Table Of ContentOn two channel flavor anisotropic and one channel compactified
Kondo models
Jinwu Ye
8
9 Physics Laboratories, Harvard University, Cambridge, MA, 02138
9
and Department of Physics and Astronomy, Johns Hopkins University, Baltimore, MD, 21218
1
(February 1, 2008)
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2 Abstract
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We reinvestigate the two channel flavor anisotropic model (2CFAK) and one
-
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t channel compacitified Kondo model (1CCK). For these two models, all the
s
. possible fixed points and their symmetries are identified; the finite size spec-
t
a tra, the electron conductivity and pairing susceptibility are calculated. It is
m
shown that the only non-fermi liquid (NFL) fixed point of the 2CFAK is the
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d NFL of the two channel Kondomodel(2CK) with the symmetryO(3) O(5).
n ×
Any flavor anisotropies between the two channels drive the system to the
o
c fermi-liquid (FL) fixed point with the symmetry O(4) O(4) where one of
[ ×
the two channels suffers the phase shift π/2 and the other remains free. The
5 NFLfixedpointofthe1CCKhasthesymmetryO(3) O(1)andhasthesame
v ×
7 thermodynamics as the NFL fixed point of the 2CK. However, in contrast to
5 the 2CK, its conductivity shows T2 behavior and there is no pairing suscepti-
0
bility divergence. Any anisotropies between the spin and isospin sectors drive
9
0 the system to the FL fixed point with the symmetryO(4) where the electrons
6
suffer the phase shift π/2. The connection and differences between the two
9
/ models are explicitly demonstrated. Therecent conjectures and claims on the
t
a NFL behaviors of the two models are commented.
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I. INTRODUCTION
Extensive attention has been lavished on the overscreened multichannel Kondo model
after the discover of its non-fermi liquid (NFL) behavior by Nozi´eres and Blandin (NB)1.
NB also pointed out that lattice effects in real metals will cause the anisotropy between the
two flavor channels and that in the low temperature limit, the impurity is totally screened
by the strong coupling channel with the weak coupling channel unaffected. Using Numerical
Renormalization Group (NRG), Ref.2 confirmed NB’s conjecture. Using Conformal Field
Theory (CFT), Ref.3 found a relevant dimension 1/2 operator in the flavor sector near the 2
channel Kondo (2CK) fixed point and suggested the system flows to the Fermi-liquid (FL)
fixed point pointed out by NB. Using Yuval-Anderson’s approach, Ref.4 found a solvable
line and calculated the exact crossover free energy function from the 2CK fixed point to the
FL fixed point along this solvable line.
It is known that in the large U limit, the ordinary one channel symmetric Anderson
impurity model(AIM) can be mapped to the one channel Kondo model. However, as shown
by Ref.6,7, if the original O(4) symmetry of the AIM is broken to O(3) O(1), in the strong
×
coupling limit, the AIM is mapped to the one channel compactified Kondo model (1CCK)
where the impurity spin couples to both the spin and the isospin(charge) currents of the one
channel conduction electrons.
Recently, Andrei and Jerez5, using Bethe Ansatz, reinvestigated the 2CFAK and conjec-
tured that the 2CFAK flow to some new NFL fixed points. Coleman and Schofield7, using
strong coupling method, reinvestigated the 1CCK and claimed the system flows to another
kind of non-Fermi liquid fixed point which, similar to 1-dim Luttinger liquid, has the same
thermodynamics as fermi liquid but different excitation spectrum. Moreover, they claimed
that the 1CCK has exactly the same low energy excitations as those of the 2CFAK, therefore
concluded that their results also apply to the 2CFAK.
Sofar,BetheAnsatzcanonlycalculatethermodynamicquantitiesofmultichannel Kondo
models, the correlation functions are needed to resolve if the fixed points are NFL or FL.
It is important to point out that the charge degrees of freedom of the original model being
removed, the 1CCK in Ref.6,7 has completely different transport properties, correlation
functions and excitation spectrums than the original 2CFAK, although it do share the same
thermodynamic properties as the 2CFAK.
