Page 1

arXiv:0801.4272v1 [cond-mat.str-el] 28 Jan 2008

Dynamical correlations in the spin-half two-channel Kondo model

A. I. T´ oth and G. Zar´ and

Theoretical Physics Department, Institute of Physics,

Budapest University of Technology and Economics, H-1521 Budapest, Hungary

(Dated: February 4, 2008)

Dynamical correlations of various local operators are studied in the spin-half two-channel Kondo

(2CK) model in the presence of channel anisotropy or external magnetic field. A conformal field

theory-based scaling approach is used to predict the analytic properties of various spectral functions

in the vicinity of the two-channel Kondo fixed point. These analytical results compare well with

highly accurate density matrix numerical renormalization group results. The universal cross-over

functions interpolating between channel-anisotropy or magnetic field-induced Fermi liquid regimes

and the two-channel Kondo, non-Fermi liquid regimes are determined numerically. The boundaries

of the real 2CK scaling regime are found to be rather restricted, and to depend both on the type

of the perturbation and on the specific operator whose correlation function is studied. In a small

magnetic field, a universal resonance is observed in the local fermion’s spectral function.

dominant superconducting instability appears in the composite superconducting channel.

The

PACS numbers: 71.10.Hf, 71.10.Pm, 71.27.+a, 72.15.Qm, 73.43.Nq, 75.20.Hr

I.INTRODUCTION

Deviations from Fermi liquid-like behavior observed

e.g. in the metallic state of high-temperature cuprate

superconductors,1,2

or in heavy fermion systems3,4

prompted physicists to look for new non-Fermi liquid

(NFL) compounds. So far a large number of such ex-

otic compounds has been found and investigated.

these systems electrons remain incoherent down to very

low temperatures and the usual Fermi liquid descrip-

tion breaks down. To our current understanding, NFL

physics may arise in many different ways: it can occur

due to some local dynamical quantum fluctuations often

described by quantum impurity models,5,6,7it can also

be attributed to the presence of the quantum fluctua-

tions of an order parameter or some collective modes,

as is the case in the vicinity of many quantum phase

transitions,6,8or for the prototypical example of a Lut-

tinger liquid,9,10,11,12,13where electrons are totally dis-

integrated into collective excitations of the electron gas.

NFL physics can also appear as a consequence of disorder

like e.g. in disordered Kondo alloys.14,15

In this paper we study a variant of the overscreened

multi-channel Kondo model:

Kondo (2CK) model, which is the simplest prototypi-

cal example of non-Fermi liquid quantum impurity mod-

els. This model has first been introduced by Nozi` eres

and Blandin,16and since has been proposed to de-

scribe a variety of systems including dilute heavy fermion

compounds,5tunneling impurities in disordered metals

and doped semiconductors.17,18,19More recently, the

2CK state has been observed in a very controlled way

in a double dot system originally proposed by Oreg and

Goldhaber-Gordon.20,2149

The two-channel Kondo model consists of a spin-1

local moment which is coupled through antiferromag-

netic exchange interactions to two channels of conduc-

tion electrons. Electrons in both channels try to screen

In

the spin-1

2, two-channel

2,

the impurity spin. If the coupling of the spin to one of

the channels is stronger than to the other then electrons

in the more strongly coupled channel screen the spin,

while the other channel becomes decoupled. However,

for equal exchange couplings, the competition between

the two channels leads to overscreening and results in a

non-Fermi liquid behavior: Among others, it is character-

ized by a non-trivial zero temperature residual entropy,

a square root-like temperature dependence of the differ-

ential conductance, a logarithmic divergence of the spin

susceptibility and the linear specific heat coefficient at

low temperatures.5This unusual and fragile ground state

cannot be described within the framework of Nozi` eres’

Fermi liquid theory.25

Being a prototypical example of non-Fermi liquid

models, the two-channel Kondo model (2CKM) has al-

ready been investigated with a number of methods.

These include non-perturbative techniques like the Bethe

Ansatz, which gives full account of the thermodynamic

properties,26,27boundary conformal field theory,28which

describes the vicinity of the fixed points, and numeri-

cal renormalization group (NRG) methods.29Further-

more other less powerful approximate methods such as

the Yuval-Anderson approach,31Abelian bosonization,32

large-f expansion,33,34and non-crossingapproximation35

have also been used to study the 2CKM successfully.

Rather surprisingly, despite this extensive work, very

little is known about dynamical correlation functions such

as the spin susceptibility, local charge and superconduct-

ing susceptibilities. Even the detailed properties of the

T-matrix, essential to understand elastic and inelastic

scattering in this non-Fermi liquid case,36have only been

computed earlier using conformal field theory (which is

rather limited in energy range) and by the non-crossing

approximation (which is not well-controlled and is un-

able to describe the Fermi liquid cross-over).37,38It was

also possible to compute some of the dynamical corre-

lation functions in case of extreme spin anisotropy us-

Page 2

2

ing Abelian bosonization results,32though these calcu-

lations reproduce only partly the generic features of the

spin-isotropic model.39Local correlations in the Ander-

son model around the non-Fermi liquid fixed point have

already been investigated with the use of NRG, although

in the absence of channel anisotropy and magnetic field.40

However, a thorough and careful NRG analysis of the

T = 0 temperature T-matrix of the 2CKM has been car-

ried out only very recently,24,36and the T ?= 0 analysis

still needs to be done.

The main purpose of this paper is to fill this gap by

giving a comprehensive analysis of the local correlation

functions at zero temperature using the numerical renor-

malization group approach. However, in the vicinity of

the rather delicate two-channel Kondo fixed point, the

conventional NRG method fails and its further developed

version, the density matrix NRG (DM-NRG)41needs to

be applied. Furthermore, a rather large number of mul-

tiplets must be kept to achieve good accuracy. We have

therefore implemented a modified version of the recently

developed spectral sum conserving DM-NRG method,

where we use non-Abelian symmetries in a flexible way

to compute the real and the imaginary parts of various

local correlation functions.42

To identify the relevant perturbations around the NFL

fixed point we apply the machinery of boundary confor-

mal field theory. Then we systematically study how the

vicinity of fixed points and the introduction of relevant

perturbations such as a finite channel anisotropy or a fi-

nite magnetic field influence the form of the dynamical

response functions at zero temperature. We mainly fo-

cus on the strong coupling regime of the 2CK model and

the universal cross-over functions in the proximity of this

region induced by an external magnetic field or channel

anisotropy. We remark that these cross-over functions,

describing the cross-over from the non-Fermi liquid fixed

point to a Fermi liquid fixed point, as well as the re-

sponse functions can currently be computed reliably at

all energy scales only with NRG. However, we shall be

able to use the results of boundary conformal field theory,

more precisely, the knowledge of the operator content of

the two-channel Kondo fixed point and the scaling di-

mensions of the various perturbations around it, to make

very general statements on the analytic properties of the

various cross-over and spectral functions.

We shall devote special attention to superconducting

fluctuations. It has been proposed that unusual super-

conducting states observed in some incoherent heavy

fermion compounds could also emerge as a result of

local superconducting correlations associated with two-

channel Kondo physics.5,43,44Here we investigate some

possible superconducting order parameters consistent

with the conformal field theoretical predictions, and find

that the dominant instability emerges in the so-called

composite superconducting channel, as it was proposed

by Coleman et al.44

The paper is organized as follows. In Section II start-

ing from the one-dimensional, continuum formulation of

the 2CKM we connect it to a dimensionless approxima-

tion of it suited to our DM-NRG calculations. We also

provide the symmetry generators used in the conformal

field theoretical and DM-NRG calculations. In Section

III we use boundary conformal field theory to classify the

boundary highest-weight fields of the electron-hole sym-

metrical 2CKM by their quantum numbers and identify

the relevant perturbations around the 2CK fixed point.

Based on this classification the fields are then expanded

in leading order in terms of the operators of the free the-

ory. In Section IV we describe the technical details of

our DM-NRG calculations. In Sections V, VI and VII we

study the real and the imaginary parts of the retarded

Green’s functions of the local fermions, the impurity spin

and the local superconducting order parameters. In each

of these sections we first discuss the analytic forms of

the susceptibilities in the asymptotic regions of the two-

channel and single channel Kondo scaling regimes, as

they follow from scaling arguments. Then we confirm

our predictions by demonstrating how the expected cor-

rections due to the relevant perturbations and the leading

irrelevant operator present themselves in the DM-NRG

data. Furthermore we determine the boundaries of the

2CK scaling regimes and derive universal scaling curves

connecting the FL and NFL fixed points for each oper-

ator under study. Finally, our conclusions are drawn in

Section VIII.

II.HAMILTONIAN AND SYMMETRIES

The two-channel Kondo model consists of an impurity

with a magnetic moment S =1

liquid (FL) of two types of electrons (labeled by the flavor

or channel indices α = 1,2), and interacting with them

through a simple exchange interaction,

2embedded into a Fermi

H =

?

?

α,µ

?DF

−DF

dk k c†

α,µ(k)cα,µ(k)(1)

+

α

?

µ,ν

Jα

2

?DF

−DF

dk

?DF

−DF

dk′?S c†

αµ(k)? σµνcαν(k′) .

Here c†

angular momentum channel with spin µ and radial mo-

mentum k measured from the Fermi momentum. In the

Hamiltonian above we allowed for a channel anisotropy

of the couplings, J1 ?= J2, and denoted the Pauli ma-

trices by ? σ.In the first, kinetic term, we assumed a

spherical Fermi surface and linearized the spectrum of

the conduction electrons, ξ(k) ≈ vFk = k, but these as-

sumptions are not crucial: Apart from irrelevant terms in

the Hamiltonian, our considerations below carry over to

essentially any local density of states with electron-hole

symmetry. The fields c†

α,µ(k) are normalized to satisfy

the anticommutation relations

α,µ(k) creates an electron of flavor α in the l = 0

?c†

α,µ(k),cβ,ν(k′)?= δα,βδµ,νδ (k − k′) ,(2)

Page 3

3

and therefore the couplings Jαare just the dimensionless

couplings, usually defined in the literature. Since we are

interested in the low-energy properties of the system, an

energy cut-off DF is introduced for the kinetic and the

interaction energies. In heavy fermion systems, this large

energy scale is in the range of the Fermi energy, DF ∼

EF, while for quantum dots, it is of the order of the single

particle level spacing of the dot, δǫ or its charging energy,

EC, whichever is smaller.

The Hamiltonian above possesses various symmetries.

To see it, it is worth to introduce the left-moving fermion

fields,

ψα,µ(x) ≡

?DF

−DF

dk e−ikxcα,µ(k) ,(3)

and to rewrite the Hamiltonian as

H =

?

α,µ

?

dx

2πψ†

α,µ(x) i∂xψα,µ(x)

+

?

α

Jα

2

?Sψ†(0)? σψ(0) . (4)

Then the total spin operators Jidefined as

Ji≡ Si+

?

: ψ†

dx

2πJi(x) ,(5)

Ji(x) ≡

1

2

?

α

α(x)σiψα(x) : (6)

commute with the Hamiltonian and satisfy the standard

SU(2) algebra,

?Ji,Jj?= iǫijkJk. (7)

In the previous equations we suppressed spin indices and

introduced the normal ordering : ... : with respect to the

non-interacting Fermi sea. In a similar way we can define

the “charge spin” density operators, for the channels α =

1,2 as

Cz

α(x) ≡

C−

α(x) ≡ ψα↑(x)ψα↓(x),

C±

1

2: ψ†

α(x)ψα(x) :

C+

α(x) ≡ ψ†

α↓(x)ψ†

α↑(x) ,

α(x) ≡ Cx

and the corresponding symmetry generators

α(x) ± i Cy

α(x) , (8)

Ci

α≡

?

dx

2πCi

α(x)(i = x,y,z) .(9)

The generators Ci

symmetry,45satisfy the same SU(2) algebra as the Ji-s,

α, which are related to the electron-hole

?

