Table Of ContentScaling and Universality in the Anisotropic Kondo Model and the Dissipative
Two–State System
T. A. Costi
Universit¨at Karlsruhe, Institut fu¨r Theorie der Kondensierten Materie, 76128 Karlsruhe, Germany
Scaling and universality in the Ohmic two–state system is investigated by exploiting the equiv-
alence of this model to the anisotropic Kondo model. For the Ohmic two–state system, we find
8
universal scaling functions for thespecific heat, C (T),static susceptibility, χ (T),and spin relax-
9 α α
ation function S (ω) dependingon the reduced temperature T/∆ (frequency ω/∆ ), with ∆ the
9 α r r r
1 renormalized tunnelingfrequency,anduniquelyspecifiedbythedissipation strengthα(0<α<1).
The scaling functions can be used to extract αand ∆r in experimentalrealizations.
n
a
J
level model [8–10]. The qualitative picture that has
1
2 PACS numbers: 71.27.+a,75.20.Hr,66.35.+a,72.15.Qm emergedcanbesummarizedasfollows. Thereisarenor-
malizedtunnelingfrequency∆ ,whichdependsonαand
r
] ∆, and which decreases monotonically with increasing α
l The low energy effective model for a large number of
e for fixed ∆. For α = 0, ∆ = ∆ (decoupled system plus
- physicalsystems correspondsto a two–state systemcou- r
r bath) and for α > 0 the renormalization of this scale
t pledtoanenvironment[1,2]. Examplesincludetwo–level
s increases dramatically as the dissipation strength is in-
atoms coupled to the electromagnetic field in quantum
at. optics, electron–transfer reactions in biological systems, creased to 1: ∆r/ωc ∼ (∆/ωc)1/1−α. Between α = 0
and α = 1, there is a range of different behaviour from
m and the tunneling of defects in metallic glasses [3]. The
coherentoscillationsat zerodissipationto damped oscil-
- simplest model for describing such systems is the spin– lations atintermediate dissipationstrengths andeventu-
d boson Hamiltonian [1],
n ally to incoherent relaxation at strong dissipation [1,2].
co HSB =−12∆σˆx+ 21ǫσˆz+Xωi(a†iai+ 12) TAt=α0>coαrcr(e∆sp)o≈nd1intghteoreaivsaani“slhoicnaglirzeantioornm”atlirzaendsittuionnnealt-
[
1 C i ing frequency ∆r = 0 [11]. Here we address one aspect
v2 + 2q0σˆzX√2miiωi(ai+a†i). (1) wlitheircahtuhraes, nnaomt ebleyenthedemaletanwiintghoinf uaniuvneirfisaedlitywaayndinsctahle-
0 i
ing inthese models,in particularfor physicalproperties,
4
Here, the Pauli matrices σˆ ,i= x,y,z describe the two–
i such as thermodynamic and dynamical quantities. We
0
level system, ∆ is the bare tunneling matrix element be-
2 apply Wilson’s numerical renormalization group (NRG)
1 tween the states σz = and σz = and ǫ is a bias. The method to the AKM to calculate the specific heat,static
↑ ↓
7 environmentisrepresentedbyaninfinite setofharmonic susceptibility and dynamical susceptibility. The equiva-
9 oscillators (labeled by the index i) with masses m and
i lence of the two models then allows us to discuss scaling
t/ frequencyspectrumωi linearlycoupledtothecoordinate and universality in the dissipative two–state system.
