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arXiv:1106.4438v1 [cond-mat.str-el] 22 Jun 2011

Numerical study of Kondo impurity models with strong potential scattering:

– reverse Kondo effect and antiresonance –

Annam´ aria Kiss,1,2, ∗Yoshio Kuramoto,3and Shintaro Hoshino3

1Budapest University of Technology and Economics, Institute of Physics and Condensed

Matter Research Group of the Hungarian Academy of Sciences, H-1521 Budapest, Hungary

2Research Institute for Solid State Physics and Optics, P.O. Box 49, H-1525 Budapest, Hungary

3Department of Physics, Tohoku University, Sendai, 980-8578, Japan

Accurate numerical results are derived for transport properties of Kondo impurity systems with potential

scattering and orbital degeneracy. Using the continuous-time quantum Monte Carlo (CT-QMC) method, static

and dynamic physical quantities are derived in a wide temperature range across the Kondo temperature TK.

With strong potential scattering, the resistivity tends to decrease with decreasing temperature, in contrast to the

ordinary Kondo effect. Correspondingly, the quasi-particle density of states obtains the antiresonance around

the Fermi level. Thermopower also shows characteristic deviation from the standard Kondo behavior, while

magnetic susceptibility follows the universal temperature dependence even with strong potential scattering. It is

found that the t-matrix in the presence of potential scattering is not a relevant quantity for the Friedel sum rule,

for which a proper limit of the f-electron Green’s function is introduced. The optical theorem is also discussed

in the context of Kondo impurity models with potential scattering. It is shown that optical theorem holds not

only in the Fermi-liquid range but also for large energies, and therefore is less restrictive than the Friedel sum

rule.

PACS numbers:

I.INTRODUCTION

A single magnetic impurity embedded into the sea of con-

duction electrons shows Kondo effect. Although this prob-

lem has been almost continuously studied during the last 40

years, dynamic and magnetic properties of the Kondo and re-

lated models still attract great interest in condensed matter

physics. Much less attention is paid to anomalous decrease

of resistivity with lowering temperature since such decrease

can be caused by many different mechanisms. The first at-

tention to this problem is traced back to the 60’s, when resis-

tivity of dilute Fe and Cu alloys in Rh matrix revealed a new

type of anomaly at low temperatures1. Namely, it was ob-

served that resistivity decreases with decreasing temperature

in these compounds. Obviously a ferromagnetic exchange in-

teraction of a localized spin and conduction electrons is the

first candidate. In fact, later study on dilute Gd and Nd impu-

rities in some La alloys such as LaAl22, and LaSn33explained

theresistivityanomalyintermsoftheferromagneticexchange

model4, and was referred to as the “reverse Kondo effect”.

In dilute Rh alloys, however, the measured susceptibility

indicates antiferromagnetic exchange interaction in these ma-

terials in contrast with the behavior of the resistivity. As al-

ternative interpretation, Fischer found that a strong potential

scattering can change sign of the Kondo logarithmic term in

the resistivity5. The treatment has been much extended and

deepened by Kondo6, who uses the scattering phase shift δv

of conduction electrons at the Fermi surface. The strength

v of the potential scattering is related to the phase shift by

tanδv= −πvρcwhere ρcis the density of states at the Fermi

surface. The leading logarithmic term in the electric resistiv-

ity changes sign when |δv| exceeds π/4, defining a critical

value for the potential scattering as vcr≡ 1/(πρc). The range

|v| > vcris calledreverseKondorange. Inthisrangetheresis-

tivity decreaseswith decreasingtemperatureas a consequence

of the strong potential scattering. Regarding thermodynamic

properties, Kondo showed that the effect of ordinary scatter-

ing is entirely absorbed into an effective exchange interaction

? J = J cos2δv. Correspondingly, the low-temperature energy

ciated with the effective exchange interaction? J.

systems, the Kondo problem with strong potential scatter-

ing might have relevance also in other compounds that show

Kondo effect.For example, the question of relevance of

ordinary scattering in URu2Si2 arises, because: (i) recent

STM experiments have found that the density of electronic

states shows Fano lineshape, i.e. antiresonance, in the nor-

mal phase,7and (ii) in the dilute system UxTh1−xRu2Si2the

resistivity decreases with decreasing temperature.8

In this paper we study the effect of strong potential scat-

tering on physical properties of the Kondo impurity. In or-

der to deal with the Kondo effect beyond the weak cou-

pling regime, the continuous-timequantum Monte Carlo (CT-

QMC) method is employed9–11.

tion it is most convenient to take the N-component Coqblin-

Schrieffer (CS) model9,12with potential scattering.

Hamiltonian is given by

?

+ Jf†

mfm′ + vCSδmm′

scale is characterized by temperature TK= De1/(2ρc?

J)asso-

Since ordinary scattering events are always present in real

In the CT-QMC simula-

The

HCS[vCS] =

km

?

εkc†

kmckm

mm′

?

?

c†

m′cm, (1)

where c†

localized electrons, respectively, at the impurity site with

SU(N) index m = 1,...,N. The constraint?

