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

Page 4

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)

Page 5

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