Ci

α,Cj

β

?

= iδαβǫijkCk

β,(10)

and they also commute with the Hamiltonian, Eq. (4).

Thus the Hamiltonian H has a symmetry SUC1(2) ×

SUC2(2) × SUS(2) in the charge and spin sectors for ar-

bitrary couplings, Jα.

To perform NRG calculations, we use the following ap-

proximation of the dimensionless Hamiltonian,29

2H

DF(1 + Λ−1)≈

?

?

α

?

µ,ν

˜Jα

2

?Sf†

0,α,µ? σµνf0,α,µ

+

∞

?

n=0

α,µ,ν

tn(f†

n,α,µfn+1,α,µ+ h.c.) ,(11)

with Λ a discretization parameter and˜Jα = 4Jα/(1 +

Λ−1). The operator, f0creates an electron right at the

impurity site, and can be expressed as

f0,α,µ=

1

√2DF

?DF

−DF

dk cα,µ(k) .(12)

The Hamiltonian Eq. (11) is also called the Wilson chain:

it describes electrons hopping along a semi-infinite chain

with a hopping amplitude tn ∼ Λ−n/2, and interacting

with the impurity only at site 0. In the NRG procedure,

this Hamiltonian is diagonalized iteratively, and its spec-

trum is used to compute the spectral functions of the

various operators.29

We remark that the Wilson Hamiltonian is not iden-

tical to H , since some terms are neglected along its

derivation.29Nevertheless, similar to H , the Wilson

Hamiltonian also possesses the symmetry SUC1(2) ×

SUC2(2) × SUS(2) for arbitrary J1 and J2 couplings.45

The corresponding symmetry generators have been enu-

merated in Table I. We can then use these symmetries to

label every multiplet in the Hilbert space and every op-

erator multiplet by the eigenvalues

?

C2

α= cα(cα+ 1). Throughout this paper, we shall use

these quantum numbers to classify states and operators.

In the presence of a magnetic field, i.e., when a term50

?

J2= j(j + 1) and

Hmagn= −gµBB Sz

(13)

is added to H , the symmetry of the system breaks down

to SUC1(2)×SUC2(2)×US(1), with the symmetry US(1)

corresponding to the conservation of the z-component of

the spin, Jz(see Table I). In the rest of the papers we

shall use units where we set gµB≡ 1.

III.THE NON-FERMI LIQUID FIXED POINT

AND ITS OPERATOR CONTENT

For J1= J2= J and in the absence of an external mag-

netic field, the Hamiltonian, H possesses a dynamically

generated energy scale, the so-called Kondo temperature,

TK≈ DFe−1/J.

The definition of TKis somewhat arbitrary. In this paper,

TKshall be defined as the energy ω at which for J1= J2

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4

Symmetry groupGenerators

SUCα(2)C+

α=

∞

X

n=0

(−1)nf†

n,α,↑f†

n,α,↓, Cz

α=1

2

∞

X

n=0

X

∞

X

µ

“

X

f†

n,α,µfn,α,µ− 1

”

, C−

α= C+†

α

SUS(2)

?J =?S +1

2

n=0

α,µ,ν

f†

n,α,µ? σµνfn,α,ν

TABLE I: Generators of the used symmetries for the two-channel Kondo model computations. Sites along the Wilson chain

are labeled by n whereas α and µ,ν are the channel and spin indices, respectively.

the spectral function of the composite fermion drops to

half of its value assumed at ω = 0 (for further details

see the end of this Section and Fig. 1). For B = 0 and

J1= J2, below this energy scale the physics is governed

by the so-called two-channel Kondo fixed point.

The physics of the two-channel Kondo fixed point and

its vicinity can be captured using conformal field the-

ory. The two-channel Kondo finite size spectrum and its

operator content has first been obtained using boundary

conformal field theory by Affleck and Ludwig.28However,

instead of charge SU(2) symmetries, Affleck and Ludwig

used flavor SU(2) and charge U(1) symmetries to obtain

the fixed point spectrum.28The use of charge SU(2) sym-

metries, however, has a clear advantage over the flavor

symmetry when it comes to performing NRG calcula-

tions: While the channel anisotropy violates the flavor

symmetry, it does not violate the charge SU(2) symme-

tries. Therefore, even in the channel anisotropic case, we

have three commuting SU(2) symmetries. If we switch

on a local magnetic field, only the spin SU(2) symmetry

is reduced to its U(1) subgroup. Using charge symme-

tries allows thus for much more precise calculations, and

in fact, using them is absolutely necessary to obtain sat-

isfactorily accurate spectral functions, especially in the

presence of a magnetic field.

To understand the fixed point spectrum and the oper-

ator content of the 2CKM, let us outline the boundary

conformal field theory in this SUC1(2)×SUC2(2)×SUS(2)

language. First, we remark that the spin density oper-

ators, Ji(x) satisfy the SU(2)k=2Kac-Moody algebra of

level k = 2,

?Ji(x),Jj(x′)?

=

k

2δijδ′(x − x′)

+ i 2π δ(x − x′)ǫijkJk(x) ,(14)

while the charge density operators, Ci

previous section satisfy the Kac-Moody algebra of level

k = 1:

α(x) defined in the

?

Ci

α(x),Cj

β(x′)

?

=

k

2δijδαβδ′(x − x′)

+ i 2π δαβδ(x − x′)ǫijkCk

α(x) .

We can use these current densities and the coset con-

struction to write the kinetic part of the Hamiltonian as

H0 = HC1+ HC2+ HS+ HI,

1

3

1

4

(15)

HCα =

?

?

dx

2π:?Cα(x)?Cα(x) : ,

dx

2π:?J(x)?J(x) : .

HS =

In H0, the first two terms describe the chargesectors, and

have central charge c = 1, while HS describes the spin

sector, and has central charge c = 3/2. The last term

corresponds to the coset space, and must have central

charge c = 1/2, since the free fermion model has central

charge c = 4, corresponding to the four combinations of

spin and channel quantum numbers. This term can thus

be identified as the Ising model, having primary fields

1 l ,σ,ǫ with scaling dimensions 0,1/16,1/2, respectively.

We can then carry out the conformal embedding in the

usual way, by comparing the finite size spectrum of the

free Hamiltonian with that of Eq. (15), and identifying

the allowed primary fields in the product space. The

the fusion rules obtained this way are listed on the left

side of Table II. The finite size spectrum at the two-

channel Kondo fixed point can be derived by fusing with

the impurity spin (which couples to the spin sector only),

following the operator product expansion of the Wess–

Zumino–Novikov–Witten model, 1/2 ⊗ 0 → 1/2, 1/2 ⊗

1/2 → 0⊕1, 1/2⊗1 → 1/2 (see RHS of Table II). Finally,

the operator content of the fixed point can be found by

performing a second fusion with the spin. The results of

this double fusion are presented in Table III. In Table III

the leading irrelevant operator,?J−1?φs, is also included.

Although it is not a primary field,28close to the 2CK

fixed point, this operator will also have impact on the

form the correlation functions.

What remains is to identify the scaling operators in

terms of the operators of the non-interacting theory. In

general, an operator of the non-interacting theory can

be written as an infinite series in terms of the scaling

operators and their descendants. Apart from the Ising

sector, which is hard to identify, we can tell by looking

at the various quantum numbers of an operator acting on

the Wilson chain, which primary fields could be present

in it. In this way, we can identify, e.g.?φs as the spin

operator?S. Thus the spin operator can be expressed as

?S = As?φs+ ...(16)

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5

c1 c2 j I Efree

0 0 0 1 l

1

2

0

0

2

1

2

1

2

0

1

2

1

2

1

1

1

2σ

1

2σ

1

1

21 1 l

1

20 ǫ

c1 c2 j I E2CKM

0 0

21 l

1

2

0 0 σ

0

20 σ

1

22

1

2

0 1 σ

0

21 σ

1

22

1

0

1

8

1

8

1

2

5

8

5

8

1

1

11

21 l

1

11

2ǫ

TABLE II: Left: Primary fields and the corresponding finite

size energies at the free fermion fixed point for anti-periodic

boundary conditions. States are classified according to the

group SUC1(2)×SUC2(2)×SUS(2) and the Ising model. The

excitation energies Efreeare given in units of 2π/L, with L the

size of the chiral fermion system. Right: Finite size spectrum

at the two-channel Kondo fixed point.

where the dots stand for all the less relevant operators

that are present in the expansion of?S, and some high-

frequency portions which are not properly captured in

the expansion above. The weight, As, can be deter-

mined from matching the decay of the spin-spin corre-

lation function at short and long times. This way we end

up with As∼ 1/√TK.

We remark that there are infinitely many operators

that contain the scaling fields in their expansion.

an example, consider the operators φτσ

σ = {↑,↓} refers to the spin components of a j = 1/2

spinor, while τ = ± refer to the charge spins of a charge

c = 1/2 spinor. To identify the corresponding operator

on the Wilson chain, we first note that f†

as a spinor under spin rotations. It can easily be seen that

the operator˜f†

0,1≡ iσyf0,1also transforms as a spinor.

We can then form a four-spinor out of these operators,

γ1≡ {f†

as a spinor under SUC1(2) rotations as well, thus φτσ

could be identified as γ1= {f†

However, we can construct another operator, F†

f†

spinor out of them: Γ1 ≡ {F†

has the same quantum numbers as γ1, and in fact, both

operators’ expansion contains φτσ

The operator φττ′

∆

is of special interest, since it is rel-

evant at the two-channel Kondo fixed point, just like

the spin. Its susceptibility therefore diverges logarith-

mically. Good candidates for these operators would be

?

τ = τ′= + component of this operator corresponds to

the superconducting order parameter

As

ψ1. Here the label

0,1,σtransforms

0,1,σ,˜f†

0,1,σ}. It is easy to show that γ1transforms

ψ1

0,1,σ,˜f†

0,1,σ}.

1≡

0,1?S? σ and its counterpart,˜F†

1≡ iσyF1, and form a four-

1,σ,˜F†

1,σ}. This operator

ψ1.

σσ′ǫσσ′γτσ

that behave as charge 1/2 spinors in both channels. The

1γτ′σ′

2

, since these are spin singlet operators

OSC≡ f†

0,1,↑f†

0,2,↓− f†

0,1,↓f†

0,2,↑,(17)

while the +− components describe simply a local opera-

tor that hybridizes the channels, ∼ f†

0,1,σf0,2,σ.

Another

?

contains the following component of the composite

superconducting order parameter

candidate

1γτ′σ′

2

wouldbethe operator,

σσ′ǫσσ′Γτσ

singlet, and has charge spins c1 = c2 = 1/2.

.This operator is also a local

It

OSCC≡ f†

0,1?S? σ iσyf†

0,2.(18)

From their transformation properties it is not obvious,

which one of the above superconducting order parame-

ters gives the leading singularity. However, NRG gives

a very solid answer and tells us that, while the suscepti-

bility of the traditional operator does not diverge as the

temperature or frequency goes to zero, that of the com-

posite order parameter does. It is thus this latter oper-

ator that can be identified as φττ′

of electron-hole symmetry, the composite hybridization

operator

∆. Note that, in case

Omix≡ f†

0,1?S? σ f0,2

(19)

has the same singular susceptibility as OSCC since they

are both components of the same tensor operator. This

is, however, not true any more away from electron-hole

symmetry. Furthermore, superconducting correlations

are usually more dangerous, since in the Cooper chan-

nel any small attraction would lead to ordering when a

regular lattice model of two-channel Kondo impurities is

considered.