The annihilation operator cmin the Wannier representation is

related to ckmby cm = N−1/2

0

kmand f†

mare creation operators of conduction and

mf†

mfm= 1

is imposed, which removes the charge degrees of freedom.

?

kckmwith N0being the

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2

number of lattice sites. We observe the relation

?

mm′

f†

mfm′c†

m′cm=

?

⇒

mm′

N=22Sf· sc+1

˜ Xmm′c†

m′cm+1

Nnc

2nc,

(2)

where˜ Xmm′ ≡ f†

scand ncare spin and charge density operators of conduction

electronsat the impuritysite. TheSU(N) KondoHamiltonian

HK[v] with potential scattering v is introducedby the relation

mfm′−δmm′/N areSU(N) generators,and

HCS[vCS] = HK[v = vCS+ J/N],

(3)

in view of eq. (2). Some typical cases of the model given by

Hamiltonian (1) are

(i) TheconventionalCSmodelwithvCS= 0, orv = J/N;

(ii) The SU(N) Kondo model with v = 0, or vCS =

−J/N;

(iii) Reverse Kondo range with |v| > vcr= 1/(πρc).

On the basis of accurate numerical results for strong po-

tential scattering, we investigate the reverse Kondo range in

detail. Furthermore,propertiesare studiedbychangingtheor-

bital degeneracyN for the CS model. This paper is organized

as follows. In Section II numerical results for the magnetic

susceptibility are presented. The characteristics of the impu-

rity t-matrixare discussed in Section III.Furthermore,numer-

ical results are givenfor transportproperties. Section IV is de-

voted to discussion of quasi-particle properties including the

Friedel sum rule and optical theorem. The summary of this

paper will be given in Section V.

II.MAGNETIC SUSCEPTIBILITY AND UNIVERSALITY

First, let us discuss the behavior of the magnetic suscepti-

bility for a givenvalue of the orbital degeneracyN. The static

susceptibility is obtained from the imaginary time data by in-

tegration as

χ(T) =

?β

0

dτχ(τ) =

?β

0

dτ?TτMH(τ)M?,

(4)

where the dipole moment M is given by M =?

script H denotes the Heisenberg picture9. We use a constant

density of states for the conductionelectrons in the simulation

as

αmαf†

αfα

with coefficients mαchosen as?

αmα = 0, and the super-

ρc(ε) = ρ0Θ(D − |ε|),

(5)

where ρ0= 1/(2D) with D = 1 as a unit of energy. In the

numerical study, we determine the Kondo temperature from

the low-temperature static susceptibility as

T−1

K

= χ(T → 0)/CN,

(6)

0

0.25

0.5

0.75

1

1.25

0.1 1 10 100

χ(T)TK/CN

T/TK

v=0 (N=2)

v=-0.48 (N=2)

v=-0.85 (N=2)

v=0.009 (N=8)

v=0.0125 (N=8)

0

0.01

0.02

0.03

0 0.25 0.5 0.75

TK

|v|

N=2

simulation

theory

FIG. 1: Upper: Temperature dependence of the static susceptibility

for theSU(N) Kondo model withpotential scattering. Potential scat-

tering v and exchange J are chosen as v = 0,−0.85 (J = 0.3), and

v = −0.48 (J = 0.44) for N = 2, and v = 0.009 (J = 0.075),

and v = 0.0125 (J = 0.0125) for N = 8. Lower: Kondo tempera-

ture for several values of potential scattering obtained in simulation.

The theoretical result TK = De1/(2ρ?

J)is also shown as dashed line.

where CN is the Curie constant. The critical strength vcris

given by vcr= 1/(πρ0) = 0.637.

Figure 1 shows χ(T) of the SU(N) Kondo (or CS) models

with several values of the potential scattering v and orbital de-

generacy N (upper), and TKfor different values of v (lower).

The result TK= De1/(2ρ?

for comparison. We observe in Fig. 1 that the susceptibility

shows universal behavior as a function of T/TKindependent

of the value of the potential scattering. Note that the data with

v = −0.85 for N = 2 as shown by blue symbols are in the re-

verse Kondo range with δvbeing larger than π/4. Even in this

case, the temperature evolution of the magnetic susceptibility

shows the universal behavior.

Now, let us turn to discuss the properties by changing the

orbital degeneracy N. For N = 2, the static susceptibil-

ity decreases monotonically with increasing temperature. On

the other hand, for large values of the orbital degeneracy like

N = 8 in Fig. 1, the susceptibility first increases as the tem-

perature is increased, and then decreases in accordance with

the free moment behavior χ ∼ 1/T for large temperatures.

