation comes from a grow- ployed for a single channel of interacting electrons coupled ing number of experimental realizations of low-dimensional to a spin-1/2 impurity.14,15 In the present case, with an arbi- electron structures—such as quasi-one-dimensional ͑1D͒ or- 2 trary number of electron channels mу2, a BCFT analysis ganic conductors, or point-contact tunneling in fractional allows for a complete classification of all possible critical 3 quantum Hall devices, where electron correlations are seen behaviors of the impurity-electron composite. Being exact, to produce manifest non-Fermi-liquid behavior. this information should prove useful as a guide to, and a test Two prominent model problems, serving as paradigms for of the validity of, other, more direct approaches to the prob- the study of non-Fermi-liquids, are the Luttinger liquid4,5 and lem ͑yet to be carried out͒. Specifically, we shall show that the ͑overscreened͒ multichannel Kondo effect.6–9 ‘‘Luttinger conformal invariance together with the internal symmetries liquid’’ is the code name for the universal low-energy behav- of the problem restrict the possible types of critical behavior ior of interacting electrons in 1D, whereas the multichannel to only two: Either the theory is the same as for noninteract- Kondo model describes a magnetic impurity coupled via a ing electrons or the electron interaction generates a specific spin exchange to several degenerate bands of noninteracting boundary operator that produces a new leading behavior of electrons in 3D. Both problems have yielded to exact solu- the impurity thermal response. In both cases the leading im- tions, exhibiting a wealth of properties not contained in the purity magnetic response is insensitive to the electron- standard Fermi-liquid picture of the metallic state. electron interaction, implying that the screening of the impu- Here we consider a spin-S magnetic impurity coupled to rity is realized in the same way as in the noninteracting mу2 degenerate bands of interacting electrons in 1D, thus problem, with over-, exact, or underscreening depending on extending the ordinary multichannel Kondo model to the the number of channels and the magnitude of the impurity case of interacting electrons. Specifically, we address the spin. While our method cannot pinpoint which of the two scenarios is actually realized, we conjecture, guided by question about the influence of electron-electron interaction 16,17 on the low-temperature thermal and magnetic response of the analogous results for the one-channel problem, that elec- impurity-electron ͑screening-cloud͒ composite. This problem tron interactions in the bulk do induce an anomalous term in is particularly interesting as it involves the interplay between the impurity specific heat in the case of exact screening (m two kinds of electron correlations; one induced by the spin ϭ2S). exchange interaction with the impurity, the other coming II. THE MODEL from the direct electron-electron interaction. Moreover, the study of a magnetic impurity in the presence of interacting As microscopic bulk model we take a multiband Hubbard electrons may shed new light on possible experimental real- chain with repulsive on-site interaction UϾ0, izations of the multichannel Kondo effect, such as certain 10 Uranium-containing heavy-fermion materials, or the cou- † HelϭϪt ͑cn,icnϩ1,iϩH.c.͒ϩU nn,inn,j , pling of conduction electrons to structural defects in metal n͚,i, n,͚ij, point contacts.11 ͑1͒