a Q= 1q σ with coupling constants C . In this paper we
m 2 0 z i The AKM [12] is given by
restrictourselvestoOhmic dissipation,forwhichthe en-
nd- vtairkoensmthenetfsoprmectJra(lωf)un=ct2ioπnαJω(,ωf)or=ωπ2 Piω(m,Ciiω2wih)eδr(eω−ωωii)s H =Xǫkc†kσckσ + J2⊥ X(c†k↑ck′↓S−+c†k↓ck′↑S+)
o a high energy cut–off and α is a dime≪nsiocnless couplcing k,σ kk′
v:c cTohnesttawnotdcihmareancstioernilzeisnsgctohueplsintrgesnogfththoefmthoededlisasriepa∆ti/oωnc. + J2k Xkk′(c†k↑ck′↑−c†k↓ck′↓)Sz +gµBhSz, (2)
i
X and α: in this paper we consider the region of parame-
where the first term represents non–interacting conduc-
ter space, ∆/ω α<1, which includes the interesting
r c
a case of a non–ad≪iabatic bath. tion electrons and the second and third terms represent
anexchangeinteractionbetweenalocalizedspin1/2and
The Ohmic spin–boson model has been intensively in-
the conduction electrons with strength J ,J . A local
vestigated over the last 10 years [1,2]. The main in- ⊥ k
magnetic field, h, coupling only to the impurity spin in
terest has been in understanding how the environment
the Kondomodel (the lasttermin Eq.2)correspondsto
influences the dynamics of the two–level system and in
a finite bias, ǫ, in the spin–boson model. The correspon-
particular how dissipation destroys quantum coherence
dence between H and H , established via bosonization
[1,2,4–7]. Insighthasbeengainedbyexploitingtheequiv- SB
[9], implies ǫ = gµ h, ∆ = ρ J and α = (1+ 2δ)2,
alence of the Ohmic two–state system to several other B ωc 0 ⊥ π
models, including the inverse square Ising model [4], the where tanδ = πρ0Jk. δ is the phase shift for scatter-
− 4
anisotropic Kondo model (AKM) [5], and the resonant ing of electrons froma potential J /4 and ρ =1/2D is
k 0 0
1
the conduction electron density of states per spin at the
Fermilevelforaflatbandofwidth2D [1,5,9]. Wechoose
0
ω =2D so that ∆=J and measure all energies rela- 2.0
c 0 ⊥
tive to D =1. Since we are interested in describing the
0
Ohmictwo–statesystemfor∆/ω α<1,thisrequires
c
intheAKMthatρ J 1 ρ J ≪forρ J 1(α 1− 1.5
0 ⊥ 0 k 0 k
≪ − ≪ →
case) and ρ J 1/(ρ J )2 for ρ J 1 (α 0 case).
0 ⊥ 0 k 0 k
≪ ≫ →
The AKM for ρ0J⊥ > ρ0Jk will be dealt with elsewhere NGS1.0
so in effect we consider only ρ J min(ρ J ,α). E
0 ⊥ ≪ 0 k N-
We solve the AKM using Wilson’s NRG method [13]. Ep
In this procedure, a separation of energy scales is made
0.5 j=1/2, S=1/2 (S=1/2)
by introducing a logarithmic mesh of k points k = z
n j=0, S=0,1,-1 (S=1)
Λ−n, Λ>1,andtransformingthec toabasisofWan- z
kσ
nier states fnσ [13] at the impurity, with f0σ = Pkckσ, 0.0
such that H = ǫ c† c is tridiagonal in k–space, 0 50 100 150
c Pkµ kµ kµ kµ N
i.e. H ∞ Λ−n/2(f† f + h.c.). The
c → PµPn=0 n+1µ nµ
Hamiltonian (2) in the new basis is now diagonalized FIG. 1. The lowest rescaled energy levels for even N it-
by defining a sequence of finite size Hamiltonians HN erations for Jk = 0.443 and J⊥ = 0.010 [5] corresponding to
containing the first N Wannier states together with the ∆ =0.01 and α= 0.8 in the spin–boson model. The energy
levels are labeled by the conserved quantum numbers, total
impurity. One diagonalizes the rescaled Hamiltonians
H¯N = ΛN2−1HN which satisfy the recursion relation pisseaucdroo–ssspoivnerjt[o18t]haensdtrtoontgalczo–ucpolminpgofinxeendt opfoisnptinatSzit.erTahtieorne
H¯ = Λ1/2H¯ + (f† f +h.c.). This gives
N+1 N Pµ N+1µ Nµ Nc corresponding toΛ−(Nc−1)/2 ≈∆r =TK. Spinrotational
the excitations and eigenstates at a corresponding set
invariance is restored at low energies (e.g., the j = 0 states
of energy scales ωN defined by ωN = Λ−N2−1 and al- with Sz = 0 and Sz = ±1 become degenerate), so the states
lows the calculationof dynamic quantities atfrequencies atthestrongcouplingfixedpointcanbelabeledbytotalspin
ω ω and thermodynamic quantities at temperatures S as indicated.