J)obtainedby Kondo6is also shown

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0

0.2

0.4

0.6

-0.3-0.2 -0.1 0

ε

0.1 0.2 0.3

-Im t(ε)

v = 0.15

β=1000

β=500

β=200

β=100

β=50

β=20

β=10

0

0.2

0.4

0.6

-0.3-0.2-0.1 0

ε

0.1 0.2 0.3

-Im t(ε)

v = 0

β=1000

β=500

β=200

β=100

β=50

β=20

β=10

0.25

0.3

0.35

0.4

0.45

0.5

0.55

-0.1 0

ε

0.1

-Im t(ε)

v = -0.85

β=1000

β=500

β=200

β=100

β=50

β=20

β=10

FIG. 2: Energy dependence of -Im t(ε) with potential scattering for N = 2 and J = 0.3. Potential scattering terms are chosen as v = 0.15

(left), v = 0 (center) and v = −0.85 (right). Values v = 0.15 and v = 0 correspond to the CS and ordinary Kondo models, respectively.

The initial increase can be understood in terms of the den-

sity of states of quasi-particles. To illustrate the mechanism

of the increase, let us consider the case of v = 0 in the non-

interactingAndersonmodel, whichsimulates qualitativelythe

quasi-particle density of states. Using the Sommerfeld expan-

sion

?∞

?

with the f-electron density of states ρf(ǫ), we obtain at low

temperatures

?

where η and ∆ are the shift and the width of the resonance

peak appearing in the density of states ρf(ε) at low tempera-

tures.

For N = 2 the resonance peak is centered at the Fermi

energy, which gives η = 0. Therefore, the coefficient of T2

in χ(T) is negative, i.e. the susceptibility decreases with in-

creasing temperatures. Increasing the value of degeneracy N,

the resonance moves to higher energy above the Fermi level,

i.e. η ∼ TK, while its width narrows as ∆ ∼ TK/N.13Thus,

the coefficient of T2in expression (8) becomes positive, so

that the susceptibility first shows increasing behavior as the

temperature is increased.

χ(T) ∼

−∞

ρf(ε)

?

−∂f(ε)

∂ε

6(kBT)2ρ′′

?

dε

= ρf(0) 1 +π2

f(0)

ρf(0)+ O(T4)

?

(7)

χ(T) ∼ ρf(0)1 +π2

3(kBT)2(3η2− ∆2)

(η2+ ∆2)2+ O(T4)

?

, (8)

III.TRANSPORT COEFFICIENTS

A.

t-matrix and Fano lineshape

We have already shown in eq. (2) that the Kondo and CS

Hamiltonians are related to each other through a potential

scattering term. In the simulation for the CS model, instead

of the bare Green’s function g, another Green’s function gCS

is used that absorbs the potential scattering vCS:

gCS= g/(1 − vCSg),

(9)

where g(z) =?

introduce a quantity tCSby the relation

k(z − εk)−1. Then the simulation gives the

renormalizedGreen’s function G of conductionelectrons. We

G = gCS+ gCStCSgCS.

(10)

On the other hand, the t-matrix t of conduction electrons is

defined by the relation

G = g + gtg.

(11)

By comparing eqs. (10) and (11), we obtain t from tCSby the

relation

t = vCS/(1 − vCSg) + tCS/(1 − vCSg)2.

(12)

In the special case of v = vCS+J/2 = 0, we recover eq. (33)

in Ref. 9.

For the moment, we concentrate on the case of N = 2.

In the CT-QMC simulation, the t-matrix is derived in the

imaginary-time domain. In order to obtain properties in the

real energy domain, analytic continuation of the numerical

data is done by using Pad´ e approximation. Figure 2 shows

the energy dependence of the impurity t-matrix for three dif-

ferent values of the potential scattering at various tempera-

tures. To simplify the notation, we take the convention in this

paper that energy including an infinitesimal imarginary part,

ε + iδ, is simply written as ε. The left panel of Fig. 2 with

the value v = 0.15 corresponds to the CS model, center panel

withv = 0totheordinarysingle-channelKondomodel,while

right panel to a strong potential scattering |v| > vcr= 0.637.

In the case of the ordinary single-channel Kondo model the

spectrum is symmetric with respect to the Fermi energy, be-

cause the model has particle-hole symmetry in this limit.

Increasing the value of the potential scattering, the Kondo

peak first moves to higher frequencies above the Fermi en-

ergy. Finally, for strong potential scattering the spectrum be-

comes highly asymmetric showing an antiresonance around

the Fermi level.

Interpretation of the asymmetric spectrum with large |v|

can be provided in terms of the Anderson model, which re-

produces the CS model (1) in the limit of deep local electron

level εf and large Coulomb repulsion U as compared with

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hybridization V . Namely we take the limits εf → −∞,

εf + U → ∞ and V2ρ0 → ∞, keeping the ratio J =

−2V2ρ0/εffinite. Now we construct the f-electron Green’s

function Gfvof the Anderson model in the presence of po-

tential scattering. Let us first consider the pure case v = 0.

Introducing the irreducible part F(z), we obtain the Green’s

function

Gf(z) = F(z)[1 + V2g(z)Gf(z)].

(13)

In the presence of potential scattering, the f-electron Green’s

function Gfv(z) satisfies the following relation

Gfv(z) = Fv(z)[1 + V2gv(z)Gfv(z)],

(14)

where gv(z) = g(z)/[1 − vg(z)].