As emphasized by AL13, although the boundary interactions only happen in the spin
sector; the spin, flavor and charge degree of freedoms are not totally decoupled, there is
a constraint( or gluing condition) to describe precisely how these degree of freedoms are
combined at different boundary fixed points, the finite size spectrum is determined by this
gluing condition. The boundary operator contents and the scaling dimensions of all the
boundary operators are also given by the gluing condition. However, in order to find the
gluing conditions at the intermediate coupling fixed points, the fusion rules should be iden-
tified which are usually difficult in Non-Abelian bosonization approach. For 4 pieces of bulk
fermions, the non-interacting theory possesses SO(8) symmetry, Maldacena and Ludwig
(MS)9 showed that finding the gluing conditions at the fixed points are exactly equivalent
to finding the boundary conditions of the fermions at the fixed points; the CFT describing
the fixed points are simply free chiral bosons with the boundary conditions. In Ref.17, the
author developed a simple and powerful method to study certain class of quantum impurity
2
models. The method can quickly identify all the possible boundary fixed points and their
maximum symmetry, therefore avoid the difficulty of finding the fusion rules, it can also
demonstrate the physical picture at the boundary explicitly. In this paper, we apply the
method to study the two models. All the possible fixed points and their symmetries are
identified; the finite size spectra, the electron conductivity and pairing susceptibility are cal-
culated. All the leading and subleading irrelevant operators are identified, their corrections
to the correlation functions are evaluated. In section II, Taking all the degrees of freedom
into account, We show that the only NFL fixed point of the 2CFAK is the NFL fixed point
of the 2CK with the symmetry O(3) O(5). Any flavor anisotropies between the two chan-
×
nels drive the system to the fermi-liquid (FL) fixed point with the symmetry O(4) O(4)
×
where one of the two channels suffers the phase shift π/2 and the other remains free. The
conventional wisdom about the 2CFAK is rigorously shown to be correct. In section III, we
repeat the same program to the 1CCK. We find that the NFL fixed point of the 1CCK has
the symmetry O(3) O(1) and has the same thermodynamics as the NFL fixed point of the
×
2CK. The finite size spectrum is listed and compared with that of the 2CK. However, in
contrast to the 2CK, its conductivity shows T2 bahaviour and there is no pairing suscepti-
bility enhancement. Any anisotropies between the spin and isospin sectors drive the system
to the FL fixed point with the symmetry O(4) where the electrons suffer the phase shift
π/2. The finite size spectrum of this FL fixed point is also listed and compared with that
of the 2CFAK. In section IV, we conclude and propose some open questions. Finally, in the
appendix, we study the stable FL fixed point of the 2CFAK using Non-Abelian bosonization
and compare with the Abelian bosonization calculations done in section II.
II. THE TWO CHANNEL FLAVOR ANISOTROPIC KONDO MODEL
The Hamiltonian of the 2CFAK is:
dψ (x)
H = ivF Z ∞ dxψi†α(x) diαx +a=Xx,y,zλa(J1a(0)+J2a(0))Sa +a=Xx,y,zαa(J1a(0)−J2a(0))Sa
−∞
+ h( dxJz(x)+Sz) (1)
Z s
where Jia(x) = 21ψi†α(x)σαaβψiβ(x) are the spin currents of the channel i = 1,2 conduction
electrons respectively. αa = 0, λa correspondtothe2CK andtheonechannel Kondomodel
±
respectively. If λa = λ,αa = α = 0 , the above Hamiltonian breaks SU(2) SU(2) U(1)
s f c
6 × ×
symmetry of the 2CK to SU(2) U(1) U(1) ( or equivalently SU(2) U(1) U(1) ,
s f c s c1 c2
× × × ×
because we have two independent U(1) charge symmetries in the channel 1 and the channel
2 ).