N
∼
k T ω . For example, the Fourier transform of
B N N
∼
χ(t,T) = iθ(t) [S (t),S (0)] , is given by χ(ω,T) =
− h z z i point at low energies (e.g. the lowest single particle
χ′(ω,T)+ iχ′′(ω,T) = 1 MN 2 e−βǫm−e−βǫn ,
ZN Pm,n| m,n| ω+i0−(ǫm−ǫn) states in Fig. (1), η1 = 0.6555,η2 = 1.976 agree with
where ǫm,ǫn are excitations of HN, ZN(T) the partition the Λ = 2 results of [13]). A detailed analysis [15] gives
function of HN, and MmN,n =hm|Sz|niN. TK ∼(∆/ωc)1/(1−α) with∆,αrelatedtoJ⊥,Jk asabove
Specifically, for the AKM we calculate: (a) the T = 0 and J J and a prefactor also depending on α, i.e.,
⊥ k
relaxation function S(ω) = −π1χ′′(ωω+iδ); (b) the impu- TK has≪the same dependence on α as the low energy
rity specific heat C(T) = ∂2Fimp/∂ T2, where the im- scale ∆r. The flow to the isotropic strong coupling fixed
−
purity free energy is given by F (T) = k T lnZ/Z point holds for any initial anisotropy, corresponding to
imp B 0
−
and Z is the conduction electron partition function; 0<α<1, with the flow being universal for each α [15]:
0
(c) the local static susceptibility χ′(ω = 0,T) corre- the energy levels for fixed α and different ∆/ωc α
≪
sponding to setting the g factor of the conduction elec- may be shifted onto each other by a translation in N
trons to zero. Under the equivalence, the operator [16],except for a small “transient”regionN 0–10 cor-
∼
σz/2 of the spin–boson model translates to Sz in the responding to high energies ω ≫∆r near the cut–off ωc.
AKM,soS(ω)givestherelaxationfunctionforthespin– We see that the energy level flow is uniquely specified
bosonproblem. Weextractthelocalstaticsusceptibility, by two parameters, ∆r (equivalently TK) which sets the
χ (T) = (1/β)(∂2lnZ /∂ǫ2) , of the Ohmic spin– crossover scale in Fig. (1), and the dissipation strength
sb sb ǫ=0
−
boson model from χ′(ω = 0,T) at finite T. Finally, the α(orequivalently the dimensionlessinitial couplingcon-
decoupling of spin and charge degrees of freedom in the stant ρ0Jk). The universal flow of energy levels is the
AKM allows identification of Fimp as the free energy originofthescaling,forfixedαandarbitrary∆/ωc ≪α,
of the spin–boson model (with bath contribution sub- in the thermodynamic quantities we discuss below [17].
tracted) and of C(T) as the corresponding specific heat. TheinsettoFig.(2)showsthatthespecificheatcurves
The energy level flow diagram for some low lying forα=0.2andseveralvaluesof∆allscaleontoauniver-
rescaled energy states is shown in Fig. (1). We see that sal curve corresponding to α = 0.2. For different α one
spin–rotational invariance, which is broken for J⊥ = Jk obtains distinct universal curves, Cα(T), Fig. (2). Scal-
6
at high energies, is restored below the low energy scale ing is valid for all temperatures in the rangekBT D0,
≪
of the model, the Kondo temperature TK(J⊥,Jk) [14], not only at low temperatures kBT ≪ ∆r = TK. In Fig.
leading to the well knownisotropicstrongcoupling fixed (2),andthroughoutthispaper,wescalethetemperature
by α/γ = 3∆ /π2k2. This follows from an exact result
r B
2
TABLE I. γ, χ and R , extracted from the fixed
akm akm
point analysis.