The t-matrix for the CS model is given by

t(z) =

v + V2Fv(z)

1 − g(z)[v + V2Fv(z)]= tv(z) +

V2Gfv(z)

[1 − vg(z)]2, (15)

where tv = v/(1 − vg). It is clear from eq. (15) that the

t-matrix reduces to

t(z) → V2Gf(z)

(16)

in the limit of v = 0 as we expect. We derive from eq. (15)

V2Gfv= (1 − vg)2(t − tv),

(17)

which is valid for any value of parameters.

As the simplest case, let us consider the non-interacting

Anderson model with U = 0.

1/(ε + iδ − εf) ≡ 1/ξ, which is not affected by potentital

scatteing. The imaginary part of the f-electron Green’s func-

tion is given by

Then we obtain F(ε) =

−ImGfv(ε) =

1

V2·

πρ0(v + V2/ξ)2

1 + π2ρ2

0(v + V2/ξ)2.

(18)

Under the transformation v → −v, the imaginary part of the

Green’s function given in eq. (18) remains the same if we put

ε−εf→ εf−ε. Physically it means that the antiresonanceis

reflected with respect to the energy εfwhen v changes sign.

Expression (18) can be put into the standard form of the Fano

lineshape14. Introducing the dimensionless parameter q by

1/q ≡ πvρ0, we rearrange the terms as follows:

−V2

vImGfv(ε) =

1

q + 1/q·(x + q)2

x2+ 1,

(19)

where x is the dimensionless energy defined by

x =

v

V2

?

q +1

q

?

ξ +1

q.

(20)

The degree of asymmetry is determined by the parameter q

that is independent of hybridization.

Expression (18) describes the characteristics of the simula-

tion results for t(ε) shown in Fig. 2, provided we put ǫf∼ 0.

We note that V2Gfv(z) is not the same as the t-matrix t(z) of

0

0.25

0.5

0.75

1

0.1 1 10 100

R(T)/R(0)

T/TK

N=2

v = 0

v = -0.48

v = -0.78

v = -0.85

v = 0.78

FIG. 3: Temperature dependence of the normalized resistivity for the

Kondo model with potential scattering for N = 2. Potential scatter-

ing v and exchange J are chosen as v = 0,−0.85 (J = 0.3), and

v = 0.78,−0.48,−0.78 (J = 0.44). The dashed line corresponds

to Kondo’s result for the resistivity given in Ref. 6.

the CS model as shown in eq. (15). However, the character-

istic lineshape comes almost from V2Gfv(z) since tv(z) and

1 − vg(z) do not have strong dependence on z. The asym-

metric spectrum for strong potential scattering has interesting

consequences with respect to transport properties such as the

resistivity or thermopower. This problem is discussed in the

next subsection.

B. Relaxation time and transport coefficients

We rely on the Boltzmann equation approach15to derive

transport coefficients. Then the relaxation time τ(ε) is related

to the t-matrix as13

τ(ε)−1= −2Imt(ε).

(21)

Let us introduce the integrals16

Ln=

?∞

−∞

dε

?

−∂f(ε)

∂ε

?

τ(ε)εn,

(22)

intermsofwhichtheconductivityσ, thermopowerS andther-

mal conductivity κ are expressed as

σ(T) = L0,

S(T) = −1

(23)

T

?

L1

L0,

L2−L2

(24)

κ(T) =

1

T

1

L0

?

.

(25)

The integrals in eq. (22) are evaluated numerically with the

CT-QMC data for the t-matrix.

Although many theoretical attempts were made to derive

the transport properties for the Kondo problem analytically in

the whole temperature range, none of these attempts was suc-

cessful. However, there are correct results in limiting cases

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such as Hamman’s formula17for T ≫ TK, Fermi-liquid

results18for T ≪ TK, or expressions for large values of the

orbital degeneracyN.19In the local Fermi-liquid range at low

temperatures, the resistivity R(T) shows the following tem-

perature dependence:

R(T)/R(0) = 1 − α(T/TK)2,

(26)

where we have used the relation R(T)/R(0) = σ(0)/σ(T),

andα is a numericalcoefficient. Forlargevalues of the orbital

degeneracy N, the 1/N expansion gives the coefficient α in

eq. (26) as19

?

This limiting result gives checkpoint of our numerical calcu-

lations.

α = π2

1 −

8

3N

?

.

(27)

C.Behavior under varying potential scattering

Figure 3 shows the temperature dependence of the normal-

ized electric resistivity across the Kondo temperature. For

|v| < vcr = 0.637, the resistivity follows universal behav-

ior as a function of T/TK. As the potential scattering in-

creases beyond the critical value, the resistivity still follows

the universal behavior in the Fermi-liquid range T ≪ TK.

As temperature increases, the resistivity starts to deviate from

the universal curve around the Kondo temperature T ∼ TK,

and shows increasing behavior as the temperature is further

increased. In the temperature range T ≫ TK, we find that

the resistivity for strong potential scattering can be described

well with Kondo’s formula6shown by dashed line in Fig. 3.