In this section, for simplicity, we take λx = λy = λ,λz = λ;αx = αy = α,αz = α. The
6 6
symmetry in the spin sector is reduced to U(1) Z O(2)10. In the following, we closely
2
× ∼
follow the notations of Emery-Kivelson11. Abelian-bosonizing the four bulk Dirac fermions
separately:
P
ψ (x) = iα e iΦiα(x) (2)
iα −
√2πa
Where Φ (x) are the real chiral bosons satisfying the commutation relations
iα
3
[Φ (x),Φ (y)] = δ δ iπsgn(x y) (3)
iα jβ ij αβ
−
The cocyle factors have been chosen as: P = P = eiπN1↑,P = P = eiπ(N1↑+N1↓+N2↑).
1 1 2 2
↑ ↓ ↑ ↓
It is convenient to introduce the following charge, spin, flavor, spin-flavor bosons:
1
Φ = (Φ +Φ +Φ +Φ )
c 1 1 2 2
2 ↑ ↓ ↑ ↓
1
Φ = (Φ Φ +Φ Φ )
s 1 1 2 2
2 ↑ − ↓ ↑ − ↓
1
Φ = (Φ +Φ Φ Φ )
f 1 1 2 2
2 ↑ ↓ − ↑ − ↓
1
Φ = (Φ Φ Φ +Φ ) (4)
sf 1 1 2 2
2 ↑ − ↓ − ↑ ↓
The spin currents Ja(x) = Ja(x)+ Ja(x) and J˜a(x) = Ja(x) Ja(x) can be expressed
1 2 1 − 2
in terms of the above chiral bosons
1 1 1 ∂Φ
s
J = cosΦ cosΦ , J = sinΦ cosΦ , J =
x s sf y s sf z
πa πa −2π ∂x
1 1 1 ∂Φ
J˜ = sinΦ sinΦ , J˜ = cosΦ sinΦ , J˜ = sf (5)
x s sf y s sf z
−πa πa −2π ∂x
After making the canonical transformation U = exp[iSzΦ (0)] and the following
s
refermionization
a b
Sx = eiπNsf, Sy = eiπNsf, Sz = iab
√2 √b2 −
b
b
1 1 b
ψsf = √2(asf −ibsf) = √2πaeiπNsfe−iΦsf
1 1
ψs,i = (as,i ibs,i) = eiπ(d†d+Nsf)e−iΦs (6)
√2 − √2πa
The transformed Hamiltonian H = UHU 1 = H +H +δH can be written in terms
′ − sf s
of the Majorana fermions14:
iv ∂a (x) ∂b (x) λ α
F sf sf
H = dx(a (x) +b (x) ) i ab (0)+i ba (0)
sf 2 Z sf ∂x sf ∂x − √2πa sf √2πa sf
b
b
iv ∂a (x) ∂b (x)
F s s
H = dx(a (x) +b (x) ) ih dxa (x)b (x)
s 2 Z s ∂x s ∂x − Z s s
δH = λ aba (0)b (0) α aba (0)b (0) (7)
− ′z s s − z sf sf
b b
b b
where λ = λz 2πv .
′z − F
It is instructive to compare the above equation with Eq.3 in Ref.24. They looks very
similar: half of the impurity spin coupled to half of the spin-flavor electrons, another half
of the impurity spin coupled to another half of the spin-flavor electrons. However the two
canonical transformations employed in the two models are different. This fact make the
boundary conditions of this model rather different from that of the two channel spin-flavor
Kondo model (2CSFK) discussed in Ref.24.
4
The above Hamiltonian was first derived by Ref.4 using Anderson-Yuval’s approach.
They found the solvable line λz = 2πv ,αz = 0 and calculated the exact crossover function
F
of free energy along this solvable line. Using EK’s method, We rederived this Hamiltonian16.
The huge advantage of EK’s method over Anderson-Yuval’s approach is that the boundary
conditions at different boundary fixed points can be identified17.
By using the Operator Product Expansion (OPE) of the various operators in Eq.720, we
get the RG flow equations near the weak coupling fixed point λ = 2πv ,λ = α = α = 0
z F z
dλ 1
= λ+αα
z
dl 2
dα 1
= α λα
z
dl 2 −
dα
z
= λα (8)
dl −
The fact that we find two relevant operators in the above equations may indicate there
are two intermediate coupling fixed points. However, in the following, the two intermediate
coupling fixed points are shown to be the same.