0163-1829/98/57͑2͒/987͑6͒/$15.0057 987 © 1998 The American Physical Society 988 MATS GRANATH AND HENRIK JOHANNESSON 57 where cn,i is the electron operator at site n, with i, j moving currents with a second species ͑labeled by ‘‘2’’͒ of 2 ϭ1,....,m and ,ϭ , band and spin indices, respec- left-moving currents: jL/R(x)ϵ jR/L(Ϫx) for xϾ0 ͓with † ↑ ↓ 1 tively, and nn,iϭcn,icn,i is the number operator. This jL/R(x)ϵ jL/R(x)͔, and analogously for spin and flavor cur- model, and its variants, has been extensively studied in the rents. This amounts to folding the system onto the positive x literature, most recently in Ref. 18 where it was argued that axis, with a boundary condition enhanced superconducting fluctuations may result when the 1/2 2/1 degeneracy of the on-site interband coupling U is properly jL ͑0͒ϭ jR ͑0͒ ͑7͒ lifted. At large wavelengths we can perform a continuum at the origin, and then analytically continuing the currents limit c ͱa/2⌿ (na) ͑with a the lattice spacing͒, n,i→ i back to the full x axis. Here Eq. ͑7͒ simulates the continuity which, for small U and away from half-filling, takes Eq. ͑1͒ at xϭ0 of the original bulk theory ͑with the analogous onto boundary conditions in spin and flavor sectors͒. The Sug- awara form of H* in Eq. ͑6͒ implies invariance under inde- 1 d el † pendent U(1)i, SU(2)i , and SU(m)i transformations ͑with Hel ϭ dx vF :L,i͑x͒i L,i͑x͒: m 2 2 ͵ ͭ ͫ dx iϭ1,2 labeling the two species͒, reflecting the chiral sym- d metry of the critical bulk theory. Ϫ:† ͑x͒i ͑x͒: We now insert a local spin S at xϭ0, and couple it to the R,i dx R,i ͬ electrons by an antiferromagnetic (Ͼ0) spin-exchange in- teraction g † † ϩ :r,i͑x͒r,i͑x͒s,j͑x͒s,j͑x͒: 2 † † 1 HKϭ:͓L,i͑0͒ϩR,i͑0͔͒ 2 ͓L,i͑0͒
† † ϩg: x x x x : . 2 ϩR,i͑0͔͒ S:. ͑8͒ L,i͑ ͒R,i͑ ͒R,j͑ ͒L,j͑ ͒ ͮ ͑ ͒ • In the low-energy limit the impurity is expected13 to renor- Here L/R,i(x) are the left/right moving components of the malize to a conformally invariant boundary condition on the electron field ⌿i(x), expanded about the Fermi points bulk theory, changing Eq. ͑7͒ into a new nontrivial boundary Ϯk : ⌿ (x)ϭeϪikFx (x)ϩeikFx (x). Summation F i L,i R,i condition on Hel* . By using BCFT to extract the set of over repeated indices for band, spin, and chirality r,sϭR,L boundary operators present for this boundary condition, is implied. The normal ordering is defined with respect to the finite-temperature effects due to the impurity can be accessed filled Dirac sea, and vF and g are given by vF via standard finite-size scaling by treating ͑Euclidean͒ time ϭ2at sin(akF) and gϭUa/, respectively. as an inverse temperature. For the purpose of implementing BCFT techniques in the presence of a Kondo impurity ͑yet to be added͒, we decouple III. IMPURITY CRITICAL BEHAVIOR charge, spin, and band (flavor) degrees of freedom in Eq. ͑2͒ by a Sugawara construction,19 using the U͑1͒͑charge͒, The set of boundary operators j and the corresponding boundary scaling dimensions ⌬ Ocan be derived from the SU(2)m ͑level m, spin͒, and SU(m)2 ͑level 2, flavor͒ Kac- j Moody currents: finite-size spectrum of Hel* through a conformal mapping from the half-plane to a semi-infinite strip. The mapping is † Jr͑x͒ϭ:r,i͑x͒r,i͑x͒:, ͑3͒ such that the boundary condition corresponding to the impu- rity on the half-plane is mapped to both sides of the strip, 1 with the boundary operators in the plane in one-to-one cor- J ͑x͒ϭ:† ͑x͒ ͑x͒:, ͑4͒ r r,i 2 r,i respondence to the eigenstates on the strip. In particular, the boundary dimensions are related to the energy spectrum A † A through the relation EϭE ϩ v / , with E the ground- J ͑x͒ϭ: T :, ͑5͒ 0 ⌬ l 0 r r,i ij r,j state energy, and l the width of the strip.12 A 2 where are the Pauli matrices, and T , A͕1,...,m Ϫ1͖, A complete set of eigenstates of Hel* are given by the are generators of the defining representation of SU(m) with charge, spin, and flavor conformal towers, each tower de- normalization trTATBϭ1/2␦AB. Diagonalizing the charge