However, Kondo’s formula cannot describe the properties for

T ≤ TK. Based on the numerical results for T ≪ TK, the

coefficient α in eq. (26) appears to be independent of v.

The upper panel of Fig. 4 shows the temperature depen-

dence of normalized thermal conductivity for N = 2 under

varyingthe potentialscattering. Thethermalconductivityalso

follows the universal behavior even for large values of the po-

tential scattering in the temperature range T ≪ TK. Namely

we obtain

?κ(T)

with γ being a numerical constant independent of v. Increas-

ing further the temperature, thermal conductivity with large

potential scattering highly deviates from the universal behav-

ior.Namely, it decreases with increasing temperature for

T ≫ TK.

The lower panel of Fig. 4 shows the temperature depen-

dence of thermopower for N = 2 under varying the potential

scattering. The asymmetry of the impurity t-matrix is most

reflected in the behavior of thermopower. This is clear if we

regard explicitly the expression for the integral L1given in

eq. (22):

?∞

κ(T)

T

/

T

?

0

= 1 + γ

?T

TK

?2

(28)

L1=1

2

−∞

dεf′(ε)ε

Imt(ε).

(29)

0

1

2

3

0.1 1

(κ/T)/(κ/T)0

T/TK

N=2

v = 0

v = -0.48

v = -0.78

v = 0.78

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

0.001 0.01 0.1 1 10

S(T)

T/TK

N=2

v = 0

v = -0.48

v= - 0.78

v = 0.78

FIG. 4: Temperature dependence of normalized thermal conductiv-

ity (upper) and thermopower (lower) for the Kondo model with po-

tential scattering for N = 2. Potential scattering v and exchange J

are chosen as v = 0 (J = 0.3), and v = 0.78,−0.48,−0.78 (J =

0.44). The dashed line in the right panel corresponds to the Fermi-

liquid behavior S(T) ∼ T.

Because of the factor ε in the numerator, the thermopower

measures the asymmetryin the energydependenceof Imt(ε).

Since the spectra is completely symmetric in the case of the

ordinary Kondo model (v = 0), the thermopower vanishes in

this case as we obtain in the simulation (see Fig. 4). On the

other hand, the thermopower acquires strong temperature de-

pendencewhen the potential scattering term is increased from

v = 0. The thermopower in the simulation shows Fermi-

liquid property

S(T) = β

?T

TK

?

(30)

for T ≪ TK. The coefficient β is negative for v < 0, while

it is positive for v > 0. The different sign of β can be simply

understood if we recall that the asymmetry in the lineshape

of Imt(ε) around the Fermi level is reversed against the sign

change v → −v. Thus, the integrals given in eq. (22) have

the properties L0(L2) → L0(L2) and L1→ −L1under the

signchange. Namely,theelectricresistivityR(T)andthermal

conductivity κ(T) remain the same under v → −v, while

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6

0

0.25

0.5

0.75

1

1.25

0.1 1

R(T)/R(0)

T/TK

v=J/N

v=0.0002 (N=50)

v=0.009 (N=8)

v=0.0125 (N=8)

0

2.5

5

7.5

10

0 10 20 30

N

40 50

α

v=J/N

1/N expansion

interpolation

0

0.25

0.5

0.75

1

0.1 1 10

R(T)/R(0)

α1/2T/TK

N=50

N=8

N=2

FIG. 5: Upper: Temperature dependence of the normalized resis-

tivity for the CS model (where v = J/N) for orbital degeneracies

N = 8 with J = 0.075, 0.1 and N = 50 with J = 0.0115.

Lower: Orbital degeneracy N-dependence of the coefficient α of the

T2term in the low-temperature resistivity defined in eq. (26). The

inset shows the scaling behavior of the resistivity for different orbital

degeneracies including α.

the thermopower changes sign as S(T) → −S(T). We have

indeed obtained these behaviors in the simulation as it can be

seen in Figs. 3 and 4.

D. Behavior under varying orbital degeneracy N

Letusnowstudytheeffectoforbitaldegeneracy. Theupper

panel of Fig. 5 shows the temperature dependence of the nor-

malized electric resistivity for large values of the orbital de-

generacy for the CS model with v = J/N across the Kondo

temperature. We observe again the universal behavior for a

given value of the orbital degeneracy N. In contrast to the

behavior of the magnetic susceptibility shown in Fig. 1, the

resistivity decreases monotonically as the temperature is in-

creased even for large N. In order to understand this feature,

we assume that the t-matrix at low temperature is determined

by the quasi-particle density of states, which is approximately

given by the effective Anderson model. Namely we assume

t(z) = V2Gf(z),

(31)

where V is the effective hybridization and Gfis the Green’s

functionof the local electron in the effectiveAndersonmodel.

Then the Sommerfeld expansion of the conductivity leads to

σ(T) ∼

?∞

−∞

1

ρf(ε)

?

?

−∂f(ε)

∂ε

?

dε

?

=

1

ρf(0)

1 +π2

6(kBT)2

2

?ρ′

f(0)

ρf(0)

?2

−ρ′′

f(0)

ρf(0)

?