The knowledge of the operator content of the two-

channel Kondo fixed point enables us to describe the

effects of small magnetic fields and small channel

anisotropies (J1 ?= J2). For energies and temperatures

below TK, the behavior of the model can be described

by the slightly perturbed two-channel Kondo fixed point

Hamiltonian. For J1≈ J2and in a small magnetic field,

B ≪ TK, this Hamiltonian can be expressed as

H = H∗

+D1/2

0

2CK+

κ0φanis+D1/2

0

?h0?φs+D−1/2

0

λ0?J−1?φs+... .

(20)

Here H∗

is the dimensionless coupling to the channel anisotropy

field, φanis, whereas the effective magnetic field,?h0, cou-

ples to the “spin field”, φs. Both of them are relevant

perturbations at the two-channel Kondo fixed point and

they must vanish to end up with the two-channel Kondo

fixed point at ω,T → 0. The third coupling, λ0, couples

to the leading irrelevant operator (see Tab. III), which

dominates the physics when κ = h = 0. The energy cut-

off D0in Eq. (20) is a somewhat arbitrary scale: it can

be though of as the energy scale below which the two-

channel Kondo physics emerges, i.e. D0 ∼ TK. Then

the dimensionless couplings κ0, λ0 and h0 are approxi-

mately related to the couplings of the original Hamilto-

2CKis the 2CK fixed point Hamiltonian, and κ0

Page 6

6

c1 c2 j I x2CK

scaling

operators

?φs

corresponding operators

0 0 1 1

1

2

?S

1

2

0

1

2σ

1

2

φτσ

ψ1

γ1 ≡

Γ1 ≡

“

“

f†

0,1,σ, (iσyf0,1)σ

F†

”

”

0,1,σ, (iσyF0,1)σ

γ2

Γ2

0

1

2

1

2σ

1

2

φτσ

ψ2

1

2

1

20 1 l

1

2

φττ′

∆

f†

0,1?S? σ iσyf†

−f0,1σy?S? σ σyf†

?S(f†

?S(f†

0,2

−f†

0,1?S? σf0,2

0,2−f0,1iσy?S? σf0,2

0,1? σf0,1− f†

0,1? σf0,1+ f†

!

0 0 0 ǫ

0 0 0 1 l

1

2

3

2

φanis

? J−1?φs

0,2? σf0,2)

0,2? σf0,2)

TABLE III: Highest-weight operators and their dimensions x2CKat the 2CK fixed point. Operators are classified by the

symmetry group SUC1(2) × SUC2(2) × SUS(2) and the scaling operators of the Ising model. The constants c1 and c2 denote

the charge spins in channels 1 and 2, respectively, while j refers to the spin, and I labels the scaling operators of the Ising

model: 1 l ,σ,ǫ. Superscripts τ,τ′= ± refer to the two components of charge spinors, while σ =↑,↓ label the components of a

spin-±1

2spinor.

nian, Eq. (11), as

κ0 ≈ KR≡ 4

h0 ≈ B/TK,

λ0 ≈ O(1) .

J1− J2

(J1+ J2)2,(21)

(22)

(23)

However, the arbitrary scale D0 in Eq. (20) can be

changed at the expense of changing the couplings: D0→

D,κ0 → κ(D), h0 → h(D) and λ0 → λ(D) in such a

way that the physics below D0remains unchanged. This

freedom translates to scaling equations, whose leading

terms follow from the conformal field theory results, and

read

dκ(D)

dx

dh(D)

dx

dλ(D)

dx

=

1

2κ(D) + ... ,

1

2h(D) + ... ,

= −1

(24)

=(25)

2λ(D) + ... ,(26)

with x = −logD. Solving these equations with the initial

conditions, D = D0∼ TK and h = h0, κ = κ0, λ = λ0,

we can read out the energy scales at which the rescaled

couplings become of the order of one,

T∗∝ TKκ2

0∼ TK(J1− J2)2

(J1+ J2)4,

(27)

(28)Th ∝ TKh2

0∼ B2/TK.

At these scales the couplings of the relevant operators

are so large that they can no longer be treated as pertur-

bations. Below T∗the single channel Kondo behavior is

recovered in the more strongly coupled channel, while Th

can be interpreted as the scale where the impurity spin

dynamics is frozen by the external field.

The prefactors in Eqs. (28) are somewhat arbitrary,

and depend slightly on the precise definition one uses

to extract these scales. In this paper, we shall use the

spectral function of the composite fermion to define the

scales TK and T∗. We define TK to be the energy at

which for KR= 0 the spectral function of the composite

fermion takes half of its fixed point value (i.e. the value

assumed at ω = 0). Whereas T∗is the energy at which

for KR > 0 it takes 75% of its fixed point value (see

Fig. 1).

It is much harder to relate Thto a physically measur-

able quantity. We defined it simply through the relation,

Th≡ ChB2

TK

,(29)

where the constant was chosen to be Ch≈ 60. This way

Thcorresponds roughly to the energy at which the NFL

finite size spectrum crosses over to the low-frequency FL

spectrum.

IV.NRG CALCULATIONS

Prior to discussing the analytic and numerical features

of the response functions, let us devote this section to the

short description of the NRG procedure used. All results

presented in this paper refer to zero temperature. The

NRG calculations were performed with a discretization

parameter Λ = 2. The sum of the dimensionless cou-

plings was˜J1+˜J2= 0.4 for each run. The NRG data were

computed with a so-called flexible DM-NRG program,42

which permits the use of an arbitrary number of Abelian

and non-Abelian symmetries (see Tab. I), and incorpo-

rates the spectral-sum conserving density matrix NRG

(DM-NRG) algorithm.41The DM-NRG method makes it

possible to generate spectral functions that satisfy spec-

tral sum rules with machine precision at T = 0 tem-

Page 7

7

10-9

10-6

10-3

100

ω

0

1

2

ρF(ω)

0.02

0.0075

0.0025

0

T *

TK

KR

FIG. 1: (color online) Spectral function ̺F of the composite

fermion operator, F0,1,↑ as a function of ω, and the definition

of the scales TK and T∗. TK is defined by the relation ̺F(ω =

TK,T = 0,KR = 0) ≡

zero KR the scale T∗is defined through ̺F(ω = T∗,T =

0,KR) ≡3

1

2̺F(ω = 0,T = 0,KR = 0). For non-

4̺F(ω = 0,T = 0,|KR|).

perature. For calculations with non-zero magnetic field

the use of the DM-NRG method represents a great ad-

vantage over conventional NRG methods,46which loose

spectral weights and violate spectral sum rules. Conven-

tional methods also lead to smaller or bigger jumps in

the spectral functions at ω = 0 which hinder the compu-

tation of the universal scaling functions provided by the

scale Th.24The DM-NRG method solves all these prob-

lems if a sufficient number of multiplets is kept. On an

ordinary desk-top computer, however, we need to use as

many symmetries as possible to keep the computation

time within reasonable limits.

In the present paper, where we study the electron-

hole symmetrical case, it is possible to use the symmetry

group SUC1(2)×SUC2(2)×SUS(2) even in case of channel

anisotropy. At these calculations the maximum number

of kept multiplets was 750 in each iteration. This cor-

responds to the diagonalization of ≈ 85 matrices with

matrix sizes ranging up to ≈ 630, acting on the vector

space of ≈ 9000 multiplets consisting of ≈ 106000 states.

In the presence of magnetic field we used the symme-

try group SUC1(2) × SUC2(2) × US(1), and retained a

maximum of 1350 multiplets in each iteration, that cor-

responds to the diagonalization of ≈ 150 matrices with

matrix sizes ranging up to ≈ 800 acting on the vector

space of ≈ 18000 multiplets consisting of ≈ 73000 states.

In the next sections, we shall see how the knowledge

of the operator content of the two-channel Kondo fixed

point can help us to understand the analytic structure of

the various dynamical correlation functions obtained by

NRG.

V.LOCAL FERMIONS’ SPECTRAL

FUNCTIONS AND SUSCEPTIBILITIES

Let us first analyse the Green’s function of the local

fermion, f†

?Γ0,α) Green’s function has already been looked into in

detail in an earlier study of ours.24We shall therefore

not discuss its analytic properties here but use it merely

as a reference to define the various energy scales in the

NRG calculations (see Fig. 1). Let us note, however, that

in the large bandwidth limit, ω,TK≪ DF, the spectral

function of the composite fermion and that of the local

fermion are simply related,

0,σ,α↔ ? γα. The composite fermion’s (F†

0,σ,α↔

̺f(ω) =

1

2DF

−π

4J2̺F(ω) . (30)

Thus, apart from a trivial constant shift and a minus

sign, the spectral function of the local fermion is that

of the composite fermion, and all features of ̺F are also

reflected in ̺f.

Before we discuss the NRG results, let us examine what

predictions we have for the retarded Green’s function of

the operator f†

0,σ,αfrom conformal field theory. By look-

ing at its quantum numbers, this operator can be identi-

fied with the operator φ+σ

ψα(see Tab. III), i.e.

f†

0,σ,α= Afφ+σ

ψα+ ... ,(31)

with the prefactor Af∝ 1/√DF. Note that Afis a com-

plex number, it does not need to be real. The dots in the

equation above indicate the series of other, less relevant

operators and their descendants, which give subleading

corrections to the correlation function of f†

more, the expansion above holds for the long time behav-

ior. The “short time part” of the correlation function of

f†

0,σ,αis not captured by Eq. (31), and gives a constant

to Gf(ω) of the order of ∼ 1/DF. Thus, apart from a

prefactor A2

f, a constant shift and subleading terms, the

Green’s function of f†

we discuss it shortly in Appendix A, the Fourier trans-

form of the Green’s function of any operator of dimension

x = 1/2 is scale invariant around the two-channel Kondo

fixed point. Since φ+σ

mension 1/2 at the 2CK fixed point, it follows that the

dimensionless retarded Green’s function, DF Gf(ω), is

also scale invariant,51

0,σ,α. Further-

0,σ,αis that of the field φ+σ

ψα. As

ψαand thus f†

0,σ,αhave a scaling di-

DFGf(ω,T) ≡ ˆ gf

Ddˆ gf

dD

From Eq. (32), we can deduce various important prop-

erties. Let us first consider the simplest case, T = 0 and

κ = h = 0. Then setting the scale D to D0 ∼ TK we

have

?ω

?ω

D,TD,κ(D),h(D),λ(D),...

?

,

= 0 .(32)

ˆ gκ,h,T=0

f

(ω) = ˆ gf

D0,λ0,...

?

.(33)

Page 8

8

Let us now rescale D → |ω|, and use the fixed point

scaling equation (26) to obtain λ(D),

ˆ gf = gf

±1,

?

|ω|

D0

λ0

.(34)

Assuming that this function is analytic in its second ar-

gument we obtain for |ω| ≪ TK

ˆ gκ,h,T=0

f

(ω) = ˆ gf

TK

?ω

?

≈ g± f+ g′

± f

?

|ω|

TK

+ ... ,(35)

with g± f and g′

Here the subscripts ± refer to the cases ω > 0 and ω < 0,

respectively. As we discussed above, the constants g± f

depend also on the short time behavior of Gf(t), and are

not universal in this sense. These constants are not inde-

pendent of each other. They are related by the constraint

that the Green’s function must be analytic in the upper

half-plane. Furthermore, electron-hole symmetry implies

that g+ f= g− f and g′

Relations similar to the ones above hold for the dimen-

sionless spectral function. It is defined as

± fsome complex expansion coefficients.