fined by a Kac-Moody primary state and its descendants.19 i sector by a Bogoliubov transformation JL/Rϭcosh()jL/R The primaries are labeled by U͑1͒ quantum numbers q in the i Ϫsinh()jR/L , with coth(2)ϭ1ϩvF /͓(2mϪ1)g͔, we obtain charge sector, SU͑2͒ quantum numbers j in the spin sector, the critical bulk Hamiltonian and SU(m) quantum numbers (Dynkin labels) i i (m1 ,...,mmϪ1) in the flavor sector: 1 vc vs * i i i i mϪ1 Helϭ dx :jL͑x͒jL͑x͒:ϩ :JL͑x͒•JL͑x͒: e eϪ 1 m 2 ͵ ͭ4m mϩ2 i i i i i q ϭC ϮD , j ϭ0, ,..., , mkϭ0,1,2, 2 2 2 2 k͚ϭ1 vf ϩ :JiA͑x͒JiA͑x͒: , ͑6͒ ͑9͒ mϩ2 L L ͮ i i i where C , D Z, and mkN, with iϭ1,2 labeling the two 1/2 where vcϭvF(1ϩ2(2mϪ1)g/vF) , vsϭv f ϭvFϪg.We species as above. We can express the complete energy spec- have here retained only exactly marginal terms in the inter- trum, and consequently the complete set of possible bound- action by removing two ͑marginally͒ irrelevant terms in the ary scaling dimensions ⌬ϭ⌬1ϩ⌬2 in terms of the quantum spin and flavor sectors.20 We have also replaced the right- numbers in Eq. ͑9͒: 57 MULTICHANNEL KONDO EFFECT IN AN INTERACTING... 989
mϪ1 ͑qi͒2 ji͑ jiϩ1͒ 1 Turning to condition ͑ii͒, we note that the Kondo interac- ⌬iϭ ϩ ϩ mi f ͑k,k͒ tion ͑8͒ couples L and R fields and thus breaks the chiral 4m mϩ2 2 mϩ2 ͚ k m ͑ ͒ kϭ1 ͩ gauge invariance in all three sectors. This implies that the mϪ1 Kac-Moody symmetries get broken down to their diagonal f ͑k,l͒ i m subgroups, i.e., U͑1͒1ϫU͑1͒2 U͑1͒ in the charge sector, ϩ ͚ ml ϩ , ͑10͒ lϭ1 m ͪ N 1 2 → SU(2)mϫSU(2)m SU(2)2m in the spin sector, and 1 2→ SU(m)2ϫSU(m)2 SU(m)4 in the flavor sector. Operators where f (k,l)ϭmin(k,l)͓mϪmax(k,l)͔ and N. → m with nonzero values of q1 and q2 may thus appear in the Each conformal boundary condition correspondsN to a se- charge sector, provided that q1ϭϪq2 as required by conser- lection rule which specifies that only certain combinations of vation of total charge. Similarly, operators in spin and flavor conformal towers are allowed. Since the trivial boundary i i sectors with nonzero quantum numbers j and m j are now condition ͑7͒ simply defines the bulk theory in terms of a allowed, provided that they transform as singlets under the boundary theory, the associated selection rule reproduces the diagonal subgroups. Remarkably, there exists precisely one * 14 bulk scaling dimensions of Hel . It is less obvious how to generic class of g-dependent operators which satisfy condi- identify the correct selection rule for the nontrivial boundary tion (i) and (ii). It is given by condition representing Eq. ͑8͒. Fortunately, we do not need the full selection rule to extract the leading impurity critical 1 2 i ͱ /2mg i ͱ /2mg s f ϳ :e ͑ m͒L:ϫ:e ͑ m͒L:ϫ ϫ ͑11͒ behavior. For this purpose it is sufficient to identify the lead- O1 ing irrelevant boundary operator8 ͑LIBO͒ that can appear in of dimension ⌬ ϭ1ϩ(gϪ1Ϫ1)/2m 1asg 0. Here the scaling Hamiltonian, as this is the operator that drives the 1 m i O → → i dominant response of the impurity. As the possible L(x) is a chiral charge boson of species i ͓i.e., L(x) i s f correction-to-scaling-operators are boundary operators con- ϭ͐dxjL(x)], while and are the singlet fields ͑under strained by the symmetries of the Hamiltonian, this sets our the diagonal subgroups͒ in the decomposition of the product strategy: We consider all selection rules for combining con- of primary fields, ( j1ϭ1/2)ϫ( j2ϭ1/2) and (1,0,...,0) formal towers in Eq. ͑9͒͑thus exhausting all conceivable ϫ(0,0,...,1) in spin and flavor sectors, respectively. These 2 boundary fixed points͒, for each selecting the corresponding carry dimensions ⌬sϭ3/͓2(mϩ2)͔ and ⌬ f ϭ(m LIBO, using Eq. ͑10͒. We then extract the possible impurity Ϫ1)/͓m(mϩ2)͔. The parameter gm is given by critical behaviors by identifying those LIBO’s that ͑i͒ pro- duce a noninteracting limit g 0 consistent with known re- Ϫ 1/2 g vF sults, and ͑ii͒ respect the symmetries→ of H*ϩH . g ϭ 1ϩ2͑2mϪ1͒ ϭ р1 ͑12͒ el K m ͩ v ͪ v F c
