One- and two-channel Kondo model with logarithmic Van Hove singularity: a numerical renormalization group solution
aa r X i v : . [ c ond - m a t . s t r- e l ] J a n One- and two-channel Kondo model with logarithmic Van Hove singularity: a numericalrenormalization group solution
A. K. Zhuravlev, A. O. Anokhin and V. Yu. Irkhin
M.N. Mikheev Institute of Metal Physics, Russian Academy of Sciences, 620108 Ekaterinburg, Russia
Abstract
Simple scaling consideration and NRG solution of the one- and two-channel Kondo model in the presence of a logarithmic VanHove singularity at the Fermi level is given. The temperature dependences of local and impurity magnetic susceptibility andimpurity entropy are calculated. The low-temperature behavior of the impurity susceptibility and impurity entropy turns out to benon-universal in the Kondo sense and independent of the s − d coupling J . The resonant level model solution in the strong couplingregime confirms the NRG results. In the two-channel case the local susceptibility demonstrates a non-Fermi-liquid power-lawbehavior. Keywords:
Kondo model, Van Hove singularities, strong correlationsAnomalous f - and d -systems possess highly unusual elec-tronic properties and magnetism. Besides the heavy-fermionbehavior, they demonstrate the non-Fermi-liquid (NFL) be-havior: logarithmic or anomalous power-law temperature de-pendences of magnetic susceptibility and electronic specificheat [1]. Their magnetism has both localized and itinerant fea-tures, being determined by both Kondo e ff ect and density ofstates (DOS) singularities which are especially important formagnetic ordering.The NFL behavior is related to peculiar features of electronand spin fluctuation spectra. In particular, the multichannelKondo model is often used which assumes existence of de-generate electron bands. This model explains power-law orlogarithmic behavior of electronic specific heat and magneticsusceptibility [2, 3, 4]. Recently, the one- and two-channelcharge Kondo e ff ect was extensively discussed for nanostruc-tures, layer systems and quantum dots [5, 6, 7, 8].In the present Letter we treat the Kondo model with the elec-tron spectrum containing a logarithmic DOS singularity. Thisvan Hove singularity is typical, in particular, for the 2D case.This is present, e.g., in the electron spectrum of graphene wherethe Kondo e ff ect is tunable with carrier density [9]. Recently, apossibility of graphene doping for the exploration of Van Hovephysics was proposed [10].Whereas the flat-band Kondo model permits exact Betheansatz solution, the model with singular DOS is a challenge foranalytical field-theoretical consideration. At present, the caseof empty conduction band was investigated by numerical renor-malization group (NRG) method [11]. A NRG and resonantlevel model treatment of the power-law and 1 / | E | ln | E | diver-gent bare DOS was performed in Refs.[12]. To investigate the Email address:
[email protected],[email protected] () case of singular DOS for the Kondo metal we use a simple scal-ing consideration and compare the results with more advancedNRG calculations.We start from the Hamiltonian of the one-center s − d ( f ) ex-change (Kondo) model H sd = X k m α ε k c † k m α c k m α − X kk ′ m αβ J kk ′ S σ αβ c † k m α c k ′ m β . (1)Here ε k is the band energy, S are spin operators with spin valuebeing S , σ are the Pauli matrices, in the case of contact coupling J kk ′ = J / N s where J is the s − d ( f ) exchange parameter, N s isthe number of lattice sites, m = ... M is the orbital degeneracyindex, α, β are spin indices.The density of states corresponding to the spectrum ε k is sup-posed to contain a Van Hove singularity near the Fermi level. Inparticular, for the square lattice with next- and next-to-nearestneighbour transfer the spectrum reads ε k = t (cos k x + cos k y ) + t ′ (cos k x cos k y +