+ f= −(g′

− f)∗.

ˆ ̺f(ω) ≡ −1

πIm ˆ gf(ω) , (36)

and assumes the following simpler form at small frequen-

cies in case of electron-hole symmetry,

ˆ ̺T,κ,h=0

f

(ω) = rf+ r′

f

?

|ω|

TK

+ ... .(37)

For ω ≫ TKthe scaling dimension of the local fermion

is xfree

f

= 1/2 corresponding to an ω-independent spec-

tral function. Perturbation theory in J amounts to log-

arithmic corrections of the form: 1/2− cst/log2(TK/ω),

as it is sketched in the upper parts of Fig.-s 2 and 3.

For T ?= 0, and κ = h = 0 using similar arguments as

before, but now rescaling D → T we find

?ω

?

ˆ gκ,h=0

f

(ω) = ˆ gf

T,

T

TK,λ0

?

≡ ˆ gf

ω

T,1,

?

T

D0

λ0,...

?

.(38)

Then by expanding ˆ gf we obtain the following scaling

form for the low temperature behavior of the spectral

function,

ˆ ̺h,κ=0

f

(ω) = Θf

?ω

T

?

+

?

T

TK

˜Θf

?ω

T

?

+ ... ,(39)

with Θf and˜Θf universal scaling functions. Note that

we made no assumption on the ratio ω/T, but both ω and

log

T

TK

0log

DF

TK

∼

cst +

?ω

T

?2

≈

1

4+ cst

?ω

TK

log

ω

TK

T > 0,KR= 0,B = 0

ˆ ̺f(ω)

≈

1

2−

cst

log2(

TK

ω)

log

T

TK

−Re ˆ gf(ω)

log

DF

TK

log

ω

TK

0

∼ω

T

∼

?

?

?

?ω

TK

T > 0,KR= 0,B = 0

FIG. 2: (color online) (top) Sketch of the dimensionless spec-

tral function ˆ ̺f = DF̺f of f†

part of its dimensionless Green’s function, Re ˆ gf = DF Re Gf

for T > 0 and KR = 0,B = 0 as a function of log(ω/TK).

Asymptotics indicated for ω < TK were derived through scal-

ing arguments. The large ω-behavior is a result of perturba-

tion theory.

0,1,σ, and (bottom) the real

T must be smaller than TK. The asymptotic properties

of Θf and˜Θf can be extracted by making use of the

facts that (i) ˆ gf(ω,T) must be analytic for ω ≪ T, (ii)

that Eq. (39) should reproduce the T → 0 results in the

limit ω ≫ T, and (iii) that by electron-hole symmetry,

ˆ ̺f must be an even function of ω. The issuing asymp-

totic properties together with those of the other scaling

functions defined later are summarized in Table V. The

asymptotic properties of the real part, Re ˆ gf, can be ex-

tracted from those of ˆ ̺f by performing a Hilbert trans-

form

Re ˆ gf(ω) = P

?

d˜ ωˆ ̺f(˜ ω)

ω − ˜ ω

(40)

with P the principal part. The obtained features are

sketched in Fig. 2 for T > 0 and κ = h = 0.

Let us now investigate the effect of channel anisotropy,

i.e. κ ?= 0 at T = 0 temperature and no magnetic field

h = 0. In this case, we can rescale D to D = |ω| to obtain

?

ˆ ̺T,h=0

f

(ω) = K±

f

?ω

T∗

?

+

|ω|

TK

˜K±

f

?ω

T∗

?

+ ... ,

(41)

Page 9

9

0

≈

1

4+

?ω

TK

∼

cst +

?

?

?

?T∗

ω

log

ω

TK

1

2log

T∗

TK

log

T∗

TK

ˆ ̺f(ω)

∼

?ω

T∗

?2

T = 0,KR> 0,B = 0

log

DF

TK

≈

1

2−

cst

log2(

TK

ω)

∼ω

T∗

∼

?

?

?

?T∗

ω

∼

?

?

?

?ω

TK

log

T∗

TK

log

DF

TK

−Re ˆ gf(ω)

0

1

2log

T∗

TK

log

ω

TK

T = 0,KR> 0,B = 0

FIG. 3: (color online) (top) Sketch of the dimensionless spec-

tral function of f†

part of its dimensionless Green’s function: Re ˆ gf = DF Re Gf

for T = 0,KR > 0 and B = 0 as a function of log(ω/TK).

Asymptotics indicated for ω < TK were derived through scal-

ing arguments. The large ω-behavior is a result of perturba-

tion theory.

0,1,σ: ˆ ̺f = DF ̺f, and (bottom) the real

with T∗the anisotropy scale defined earlier. The su-

perscripts ± refer to the cases of positive or negative

anisotropies: the superscript “+” is used when the cou-

pling is larger in the channel where we measure the

Green’s function of f†

0,α,σ. The asymptotics of the univer-

sal functions K±

scaling arguments as before and they differ only slightly

from those of Θf and˜Θf (see Table V for a summary).

The properties of ˆ ̺T,h=0

f

(ω) are summarized in Fig. 3.

A remarkable feature of the spectral function is that it

contains a correction ∼

be obtained by doing perturbation theory in the small

parameter κ(ω) at the two-channel Kondo fixed point.

From the asymptotic forms in Table V we find that

in the local fermion’s susceptibility a new scale, T∗∗

√T∗TK appears as a result of the competition be-

tween the leading irrelevant operator and the channel

anisotropy:24It is only in the regime T∗∗

the leading irrelevant operator determines the dominant

scaling behavior of the local fermion’s susceptibility, i.e.,

we observe the true two-channel Kondo physics.

expected properties of ˆ ρf and the real part of its dimen-

sionless Green’s function ˆ gf in the presence of channel

asymmetry are summarized in Fig. 3. These analytic ex-

pectations are indeed met by our NRG results.

fand˜K±

fcan be obtained through similar

?T∗/|ω|. This correction can

f

∼

f

< ω < TKthat

The

Fig. 4.(a) depicts the spectral function of f†

several values of KR as a function of ω/TK on a loga-

rithmic scale. The overall scaling is very similar to the

one sketched in Fig. 3, except that the high tempera-

ture plateau is missing; this is due to the relatively large

value of TK, which is only one decade smaller than the

bandwidth cut-off. Figures 4.(b − e) are the numerical

confirmations of the asymptotics stated. In all these fig-

ures dashed straight lines are to demonstrate deviations

from the expected behavior. In Fig. 4.(b) we show the

square root-like asymptotics in the 2CK scaling regime

for the channel symmetric case. This behavior is a con-

sequence of the dimension of the leading irrelevant op-

erator as it has just been discussed. In Fig. 4.(c) the

same asymptotics is shown in the same region in case

of a finite channel anisotropy, whereas below them Fig.

4.(d) demonstrates (1/ω)1/2-like behavior resulting from

the relevant perturbation of the 2CK fixed point Hamil-

tonian with channel anisotropy. In Fig. 4.(e) the FL-like

ω2-behavior is recovered below T∗, which is typical of

fermionic operators in the 1CK scaling regimes.

In Figs. 5.(a−b) we show the universal scaling curves,

K±that connect the two-channel and single channel fixed

points at low-frequencies as a function of ω/T∗. They

were computed from runs with negative and positive val-

ues of KR. This universal behavior is violated for values

0,1,σfor

10-6

10-3

100

103

ω / TK

0

0.25

0.5

ρf(ω)

0.001

0.0075

0.05

0.25

- 0.001

- 0.0075

- 0.05

- 0.25

0

00.3 0.6

(ω / TK)1/2

KR = 0.02

0.28

0.32

ρf(ω)

KR = 0

1.6

(ω / TK)1/2

KR = 0.02

2

0.42

0.435

ρf(ω)

KR = 0.02

1.8 1.9

(ω / T* )-1/2

0.065

0.07

ρf(ω)

0 0.0002

(ω / T* )2

0.0004

0

7.5×10- 4

1.5×10- 3

ρf(ω)

(a)

(b)(c)

(d)(e)

<

<

<<

<

KR< 0

KR> 0

KR

FIG. 4: (color online) (a) Dimensionless spectral function of

f0,1,σ: ˆ ̺f(ω) = DF ̺f(ω) as a function of ω/TK for different

values of KR. (b − e) Numerical confirmations of the low-

frequency asymptotics derived through scaling arguments in

Sec. V. Dashed straight lines are to demonstrate deviations

from the expected√ω-like (b − c), 1/√ω-like (d) and ω2-like

(e) behavior. In plots (c − e) T∗/TK = 2.4 × 10−4.

Page 10

10

Scaling Function

Asymptotic Form

x ≪ 1 ,

θ0

˜θ0

f,0+ κ± ′

˜ κ±

β0

˜β0

θ0

˜θ0

κ0

Ssgn(x) |x|1/2,

β0

S,zx ,

˜β0

1 ≪ x

θ∞

˜θ∞

Scaling Variable

x

2CK

Scaling Regime

Θf (x)

˜Θf (x)

K±

˜K±

Bf,σ (x)

˜Bf,σ (x)

ΘS (x)

˜ΘS (x)

KS (x)

˜KS (x)

BS,z (x)

˜BS,z(x)

f+ θ0 ′

f+˜θ0 ′

f,0x2,

f x2,

f x2,

f

f x1/2

ω/TT ? ω

f(x)

f(x)

κ±

κ±

f,∞+ κ± ′

f,∞

˛˛1

x

x

˛˛1/2

˛˛1/2

f,0|x|3/2,

f,σ+ β0 ′

f,σ|x|3/2,

Sx ,

Sx ,

Sx ,

˜ κ±

f,∞

ω/T∗

T∗∗

f

? ω ,T∗∗

f

∝√T∗TK

f,σx2, β∞

f,σ+ β∞ ′

f,σ

˜β∞

f,σ

˛˛1

ω/Th

T∗∗

h ? ω ,T∗∗

h ∝√ThTK

θ∞

S sgn(x)

S sgn(x) |x|1/2

κ∞

˜ κ∞

β∞

˜θ∞

ω/TT ? ω

S sgn(x) + κ∞ ′′

S sgn(x)

S,z+ β∞ ′

S,z

˜β∞

S,z

S

1

x

˜ κ0

ω/T∗

T∗∗

s

? ω ,T∗∗

s

∝`T∗2TK

T∗∗

´1/3

˛˛1

x

˛˛1/2

S,z|x|1/2,

ω/Th

T∗∗

h ? ω ,

h ∝√ThTK

TABLE IV: Asymptotic behavior of the universal cross-over functions. At finite temperature, the boundary of the two-channel

Kondo scaling regime is set by the temperature. At zero temperature, the various boundaries of the 2CK scaling regime derive

from the competition between the leading irrelevant operator and the relevant perturbation.

10-3

100

103

106

ω / T*

0

0.25

0.5

ρf(ω)

KR = 0.05

KR = 0.02

KR = 0.0075

KR = 0.0025

KR = 0.001

Κ+

f

0.25

0.5

ρf(ω)

KR = - 0.05

KR = - 0.02

KR = - 0.0075

KR = - 0.0025

KR = - 0.001

Κ−

f

(a)

(b)

<

<

FIG. 5: (color online) Universal collapse of the dimensionless

spectral functions, ˆ ̺f = DF ̺f (with f in channel 1) to two

scaling curves, K±

negative (a) values of KR.

fas a function of ω/T∗for positive (b) and

of KRhigher than the highest ones shown in Fig. 5, where

T∗becomes comparable to TK.