A. Overscreening: m>2S and plays the role of a generalized ͑channel-dependent͒ Luttinger-liquid charge parameter.5 Let us first focus on the case mϾ2S. Here the noninter- The next-leading generic irrelevant boundary operator sat- ϭ acting (g 0) problem renormalizes to a nontrivial fixed isfying ͑i͒ and ͑ii͒ is independent of g, and given by point, as can be seen by passing to a basis of definite-parity 21 (PϭϮ) fields Ϯ,i(x)ϭ(1/&)͓L,i(x)ϮR,i(Ϫx)͔. 1 1 2 1 2 2 2ϳJϪ1• ϫ1 ϩ1 ϫJϪ1• , ͑13͒ In this basis Hel*͓gϭ0͔ϩHK becomes identical to the Hamil- O tonian representing 3D noninteracting electrons in 2m chan- with Ji i the first Kac-Moody descendant of the spin-1 nels ͑PϭϮ, iϭ1,...m͒, coupled to a local spin in the m Ϫ1• primary field i, obtained by contraction with the Fourier positive parity channels only. At low temperatures this sys- mode Ji of the SU(2) currents. It carries dimension tem flows to the overscreened m-channel Kondo fixed point Ϫ1 m 22 ⌬ ϭ(mϩ4)/(mϩ2), and is the same operator that drives with a LIBO of dimension ⌬ϭ(4ϩm)/(2ϩm). Consider O2 first the case that this fixed point is stable against perturba- the leading impurity response in the noninteracting problem. tions in g ͑or connected to a line of gϾ0 boundary fixed For certain special values of m additional g-dependent points via an exactly marginal operator͒. To search for a boundary operators satisfying ͑i͒ and ͑ii͒ appear, but as these novel leading scaling behavior for gÞ0 it is then sufficient are nongeneric and, for given m, of higher dimensions than 24 to search for boundary operators with dimensions in the in- ⌬ , we do not consider them here. O1 terval 1р⌬р(4ϩm)/(2ϩm), which produce an impurity Piecing together the results, it follows that either in O1 response analytically connected to that of the noninteracting Eq. ͑11͒ or 2 in Eq. ͑13͒ play the role of a LIBO at the g theory. An operator with 1р⌬Ͻ3/2 contributes an impurity Ͼ0 boundaryO fixed point. We may thus define a scaling specific heat scaling as (⌬Ϫ1)2T2⌬Ϫ2,8 and condition ͑i͒ Hamiltonian then requires that, as g 0, ⌬ (4ϩm)/(2ϩm) or that → LIBO→ ⌬LIBO 1 ͓with in this case next-leading dimension ⌬ (4 H ϭH*ϩ 0͒ϩ 0͒ϩ 0͒ ..., → → scaling el 1 1͑ 2 2͑ 3 3͑ ϩm)/(2ϩm)͔. On the other hand, if the gϭ0 fixed point O O O ͑14͒ gets destabilized as g is switched on, condition ͑i͒ enforces the LIBO at the new gϾ0 boundary fixed point to become with conjugate scaling fields, and less relevant op- j OjϾ2 marginally relevant for gϭ0 ͑so as to produce the necessary erators. In the case that 1 in Eq. ͑11͒ does not appear, 1 flow back to the known gϭ0 overscreened m-channel fixed ϵ0. Using Eq. ͑14͒, theO thermal response may now be cal- 23 point͒. Thus, for this case condition ͑i͒ unambiguously re- culated perturbatively in the scaling fields j , using standard quires that ⌬ 1asg 0. techniques.8 We thus obtain for the impurity specific heat: LIBO→ → 990 MATS GRANATH AND HENRIK JOHANNESSON 57
