Entanglement of an impurity and conduction spins in the Kondo model

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by  Sangchul Oh

Entanglement of an impurity and conduction spins in the Kondo model
Sangchul Oh and Jaewan Kim
∗ †
School of Computational Sciences, Korea Institute for Advanced Study, Seoul 130-722, Korea
(Dated: February 9, 2008)
Based on Yosida’s ground state of the single-impurity Kondo Hamiltonian, we study three kinds
of entanglement between an impurity and conduction electron spins. First, it is shown that the
impurity spin ismaximally entangled with all theconduction electrons. Second,a two-spin density
matrix of the impurity spin and one conduction electron spin is given by a Werner state. We find
that the impurity spin is not entangled with one conduction electron spin even within the Kondo
6
0 screening length ξK, although there is thespin-spin correlation between them. Third, we show the
density matrix of two conduction electron spins is nearly same to that of a free electron gas. The
0
single impurity does not change theentanglement structure of theconduction electrons in contrast
2
to thedramatic change in electrical resistance.
n
a PACSnumbers: 03.67.Mn,03.65.Ud,75.20.Hr,72.15.Qm
J

5
2 Entanglement, the quantum correlation between sub- (a)

systems, is considered to be one of the key concepts
1
in quantum mechanics and quantum information sci-
v
6 ence [1]. A study of entangled structures of quantum
6 many-body systems is of great importance for provid-
1 ing us not only a new view on their physical properties (b) (c)
1 r
butalsothe basicknowledgeforfabricatingquantumin- 1
0 r
formation processors. For example, entanglement of the
6
ground states of one-dimensional quantum spin lattice r
0 2
/ models has been intensively investigated in context of
h
quantum phase transition [2, 3]. In a non-interacting
p
- electron gas, entanglement of two electron spins due to FIG. 1: (color online). Schematic of three kinds of entan-
t the Pauliexclusionprinciple hasbeen studied inconnec- glement: entanglement(a)betweentheimpurityspinandall
n
a tion with its characteristic length scale, the Fermi wave- the conduction electrons, (b) between the impurity spin and
u lengthλ [4,5,6]. Entanglementoftwoelectronspinsof one conduction electron spin at r, and (c) between two con-
q F duction electrons at r1 and r2. The rectangular box is the
a BCS superconductor has also examined with relation
: conduction electrons, thearrow with afilled circle represents
v to the coherence length ξ [7].
0 the impurity spin, and the red and blue arrows denote the
i
X conduction electron spins.
r
a
The Kondo model describing the exchange interac-
Let us consider the spin-1 Kondo Hamiltonian
tion between the impurity spin and the conduction elec- 2
trons has been one of challenging quantum many-body
problems in condensed matter physics [8, 9, 13]. Vari- H = ǫkc†kσckσ−2JS·s(0), (1)
ous theoretical tools such as Anderson’s scaling theory, Xkσ
Wilson’snumericalrenormalizationgroupapproach,and
whereSisthespinofthemagneticimpurity,s(0)thespin
the Bethe ansatz, etc. have been applied to the Kondo
model [8]. Recently, Kondo effects in nanodevices have of the conduction electrons at r = 0, and c†kσ creates a
been revisited with potential applications to spin de- conduction electron with momentum k and spin σ. The
vices or spin qubits [9, 10, 11]. Although the Kondo exchange interaction J is assumedto be negative, so the
model have been very intensively studied, its entangle- impurity spin is anti-parallel to the conduction spins.
ment structure remains to be explored. In this paper YosidapresentedthevariationalgroundstateofEq.(1)
we investigate three kinds of entanglement between the given by a singlet state [8, 13, 14, 15]
impurity spin and conduction electron spins as shown
in Fig. 1. From this study on the entanglement struc- 1
Ψ = ( φ χ φ χ ) , (2)
s
ture of the Kondo model we provide a clear view on | i √2 | ↓i| ↑i−| ↑i| ↓i
the Kondo screening cloud with the size of the Kondo
screeninglengthξ ,whichistheholygrailintheKondo where χ is the spin-up state of the impurity spin and
K
| ↑i
physics [9, 12]. Also we discuss a possibility for the use φ denotes the state of conduction electrons with one
| ↓i
of an electron gas in coupling spin qubits [10, 11]. excess down spin. The simplest form of φ is the state
| ↓i
2
withonedownspinaddedoutsidethefilledFermisphere conduction spin at r. In the second quantization, it is
given as follows
1
|φ↓i= √N kX>kF Γkc†k↓|Fi, (3) ρ(α2β);α′β′(r)= 21hΨs|ψˆβ†′(r)φˆ†α′(0)φˆα(0)ψˆβ(r)|Ψsi, (7)
where Γk are variational parameters, N the normaliza- where an operator ψˆβ†(r) creates a conduction electron
stipohnerfea.ctTohre, vaanrdiat|Fioina=l paQrakm≤keFtecr†ksσΓ|0kiatrheegifivlelendbyFermi with spin β at r and φˆα is a creation operator of the
impurity spin α at origin. Here α,α,β,β refer to the
′ ′
1 spin indices, i.e., or .
Γk = , (4) From Eqs. (2) a↑nd (↓7), we obtain
ǫk+EB
ρ 0 0 0
where E = k T = D exp[2/3JN(0)]. Here T is ;
B B K K ↑↑↑↑
 