1) (2)and we have the density of states ρ ( E ) ≃ π √ t − t ′ ln 16 √ t − t ′ | E | (3)where the bandwidth is determined by | E − t ′ | < | t | . Belowwe use the approximate density of states for t ′ = ρ ( E ) = ̺ F ( E ) , F ( E ) = ln 4 D | E | , ̺ = π D , D = t . (4)We apply the “poor man scaling” approach [13]. This consid-ers the dependence of e ff ective (renormalized) coupling J e f ( C )on the flow cuto ff parameter C → − Preprint submitted to Elsevier September 18, 2018 o find the scaling equation we pick out in the sums for theKondo terms the contribution of intermediate electron statesnear the Fermi level with C < ε k < C + δ C to obtain to next-leading order in J (see details in [14, 15]) δ J e f ( C ) = ̺ J [ F ( C ) + J ̺ MF ( C / F ( − C / δ C / C (5)where the factor of F comes from the singularity of DOS. Forthe correction to the localized magnetic moment S e f determin-ing the Curie constant, as determined from the Kondo contribu-tion to local magnetic susceptibility [16, 17], we have (cf. [14]) δ S e f ( C ) / S = ̺ J MF ( C / F ( − C / δ C / C . (6)The lowest-order scaling calculation according to (5) (seealso Refs. [18, 14]) yields for the boundary of the strong-coupling region T K ∝ D exp − π D | J | ! / . (7)However to describe a possible NFL-type behavior(intermediate-coupling fixed point), we have to use the next-order scaling equations ∂ g e f ( ξ ) ∂ξ = [ ξ − M ξ + ln 2) g e f ( ξ )] g e f ( ξ ) (8)where ξ = ln | D / C | and we have introduced the dimensionlesse ff ective s − d coupling constant g e f ( C ) = − ̺ J e f ( C ) . (9)In the flat-band case (where the singular factors ξ + ln 2 are re-placed by unity) such equations give a finite fixed point g e f ( ξ →∞ ) = / M . It is known that this point is unphysical (unreach-able) for M =
1, but for M > M = G e f ( ξ ) ≡ g e f ( ξ ) ξ at large ξ can be obtained, which has the form G e f ( ξ ) = M + − M ! ξ . (10)The second term can change sign and is positive for large M , sothat the derivative of G e f ( ξ ) changes its sign. Thus the detailsof scaling behavior are rather sensitive to parameters. Besidesthat, the factors in (8) is well determined only within the 1 / M -expansion. Nevertheless, these results demonstrate existence ofthe “fixed point” G e f ( ξ ) → / M By analogy with Refs. [14, 15] we can write down the scalingequation for the e ff ective localized magnetic moment ∂ ln S e f ( ξ ) ∂ξ = − M G e f ( ξ ) (11)so that to leading approximation S e f ( C ) ≃ ( | C | / T K ) ∆ , ∆ = / M (12) and for the local magnetic susceptibility, χ loc ( T ) = / T Z h S z ( τ ) S z i d τ , (13)we obtain the power-law dependence χ loc ( T ) ∝ S e f ( T ) / T ∝ ( T / T K ) ∆ − (14)where we have taken into account that G e f ( C ) reaches the valueabout 2 / M at | C | ∼ T K . The dependence χ loc ( T ) follows at hightemperatures the Curie–Weiss law, has a maximum at T ∼ T K and decreases with further increasing T for small M (however,in this case the scaling results for T < T K are not reliable). Forlarge M , χ loc ( T ) is divergent at low temperatures.Note that the exponent in (14) becomes modified in higherorders in 1 / M , and in the flat-band case one has ∆ = / ( M + ffi cient informationprovided by 1 / M expansion. In particular the result for sus-ceptibility (14) does not di ff er from the corresponding flat-bandresult and does not take into account the logarithmic factor at M = ff erences in the perturbation expansionfor the singular DOS and flat-band cases. Leaving the alge-braical structure of the perturbation series the same, it leads tothe expansion in terms of G e f ( ξ ) for the singular