The original impurity spin in H are related to those in H by
′
SH = US U 1 = S cosΦ (0) S sinΦ (0)
x x − x s − y s
SH = US U 1 = S sinΦ (0)+S cosΦ (0)
y y − x s y s
SH = US U 1 = S (9)
z z − z
Using the refermionization Eq.6, the original impurity spin in H can be written in terms
of fermions
SH = i(ba +ab )
x s,i s,i
SyH = i(bbbs,i −bbas,i)
SH = biab b (10)
z −
b
b
At λ = 0, the spin boson Φ completely decouples from the impurity in H , there-
′z s ′
fore χ = 0. Because the canonical transformation U is a boundary condition changing
imp
operator12,17, at λ = 0, this leads to
′z
as(0) = as (0), bs(0) = bs (0) (11)
L − R L − R
FollowingRef.17,inordertoidentifythefixedpointsalongthesolvablelineλ = 0,α = 0
′z z
(we also set h = 0), we write H in the action form
sf
γ ∂a(τ) γ ∂b(τ)
1 2
S = S + dτa(τ) + dτb(τ)
0 2 Z ∂τ 2 Z b∂τ
b
b
λ b α
i dτa(τ)b (0,τ)+i dτb(τ)a (0,τ) (12)
− √2πa Z sf √2πa Z sf
b
b
When performing the RG analysis of the action S, we keep21 1: γ = 1,λ fixed, 2:
2
γ = 1,α fixed, 3: λ,α fixed; three fixed points of Eq.7 can be identified
1
5
A. Fixed point 1
This fixed point is located at γ = 0,γ = 1 where b decouples, but a loses its kinetic
1 2
energy and becomes a Grassmann Lagrangian multipliebr. Integrating a out leads to the
b
following boundary conditions18:
b
bsf(0) = bsf(0) (13)
L − R
Eqs.11,13 can be expressed in terms of bosons:
Φ (0) = Φ (0)+π, Φ (0) = Φ (0)+π (14)
s,L s,R sf,L sf,R
−
This is just the non-fermi liquid fixed point of the 2CK. The three Majorana fermions
in the spin sector being twisted, this fixed point possesses the symmetry O(3) O(5). The
×
finite size spectrum of this fixed point was listed in Ref.17.
The local correlation functions at the 2CK fixed point are17:
1 γ2
a(τ)a(0) = , b (τ)b (0) = 1 (15)
h i τ h sf sf i τ3
b b
From the above equation, we can read the scaling dimensions of the various fields [b] =
0,[a] = [as] = [bs] = [asf] = 1/2,[bsf] = 3/2. b
As shown in Ref.17, at the fixed point, the impurity degree of freedoms completely dis-
b
appear: b decouples and a turns into the non-interacting scaling field at the fixed point19
b
b
a b (0,τ) (16)
sf
∼
b
Using Eq.10, the impurity spin turns into
SH(τ) = i(ba (0,τ)+b (0,τ)b (0,τ))
x s,i sf s,i
SH(τ) = i(bbb (0,τ) b (0,τ)a (0,τ))
y s,i − sf s,i
SH(τ) = ibbb (0,τ) (17)
z sf
b
Using the relation
ψH(x) = Uψ (x)U 1 = isgnxψ (x) (18)
s s − s,i
We get22
S (τ) = i( bb (0,τ)+b (0,τ)a (0,τ))
x s sf s
−
S (τ) = i(bab(0,τ)+b (0,τ)b (0,τ))
y s sf s
S (τ) = i(bbb (0,τ)+a (0,τ)b (0,τ)) (19)
z sf s s
b
The impurity spin-spin correlation function Sa(τ)Sa(0) = 1.