called the Kondo temperature, 2D the bandwidth of the 1 0 ρ ; ρ ; 0
conduction electrons, and N(0) the density of states of ρ(2)(r)=  ↑↓↑↓ ↑↓↓↑  , (8)
2 0 ρ ρ 0 
the conductionelectrons. The characteristiclength scale  ↓↑;↑↓ ↓↑;↓↑ 
 
of the Kondo model is the Kondo screening length ξK,  0 0 0 ρ ; 
the size of the Kondo screening cloud  ↓↓↓↓
where the diagonal elements are given by
~v E 2
ξK = kBTFK = kBTFK kF , (5) ρ↑↑;↑↑ =hφ↓|ψˆ↑†(r)ψˆ↑(r)|φ↓i, (9a)
ρ ; = φ ψˆ†(r)ψˆ (r)φ , (9b)
where vF is the Fermi velocity, EF the Fermi energy, ↑↓↑↓ h ↓| ↓ ↓ | ↓i
amnudchkFlatrhgeerFtehramnitmheomFeernmtuimw.avSeinlecnegEthFλ≫F =kB2TπK/k,FξK. is ρρ↓↑;;↓↑ ==hφφ↑|ψψˆˆ↑††((rr))ψψˆˆ↑((rr))|φφ↑i,, ((99dc))
(i) Entanglement between the impurity spin and the ↓↓↓↓ h ↑| ↓ ↓ | ↑i
conduction electrons. As depicted in Fig. 1-(a), let us and the off-diagonal elements are given by
investigate entanglement between the impurity spin and
alltheconductionelectrons. Sincethetotalstate,Eq.(2) ρ↑↓;↓↑ =−hφ↓|ψˆ↓†(r)ψˆ↑(r)|φ↑i=ρ∗↓↑;↑↓. (9e)
is a pure state, entanglementbetween them is quantified Sincethestate φ hasN/2upandN/2+1downspins
bythe vonNeumann entropyof the reduceddensity ma- of the conduction|e↓leictrons, the matrix element ρ is
;
trix ρim of the impurity spin, S(ρim)= Tr[ρimlogρim]. given by the electron density with up spin ↑↑↑↑
−
Herethe subscript‘im’ representsthe impurity. Bytrac-
N n
ingoutthedegreesoffreedomoftheconductionelectrons ρ = = , (10)
;
in Eq. (2), one easily obtains ρ = Tr Ψ Ψ . It is ↑↑↑↑ 2V 2
im con s s
| ih |
given by the fully mixed state where V is the volumeofthe conductionelectronsandn
is the electron density. Similarly we have ρ = n/2.
;
1 1 0 ↓↓↓↓
ρim = 2(cid:20)0 1(cid:21) , (6) The matrix element ρ↑↓;↑↓ is given by
n
ρ = +f(r), (11)
;
which gives the maximum entropy S(ρ )=1. Thus we ↑↓↑↓ 2
im
find that the impurity spin is maximally entangled with where f(r) is defined by
the conduction electrons.
(ii) Entanglement of the impurity spin and one con- f(r) 1 ΓkΓk′e−i(k′−k)·r. (12)
≡ V
duction electron. Ifa magneticimpurity is immersedin N kkX′>kF
a Fermi sea, conduction electrons will align in order to
The systemis isotropic,so f(r) depends only onthe dis-
screenthe magnetic field by the impurity. To put itsim-
tancer. Wegettheoff-diagonalelementρ = f(r).
ply,thismagneticdisturbanceiscalledtheKondoscreen- Thus ρ(2) is given by the sum of the full↑y↓;↓m↑ixed−state
ing cloud of electrons (or an electron) with the size of
and a spin-singlet state
orderξ ,formingasingletwithanimpurity[12]. Soone
K
may expect some correlation or entanglement between n 0 0 0
2
theimpurityspinandaconductionelectronintheKondo  
screening cloud. ρ(2) = 1 0 n2 +f(r) −f(r) 0 , (13a)
 