DOS ratherthan g e f ( ξ ) for the flat-band case. Moreover, it gives the in-verse logarithmic contributions to the impurity entropy and thespecific heat (see below).Therefore, we calculate impurity magnetic susceptibility, en-tropy and specific heat by using numerical renormalizationgroup (NRG) approach [20] in the one- and two-channel cases.The NRG procedure starts from the solution of the isolated-impurity problem (sites “imp” and ǫ in Fig. 1). At the ini- t ❞ ❞ ❞ ❞ ✲✛ ✲✛ ✲✛ ✲✛ imp J ǫ γ ǫ γ ǫ γ ǫ . . . Figure 1: Representation of the Kondo model in the form of a semiinfiniteWilson chain tial step, we add a first conducting electronic site ǫ , and con-struct and diagonalize a Hamiltonian matrix on this Hilbertspace (with a 4 M –fold higher dimensionality). This procedureis multiply repeated. However, since the dimensionality of theHilbert space grows as 4 MN ( N is the number of an iteration), itis impossible to store all the eigenstates during the calculation.Therefore, it is necessary to retain after each iteration only thestates with the lowest energies. If we restrict ourselves to a cer-tain maximum number of stored states (determined by the com-putational possibilities), it is necessary, starting from a certainiteration, to retain of the order of 1 / M of states at each step.2ractically, the number of the states is reasonable in the one-and two-channel cases. To take into account the disturbanceintroduced by the elimination of the high-lying states we useWilson’s logarithmic discretization of the conduction band [20](see also Ref. [18]). In real calculations for M = states at each NRG iterationwith Wilson’s logarithmic discretization factor being Λ = S ( T =
0) with the Betheansatz results for the flat-band DOS, which is ln 2 /
2, [4] (seebelow Fig. 7) demonstrates su ffi cient of NRG calculations. Forthe case M = Λ =
Λ = . Λ = t = / , t ′ = D ′ from the normal-ization condition, which gave D ′ = . D . E N ( N - ) / N (even) SD J=-0.2 FB J=-0.3 a E N ( N - ) / N (even) SD J=-0.15 FB J=-0.25 b Figure 2: The picture of lowest energy levels for flat-band (FB) DOS and forDOS with Van Hove singularity (SD): (a) the one-channel model,
Λ =
2, (b)the two-channel model,
Λ = The picture of energy levels depending on the NRG step N (multiplied by Λ ( N − / ) is presented in Figs. 2. The energiesare referred to the energy of the ground state. For flat-band andsingular DOS cases the values of the corresponding parameters J were chosen from the approximate equality of Kondo temper-atures T K for the cases addressed. An important di ff erence be-tween the flat-band and singular DOS situations is considerablyslower tending of the curves E ( N ) to the asymptotic values inthe latter case (cf. Ref. [21]). The slow fall o ff in the Van Hove case is somewhat similar to the situation for an underscreenedKondo model [22].Because of retaining only part of the energy spectrum atthe N -th step of the NRG procedure, thermodynamic aver-ages should be calculated at a temperature that depends on Λ , T N = Λ − N / T [20]. Here the starting temperature T should benot too small to avoid the problem of discreteness of the energyspectrum. The total entropy and specific heat read [23] S tot = h H i tot / T + ln Z tot , C tot = h h H i tot − h H i i / T , (15)where Z is partition function. On di ff erentiating h S z i tot withrespect to magnetic field one obtains [20] T χ tot ( T ) = h S z i tot − h S z i , (16)The quantities T χ band ( T ), S band and C band are calculated in asimilar way, and the corresponding impurity contributions areobtained by subtracting them from (15)-(16).First we consider the results for M = χ loc (13). This is the susceptibilityof a single impurity in a magnetic field that acts locally onlyon this impurity; this can be measured experimentally from theimpurity spin correlation function and is obtained in simple per-turbation calculations [16].Instead of calculating (13) directly we use the following pro-cedure. One can apply small magnetic field h to impurity spinonly and then calculate numerically derivative of the local mag-netization induced, d h S z i / dh , in the limit h →