The real parts of the local fermion susceptibilities are

plotted in Fig. 6 for several values of KR. They were ob-

tained by performing the Hilbert transformations numer-

ically. They should show a three-peak structure based on

the analytic considerations (see Fig. 3). There are two

low-frequency peaks clearly visible, associated with the

cross-overs at T∗and TK. Furthermore there should be

a non-universal peak at the cut-off. For relatively large

channel anisotropies, where T∗∼ TK, the former two

peaks cannot be clearly separated in Fig. 6. Also, due to

the large value of TK∼ the band cut-off, DF, the peak

at ω ∼ TKand the smeared singularity at ω = DFmerge

to a single non-universal feature in our NRG curves.

Let us now turn to the effect of a finite magnetic field,

B ?= 0 for the case T = 0,KR = 0. As h and κ scale

the same way in the 2CK scaling regime, the argument

concerning the κ ?= 0 case can be repeated with minor

modifications. Now, however, the spin SUS(2) symmetry

is violated, and therefore the spectral functions of f†

and f†

0,α,↓become different, and they are no longer even

either. Nevertheless, due to particle-hole symmetry, they

are still related through the relations

0,α,↑

ˆ ρf,↑(ω,T,κ,h,...) = ˆ ρf,↓(−ω,T,κ,h,...) ,

ˆ ρf,↑(ω,T,κ,h,...) = ˆ ρf,↓(ω,T,κ,−h,...) . (42)

We are thus free to choose the orientation of the magnetic

10-6

10-3

100

103

ω / TK

-0.1

0

0.1

0.2

-Re gf(ω)

0

0.001

0.0025

0.0075

0.02

0.05

- 0.001

- 0.0025

- 0.0075

- 0.02

- 0.05

<

KR

FIG. 6: (color online) Real part of the dimensionless Green’s

function: Re ˆ gf = DF Re Gf (with f†in channel 1) as a

function of ω/TK for different values of KR. From among the

three peaks sketched in Fig. 3 only the two peaks around T∗

and TK are shown.

Page 11

11

-40-2002040

ω / TK

0

0.25

0.5

ρf,↑(ω)

<

5.6 × 10 -1

1.1 × 10 -1

5.6 × 10 - 2

1.1 × 10 - 2

5.6 × 10 - 3

0

B / TK

10-3

100

103

ω / TK

0

0.25

0.5

ρf,↓(ω)

<

5.6 × 10 - 1

1.1 × 10 - 1

5.6 × 10 - 2

1.1 × 10 - 2

5.6 × 10 - 3

1.1 × 10 - 3

0

0

0.25

0.5

ρf,↑(ω)

<

B / TK

FIG. 7: (color online) Top: Dimensionless spectral function

of f0,1,↑ : ˆ ̺f,↑ = DF ̺f,↑ for different values of B as a func-

tion of ω/TK on linear scale. Bottom: Dimensionless spectral

function of f0,1,↑ (a) and of f0,1,↓ (b) for different values of B

as a function of ω/TK on logarithmic scale.

field downwards. Then, after rescaling D → |ω| we get

ˆ ̺κ,T=0

f,σ

(ω) = Bf,σ

?ω

Th

?

+

?

|ω|

TK

˜Bf,σ

?ω

Th

?

+... ,

(43)

where the label σ refers to the different spin components

and Bf,σand˜Bf,σare yet another pair of universal cross-

over functions. The asymptotic properties of the func-

tions Bf,σand˜Bf,σare summarized in Table V.

Fig. 7 shows the spectral functions ˆ ̺f,σ as a function

of ω/TK on linear and logarithmic scales for different

magnetic field values. The same curves are depicted as

a function of ω/Thin Fig. 8, which demonstrates the ex-

istence of the universal scaling curves, Bf,σ, i.e. that by

using the scale, Ththe local fermion’s spectral functions

can be scaled on top of each other for small enough mag-

netic fields. In this magnetic field region, we find a peak

at Thfor the spin-↑ component of f†, while at the same

place there is a dip for the spin-↓ component. This is a

remarkable feature that is associated with inelastic scat-

tering off the slightly polarized impurity spin. In fact,

10-3

100

103

ω / Th

106

109

0

0.25

0.5

ρf,↓(ω)

<

B / T K = 1.1 × 10 - 5

B / T K = 5.6 × 10 - 5

B / T K = 1.1 × 10 - 4

B / T K = 5.6 × 10 - 4

B / T K = 1.1 × 10 - 3

0.25

0.5

ρf,↑(ω)

<

(a)

(b)

Bf, ↓

Bf, ↑

FIG. 8: (color online) Universal collapse of the dimensionless

spectral functions: ˆ ̺f,↑= DF ̺f,↑ and ˆ ̺f,↓= DF ̺f,↓ to two

scaling curves: Bf,↑ and Bf,↓ for sufficiently small, non-zero

values of B as a function of ω/Th.

the same uinversal features also appear in the spectral

functions of the composite fermions, which we compute

independently and which are directly related to those of

the conduction electrons by Eqn. (30).46The rescaled

spectral functions ˆ ̺F,σ(ω) are shown in Fig.9.

Although this numerical evidence can be obtained by

conventional NRG methods not using the density matrix,

this is no longer true for the sum of the local fermions

spectral function over the different spin components. In

fact, for this quantity universal scaling curves in the pres-

ence of magnetic field cannot be obtained using NRG be-

cause of the increase in the size of the numerical errors at

0

1

ρF,↓(ω)

10-3

100

103

ω / TB

106

109

0

1

ρF,↑(ω)

<

B / T K = 1.1 × 10 - 5

B / T K = 5.6 × 10 - 5

B / T K = 1.1 × 10 - 4

B / T K = 5.6 × 10 - 4

B / T K = 1.1 × 10 - 3

(a)

(b)

BF, ↑

BF, ↓

<

FIG. 9: (color online) Universal collapse of the dimensionless

spectral functions of the composite fermion operator: ˆ ̺F,↑=

DF ̺f,↑ and ˆ ̺F,↓= DF ̺F,↓ to two scaling curves: BF,↑ and

BF,↓ for sufficiently small, non-zero values of B as a function

of ω/Th.

Page 12

12

10-6

10-3

100

103

ω / TK

0

0.5

1

Σσ ρf,σ(ω)

5.6 × 10 - 1

1.1 × 10 - 1

5.6 × 10 - 2

1.1 × 10 - 2

5.6 × 10 - 3

1.1 × 10 - 3

1.1 × 10 - 4

0

B / TK

<

FIG. 10: (color online) Sum of the dimensionless spectral

functions: ˆ ̺f,↑ = DF ̺f,↑ and ˆ ̺f,↓ = DF ̺f,↑ for different

values of B as a function of ω/TK.

low-frequencies and the mismatch between the positive

and negative frequency parts of the spectral functions.

The sum of the local fermion’s spectral function over the

two spin components is depicted in Fig. 10 as a function

of ω/TK. Here the splitting of the Kondo resonance in

the energy-dependent scattering cross section appears as

a minimum at ω ∼ Th. Unfortunately, for even smaller

magnetic fields the accuracy of our numerical data is in-

sufficient to tell if the splitting of the Kondo resonance

survives in the limit B → 0, as conjectured in Ref. 24. In

the data with B/TK> 1.1 × 10−4, there seems to be al-

ways a shallow minimum in the spectral function, and we

see no indication for crossing of the curves as the magni-

tude of the field is reduced. If there is indeed no crossing

of the spectral functions and if the deviation from the

?|ω|-behavior indeed starts at ω ≈ Th∼ B2/TK, which

Ansatz results it would immediately follow that there

must always be a splitting of the Kondo resonance, since

?

follow from the pure

?|ω|-dependence of the spectral

do not constitute a real proof.

is the only natural assumption, then, from exact Bethe

σ[̺fσ(ω = 0,B) − ̺fσ(ω = 0,0)] ∼ B ln(TK/B),22

while?

function at B = 0. However, these analytical arguments

σ[̺fσ(ω = Th,B) − ̺fσ(ω = 0,0)] ∼ |B| would

With small modifications, the analysis presented in

this subsection carries over to essentially any fermionic

operator that has quantum numbers c1 = j = 1/2 or

c2 = j = 1/2 and has a finite overlap with the pri-

mary fields φψ1 and φψ2, only the high-frequency be-

havior (ω > TK) and the normalization factors become

different. Typically, a local operator having the same

charge and spin quantum numbers as φψα will have a

finite overlap with them. However, in some cases the in-

ternal Ising quantum number of an operator may prevent

an overlap and, of course, one can also construct opera-

tors by, say, differentiating with respect to the time, that

would correspond to descendant fields.

VI. SPIN SPECTRAL FUNCTIONS AND

SUSCEPTIBILITIES

In this section, we shall discuss the properties of the

spin operator,?S, which is the most obvious example of a

bosonic operator of spin j = 1 and charge quantum num-

bers c1= c2= 0 that overlaps with the scaling operator

φs. There are, however, many operators that have the

same quantum numbers: two examples are the so-called

channel spin operator,

?SC≡ f†

0,1? σf0,1− f†

0,2? σf0,2,(44)

or a composite channel spin operator

?SCC≡ F†

0,1? σf0,1− F†

0,2? σf0,2.(45)

Our discussion can be easily generalized to these opera-

tors with minor modifications.

The analysis of the spin spectral function goes along

the lines of the previous subsection. First we recall that

the field?φsappears in the expansion of the spin operator,

?S = As?φs+ ... ,(46)

with As∼ 1/√TK∼ 1/√D0. Therefore, the appropriate

dimensionless scale invariant Green’s function (usually

referred to as the dynamical spin susceptibility) is defined

as

ˆ gS

?ω

D,TD,κ0,h0,...

?

≡ TKGS(ω,T,κ0,...,D0) .

(47)

We shall not repeat here all the steps of the deriva-

tion, only summarize the main results. In the absence

of a magnetic field, h = 0 the spectral function of the

spin operator is odd. Furthermore, at T = 0 and for no

anisotropy, κ = 0, the spectral function has a jump at

ω = 0,5

ˆ ̺T,h,κ=0

S

(ω) ≈ sgn(ω)

?

rS+ r′

S

?

|ω|

TK

+ ...

?

.(48)

This jump corresponds to a logarithmically divergent

dynamical susceptibility, Re χS(ω) = −Re GS(ω) ∝

ln(TK/ω)/TK.

For ω ≫ TKthe impurity spin becomes asymptotically

free, decoupled from the conduction electrons, therefore

its ω-dependence is set by its scaling dimension at the

free fermion fixed point where xfree

plication that its correlation fuction decays as ω−1cor-

responding to the Curie-Weiss susceptibility with loga-

rithmic corrections present, known from Bethe Ansatz

results and from perturbation theory.

At finite temperatures T ?= 0, but for κ = h = 0, we

obtain the following scaling form for T,ω ≪ TK:

?

S

= 0. It has the im-

ˆ ̺h,κ=0

S

(ω) ≡ ΘS

?ω

T

?

+

T

TK

˜ΘS

?ω

T

?

+ ....(49)

Page 13

13

ˆ ̺S(ω)

∼

cst +

?

?

?

?ω

TK

T > 0,KR= 0,B = 0

∼

TK

ω log2

ω

TK

log

ω

TK

log

T

TK

0

∼ω

T

ˆ ̺S(ω)

∼

cst +

?

?

?