h i τ
The above equations23 are consistent with the CFT identifications30
S~ φ~ +J~+ (20)
∼ ···
6
The 2CK fixed point is unstable, because there is a dimension 1/2 relevant operator ba ,
sf
the OPE of asf with itself will generate the dimension 2 energy momentum tensor ofbthis
Majorana fermion T(τ) = 1a (0,τ)∂asf(0,τ), The OPE of the energy momentum tensor with
2 sf ∂τ
the primary field a is
sf
1a (τ ) L a (τ )
T(τ1)asf(τ2) = (τ21sf τ22)2 + −τ11 sfτ22 +L−2asf(τ2)+··· (21)
− −
First order descendant field of this primary field L a (0,τ) = ∂asf(0,τ) with dimension
1 sf ∂τ
−
3/2 is generated. λ term in δH has scaling dimension 3/2, it will generate a dimension 2
′z
operator a (0,τ)∂as(0,τ) +b (0,τ)∂bs(0,τ). γ term has dimension 2 also.
s ∂τ s ∂τ 2
From Eq.15, we can see α term has scaling dimension 5/2, it can be written as
z
∂a(τ) ∂b (0,τ)
sf
: a(τ) : a (0,τ) =: b (0,τ) : a (0,τ) (22)
sf sf sf
∂τ ∂τ
b
b
The bosonized form of this operator is
1
: (cos2Φ (0,τ) (∂Φ (0,τ))2) : sinΦ (0,τ) (23)
sf sf sf
− 2
Using CFT, Ref.3 predicted a dimension 1/2 relevant operator φ3 in the flavor sector.
f
Ref.30 classified all the first order descendants of the primary operator in the spin sector. In
the flavor sector, the same classification apply, J~ φ~ is Charge-Time Reversal (CT) odd,
1 f
therefore is not allowed, but L φ3 is CT even. T−h·e CFT analysis is completely consistent
1 f
−
with the above EK’s solution.
In order to make this fixed point stable, we have to tune α = α = 0, namely the channel
z
anisotropy is strictly prohibited. If α = 0, but α = 0, because α is highly irrelevant, it
z z
6
seems the 2CK fixed point is stable. However, this is not true. From the RG flow Eq.8, it
is easy to see that even initialy α = 0, it will be generated, α is ’dangerously’ irrelevant.
z
B. Fixed point 2
This fixed point is located at γ = 1,γ = 0 where a decouples, but b loses its kinetic
1 2
energy and becomes a Grassmann Lagrangian multiplier. Integrating bbout leads to the
b
following boundary conditions: b
asf(0) = asf(0) (24)
L − R
Eqs.11,24 can be expressed in terms of bosons:
Φ (0) = Φ (0)+π, Φ (0) = Φ (0) (25)
s,L s,R sf,L sf,R
−
This fixed point also possesses the symmetry O(3) O(5). In fixed points 1 and 2, a and
×
b, b and a exchange roles.
sf sf b
b As discussed in the fixed point 1, αz is ’dangerously’ irrelevant. In order to make this
fixed point stable, we have to tune λ = α = 0. This fixed point is actually the same with
z
7
the 2CK fixed point. This can be seen most clearly from the original Eq.1: if λ = α = 0,
z
under the SU(2) transformation on the channel 2 fermions ψ iψ ,ψ iψ , the
2 2 2 2
spin currents of channel 2 transform as Jx Jx,Jy Jy,J↑z → Jz,↑Eq.1↑i→s tr−ansfo↑rmed
2 → − 2 2 → − 2 2 → 2
back to the 2 channel flavor symmetric Kondo model. This can also be seen from Eq.5,
˜ ˜
J ,J ,J also satisfy the SU (2) algebra.
x y z 2
d
C. Fixed point 3
This fixed point is located at γ = γ = 0 where both a and b lose their kinetic energies
1 2
and become two Grassmann Lagrangian multipliers. Integratibng them out leads to the
b
following boundary conditions:
bsf(0) = bsf(0), asf(0) = asf(0) (26)
L − R L − R
Eqs.11, 26 can be expressed in term of bosons:
Φs = Φs +π, Φsf = Φsf +π (27)
L R L R
Substituting the above equation to Eqs. 2 4 and paying attention to the spinor nature
of the representation28, it is easy to see that depending on the sign of α, one of the two
channels suffer π phase shift, the other remains free. The four Majorana fermions being
2
twisted, this fixed point has the symmetry O(4) O(4) with g = 1. The finite size spectrum
×
of this fixed point is listed in Table I, it is the sum of that with phase shift π/2 and that of
free electrons. This scenario is completely consistent with NRG results of Ref.2.