3.0 α ∆=J χ γ R
⊥ akm akm
1.5 10−3 10−6 489.2 3214.6 2.003
1.0 ∆∆==1100--23 10−3 10−4 4.78 30.76 2.04
∆=10-4 1/3 0.01 275.1 1806.1 2.005
2.0 1/3 0.1 8.61 56.4 2.01
0.5
T 0.7 0.01 1.90×106 1.25×107 2.002
γ
T)/ 0.0 0.9 0.1 1.81×106 1.19×107 2.003
(α -1 0 1 2 3
C α=0.95
1.0
α=0.4
α=1/3
α=0.2 case are small for weak dissipation and only set in when
α=0.1 α 1−. In this limit and for ∆/ω 1, the scaling
c
→ ≪
0.0 functions for the Kondo problem are recovered for the
-4.0 -2.0 0.0 2.0 4.0 6.0
specific heat and static susceptibility [15].
log (γT/α)
10
FIG. 2. Universal specific heat curves, Cα(T),
γT
for the Ohmic two–state system for 0 < α < 1. α=0.95
γ = limT→0C(T)/T ∼ α/∆r is extracted from the fixed α=0.7
point analysis [15]. The symbols represent the temperatures α=1/2
at which the numerical second derivative of F has been α=1/3
imp
calculated. Theinset is for α=0.2. 0.2 α=0.2
) α=0.1
T
(
α
χ
for the Wilson ratio [8,19]discussedbelow together with T
B
ourdefinitionχ (0)=1/2∆ . Thespecificheatislinear k 0.2
sb r 0.1
intemperaturefork T ∆ and0<α<1,withalin-
B r
≪ 0.1
earcoefficientγ ingoodagreementwithvaluesextracted
from an analysis of the strong coupling fixed point [15]. 0.0
This is also expected from the Fermi liquid groundstate -2.0 0.0 2.0 4.0 6.0
0.0
of the model. We note thatthe T3 coefficientof the spe- -3.0 -1.0 1.0 3.0 5.0 7.0
cificheatchangessignclosetoα=1/3,correspondingto log (γΤ/α)
10
the appearance, for weak dissipation, of damped oscilla-
tions at frequency ∆ [5,6]. The peak in C(T)/T can be FIG. 3. Universal curves for the static susceptibility of
r
theOhmictwo–state system,k Tχ (T),for 0<α<1, with
taken as a signature of a two–level system weakly cou- B α
γ ∼α/∆ asinFig. 2. Theinsetisforα=0.8and∆=0.005
pledtobosonicexcitations-itisabsentforstrongOhmic r
(◦), ∆=0.05 ((cid:3)) and ∆=0.01 (⋄).
dissipation. The strong dependence of C(T)/T on α for
weak dissipation is also seen in other quantities, such as
A universal Wilson ratio for the AKM [8] has been
in the spin response [5].
proven also for the Ohmic and non–Ohmic spin–boson
The universal curves for the static susceptibility,
models [19]. For the Ohmic case, the Wilson ratio for
χα(T), parametrized by the dimensionless dissipation thespin–bosonmodel,R =lim 4 π2kB2 Tχsb =2/α
strength, α, are shown in Fig. (3). The inset shows that sb T→0 3(gµB)2 C
curves with the same α=0.8 and different ∆ scale onto [19]. This is related to the Wilson ratio, Rakm =
the same curve for kBT ≪D0. The thermodynamic cal- limT→0 34(gπµ2kBB2)2TχCakm = 2, for the AKM [8] by Rsb =
culationfor χα becomes inaccurate atlow temperatures, Rakm/αsinceχsb =χakm/α[8]andχakm isthesuscepti-
kBT ∆r (discussed in detail in [5,15]), and we have bilityoftheAKM(withag factorof2fortheconduction
≪
to resort to an analysis about the strong coupling fixed electrons). Table I shows that R = 2 (R = 2/α) is
akm sb
point. This yields a finite susceptibility at T = 0 which recoveredfor ∆/ω α and 0<α<1.