?ω

TK

∼

TK

ω log2

ω

TK

log

ω

TK

0

∼

cst +T∗

ω

∼ω

T∗

T = 0,KR?= 0,B = 0

log

T∗

TK

2

3log

T∗

TK

FIG. 11: (color online) Top: Sketch of the dimensionless spec-

tral function of?S: ˆ ̺S = TK̺S = −TKIm χS(ω)/π for T > 0

and KR = 0,B = 0 as a function of log(ω/TK). Bottom:

Sketch of ˆ ̺S = TK̺S = −TKIm χS(ω)/π for T = 0 and

KR ?= 0,B = 0 as a function of log(ω/TK). Asymptotics in-

dicated for ω < TK were derived through scaling arguments.

The large ω-behavior is a result of perturbation theory.48

The asymptotic properties of the scaling functions ΘS

and˜ΘSare listed in Table V.

In case of finite channel anisotropy but zero tempera-

ture we obtain for ω ≪ TKthe scaling form

ˆ ̺T,h=0

S

(ω) ≈ KS

?ω

T∗

?

+

?

|ω|

TK

˜KS

?ω

T∗

?

+... .(50)

The asymptotic properties of KS,˜KSare only slightly dif-

ferent from those of ΘS,˜ΘS(see Table V): below T∗the

spectral function displays analytic behavior, while the

regime ω > T∗is governed by non-analytical corrections

associated with the 2CK fixed point. In this regime a fea-

ture worth mentioning is the appearance of a correction,

∼ T∗/ω to KS, more precisely, the lack of a

correction. This is due to the fact that the anisotropy

operator is odd, while the spin operator is even with re-

spect to swapping the channel labels. Therefore there is

no first order correction to the spin-spin correlation func-

tion in κ, and the leading corrections are only of second

order, i.e., of the form κ2/ω. From the comparison of

the terms in KS and˜KS it also follows the existence of

?|T∗/ω|

0

−Re ˆ gS(ω)

∼ log

TK

max{T,T∗}

∼

TK

ω

1+cstlog

ω

TK

log

ω

TK

log

max{T,T∗}

TK

∼ log

TK

ω

FIG. 12: (color online) Sketch of the real part of the di-

mensionless Green’s function of?S, Re ˆ gS = TK Re χS(ω) ≡

TK Re GS(ω) for T,T∗> 0 as a function of log(ω/TK).

another cross-over scale,

T∗∗

s

∼

?

T∗2TK

?1/3

, (51)

that separates the regimes governed by the leading rel-

evant and leading irrelevant operators.

the subscript s to indicate that this scale T∗∗

ent from the scale T∗∗

f

introduced in relation to the local

fermion’s spectral function. The asymptotic properties

of ˆ ̺S∝ χS(ω) for T > 0,KR= 0 and T = 0,KR?= 0 are

sketched in the upper and lower parts of Fig. 11, while

the behavior of the real part is presented in Fig. 12.

The expectations above are indeed nicely born out by

the NRG calculations: Fig. 13 shows the impurity spin

spectral functions as a function of ω/TKfor various KR-

s and their asymptotic properties. First, in Fig. 13.(b)

we show a very small logarithmic ω-dependence that we

observed below TK at the 2CK fixed point. The am-

plitude of this log(ω)-dependence was reduced as we in-

creased the number of multiplets. It appears that this

behavior is not derived from the lognormal smoothing

of the NRG data, and it may be due to some approxi-

mations used in the spectral sum-conserving DM-NRG

procedure. In Fig. 13.(c) we show the square root-like

behavior around the 2CK Kondo fixed point which is

attributed to the leading irrelevant operator, while Fig.

13.(d) shows that first order corrections coming from the

scaling of the channel anisotropy are indeed absent just

as we stated above, and only second order terms appear,

resulting in an 1/ω-like behavior.

demonstrates the linear ω-dependence, which is charac-

teristic of most bosonic operators in the proximity of an

FL fixed point. All these findings support very nicely the

analytical properties summarized in Table V.

The spin spectral functions also collapse to a univer-

sal scaling curve describing the cross-over from the two-

channel Kondo to the single channel Kondo fixed points,

when they are plotted against ω/T∗∗. This universal data

collapse is demonstrated in Fig. 14 where the impurity

Here we used

is differ-

s

Finally, Fig. 13.(e)

Page 14

14

10-6

10-3

100

103

ω / TK

ρS(ω)

<

0

0.04

0.08

ρS(ω)

<

0.25

0.15

0.05

0.02

0.0075

0.0025

0.001

0

-16

ln(ω / TK)

-14

0.0885

0.089

0.0895

ρS(ω)

<

750 multiplets

1000 multiplets

KR = 0

0.40.60.8

(ω / TK)1/2

0.06

0.07

KR = 0

00.005

ω / T *

0.01

0

0.005

0.01

0.015

ρS(ω)

<

KR = 0.25

48

T * / ω

0.06

0.07

0.08

ρS(ω)

<

KR = 0.25

(a)

(b)

(c)

(d)

(e)

KR

FIG. 13: (color online) (a) Dimensionless spectral function

of?S: ˆ ̺S = TK̺S = −TKIm χS(ω)/π as a function of ω/TK

for different values of KR. (b) Minute log(ω)-dependence at

the lowest frequencies diminishing as a function of the num-

ber of kept multiplets. (c−e) Numerical confirmations of the

low-frequency asymptotics derived from scaling arguments in

Section VI. Straight dashed lines are to demonstrate devia-

tions from the expected√ω-like (c), ω-like (d), and 1/ω-like

behavior (e). In plots (d − e) T∗/TK = 7 × 10−2.

spin spectral functions are plotted for various KR val-

ues.The data collapse works up to somewhat higher

anisotropy values than for the local fermions’ spectral

functions as it is indicated by the KR-dependence of the

scales T∗∗

s

and T∗∗

The real part of the spin susceptibility was obtained

through numerical Hilbert transformation, and is shown

in Fig. 15 as a function of ω/TK for various values of

KR. These curves meet the expected behavior sketched

in Fig. 12: they display a logarithmic increase at high-

frequencies and saturate at values that correspond to

Re χS∼ ln(TK/T∗) / TK.

Let us finally discuss the case, T = κ = 0 but h ?=

0. Then the components of?S are distinguished by the

magnetic field: The spectral function of Szhas almost

the same features as for finite channel anisotropies. Since

Szis a hermitian operator, its spectral function remains

odd and acquires the following corrections in the different

scaling regimes

f.

ˆ ̺S,z ≡ BS,z

?ω

Th

?

+

?

|ω|

TK

˜BS,z

?ω

Th

?

+ ...(52)

with the scaling functions BS,z,˜BS,z having the asymp-

100

103

106

ω / T *

0

0.04

0.08

ρS(ω)

KR = 0.05

KR = 0.02

KR = 0.0075

KR = 0.0025

KR = 0.001

ΚS

<

FIG. 14: (color online) Universal collapse of the dimensionless

spectral function of?S: ˆ ̺S = TK̺S to the scaling curve, KS

as a function of ω/T∗for sufficiently small, non-zero values of

KR.

totic properties listed in Table V.

Note that in this case the first order correction coming

from the magnetic field does not vanish, and leads to the

appearance of a cross-over scale ∼√ThTK.

The perpendicular components of the impurity spin

have somewhat different properties. First of all, the op-

eratorsS±are not Hermitian, and therefore their spectral

functions are not symmetrical. The spectral functions of

the operators Sxand Syare, however, symmetrical, and

their Green’s functions (and susceptibilities) are related

through

Gx

S= Gy

S=1

4(G+−

S

+ G−+

S

) .(53)

The corresponding dimensionless spectral functions, ˆ ̺z

and ˆ ̺±

sas computed by our DM-NRG calculations are

shown in Fig. 16 as a function of ω/TK, while the uni-

versal scaling with ω/This confirmed for low-frequencies

s

10-6

10-4

10-2

100

ω / TK

0

1

2

-Re gS(ω)

0.25

0.15

0.05

0.02

0.0075

0.0025

0.001

0

KR

<

FIG. 15: (color online) Real part of the dimensionless Green’s

function (susceptibility) of?S, Re ˆ gS = −TK Re χS(ω), as a

function of ω/TK, for different values of KR.

Page 15

15

0

0.1

0.2

ρS,+(ω) / 2

<

1.1 × 10 - 1

5.6 × 10 - 2

1.1 × 10 - 2

5.6 × 10 - 3

1.1 × 10 - 3

5.6 × 10 - 4

1.1 × 10 - 4

0

0

0.1

0.2

ρS,z(ω)

<

10-6

10-3

100

103

ω / TK

0

0.1

0.2

ρS,-(ω) / 2

<

(a)

(b)

(c)

B / TK

FIG. 16: (color online) (a) Dimensionless spectral function

of S+: ˆ ̺S,+ = TK̺S,+, (b) of Sz: ˆ ̺S,z = TK̺S,z and (c) of

S−: ˆ ̺S,− = TK̺S,− for different values of B as a function of

ω/TK.

in Fig. 17. This scaling also turned out to be valid for

values of B higher than the ones for fermions (see Fig.

17). The scaling functions BS,z and BS,± behave very

similarly. This is somewhat surprising, since the naive

expectation would be to have a resonance in BS,+, just

as in the local fermion’s spectral function, that would

correspond to a spin-flip excitation at the renormalized

spin splitting, Th.

excitations”.

However, quite remarkably, a resonance seems to ap-

pear in χ”

S,z(ω)/ω at a frequency ω ∼ Th, while we find

no resonance in χ”

S,±(ω)/ω. This can be seen in Fig. 18,

where T2

K̺(ω)/ω is plotted for the different spin compo-

nents as a function of ω/TK for various magnetic field

values. This seems to indicate that the spin coherently

oscillates between the spin up and spin down compo-

nents, while its x,y components simply relax to their

equilibrium value.

VII.SUPERCONDUCTING CORRELATIONS

In the last section, let us investigate the local super-

conducting correlation functions. These deserve special

attention, since many heavy fermion compounds display

exotic superconducting phases that may possibly be in-

duced by local two-channel Kondo physics.5The most

obvious candidates for the corresponding local operators

have been identified in Section III, and are the local

channel-asymmetric superconducting operator, OSC =

f†

0,2,↑, and the composite fermion su-

0,1,↑f†

0,2,↓− f†

0,1,↓f†

0

0.1

0.2

ρS,+(ω) / 2

<

5.6 × 10 - 2

1.1 × 10 - 2

5.6 × 10 - 3

1.1 × 10 - 3

5.6 × 10 - 4

1.1 × 10 - 4

5.6 × 10 - 5

1.1 × 10 - 5

0

0.1

0.2

ρS,z(ω)

<

10-3

100

103

106

109

1012

ω / Th

0

0.1

0.2

ρS,-(ω) / 2

<

(a)

(b)

(c)

B / TK

BS,+

BS,z

BS,-

FIG. 17: (color online) Universal collapse of ˆ ̺S,+ = TK̺S,+,

ˆ ̺S,z = TK̺S,z and ˆ ̺S,− = TK̺S,− to the three scaling curves:

BS,+,BS,z and BS,− for sufficiently small, non-zero values of

B as a function of ω/Th.

10-3

106

100

103

106

TK ρS,+(ω) / 2ω

1.1 × 10 - 1

1.1 × 10 - 2

1.1 × 10 - 3

1.1 × 10 - 4

0

10-3

106

100

103

TK ρS,z(ω) / ω

10-6

10-3

100

103

ω / TK

10-3

100

103

TK ρS,-(ω) / 2ω

(a)

(b)

(c)

B / TK

<

<

<

FIG. 18: (color online) (a) ̺(ω)/ω of S+: TKˆ ̺S,+/2ω =

T2

TKˆ ̺S,−/2ω = T2

tion of ω/TK.