The local correlation functions at the FL fixed point are17:
1 γ2
a(τ)a(0) = , b (τ)b (0) = 1
h i τ h sf sf i τ3
b b 1 γ2
b(τ)b(0) = , a (τ)a (0) = 2 (28)
h i τ h sf sf i τ3
b b
From the above equation, We can read the scaling dimensions of the various fields:
[a] = [b] = [a ] = [b ] = 1/2,[a ] = [b ] = 3/2.
s s sf sf
Atbthe fixed point, the impurity degree of freedoms completely disappear: a,b turn into
b
the non-interacting scaling fields at the fixed point b
b
a b (0,τ), b a (0,τ) (29)
sf sf
∼ ∼
b
b
Using Eqs.10, 18, the impurity spin turns into
S (τ) = i( a (0,τ)b (0,τ)+b (0,τ)a (0,τ))
x sf s sf s
−
S (τ) = i(a (0,τ)a (0,τ)+b (0,τ)b (0,τ))
y sf s sf s
S (τ) = i(a (0,τ)b (0,τ)+a (0,τ)b (0,τ)) (30)
z sf sf s s
The impurity spin-spin correlation function show typical FL behavior
1
Sz(τ)Sz(0) = (31)
h i τ2
8
Using the fermionized form of the Eq.5 and paying attention to the spinor nature of the
representation, it is easy to see the impurity spin renormalizs into either J~ (0,τ) or J~ (0,τ)
1 2
depending on the sign of α. This is consistent with the CFT analysis in the Appendix.
There are 4 leading irrelevant operators with dimension 2 in the action S : γ and γ
1 2
terms, λ term and a (0,τ)∂as(0,τ) +b (0,τ)∂bs(0,τ) which will be generated by the λ term.
′z s ∂τ s ∂τ ′z
The α term has dimension 4, it can be written as : a(τ)∂a(τ) :: b(τ)∂b(τ) :.
z ∂τ ∂τ
The bosonized forms of the 4 leading irrelevant operatorsbare25 b b
b
∂a(τ) 1
a(τ) = cos2Φ (∂Φ (0))2
sf sf
∂τ − 2
b
b
∂b(τ) 1
b(τ) = cos2Φ (∂Φ (0))2
sf sf
b∂τ − − 2
b
aba (0)b (0) = ∂Φ (0,τ)∂Φ (0,τ)
s s sf s
b ∂a (0,τ) ∂b (0,τ)
a (0,bτ) s + b (0,τ) s = (∂Φ (0,τ))2 (32)
s s s
∂τ ∂τ
Following themethod developed inRef.17, their contributions to thesingle particle Green
functions can be calculated. The first order correction to the single particle L-R Green
function ( x > 0,x < 0 ) due to the first operator in the above Eq. is
1 2
h ψ1↑(x1,τ1)ψ1†↑(x2,τ2)i = Z dτhe−2iΦc(x1,τ1)e2iΦc(x2,τ2)i
e−2iΦs(x1,τ1)e2i(Φs(x2,τ2)+π) e−2iΦf(x1,τ1)e2iΦf(x2,τ2)
× h ih i
1
e−2iΦsf(x1,τ1)(: cos2Φsf(0,τ) : : (∂Φsf(0,τ))2 :)e2i(Φsf(x2,τ2)+π)
× h −2 i
(z z¯ ) 2 (33)
1 2 −
∼ −
Where z = τ +ix is in the upper half plane, z¯ = τ +ix is in the lower half plane.