0. To be sure thatthis limit with a linear dependence h S z i ∝ h has been reachedwe performed calculations for a series of h values.Numerical results for χ loc are shown in Figs. 3–5. They l o c T K l o c T / T K SD J=-0.2 SD J=-0.1 SD J=-0.06 FB J=-0.1 loc
Figure 3: NRG results for T loc K χ loc extrapolated for Λ = clearly demonstrate that there exists characteristic crossovertemperature T loc K for χ loc . Similar to Wilson [20] we use (some-what ambiguously) the definition T loc K χ loc ( T loc K ) = . T loc K arepresented in Table 1.However, unlike flat-band case, we did not observe exactuniversal behavior of T loc K χ loc ( T ) as a function of T / T loc K (seeFig. 3).3 able 1: NRG calculations for T loc K with J varying J M = , Λ = M = , Λ = · − · − -0.06 1.68 · − · − -0.1 1.39 · − · − -0.15 5.32 · − · − -0.2 1.19 · − · − Table 2: χ loc (0) with varying J ; M =
1, extrapolation to
Λ = J -0.06 -0.1 -0.2 χ loc (0) 454.488 52.9607 5.93006The behavior of χ loc ( T ) at low temperatures is shown inFig. 5. The empirical linear dependence χ loc ( T ) χ loc (0) = + χ loc (0) T (17)describes the low-temperature behavior rather well, with χ loc (0)presented in Table 2.Generally, the low temperature dependence of χ loc is ratherunusual: instead of nearly constant value below T K one ob-serves a linear behavior. To confirm that this behavior is not anartifact, NRG calculations were performed for a series of valuesof Λ (see Fig. 4).Thus the situation in the singular DOS case di ff ers from thatin the flat band case where typical parabolic Fermi liquid de-pendence T loc K χ loc ( T ) = . − . T / T loc K ) takes place (seeFig. 3). T/D l o c l o c T/D =1.5 =2 =2.5 = 0 = 0.01=2
Figure 4: NRG results for χ loc . One-channel case with singular DOS, J = − . Λ varies. µ is the distance between singularity and the Fermi level. If the singularity is slightly shifted from the Fermi level, with µ being the value of the shift, we should have the crossover tothe typical for the flat-band temperature behavior. However atsmall µ the dependence of χ loc ( T ) acquires at lowering tem-perature a linear increase instead of a maximum (see inset inFig. 4). l o c ( T ) / l o c ( ) T loc (0) J=-0.06 J=-0.1 J=-0.2 1+(2/3)T loc (0) Figure 5: NRG results for χ loc extrapolated to Λ =
For completeness we introduce an alternatively defined sus-ceptibility χ imp , which can be expressed as a di ff erence of mag-netic susceptibilities of the whole system and the system with-out impurity: χ imp ( T ) = χ tot ( T ) − χ band ( T ) , (18)where χ tot is the total magnetic susceptibility, and χ band is thesusceptibility of non-interacting band electrons. Since in thisdefinition magnetic field acts on the whole system, this quan-tity can be experimentally determined from magnetic measure-ments; it is also usually treated in Bethe ansatz solutions. In theflat-band case we have at low temperatures the standard Fermi-liquid behavior T imp K χ imp ( T ) = . − . T / T imp K ) , where ac-cording to Wilson [20] we use the definition T imp K χ imp ( T imp K ) = . χ imp and χ loc can be di ff erent. In our case χ imp ( T ) at not too low T demonstrates the usual universal Kondo behavior with T imp K being defined according to Wilson.However, with decreasing temperature χ imp ( T ) deviates fromthe universal behavior and changes its sign. At very