+ O(T4)?,

(32)

where

ρf(ε) = −π−1ImGf(ε).

(33)

We obtain the resistivity R(T) = σ(T)−1from eq. (32). Us-

ing the quasi-particle density of states for the non-interacting

Anderson model with v = 0, we obtain

?

where η and ∆ are the shift and the width of the resonance

peak appearing in ρf(ε) at low temperatures. Irrespective of

themagnitudeofparametersη and∆, the coefficientoftheT2

term in the low-T resistivity is always negative. Hence R(T)

given by eq.(34) decreases as temperature increases for any

value of N. Namely, the quasi-particle picture is consistent

with the monotonous change obtained in the simulation.

TheresistivityobtainednumericallyhasT2temperaturede-

pendenceat low temperatures,whichis expressedgenerallyin

eq. (26). The lower panel of Fig. 5 shows the coefficient α of

the T2term for several values of the orbital degeneracyN ob-

tained in the simulation. The result of 1/N expansion19given

in eq. (27) is also shown in the figure. We find that the nu-

merical data coincide with the 1/N expansion result for large

values of the degeneracy N. Thus, our CT-QMC simulation

has produced accurate numerical results for large values of

orbital degeneracies N → ∞, which might be difficult in the

case of other numerical techniques. For small values of de-

generacy N, the coefficient α shows linear-N dependence in

our simulation.

Now we fit the simulated N-dependence of α by a rational

function as α(N) = F(l)

ck1N +...cklNl. We find that the minimal functionwhich can

give a good fit to the numerical data has the form

R(T) ∼ ρf(0)1 −π2

3(kBT)2

1

(η2+ ∆2)+ O(T4)

?

,(34)

1(N)/F(m)

2

(N), where F(l)

k

= ck0+

α(N) =c10+ c11N + c12N2

1 + c21N + c22N2.

(35)

The limiting cases N → 0 and N → ∞ are obtained from the

formula (35) as

α(N → 0) ∼ c10+ (c11− c10c21)N

(36)

Page 7

7

0

1

2

3

4

5

6

7

0.1 1

(κ/T)/(κ/T)0

T/TK

N=8

v=J/8

v=0.009

v=0.0125

0

0.25

0.5

0.75

1

1.25

1.5

1.75

0.1 1

S(T)

T/TK

N=8

v=J/8

v=0.009

v=0.0125

FIG. 6: Temperature dependence of normalized thermal conduc-

tivity (upper) and thermopower (lower) for the CS model (where

v = J/N) for orbital degeneracy N = 8. Potential scattering terms

are chosen as v = 0.009 (J = 0.075) and v = 0.0125 J = 0.1).

The dashed line corresponds to the Fermi-liquid behavior given in

eqs. (30) and (28).

and

α(N → ∞) ∼c12

c22

?

1 −1

N

?c11

c12

−c21

c22

??

.

(37)

Choosingthe coefficientsin eq.(35)as c10= 0.3; c11= 0.53;

c12= 0.17; c21= 0.1; c22= 0.0173, the numerical data can

be fitted well (see Fig. 5). The result of 1/N expansion19

given in eq. (27) is also reproduced in the large-N range. In

the inset of right part of Fig. 5 we plot the normalizedresistiv-

ity for different values of orbital degeneracy N as a function

of α(N)1/2T/TK. The results show the scaling behavior of

the resistivity in the Fermi-liquid range. For T ≥ TK, the

scaling property breaks down.

In Fig. 6, thermal conductivity κ and thermopower S are

shown for the CS model with orbital degeneracy N = 8.

The low-T behaviors of κ(T) and S(T) are consistent with

the Fermi-liquid result given in eqs. (28) and (30). As the

temperature is further increased, the thermopowerS(T) has a

peak, while the thermal conductivity κ(T) monotonously in-

creases. Both quantities show universalbehavioras a function

of T/TK.

Note that the coefficient β for S(T) has a positive sign for

large N. This behavior is explained as follows. In a manner

similar to eq. (32), L1is given by

?∞

Using the result for L0given by eq. (34), we obtain S(T) in

the lowest order of T as

S(T) =2π2

3

Since η ∼ TK> 0 for largeN, we obtainβ > 0. On the other

hand, we obtain η = 0 for the symmetric Anderson model

with N = 2. In this case, the sign of β depends on the sign of

v as it can be observed in Fig. 4.

L1(T) =

−∞

ε

ρf(ε)

?

−∂f

∂ε

?

= −2π3

3∆ηT2.

(38)

η

η2+ ∆2T = βT.

(39)

IV. QUASI-PARTICLE PROPERTIES

A.Friedel sum rule

At temperatures T ≪ TK, the conduction electrons screen

the magnetic impurity and they together form a local singlet.

In this range the ground state is a local Fermi-liquid. The

Friedel sum rule (FSR) relates the phase shift for scattering of

the conduction electrons by the impurity to its charge. In the

case of the CS model, the f-electron Green’s function cannot

be defined because there is no charge degrees of freedom in

this localized model since it is eliminated. Instead, the im-

purity t-matrix is used to describe the effect of exchange and

potential scatterings. In Section III, we have related the t-

matrix to the Green’s function Gfv(z) of localized electrons

with potential scattering v. It is the quantity Gfv(z) that is

expected to keep the FRS in the presence of v.