Ϫ1 2 gϪ1Ϫ1 /m problem with impurity spin 1/2: the electron screening cloud C ϭc ͑1Ϫg ͒ T͑ m ͒ imp 1 m behaves as a local Fermi liquid with a /2 phase shift of the TK single-electron wave functions. Analogous to the single- 1 14 c2T ln ϩ••• mϭ2, Sϭ 2 channel problem, there are three degenerate LIBO’s for ϩ ͩ T ͪ T 0. this case, given by the energy-momentum tensors ͑of dimen- 4/͑mϩ2͒ → ͭ c2ЈT ϩ••• mϾ2, mϾ2S sion ⌬ϭ2͒ in charge, spin, and flavor sectors. When m ͑15͒ ϭ2S these produce the leading term in the impurity specific heat, Cimpϭb1Tϩ ..., aswell as in the susceptibility, imp 2 Here c1 , c2 , and c2Ј are amplitudes of second order in the ϭb2Ϫb3T ϩ ..., with b1,2,3Ͼ0 amplitudes linear in the scaling fields with corresponding indices, TK plays the role scaling fields, and with higher powers in temperature coming of a Kondo temperature, and ‘‘...’’ denotes subleading terms. from higher-order descendants of the identity operator. Note that the amplitude of the leading term in Eq. ͑15͒ al- When mϽ2S the impurity spin is only partially screened as ways vanishes when gϭ0, thus making the second there are not enough conduction-electron channels to yield a g-independent term dominant. singlet ground state. This leaves an asymptotically decoupled The result in Eq. ͑15͒ is exact and independent of the spin SϪm/2Ͼ0, adding a Curie-like contribution to Cimp and precise nature of the boundary fixed point. In the case that imp , in addition to logarithmic corrections characteristic of 9 1ϵ0, and hence c1ϵ0, the gϾ0 fixed point is the same as asymptotic freedom. for the ordinary ͑noninteracting͒ overscreened problem, al- Let us study the case mϭ2S and explore what happens though the content of subleading irrelevant operators when turning on the electron interaction. Implementing con- OjϾ2 may differ. In the alternative case, with 1Þ0(c1Þ0), the dition ͑i͒ from the previous section, possible LIBO’s appear- situation is more intricate, with, in principle, three possibili- ing for gϾ0 must have dimensions ⌬ with the property ties: ͑a͒ the gϭ0 ͑ordinary overscreened Kondo͒ and gÞ0 ⌬ 1or⌬ 2asg 0. Using condition ͑ii͒, we find that in → → → fixed points are the same, but with different contents of ir- addition to 1 in Eq. ͑11͒ there are several new allowed relevant operators, ͑b͒ the gϭ0 and gϾ0 fixed points are classes of g-dependentO generic boundary operators, all with continuously connected via a critical line by an exactly mar- ⌬ 2asg 0. As any boundary operator with dimension ginal boundary operator with a scaling field parametrized by ⌬Ͼ→3/2 produces→ the same leading scaling in temperature as g,or͑c͒the gϭ0 and gϾ0 fixed points are distinct and the a ⌬ϭ2 operator ͑although of different amplitudes͒,14 the flow between them is governed by a marginally relevant only possible leading term with an interaction-dependent ex- 23 operator. We postpone a discussion of the various possi- ponent is again generated by 1 , as in the overscreened bilities to the next section. case. Thus, since does not contributeO to the impurity sus- O1 Turning to the impurity magnetic susceptibility imp , its ceptibility, its leading behavior remains the same as in the leading scaling behavior is produced by the lowest- noninteracting problem, exhibiting exact screening with a dimension boundary operator which contains a nontrivial constant zero-temperature contribution, singlet SU(2)2m factor that couples to the total spin 14 2 density. Since 1 in Eq. ͑11͒ only contains the identity in ϭb ϩ•••Ϫb T ϩ•••, mϭ2S, gу0, T 0, O imp 2 3 → the SU(2)2m sector the desired operator is identified as 2 in ͑17͒ Eq. ͑13͒. Thus, the leading term in the magnetic susceptibil-O where ••• denotes possible second-order contributions in ity imp due to the impurity is independent of the electron- electron interaction, and remains the same as for the nonin- scaling fields. The amplitude b2 is the same as in the nonin- teracting overscreened m-channel Kondo problem:7,8 teracting problem, while b3 may pick up second-order interaction-dependent terms contributed by subleading op- erators. The leading possible interaction-dependent correc- TK c2 ln ϩ••• mϭ2, Sϭ1/2 tion scales as ϭ ͩ T ͪ T 0, imp → ͑2Ϫm͒/͑mϩ2͒ corr Ϫ1 Ϫ1 ͭ c2ЈT ϩ••• mϾ2, mϾ2S 1ϩ͑gm Ϫ1͒/m impϳ͑gm Ϫ1͒T ͑18͒ ͑16͒ and is produced at second order by the composite boundary where c and c are second order in scaling fields and 2 2Ј ••• operator denotes subleading terms. We find that there are no sublead- ing interaction-dependent divergent contributions possible, 1 2 i ͱ /2mg i ͱ /2mg diag s f ϳ:e ͑ m͒L:ϫ:e ͑ m͒L:ϫJ ϫ . as these would give rise to new divergences also in the non- O3 Ϫ1 • interacting limit. ͑19͒ diag 1 2 Here J ϵJLϩJL is the generator of the diagonal SU(2)2m B. Exact screening and underscreening: mр2S subgroup in the spin sector, s is a diagonal spin-1 field in An analysis analogous to the one above can be carried out the product of primaries ( j1ϭ1/2)ϫ( j2ϭ1/2), and the for mр2S as well. Again passing to a definite-parity basis charge and flavor factors are the same as for 1 above. The 9 O and exploring known results, one verifies that the noninter- operator 3 has scaling dimension ⌬3ϭ1ϩ⌬1ϭ2 Ϫ1 O acting 1D electron ground state carries a spin SϪm/2 corre- ϩ(gm Ϫ1)/2m and, as seen in Eq. ͑18͒, gives a vanishing sponding to a strong-coupling fixed point. When mϭ2S the amplitude at gϭ0, thus ensuring the correct behavior in the impurity is completely screened and the situation is essen- noninteracting limit. Whether 3 appears in the spectrum or tially the same as for the ordinary single-channel Kondo not, however, must be checkedO by an independent method. 57 MULTICHANNEL KONDO EFFECT IN AN INTERACTING... 991
Two possible scenarios again emerge for the scaling of d1 ϭϪ͑q ϩq ͒͑ ϩ ͒ , ͑22͒ the impurity specific heat: Either it remains the same as in d ln 1 2 1 2 1 the noninteracting exactly screened problem or, in the case 0 that 1 appears as a LIBO: O with 0 a short-time cutoff, q1 and q2 the charge quantum Ϫ Ϫ1 numbers of , and and the scaling fields of the 1 2 ͑gm Ϫ1͒/m 1 1 2 Cimpϭc1͑1Ϫgm ͒ T ϩb1Tϩ••• , O 1 2 exactly marginal charge currents jL and jL ͓allowed due to mϭ2S, gу0, T 0, ͑20͒ breaking of particle-hole symmetry in the microscopic → Hamiltonian ͑1͒͑Ref. 26͔͒. Conservation of total charge q1 where the amplitude c1 is independent of gm, and b1 may ϩq2ϭ0 thus gives d1 /dln0ϭ0 to second order in the differ by second-order additive terms coming from scaling fields. If this property does persist to higher orders interaction-dependent subleading corrections. Notably, by ͑as suggested by the cancellation of the one-loop contribu- putting mϭ1 in Eq. ͑20͒ we recover the critical exponent tion due to a symmetry͒, the gÞ0 and gϭ0 boundary fixed 16 conjectured by Furusaki and Nagaosa for the impurity spe- points are either ͑a͒ the same ͑but with different contents of cific heat in the ͑exactly screened͒ single-channel problem. irrelevant operators͒ or ͑b͒ connected via a line of fixed The leading possible interaction-dependent correction to ͑20͒ 1 2 points by the exactly marginal charge currents jL and jL is also produced by 3 and scales as with scaling fields and parametrized by g .Asweare O ͑ 1 2 ͒ unable to determine the actual scenario, the question about Ϫ Ϫ1 corr 1 2ϩ͑gm Ϫ1͒/m Cimpϳ͑gm Ϫ1͒T , ͑21͒ the relation of the gϾ0 fixed point to that of the noninter- with a vanishing amplitude in the noninteracting limit. acting problem remains open. For mϽ2S, the influence of the electron-electron interac- A second open issue is whether the electron-electron in- tions on the impurity-electron composite corresponding to teraction influences the impurity-electron ͑screening cloud͒ the screened part of the impurity spin is the same as for the composite differently when the impurity is overscreened exact screening, with the same scenarios for the critical be- (mϾ2S) as compared to under or exact screening (m havior. However, the weak coupling between the uncompen- р2S). In