low T wehave irrespective of J (Fig. 6) T χ imp ( T ) ≈ − . D / T ) . (19)The temperature of loss of universality for χ imp corresponds tothat for χ loc ( T ): both the sign change in the former and maxi-mum in the latter are connected with overcompensation of im-purity spin by conduction electrons. For the case M = χ imp was discussed in details in Ref. [18].To compare these results with above perturbative considera-tion, we remember that in our case, as follows from the structureof perturbation theory, g e f ( T ) → G e f ( T ) = g e f ( T ) ln( D / T ), and G ∗ − G e f ∼ / ln( D / T ) (20)according to (10). Thus the terms, proportional to G ∗ − G e f ,should give 1 / ln( D / T )-contribution to thermodynamic charac-teristics. In particular, the 1 / ln( D / T ) terms in χ imp ( T ) are ex-pected. However, the calculation of χ imp ( T ) requires a carefulcollection of all the contributions to susceptibility.4 E-7 1E-6 1E-5 1E-4 1E-3 0.01 0.10.000.050.100.150.20 T i m p T/D
M=1 J=-0.1 M=1 J=-0.15 M=1 J=-0.2 M=2 J=-0.1 M=2 J=-0.15 M=2 J=-0.2
Figure 6: NRG results for T χ imp . One-channel ( M =
1) and two-channel ( M =
2) cases with singular DOS.
A similar behavior is obtained for the impurity entropy(Fig. 7) and for specific heat, S imp ( T ) ≈ − . D / T ) , C imp ( T ) ≈ − . ( D / T ) . (21)According to Ref. [19] (Eq.44), the entropy in the flat bandmultichannel model contains the contribution ( π / Mg e f ( T ) − (3 / M g e f ( T )) with g ∗ − g e f ( T ) ∼ T ∆ . To leading orderin 1 / M , this yields the contribution, proportional to [ g ∗ − g e f ( T )] ∼ T ∆ . One can expect that in our case terms lin-ear in g e f ( T ) → G e f ( T ) = g e f ( T ) ln( D / T ) will occur in theentropy. However, to obtain a correct description of strong cou-pling regime, a more accurate analysis is required. S i m p T/D M=2 SD J=-0.1 M=2 SD J=-0.15 M=2 SD J=-0.2 M=2 FB J=-0.2 M=1 SD J=-0.2
Figure 7: NRG results for the impurity entropy S imp in the two-channel casefor singular DOS (SD) and non-singular DOS (FB, flat band) logarithmic sin-gularity, and for the one-channel case with singular DOS. The latter curve isshown with the shift by ln 2 / These results correct somewhat our previous calcula-tions [18]. The occurrence of the inverse-logarithm contribu-tions is in a qualitative agreement with the above scaling con-sideration.The physical picture can be explained as follows. At lowtemperatures the impurity spin is completely screened and wecome to the situation where χ imp is determined by the contri-bution of conduction electrons which is independent of J . The e ff ective bandwidth in the singular case decreases, and the sit-uation is close to that in the model with a hole at the magneticimpurity site ( J → −∞ ): here the total magnetic susceptibilityof the system is smaller than that of bare electrons, so that thecontribution χ imp is evidently negative. l o c loc T K l o c T/T K SD J=-0.5 SD J=-0.3 SD J=-0.2 SD J=-0.15 SD J=-0.1 FB J=-0.2
Figure 8: NRG results for χ loc . The case of the two-channel Kondo model with(SD) and without (FB) logarithmic singularity in one-electron DOS. Thus in the strong coupling regime (at low temperatures) theproblem is reduced to the resonant level model. A quantitativeanalytical solution in this model can be obtained similar to [12].In terms of the impurity free energy one obtains S imp = − ∂ F imp ∂ T = − X σ Z ∞−∞ d z π z cosh z ℑ ln( − G − r ,σ (2 T z ))(22) T χ imp = − X σ Z ∞−∞ d z π sinh z cosh z ℑ ln( − G − r ,σ (2 T z )) (23)where G r ,σ ( z ) is the retarded Green’s function of the resonantlevel hybridized with the band.Calculating the Green’s function of the level hybridized withthe singular DOS we have G − r ,σ ( z ) = z + i + − v G r ,σ ( z ) (24)where