The FSR reads13

V2Gf(0) = −

i

πρ0sin2?π

N

?

,

(40)

since the occupation number is unity in the CS and Kondo

models.

Figure 7 shows the Green’s function Gfv(0) obtained by

simulation at finite temperatures. For small potential scatter-

ing the simulation results show good agreement with the ex-

pectation given in eq. (40). As the value of v is increased,

ReV2Gfv(0) is still close to zero, but −ImV2Gfv(0) highly

deviates from the theoretical result. The reason of this devia-

tion is the following. The theoretical result given in eq. (40)

is realized at T = 0, which is almost realized for T ≪ TK

in the simulation. As the value of the potential scattering

is increased, however, the corresponding Kondo temperature

TK = De1/(2ρ?

pling?J decreases rapidly6. Therefore, we have to go at lower

tionT ≪ TK, whichisnotfulfilledinFig.7forlargevaluesof

v since the results are obtained for a fixed temperature value

β = 1/T = 1000. In principle it is possible to go at lower

temperatures in the simulation, but it becomes computation-

ally harder.

J)rapidly decreases since the effective cou-

and lower temperaturesin the simulation to achieve the condi-

Page 8

8

0

0.5

1

-3-2-1 0 0.5

-0.6

-0.4

-0.2

0

V2Gfv(0)

σinel

v

T>>TK

T=TK

T<<TK

-Im V2Gfv

Re V2Gfv

FIG. 7: Imaginary and real parts of the Green’s function Gfv ex-

pressed ineq. (17) atthe Fermilevel asafunction of potential scatter-

ing v together with the analytical result for −ImV2Gfv(0) obtained

from the FSR (dashed line). The figure also shows the inelastic scat-

tering cross section σinel as blue triangles. The numerical data are

obtained at temperature β = 1/T = 1000.

B.Optical theorem

The optical theorem is less restrictive than the FSR since

the formerdoes not require the Fermi liquid groundstate. Op-

tical theoremis relatedto the unitarityof theS matrix,20andit

follows when the scattering of the conduction electrons from

the impurity is totally elastic at the Fermi level. Optical the-

orem was originally formulated for problems of scattering of

a single-particle. When the scattering event happens without

energy loss, i.e. it is totally elastic, there is a relation between

the square and the imaginary part of the t-matrix. To express

this relation, the S-matrix is decomposed as21

S = 1 + iT,

(41)

where we write the matrix element of the t-matrix T as

?n|T|n′? = 2πδ(εn− εn′)?n|t|n′?. Here, a state |n? repre-

sents a single-particle state with momentum k and spin σ as

|n? = |kσ?. After some manipulations we obtain from rela-

tion (41) that

?k|SS†|k? − 1 = 2π

?

2π

?

n

δ(εk− εn)|?n|t|k?|2

+ 2Im?k|t|k?].

(42)

The first term of eq. (42) in the right-hand side is related to

the elastic scattering cross section σel, while the second term

to the total scattering cross section σtotalas21,22

σel = −2π

?

n

δ(εk− εn)|?n|t|k?|2,

(43)

σtotal = 2Im?k|t|k?.

(44)

The inelastic scattering cross section σinelis the difference of

σtotaland σelas

σinel= σtotal− σel.

(45)

For scattering only in the s channel and assuming spin con-

servation, the inelastic cross section is expressed as22

σinel(ω) = (|s(ω)|2− 1)/(2π) = 2Imt(ω) + 2πρ0|t(ω)|2

?

where s and t are eigenvalues of the S-matrix and t-matrix,

respectively.

The eigenvalues s of the S-matrix lie within the complex

unit circle. The scattering is completely elastic, i.e. σinel =

0, when the unitary condition S S†= 1 is satisfied, which

means |s|2= 1. In this case we have the relation

= 2πρ0

1

πρ0Imt(ω) + |t(ω)|2

?

,

(46)

|t|2= −

1

πρ0Imt

(47)

from eq. (46).

The problem is formulated so far for single-particle scat-

tering, but it is more general and can be applied for many-

particle problems as well such as the single-channel Kondo

model. It is the most easily understoodin the limit of ε → ∞,

whenthe magneticimpurityis completelydecoupledfrom the

conduction electrons. In this case the conduction electrons

scatter without energy loss, and the optical theorem is held.

Although the t-matrix is complicated and contains many scat-

teringeventsat low temperatures,strictly in the limit of ε → 0

the optical theorem holds again. If we express the t-matrix as

t = |t|eiθ,

(48)

where θ is the phase of the t-matrix t. Equation (47) gives

− Imt(ε) =sin2θ

πρ0

(49)

for the Kondo model with energy not only ε = 0 but also

ε → ∞. Namely, relation (49) is satisfied in the case when

the scattering is totally elastic. In the case of ordinary single-

channelKondomodel(v = 0),θ = −π/2at theFermienergy,

so from eq. (49) we recover the FSR

− Imt(0) =

1

πρ0.