1 1 1 2 2 2
By using the following OPE:
i
: e−2iΦsf(z1) :: e2iΦsf(z2) := (z1 z2)−1/4 (z1 z2)3/4 : ∂Φsf(z2) :
− − 2 −
i 1
(z z )7/4 : ∂2Φ (z ) : (z z )7/4 : (∂Φ (z ))2 : + (34)
1 2 sf 2 1 2 sf 2
−4 − −8 − ···
Itiseasetoseethattheprimaryfield: cos2Φ (0,τ) :makesnocontributionstothethree
sf
point function. It was shown by the detailed calculations in Ref.27 that only the part of the
self-energy which is both imaginary and even function of ω contributes to the conductivity.
Although the energy momentum tensor : (∂Φ (0,τ))2 : do make ω contribution to the
sf
∼
self-energy in the first order26, because it is a odd function, it does not contribute to the
electron conductivity in this order. Same arguments apply to the other operators in Eq.32.
Second order perturbations in these operators lead to the generic T2 fermi liquid bahaviour
of the electron conductivity.
The results of this section were applied to a two level tunneling system with slight
modifications in Ref.28. The universal scaling functions in the presence of external magnetic
field which breaks the channel symmetry were also discussed there.
9
III. COMPACTIFIED ONE CHANNEL KONDO MODEL
Assuming Particle-Holesymmetry, thenon-interactingonechannelKondomodelhastwo
commuting SU(2) symmetry, one is the usual spin symmetry with the generators Ja(a =
x,y,z) another is the isospin symmetry with the generators Ia(a = x,y,z).
1 1 1
Jx = (ψ†ψ +ψ†ψ ), Jy = (ψ†ψ ψ†ψ ), Jz = (ψ†ψ ψ†ψ )
2 ↑ ↓ ↓ ↑ 2i ↑ ↓ − ↓ ↑ 2 ↑ ↑ − ↓ ↓
1 1 1
Ix = (ψ†ψ† +ψ ψ ), Iy = (ψ†ψ† ψ ψ ), Iz = (ψ†ψ +ψ†ψ ) (35)
2 ↑ ↓ ↓ ↑ 2i ↑ ↓ − ↓ ↑ 2 ↑ ↑ ↓ ↓
The diagonal and off-diagonal components of the isospin currents represent respectively
the charge and pairing density at the site x.
The one channel compactified model proposed by Ref.7 is a model where the impurity
spincouplestoboththespinandtheisospincurrentsoftheonechannelconductionelectrons
dψ (x)
H = iv ∞ dxψ (x) α + λa(Ia(0)+Ja(0))Sa + αa(Ia(0) Ja(0))Sa
c F Z α† dx a=Xx,y,z a=Xx,y,z −
−∞
+ h( dx(Iz(x)+Jz(x))+Sz) (36)
Z
The ordinary symmetric Anderson impurity model in a one dimensional lattice is
H = it (ψ (n+1)ψ (n) h.c.)
Xn,α α† α −
1 1
+ iV (ψ (0)d h.c.)+U(n )(n ) (37)
Xα α† α − d↑ − 2 d↓ − 2
The O(4) symmetry of the AIM can be clearly displayed in terms of the Majorana
fermions
1 1
ψ (n) = (χ (n) iχ (n)), d = (d id )
1 2 1 2
↑ √2 − ↑ √2 −
1 1
ψ (n) = ( χ (n) iχ (n)), d = ( d id ) (38)
3 0 3 0
↓ √2 − − ↓ √2 − −
Breaking thesymmetry fromO(4)toO(3) O(1)inthehybridization15, theHamiltonian
×
37 becomes:
3
H = it χ (n+1)χ (n)+iV χ (0)d
α α 0 0 0
Xn αX=0
3
+ iV χ (0)d +Ud d d d (39)
α α 1 2 3 0
X
α=1
In the large U limit, projecting out the excited impurity states, we can map the Hamil-
tonian 39 to the 1CCK Hamiltonian 36 with
2V2 2V V
0
λ = , α = (40)
U − U
10