the overscreened case with free electrons the im- sated ͑asymptotically free͒ part of the impurity spin and the purity induces a critical behavior where the size of the conduction electrons produce corrections at finite T ͑Ref. 9͒ screening cloud diverges as one approaches zero tempera- that may get modified by the electron-electron interaction. ture. As a consequence, all conduction electrons become cor- We have not attempted to include these effects here. related due to the presence of the impurity. By turning on a weak ͑screened͒ Coulomb interaction among the electrons ͓in 1D simulated by the local e-e interaction in Eq. ͑2͔͒ these IV. DISCUSSION correlations may change. Will the change be such as to pro- To conclude, we have presented an analysis of the pos- duce a novel impurity critical behavior? While our analysis sible low-temperature thermodynamics of the multichannel does not provide an answer, it predicts its exact form if it Kondo problem for an interacting electron system in 1D. does appear. In the case of exact screening there is strong While the leading term in the impurity susceptibility remains evidence that the impurity scaling behavior is indeed gov- the same as for noninteracting electrons ͑for any number of erned by interaction-dependent exponents. In this case the channels m and impurity spin S͒, there are two possible be- screening cloud has a finite extent. Turning on the Coulomb haviors for the impurity specific heat consistent with the interaction, electrons ‘‘outside’’ of the cloud will become symmetries of the problem: Either it remains the same as in correlated, most likely influencing the rate with which they the noninteracting problem or it acquires a new leading term, tunnel into and out of the cloud, hence influencing its prop- Ϫ1 erties. As this impurity-electron composite is described by a scaling with a non-Fermi-liquid exponent ␣mϭ(gm Fermi-liquid fixed point where the electrons simply acquire Ϫ1)/m, with gmр1 in Eq. ͑12͒ measuring the strength of ͑ the repulsive electron-electron interaction. These results are a phase shift͒ in exact analogy with the single-channel exact, given the existence of a stable boundary fixed point, Kondo problem, we expect that the effect of turning on elec- an assumption common to all applications of BCFT to a tron interactions will indeed be similar to the single-channel quantum impurity problem.13 As we discussed in Sec. III A, case. Considering the recent Monte Carlo data by Egger and 17 this fixed point ͑for given m and S͒ is disconnected from that Komnik supporting the single-channel Furusaki-Nagaosa scaling,16 this strongly favors the appearance of the of the noninteracting problem only if the LIBO 1 in Eq. ͑11͒ turns marginally relevant as g is set to zero.O Does this interaction-dependent exponents in Eqs. ͑18͒, ͑20͒, and ͑21͒ happen? A conclusive answer would require a construction when mϭ2S. of the corresponding exact renormalization-group ͑RG͒ equations. This is a nontrivial task, and we have only carried ACKNOWLEDGMENTS out a perturbative analysis to second order in the scaling fields. Turning Eq. ͑14͒ into a Lagrangian, and integrating We thank I. Affleck, N. Andrei, R. Egger, P. Fro¨jdh, and out the short-time degrees of freedom using the operator A.W.W. Ludwig for valuable comments and suggestions. product expansion,25 one obtains the one-loop RG equation H.J. acknowledges support from the Swedish Natural Sci- for the scaling field conjugate to : ence Research Council. 1 O1 992 MATS GRANATH AND HENRIK JOHANNESSON 57