(50)

Let us discuss the optical theorem in the context of our nu-

merical data. Figure 7 shows the obtained inelastic scattering

cross section σinelas a function of potential scattering. Note

that σinelshows non-monotonousbehavior as a function of v.

Namely, σinelis almost zero for small values of the potential

scattering, but becomes non-zeroas |v| is increased. With fur-

ther increase of |v|, however, σinelapproaches to zero again.

Since TKdecreases as the potentialscattering increases, in the

large-v range the condition T ≫ TKis satisfied at T = 10−3

used in the simulation. On the other hand, in the small-v

range we have the condition T ≪ TKwith the same value:

T = 10−3. It is confirmed in the simulation that the opti-

cal theorem holds both at T ≪ TKand T ≫ TK, but not

for T ∼ TK. This happens because the ranges T ≪ TKand

T ≫ TKcorrespond to the limits ε/TK≪ 1 and ε/TK≫ 1,

respectively, where expression (49) is satisfied as explained

above.

Page 9

9

V.SUMMARY

In this paper we have studied Kondo impurity models with

potential scattering and orbital degeneracy by using CT-QMC

numerical technique. We have derived accurate numerical re-

sults for the impurity t-matrix, thermal, and transport proper-

ties in a wide temperature range across the Kondo tempera-

ture TK. Properties in the reverse Kondo range has been in-

vestigated in detail. The results shown in this paper are nu-

merically exact since CT-QMC does not use any approxima-

tion. We have explicitly demonstrated that CT-QMC simula-

tion technique gives numerically exact results for large val-

ues of the orbital degeneracy N, which might be difficult to

achieve in the case of other numerical techniques.

For large values of the potential scattering, non-trivial

physics appears even in the impurity problem. Namely, the

resistivity shows anomalous increase with increasing temper-

ature in contrast to the ordinary Kondo effect. This unusual

behavior is caused by an antiresonance developingaround the

Fermienergyinthequasi-particledensityofstatesasthevalue

of the potential scattering is increased. This antiresonance

does not influence the universal behavior of the magnetic sus-

ceptibility. However, the sign of the Kondo logarithmic term

changesintheresistivitywhenthepotentialscatteringexceeds

a critical value, i.e. in the reverse Kondo range, which causes

the resistivity decrease with decreasing temperature.

We have studied the effect of strong potential scattering on

thermal and transport properties of the Kondo impurity, and

obtained that

(i) the magnetic susceptibility follows the universal tempera-

ture dependence even with strong potential scattering;

(ii) the resistivity also follows the universal temperature de-

pendence for small values of the potential scattering;

(iii) when the potential scattering exceeds a critical value, the

resistivity still shows universal behavior in the Fermi-liquid

range, but starts to deviate from the universal curve around

the Kondo temperature, and increases as the temperature is

furtherincreased; (iv)theobtainedtemperaturedependenceof

the resistivity for T ≫ TKin the reverse Kondo range agrees

quantitatively with Kondo’s theoretical result;

(v) the thermal conductivity also shows universal behavior in

the Fermi-liquid range, but highly deviates from the universal

curve for T ≫ TKin the reverse Kondo range;

(vi) the asymmetry of the t-matrix developingwith increasing

value of the potential scattering is most reflected in the tem-

perature dependence of the thermopower;

(vii) the sign of the thermopower depends on the sign of the

potential scattering.

In addition to the study of thermal and transport properties,

we have discussed the Friedel sum rule and optical theoremas

well. We have shown that the t-matrix of the Kondo model in

the presence of potential scattering is not the relevant quantity

for the Friedel sum rule. Instead, the Friedel sum rule is sat-

isfied with a proper limit of the f-electron Green’s function.

We have demonstrated that optical theorem is less restrictive

than the Friedel sum rule, because the former holds not only

in the Fermi-liquid range, but for large energies as well.

Finally we mention an interesting question whether the be-

havior found for strong potential scattering has relevance in

real systems. Note that recent STM experiments on URu2Si2

have found that the density of states shows Fano lineshape in

the normal phase, and in the dilute system UxTh1−xRu2Si2

the resistivity decreases with decreasing temperature. Since

some important aspect of the U ion with non-Kramers con-

figuration 5f2may not be described by a localized spin with

S = 1/2, account of the strong potential scattering in more

realistic models is desirable. We hope that our results in this

paper will stimulate further study concerning the Fano line-

shape and other aspects, which reflect interplay of the Kondo

effect and potential scattering.

Acknowledgments

The authors are grateful to Dr. J. Otsuki for his guidanceon

the details ofCT-QMC simulationtechnique,and also foruse-

ful discussions. AK acknowledges the Magyary programme

and the EGT Norway Grants, and also the financial support

from the European Union Seventh Framework Programme

through the Marie Curie Grant PIRG-GA-2010-276834.

∗Electronic address: akiss@szfki.hu

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