Let us consider the impurity spin and one conduction 20 f(r) n +f(r) 0
spin at r from the impurity as shown in Fig. 1-(b). En-  − 2 
0 0 0 n
tanglementbetweenthemcanbemeasuredifoneobtains  2
I
thetwo-spindensitymatrixbytracingoutallthedegrees
=n +f(r)Ψ(−) Ψ(−) , (13b)
of freedom of Eq. (2) except the impurity spin and one 4 | ih |
3
where Ψ( ) = 1 ( ) and I is the 4 4 unit
| − i √2 |↑↓i−|↓↑i × 1
matrix. The meaning of Eq. (13) is that the first term
0.8
is the uniform back ground and the second is the singlet 0)
(N 0.6
staStiencfoerTmriρn(g2)a=Kno+ndfo(rs)c,rleeetnuinsginctlroouddu.cethenormalized (r)/fN 0.4
two-spin density matrix ρ defined by ρ(2) = (n+f)ρ. f 0.2
Then we find that ρ is given by the Werner state [4, 5] 0
I -0.2
ρ=(1 p) +pΨ( ) Ψ( ) , (14)
− 4 | − ih − | -0.4
0 5 10 15 20 25 30
where p = f/(n+f). The impurity spin is entangled k r
F
with one conduction electron spin at r if p > 1/3, that
is, f >n/2. FIG. 2: fN(r)/fN(0) as a function of kFr. The numerical
Letusexaminewhetherρisentangledornotbycalcu- values, EF,EB and D are taken in thetext.
latingf(r). SinceΓkisreal,werewritef(r)=Vf˜(r)2/
where the function f˜(r) is defined by N
Fig. (2). Then f/n could not be greater than the half,
1
f˜(r) Γkeik·r. (15) that is, Eq. (19) is not satisfied. Therefore the impurity
≡ V kX>kF spin is not entangled with one conduction electron spin
even within the Kondo screening length, i.e., r <ξ .
K
After some calculations, f˜(r) becomes the integral form Fig. (1) shows the numerical calculation of
f (r)/f (0) as a function of k r. f (r) extends
N N F N
f˜(r)= N(0) D/EF sinhkFrp1+tEB/EFidt, (16) oisvenrottihnegKbountdtohescsrpeienn-isnpginlecnogrtrhelaξtKio.nWfuenfictnidonthbaettwfe(ern)
k r Z t+1
F 0 the impurity spin and the conduction spin at r,
where N(0) is the density of state atFermi leveland be-
gcoams.eTshNe(n0o)r=mamlikzFat/i2oπn2f~a2ct=or3n/4isEgFivfeonrbayfree electron hσizmσz(r)i=ρ↑↑;↑↑+ρ↓↓;↓↓−ρ↑↓;↑↓−ρ↓↑;↓↑
N
= 2f(r). (20)
VN(0) −
= Γ2k = y(D,EB,EF), (17a)
N E
kX>kF B Notice that f(r) is qualitatively similar to the spatial
spin-spin correlation function S s(r) = 3 σz σz(r) ,
where the definite integral y(D,EB,EF) is defined by which was calculated by Chenhet·al [16i] and4hbyimGuberi-
natis et al for the Anderson model [17]. It is interesting
D/EF 1+tE /E
y(D,E ,E ) B F dt. (17b) that ρ(2)is given by a pseudo entangled state in liquid-
B F ≡Z0 p (1+t)2 state NMR [1, 18]. The Kondo screening cloud [9, 12]
could be detected throughthe extra Knight shift experi-
Thus we obtain
ment which measures f(r).
3nEB fN(r)2 Our result shows that even if there is the spin-spin
f(r)= , (18)
4 EF y(D,EB,EF) correlation between the impurity and a conduction elec-
tronintheKondoscreeningcloud,entanglementbetween
where fN(r) f˜(r)/N(0). The condition of entangle- themvanishes. Asimpleexplanationforthisisasfollows.
≡
ment of ρ, f >n/2, becomes Since many conduction electrons are coupled to a single
impurity, the quantum correlationbetween the impurity
3 E f (r)2
B N >1. (19) and each conduction electron is very tiny so there is no
2(cid:18)E (cid:19)y(D,E ,E )
F B F entanglement between them.