K̺S,+/2ω, (b) of Sz: TKˆ ̺S,z/ω = T2

K̺S,−/2ω for different values of B as a func-

K̺S,z/ω and (c) of S−:

perconductor field, OSCC= f†

For the composite superconductor we find the expan-

sion,

0,1?S? σ iσyf†

0,2.

OSCC= ASCCφ++

∆+ ...(54)

where the expansion coefficient ASCC can be estimated

from the high-frequency behavior of the correlation func-

Page 16

16

tion up to logarithmic prefactors as ASCC∼√TK/DF.

While for the impurity spin, one can exclude logarith-

mic corrections to the expansion coefficient ASin Eq. 46

based upon the exact Bethe Ansatz results, this is not

possible for the superconducting correlation function. In

fact, we know that in the expansion of the composite

fermion itself the correct prefactor is AF ∼ J/√TK ∼

1/?√TKln(DF/TK)?.46Therefore, similar logarithmic

less, in the following, we shall disregard possible logarith-

mic corrections, and define the normalized dimensionless

and scale-invariant correlation function through the rela-

tion,

factors could appear in the prefactor ASCC. Neverthe-

D2

TK

F

GSCC(ω) = ˆ gSCC(ω) .(55)

Apart from its overall amplitude and its high-frequency

behavior, in the low-frequency scaling regimes the spec-

tral function of the composite superconductor operator

behaves the same way as that of Sz(see Tab. III). There-

fore we merely state its asymptotics without further ex-

planation.

In the absence of anisotropy and magnetic field, κ =

h = 0, for ω ≪ TK the spectral function becomes a uni-

versal function, ˆ ρSCC(ω/T), whose behavior is described

by the scaling form,

ˆ ̺h,κ=0

SCC(ω) ≈ ΘSCC

?ω

T

?

+

?

T

TK

˜ΘSCC

?ω

T

?

+ ... ,

(56)

while in the presence of anisotropy, but at T = 0 tem-

perature and for h = 0, the spectral functions behave

as

ˆ ̺T,h=0

SCC(ω) ≈ KSCC

?ω

T∗

?

+

?

|ω|

TK

˜KSCC

?ω

T∗

?

+... .

(57)

Finally, in a finite magnetic field but for κ = 0

anisotropy and T = 0 temperature the spectral function

assumes the following scaling form,

ˆ ̺κ=T=0

SCC

≡ BSCC

?ω

Th

?

+

?

|ω|

TK

˜BSCC

?ω

Th

?

+ ... .

(58)

The properties of the the various scaling functions de-

fined above are identical to those of the corresponding

spectral functions of the Sz, which were detailed in Ta-

ble V, therefore they have not been included in Table V.

The asymptotic properties are nicely confirmed by our

NRG calculations. The dependence on the anisotropy,

together with the ∼

regimes are plotted in Fig. 19. Here the high-frequency

region, ω > TK, is also displayed, where the spectral

function is roughly linear in the frequency, as dictated

by the free fermion fixed point.

?|ω|, the ∼ 1/ω and the ∼ ω scaling

10-6

10-3

100

103

106

ω / TK

ρSCC(ω)

<

0

3

6

9

ρSCC(ω)

<

0.25

0.15

0.05

0.02

0.0075

0.0025

0.001

0

-16-12

ln(ω / TK)

KR = 0.25

8.28

8.31

ρSCC(ω)

<

KR = 0

11.5

(ω / TK)1/2

KR = 0.25

9

9.6

KR = 0

00.005

ω / T *

0.01

0

0.3

0.6

0.9

ρSCC(ω)

<

2

T * / ω

4

7.2

7.6

ρSCC(ω)

<

(a)

(b)(c)

(d)(e)

KR

FIG. 19: (color online) (a) Dimensionless spectral function,

ˆ ̺SCC = D2

ω/TK for different values of KR. (b). The very weak log(ω)-

dependence at the lowest frequencies. This dependence is sup-

pressed as we increased the number of kept multiplets. (c−e)

Numerical confirmations of the low-frequency asymptotics de-

rived from scaling arguments in Section VII. Dashed straight

lines are to demonstrate deviations from the expected√ω-

like (c), ω-like (d) and 1/ω-like (e) behavior. In plots (d − e)

T∗/TK = 7 × 10−2.

F/TK ̺SCC of the operator OSCC, as a function of

Fig. 20 displays the real part of the dimensionless

Green’s function, that is essentially the real part of the

superconducting susceptibility. This diverges logarithmi-

cally for T∗= 0, but for finite T∗’s it saturates, corre-

sponding to a susceptibility value

χSCC∼TK

D2

F

ln

?TK

T∗

?

.

Notice that there is a small prefactor in front of the log-

arithm that arises from the asymptotically free behavior

at large frequencies.

The universal collapse of the low-frequency part of the

curves in terms of ω/T∗is shown in Fig. 21. The cross-

over curve, KSCC

over function, KS, and displays a plateau at large fre-

quencies from which it deviates as 1/ω, until it finally

reaches the linear frequency regime below T∗.

Application of a magnetic field has effects very similar

to the anisotropy, as shown in the upper part of Fig. 22.

In Fig. 22 the small logarithmic increase at small fre-

quencies is more visible. As mentioned before, this in-

crease is most likely an artifact of the spectral sum con-

serving approximation of Ref. 41 and it is due to the

?ω

T∗

?is very similar to the spin cross-

Page 17

17

10-6

10-3

100

ω / TK

50

100

150

200

250

-Re gSCC(ω)

0.25

0.15

0.05

0.02

0.0075

0.0025

0.001

0

KR

<

FIG. 20: (color online) Real part of the dimensionless Green’s

function of OSCC: Re ˆ gSCC = D2

of ω/TK for different values of KR.

F/TK Re GSCC as a function

way this method redistributes spectral weights. This is

based on the observation that the slope of the logarithm

gets smaller if we increase the number of multiplets kept.

These curves also collapse to a single universal curve as

a function of ω/Th, as shown in the lower part of Fig. 22.

Finally, in Fig. 23, we show the numerically ob-

tained spectral function and the corresponding dimen-

sionless susceptibility of the non-composite superconduc-

tor, OSC= f†

function displays no plateau below TK, but it exhibits a

linear in ω behavior below TK, and correspondingly, the

susceptibility Re χSC remains finite for ω → 0 even in

the absence of anisotropy and an external magnetic field,

i.e., at the 2CK fixed point.

This implies that, although its charge and spin quan-

tum numbers would allow it, the expansion of this op-

erator does not contain the scaling operator φττ′

may be due to the difference in the Ising quantum num-

bers, which we did not identify. Thus the dimension of

the highest-weight scaling operator that appears in the

0,1,↑f†

0,2,↓−f†

0,1,↓f†

0,2,↑. Clearly, this spectral

∆. This

100

103

106

ω / T *

0

2

4

6

8

10

ρSCC(ω)

<

KR = 0.05

KR = 0.02

KR = 0.0075

KR = 0.0025

KR = 0.001

ΚSCC

FIG. 21: (color online) Universal collapse of ˆ ̺SCC to the scal-

ing curve KSCC as a function of ω/T∗for sufficiently small,

non-zero values of KR.

10-6

10-3

100

103

ω / TK

0

3

6

ρSCC(ω)

1.1 × 10 - 1

5.6 × 10 - 2

1.1 × 10 - 2

5.6 × 10 - 3

1.1 × 10 - 3

5.6 × 10 - 4

1.1 × 10 - 4

0

B / TK

<

10-6

10-3

100

103

106

109

ω / Th

0

3

6

ρSCC(ω)

5.6 × 10 - 2

1.1 × 10 - 2

5.6 × 10 - 3

1.1 × 10 - 3

5.6 × 10 - 4

1.1 × 10 - 4

5.6 × 10 - 5

1.1 × 10 - 5

BSCC

B / TK

<

FIG. 22: (color online) Top: Dimensionless spectral func-

tion ˆ ̺SCC = −(D2

of the composite superconductor operator OSCC for different

values of B, as a function of ω/TK. Bottom: Universal col-

lapse of ˆ ̺SCC to the scaling curve BSCC for sufficiently small,

non-zero values of B as a function of ω/Th.

F/πTK) Im χSCC = −(D2

F/πTK) Im GSCC

expansion of OSC is x = 1 and not 1/2, as one would

naively expect based upon a simple comparison of quan-

tum numbers. Turning on a small anisotropy or magnetic

field does not influence substantially the spectral proper-

ties of the corresponding Green’s function, either.

VIII.CONCLUSIONS

In the present paper we gave a detailed discussion of

the spectral properties of the two-channel Kondo model.

We analyzed the properties of the correlation functions

of various local operators in the presence of a channel

anisotropy and an external magnetic field. In particular,

we studied numerically and analytically the correlation

functions of the local fermions, fα,σ≡ f0,α,σ, the compo-

nents of the impurity spin,?S, the local superconductivity

operator, OSC≡ f†

ductor operator, f†

2. The selection of these op-

1iσyf†

1?S? σ iσyf†

2, and the composite supercon-

Page 18

18

0 4080

ω / TK

0

0.1

0.2

0.3

ρSC(ω)

0

0.001

0.0025

0.0075

0.02

0.05

0.15

10-9

10-6

10-3

100

ω / TK

0.8

0.84

-Re gSC(ω)

0

0.001

0.0025

0.0075

0.02

0.05

0.15

<

(a) (b)

KR

KR

<

FIG. 23: (color online) (a) Dimensionless spectral function

of OSC: ˆ ̺SC = DF ̺SC as a function of ω/TK for different

values of KR, and (b) the real part of its dimensionless Green’s

function: Re ˆ gSC = DF Re GSC.

erators was partially motivated by conformal field theory,

which tells us the quantum numbers and scaling dimen-

sions of the various scaling operators at the two-channel

Kondo fixed point.28There are, however, many operators

that have quantum numbers identical with the scaling

fields. Here we picked operators having the right quan-

tum numbers, and at the same time having the largest

possible scaling dimension at the free fermion fixed point,

where J1,J2→ 0. These are the operators, whose spec-

tral functions are expected to have the largest spectral

weight at small temperatures (among those having the

same quantum numbers), and which are therefore the

primary candidates for an order parameter, when a lat-

tice of 2CK impurities is formed, as is the case in some

Uranium and Cerium-based compounds. The operators

above are, of course, also of physical interest on their

own: the spectral function of fα,σ is related to the tun-

neling spectrum into the conduction electron see at the

impurity site, the Green’s function of?S is just the dy-

namical spin susceptibility that can be measured under

inelastic neutron scattering, and finally the local super-

conducting operators are candidates for superconducting

ordering in heavy fermion materials. We remark that, in

the electron-hole symmetrical case, the other components

of the operator multiplet that contains the composite su-

perconducting order parameter OSCC would correspond

to a composite channel-mixing charge density ordering.

Of course, the susceptibilities of this operator has the

same properties as that of χSCC(ω).