v is an e ff ective hybridization matrix element and G r ,σ ( z ) = ρ " P Z D − D d ǫ ln | D /ǫ | z − ǫ − i π ln | D / z | θ ( D − z ) (25)is the band Green’s function for the singular DOS at the reso-nant level site, θ ( x ) is the Heaviside step function. Since thelow T behavior is dominated by small | z | ≪ D one has ℑ ln( − G − r ,σ ( z )) ≈ − π + sign( z ) arctan π | D / z | (26)and we derive for leading corrections irrespective of vS imp ≈ − ln 4ln | D / T | , T χ imp ≈ − / | D / T | (27)5n a fair agreement with the NRG results (19),(21). The aboveconsideration shows that the unusual low temperature behavioris solely determined by the logarithmic van Hove singularity ofthe band spectrum (cf. [12]), the singular contribution havingessentially one-electron nature.Now we pass to the two-channel situation. In the flat-bandcase χ loc ( T ) is known to behave as ln( T K / T ) [4] (such a behav-ior was also reproduced by our test calculation, Fig. 8). How-ever, for the logarithmic DOS the local susceptibility χ loc ( T )demonstrates a power-law non-Fermi-liquid behavior (Fig. 8),which can be fitted at low T as T loc K χ loc ( T ) ∼ ( T loc K / T ) α (28)where α slightly decreases with decreasing | J | (see Table 3). Table 3: Estimated values for the exponent α in (28). Note that the accuracy atsmall | J | is low J -0.5 -0.3 -0.2 -0.15 -0.1 α M = M = ff ective increase ofthe number of scattering channels. Although it is often di ffi cultto distinguish between the experimental logarithmic and power-law dependences, especially in the case of small exponents, thisconclusion is important from the theoretical point of view.As demonstrated in Ref. [19] from the 1 / M -expansion, in theflat-band case the change in the gyromagnetic ratio (which en-ters total magnetic susceptibility) leads to a change in numer-ical factors only, so that both χ loc ( T ) and χ imp ( T ) are positiveand behave in similar way. However, in our case the behaviorof χ imp ( T ) is again qualitatively di ff erent: we have the depen-dence (see Fig. 6) T χ imp ( T ) ≈ − . D / T ) . (29)For the M = / T theentropy approaches this value from below according to the law S imp ( T ) ≈ ln 22 − . D / T ) ≈ S channel imp + ln 22 . (30)Thus the temperature dependence in (30) is the same as in theresonant level model (27).We demonstrated that the behavior in the Kondo model withsingular logarithmic DOS di ff ers radically from that in thesmooth DOS model. This has unusual behavior with nega-tive impurity magnetic susceptibility and entropy at low T . Onthe other hand, the local magnetic susceptibility which is de-termined by the linear response remains positive. This has a shallow maximum in the one-channel case and demonstrates apower-law NFL behavior for two-channel case. As for the 1 / M -expansion, this yields the results which di ff er from those forsmooth DOS case by the replacement of the e ff ective couplingconstant by G e f . However, the NRG method enables one to ob-tain a more detailed information (although the calculations arerather cumbersome because of slow convergence in the singu-lar case). In particular, NRG calculations reproduce the Betheansatz results for the flat band M = M = ff erent (weak power-law di-vergence). We note also absence of Wilson’s self-similarity interms of T / T K . When shifting the singularity from the Fermilevel, the system can demonstrate non-trivial crossovers at low-ering T , which can be observed experimentally.The research was carried out within the state assignment ofFASO of Russia (theme “Quantum” No. 01201463332) andsupported in part by Ural Branch of Russian Academy of Sci-ence (project no. 15-8-2-10). References [1] G. R. Stewart,
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