For usual Kondo systems, the Kondo temperature T (iii) Entanglement between two conduction electron
K
ranges from few to hundreds Kelvin, T 1 300K. spins. It has been believed that conduction electrons
K
The Fermi temperature T = E /k is≃abou∼t T within the Kondo screening cloud are mutually corre-
F F B F
104K. This implies E /E 10 2 10 4. How≃- lated because they have information on the same impu-
B F − −
≃ ∼
ever, according to Eq. (5), the small ratio of E /E = rity [9]. We examine whether the impurity spin induces
B F
k T /E indicates the Kondo screening length ξ is the non-classical correlation, i.e., entanglement between
B K F K
muchlargerthan the Fermi wavelengthλ . If the band- conduction electrons.
F
width D is assumed to be E D E , we have Considertwoconductionspinsatr andr asshownin
B F 1 2
≪ ≪
y(D,E ,E ) 1. Also f (r) (1) as shown in Fig 1-(c). The density matrix ρ of all the conduction
B F N con
≃ ≃ O
4
electrons is given by the mixture of φ and φ Our analysis is based on Yosida’s ground state which
| ↑i | ↓i
describes the Kondo physics in an simple but effective
1
ρ =Tr Ψ Ψ = ( φ φ + φ φ ) . (21) way. There are another ways to solve the problem:
con im s s
| ih | 2 | ↑ih ↑| | ↓ih ↓| the variational ground state of the Anderson Hamilto-
FromEq.(21),thedensitymatrixρ˜(2) oftwoconduction nian[20,21],thequantumMonteCarlomethod[17],the
electron spins is written by perturbativerenormalization[16],andthedensitymatrix
renormalization group for a tight-binding model [12].
ρ˜(σ21′),σ2′;σ1,σ2(r1,r2) In conclusion, we have studied entanglement of the
impurity spin and the conduction electron spins in the
1
= 2Trhψˆσ†2′(r2)ψˆσ†1′(r1)ψˆσ1(r1)ψˆσ2(r2)ρconi . (22) Kondo model based on Yosida’s ground state. First, it
has been shown that the impurity spin is maximally en-
After a lengthy calculation, we obtain tangled with all the conduction electrons. Second, the
two-spindensitymatrixoftheimpurityspinandonecon-
ρ˜(2)(r1,r2)=ρ(fr2e)e(r′)+∆ρ(r1,r2), (23) duction electron spin at r is given by a Werner state. It
has been found that the impurity spin is not entangled
where r r r and ρ(2) (r ) is the density matrix of
′ ≡ 1− 2 free ′ withaconductionelectronspinwithintheKondoscreen-
two electron spins of a free electron gas [4, 5]
ing cloud even though there exists the spin-spin correla-
1 g2 0 0 0 tionbetweenthem. Third,theentanglementstructureof
ρ(fr2e)e(r′)= n82  −00 1g2 −1g2 00  (24) tinhecoconntrdausctttioontheleecsttrroonnsgiesffleitcttleoanffeelcetcetdricbayltrheesisimtapnucrei.ty
 − 
 0 0 0 1 g2 We thank Prof. K. Yamada of University of Tokyo
 − 
for helpful comments. This work was supported by
with g(r ) 2 eikr′. The small change is Korean Research Foundation Grant (KRF-2004-041-
′ ≡ N k kF ·
P ≤ C00089). J.K.wassupportedbyaGrant(TRQCQ)from
a+b 0 0 0 the Ministry of Science and Technology of Korea.
∆ρ(r ,r )= n2 3EB  0 a b 0  , (25)
1 2 8 2E 0 b a 0
F  
 0 0 0 a+b
 
where a g(0) fN(r1)2+fN(r2)2 /2 and b ∗ Electronic address: [email protected]
g(r′)fN(r1)≡fN(r2). (cid:2)Since EB/EF is very(cid:3) small,we hav≡e † Electronic address: [email protected]
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