In addition to these operators, there are two more

operators of possible interest: the so-called composite

Fermion’s Green’s function is related to the T-matrix,

T(ω) that describes the scattering properties off a two-

channel impurity (or the conductance through it in case

of a quantum dot), and was already studied in detail in

Ref.24. A further candidate is the channel anisotropy op-

erator. This has also a logarithmically divergent suscep-

tibility, and would also be associated with a composite

orbital ordering in case of a two-channel Kondo lattice

system. However, the spectral properties of this latter

operator are so similar to those of the composite super-

conductor that we have decided no to show data about

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T∗∗

s

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f

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T∗∗

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T∗

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KR

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h

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B

FIG. 24: (color online) Top: Sketch of the various 2CK scal-

ing regimes in the presence of channel anisotropy for the lo-

cal fermions bounded by T∗∗

f

bounded by T∗∗

Bottom: Sketch of the 2CK scaling regime for the suscepti-

bilities of the highest-weight fields bounded by T∗∗

crossover scale Th besides.

from below and for the spin

s , the crossover scale T∗∗is also indicated.

h, and the

them.

For the numerical calculations we used a flexible DM-

NRG method, where we exploited the hidden charge

SU(2) symmetries45as well as the invariance under spin

rotations to obtain high precision data.

the scaling operators in this case, we reconstructed the

boundary conformal field theory of Affleck and Ludwig

for this symmetry classification.

the scaling properties of the various dynamical correla-

tion functions and identified the corresponding univer-

sal cross-over functions and their asymptotic properties,

based upon simple but robust scaling arguments. In this

way, universal scaling functions describing the cross-over

from the two-channel Kondo fixed point to the single

channel Kondo fixed point (for J1?= J2) and to the mag-

netically polarized fixed point (for B ?= 0) have been

introduced, which we then determined numerically. We

emphasize again that presently these universal cross-over

functions can only be determined through the applica-

tion of DM-NRG, and in fact, for the scaling curves in

the presence of a magnetic field the application of the

DM-NRG method was absolutely necessary.

Our numerical calculations confirmed all our analyt-

ical expectations, and they confirmed that actually, in

the presence of an applied magnetic field, or channel

anisotropy, the two-channel Kondo scaling regime is

To identify

We then established

Page 19

19

rather restricted, and it may also depend on the physical

quantity considered. In Fig.24 we sketched the regimes

where the pure two-channel Kondo behavior can be ob-

served. Notice that in the presence of anisotropy the

two-channel Kondo scaling regime of the spin suscepti-

bility has a boundary that differs from the boundary of

the two-channel Kondo scaling regime of the T-matrix.

Some of the spectral functions show rather remarkable

features: In a magnetic field, e.g., the spectral function

of the composite fermion, F†

at a frequency ω = Th. This peak corresponds to spin-

flip excitations of the impurity spin at the renormalized

magnetic field. Remarkably, this peak is accompanied

by a dip of the same size at the same frequency for spin

down electrons. This dip is actually very surprising and

is much harder to explain. Similar features appear but

with opposite sign in the local fermions’ spectral func-

tions. Even more surprisingly, this resonant feature is

completely absent in the spectral function of the spin op-

erators, S±.

One of the interesting results of our numerical anal-

ysis was that only the composite superconductor OSC

has a logarithmically divergent susceptibility.

thus the primary candidate for superconducting ordering

for a 2CK lattice system. We remark here that while

for a single impurity the superconducting susceptibil-

ity seems to have a rather small amplitude, Re χSC ∼

TK/D2

carriers is also renormalized, and therefore the band-

width is expected to get renormalized as DF→ TK.5As

a result, the corresponding susceptibility can be rather

large, and drive, in principle, a superconducting instabil-

ity. Interestingly, although the results are still somewhat

controversial,43in the two-channel Kondo lattice these

local superconducting correlations do not seem to induce

a superconducting transition.47This may be, however, an

artifact of the standard two-channel Kondo lattice model,

which does not account properly for the orbital and band

structure of an f-electron material.44We believe, that in

a more realistic lattice of two-channel Kondo impurities a

composite superconducting order develops, similar to the

one suggested in Ref. 44. However, DMFT + DM-NRG

calculations would be needed to confirm this belief.

Acknowledgement: We are especially grateful to L.

Borda for making his code available for the Hilbert trans-

formations and for the lot of valuable discussions. Use-

ful comments on the manuscript from Z. Bajnok and I.

Cseppk¨ ovi are highly appreciated. This research has been

supported by Hungarian grants OTKA Nos. NF061726,

T046267, T046303, D048665, NK63066.

α,↓shows a universal peak

This is

Fln(TK/T), in a lattice model the mass of the

APPENDIX A: SCALING PROPERTIES OF

TWO-POINT FUNCTIONS

In this appendix, we discuss the scaling properties of

various scaling functions. Essentially, we use the gener-

alized Callan-Symanzik equations. For the sake of sim-

plicity, let us first focus on the retarded Green’s function

of the z-component of the operator φs,

G (t,H) ≡ −i? [φz

s(t),φz

s(0)] ?Hθ(t) , (A1)

and its Fourier transform, G(ω,T). Let us investigate the

scaling properties of this function in the absence of mag-

netic field. From the fact that φs is the field conjugate

to the external “magnetic field”, h, and that the parti-

tion function (generating function) must be scale invari-

ant under the renormalization group, we easily get the

following differential equation

D∂G

∂D+

?

µ

βµ

∂G

∂uµuµ+ (2βh− 1)G ≈ 0 ,(A2)

with uµa shorthand notation for the dimensionless cou-

plings, {uµ} = {κ,λ,...} that occur in H, and βµ the

corresponding β-functions,

d ln uµ

dx

= βµ({uν}) ,(A3)

with x = −ln(D) the scaling variable. In the vicinity of

the two-channel Kondo fixed point the β-functions just

assume their fixed point value, which are just the renor-

malization group eigenvalues, yµ= d − xµ, with the di-

mension d = 1, since all operators are local and live in

time only. Since, for φs we have yh= 1/2, in the close

vicinity of the two-channel Kondo fixed point we obtain

dG

dD≈ 0 .(A4)

One can also easily show that

DdG

dD= −ωdG

dω.

(A5)

These relations imply that, G(ω,T,D) is scale invariant,

and is only a function of ω/D and T/D. Clearly, similar

equations hold for the correlation functions of all opera-

tors with dimension 1/2. Furthermore, the above scaling

property can easily be modified for operators having di-

mensions yµ?= 1/2.

1For a recent review see P. A. Lee, From high temper-

ature superconductivity to quantum spin liquid: progress

in strong correlation physics, submitted for Report of

Progress in Physics (2007).

2E. W. Carlson et al., Concepts in High Temperature Super-

conductivity in The Physics of Conventional and Uncon-

ventional Superconductors, Vol. II. ed. K. H. Bennemann

and J. B. Ketterson, Springer-Verlag (2004).

3H. von L¨ ohneysen, A. Rosch, M. Vojta, P. W¨ olfle, Rev.

Mod. Phys. 79, 1015 (2007).

Page 20

20

4P. Coleman, Heavy Fermions: electrons at the edge of mag-

netism, in Handbook of Magnetism and Advanced Mag-

netic Materials, J. Wiley and Sons (2007).

5For a review see D. L. Cox, A. Zawadowski, Adv. in Phys.,

47, 599 (1998).

6Q. Si et al., Nature 413, 804-808 (2001).

7M. Vojta, Phil. Mag. 86, 1807 (2006).

8M. Vojta, Rep. Prog. Phys. 66, 2069 (2003).

9J. M. Luttinger, J. Math. Phys. 4, 1154 (1963).

10M. Bockrath et al., Nature 397, 598 (1999).

11H. Ishii et al., Nature 426, 540 (2003).

12P. M. Singer et al., Phys. Rev. Lett. 95, 236403 (2005).

13B. D´ ora, M. Gul´ acsi, F. Simon and H. Kuzmany, Phys.

Rev. Lett. 99, 166402 (2007).

14M. Milovanovi´ c et al., Phys. Rev. Lett. 63, 82 (1989).

15V. Dobrosavljevi´ c et al., Phys. Rev. Lett. 69, 1113 (1992).

16Ph. Nozi` eres and A. Blandin, J. Phys. Paris, 41, 193

(1980).

17S. Katayama, S. Maekawa and H. Fukuyama, J. Phys. Soc.

Jpn. 50, 694 (1987).

18J. von Delft et al., Ann. Phys. 263, 1 (1998).

19T. Cichorek et al., Phys. Rev. Lett. 94, 236603 (2005).

20Y. Oreg and D. Goldhaber-Gordon, Phys. Rev. Lett. 90,

136602 (2003).

21R. M. Potok et al., Nature 446, 167 (2007).

22M. Pustilnik, Phys. Rev. B, 69, 115316 (2004).

23F. B. Anders, E. Lebanon and A. Schiller, Phys. Rev. B

70, 201306 (2004).

24A. I. T´ oth, L. Borda, J. von Delft and G. Zar´ and,

Phys. Rev. B 76, 155318 (2007).

25Ph. Nozi` eres, J. Low Temp. Phys. 17, 31 (1974).

26N. Andrei and C. Destri, Phys. Rev. Lett. 52, 364 (1984).

27A. M. Tsvelick, P. B. Wiegmann, J. Stat. Phys. 38, 125

(1985).

28I. Affleck, A. W. W. Ludwig, Nucl. Phys. B 352, 849

(1991); ibid 360, 641 (1991), I. Affleck et al., Phys. Rev.

B 45, 7918 (1991).

29K. G. Wilson, Rev. Mod. Phys. 47, 773 (1975).

30H. B. Pang and D. L. Cox, Phys. Rev. B 44, 9454 (1991).

31K. Vlad´ ar, A. Zawadowski, G. T. Zim´ anyi, Phys. Rev. B,

37, 2001 (1988); ibid 37, 2015 (1988)

32V. J. Emery and S. Kivelson, Phys. Rev. B 47, 10812

(1992).

33J. Gan, N. Andrei and P. Coleman, Phys. Rev. Lett. 70,

686 (1993).

34G. Zar´ and and K. Vlad´ ar, Phys. Rev. Lett. 76, 2133

(1996).

35D. L. Cox and A. E. Ruckenstein, Phys. Rev. Lett. 71,

1613 (1993).

36L. Borda et al., Phys. Rev. B 75, 205125 (2005).

37I. Affleck and A. W. W. Ludwig, Phys. Rev. B 48, 7297

(1993).

38J. Kroha, P. W¨ olfle and T. A. Costi, Phys. Rev. Lett. 79,

261 (1997).

39A. M. Sengupta, A. Georges, Phys. Rev. B 49, 10020

(1994).

40S. Suzuki, O. Sakai and Y. Shimizu, Solid State Comm.

104 429 (1997).

41W. Hofstetter, Phys. Rev. Lett. 85, 1508 (2000); R. Pe-

ters, T. Pruschke, F. B. Anders, Phys. Rev. B 74, 245114

(2006); A. Weichselbaum, J. von Delft, Phys. Rev. Lett.

99, 076402 (2007).

42A. I. T´ oth et al., unpublished.

43M. Jarrell et al., Phys. Rev. Lett. 77, 1612 (1996).

44N. Andrei et al., J. Phys. Cond. Mat., 10, L239 (1998); P.

Coleman et al., Phys. Rev. B 60, 3608 (1999).

45B.A.Jones,C.M.Varma

Phys. Rev. Lett. 61, 125 (1987).

46T. Costi, Phys. Rev. Lett. 85, 1504 (2000).

47F. B. Anders, M. Jarrell and D. L. Cox, Phys. Rev. Lett.

78, 2000 (1997).

48M. Garst et al., Phys. Rev. B 72, 205125 (2005).

49For further theoretical studies see Ref.-s 22,23,24.

50Throughout the paper we use units of ? = kB = vF = 1.

51Throughout this paper we discuss only retarded Green’s

functions. The other Green’s functions are related to them

by simple analytic relations in equilibrium.

andJ.W.Wilkins,