1 For a review, see, e.g., H. J. Schulz, in Proceedings of Les 20 The irrelevance ͑in renormalization-group sense͒ of the † † Houches Summer School LXI, edited by E. Akkermans et al. excluded piece Hexclϭg(:L,iR,iR,jL,j :ϩ 1/2m : † † ͑Elsevier, Amsterdam, 1995͒. L,iL,iR, jR, j :) in Eq. ͑6͒ can be seen by using the fact 2 See, e.g., Low-Dimensional Conductors and Superconductors, that Hel in Eq. ͑2͒ hasaU͑1͒ϫSU(2m) symmetry, and hence edited by D. Je´roˆme and L. G. Caron ͑Plenum, New York, can be expressed in terms of U͑1͒ and SU(2m)1 Kac-Moody A A 2 1987͒. currents. In terms of these HexclϭϪ2gJLJR , A͕1,...,(2m) 3 F. P. Milliken, C. P. Umbach, and R. A. Webb, Solid State Com- Ϫ1͖, which is ͑marginally͒ irrelevant for gϾ0. ͓For a proof of mun. 97, 309 ͑1996͒. this in the analogous SU(2)1 case, see I. Affleck, D. Gepner, H. 4 F. D. M. Haldane, J. Phys. C 14, 2585 ͑1981͒. J. Schulz, and T. Ziman, J. Phys. A 22, 511 ͑1989͒.͔ 5 21 For a review, see J. Voit, Rep. Prog. Phys. 58, 977 ͑1995͒. Note that the interacting part of the bulk Hamiltonian Hel be- 6 P. Nozie`res and A. Blandin, J. Phys. ͑Paris͒ 41, 193 ͑1980͒. comes nonlocal in the definite-parity basis, thus precluding its 7 N. Andrei and C. Destri, Phys. Rev. Lett. 52, 364 ͑1984͒;P.B. use when gÞ0. Wiegmann and A. M. Tsvelik, Z. Phys. B 54, 201 ͑1985͒. 22 I. Affleck, A. W. W. Ludwig, H.-B. Pang, and D. L. Cox, Phys. 8 I. Affleck and A. W. W. Ludwig, Nucl. Phys. B 352, 849 ͑1991͒; Rev. B 45, 7918 ͑1992͒; N. Andrei and A. Jerez, Phys. Rev. 360, 641 ͑1991͒; A. W. W. Ludwig and I. Affleck, Phys. Rev. Lett. 74, 4507 ͑1995͒. Lett. 67, 3160 ͑1991͒. 23 An inspection of possible selection rules for a gϾ0 stable bound- 9 For a review, see P. Schlottmann and P. D. Sacramento, Adv. ary fixed point, using Eq. ͑10͒, shows that only dimensions ⌬ Phys. 42, 641 ͑1993͒; D. L. Cox and A. A. Zawadovski, у1 are allowed in the limit g 0. As no symmetry is broken at → cond-mat/9704103 ͑unpublished͒. gϭ0, a destabilizing operator can hence be at most marginally 10 D. L. Cox, Phys. Rev. Lett. 59, 1240 ͑1987͒. relevant. 11 D. C. Ralph, A. W. W. Ludwig, J. von Delft, and R. A. Buhrman, 24 A computer check of the spectra with 2рmр50, using Eq. ͑10͒, Phys. Rev. Lett. 72, 1064 ͑1994͒. reveals that additional operators satisfying ͑i͒ and ͑ii͒ occur at 12 J. L. Cardy, Nucl. Phys. B 324, 581 ͑1989͒. mϭ2, 4, 6, 8, 9, 18, 32, and 50. As there is no change in the 13 For a review, see A. W. W. Ludwig, in Quantum Field Theory structure of the Hamiltonian for these particular channel num- and Condensed Matter Physics, edited by S. Randjbar-Daemi bers, we expect these operators to be excluded by the selection and L. Yu ͑World Scientific, Singapore 1994͒; and I. Affleck, rule defining the boundary fixed point. Acta Phys. Pol. B 26, 1869 ͑1995͒. 25 For a review of the method, see e.g., J. L. Cardy, Scaling and 14 P. Fro¨jdh and H. Johannesson, Phys. Rev. Lett. 75, 300 ͑1995͒; Renormalization in Statistical Physics ͑Cambridge University Phys. Rev. B 53, 3211 ͑1996͒. Press, Cambridge, 1996͒. 15 P. Durganandini, Phys. Rev. B 53, R8832 ͑1996͒. 26 Although the multiband Hubbard model in Eq. ͑1͒ is not invariant 16 A. Furusaki and N. Nagaosa, Phys. Rev. Lett. 72, 892 ͑1994͒. See under charge conjugation ͑particle-hole transformation͒ away also Refs. 14 and 15. from half-filling, the low-energy effective theory in Eq. ͑2͒ is 17 R. Egger and A. Komnik, cond-mat/9709139 ͑unpublished͒. invariant under the corresponding transformation. The reason for 18 M. I. Salkola and A. V. Balatsky, Phys. Rev. B 51,15267͑1995͒. this is that the normal ordering of the latter subtracts the 19 For a review see, e.g., J. Fuchs, Affine Lie Algebras and Quantum symmetry-breaking terms. We may remove this accidental sym- Groups ͑Cambridge University Press, Cambridge, 1992͒, and metry by explicitly introducing operators that break charge con- references therein. jugation.