c
isaccuratetowithin1%for0<α<1(seeTableIbelow Dynamical quant≪ities also show the universality dis-
and[15]). Athightemperatures,∆r kBT D0,there cussed above for thermodynamic quantities. The T = 0
≪ ≪
is a dramatic difference in the approachof kBTχα(T) to relaxation functions, S(ω), for dissipation strength 0 <
its free spin value of 1/4 between the cases of weak and α < 1 have been given in [5]. A detailed analysis shows
strong dissipation. In the former, the free spin value is that these are universal functions of ω/∆ parametrized
r
reachedveryrapidlyonincreasingthetemperatureabove by α: S = S (ω/∆ ). The case of α=1/3, correspond-
α r
∆r. The logarithmic terms characterizing the slow ap- ing to the crossoverbetween damped oscillations andin-
proachof kBTχα(T) to the free spin value in the Kondo coherentrelaxation[5,6,20],is shownin Fig.(4). Scaling
3
universalityandscalinginthesemodels. Foranisotropies
in the AKM, 0 < J J < + , corresponding to
⊥ k
≪ ∞
5 dissipation strengths 0 < α < 1 and bare tunneling fre-
quencies ∆/ω α in the Ohmic spin–boson model,
c
≪
thethermodynamic(dynamic)propertiesofthesemodels
0 are characterizedby universal scaling functions of T/∆
r
0)] (ω/∆r) which are distinct functions for different α and
ωS()/S( -5 ∆∆==00..0000011 α=1/3 a∆rre.inAdseipnenthdeenKtoonfd∆o—prtohbelelmat,tetrheenstcearliinngg ofunnlycttiohnrosuagrhe
g[10 ∆=0.01 universal with deviations from scaling at high frequen-
o cies andtemperatures arisingfromfinite ∆/ω . The dis-
l -10 ∆=0.1 sipation strength in the Ohmic two–state sycstem, just
as the anisotropy in the AKM, determines the essential
-15 physics, in particular the renormalization of the low en-
-10 -5 0 5 10 ergy scale ∆ /ω (∆/ω )1/(1−α) and the form of the
r c c
log10(γω) scalingfunctions. T∼heperspectivegainedabovemayalso
be useful in understanding the highly anisotropic multi–
FIG.4. Universal curvefor the relaxation function Sα(ω). channelKondomodelswhichariseinthecontextofsingle
electrondevices,andtwo–levelsystemsinsolidsinteract-
ing with electrons [21].
in S (ω) extends to all frequencies ω D and is not
α 0
≪ We acknowledge useful discussions with P. W¨olfle,
restricted to ω . ∆ . At high frequencies, ω ∆ , we
r r
≫ A. Rosch, C. Roth, T. Kopp, J. von Delft, J. Kroha
obtain S (ω) ω−(4−2α), thus χ′′(ω) ω−(3−2α) and
α
C (t) [σ (t)∼,σ (0)] 1 ct2(1−α)∼for D−1 t and A. E. Ruckenstein. This work was supported by
s ≡ h z z −i ∼ − 0 ≪ ≪ the Deutsche Forschungsgemeinschaft and the German-
∆−1 with the α dependent exponents being accurate to
r Israeli Foundation.
within 0.1% for 0 < α < 1. These results agree with
short–timeapproximations[1]andperturbativemethods
[9] in this limit. More importantly, they indicate, that
scaling, in the sense discussed in this paper, can only
be expected for fixed α. At low frequencies, ω ∆ ,
r
≪
the Fermi liquid behaviour of χ′′(ω,T = 0) αω, gives
∼
C (t) α/t2, with an α independent exponent for [1] A. J. Leggett et al., Rev. Mod. Phys. 59,1 (1987);ibid.
s
t 1/∼∆ −[4,5,19]. 67, 725 (1995).
r
≫
Deviationsfromscaling,startingathightemperatures [2] U. Weiss, in Series Modern Condensed Matter Physics,
and frequencies, set in on increasing ∆/ω for our finite (World Scientific, Singapore 1993), Vol. 2.
c
[3] A. LeClair et al.,preprint hep-th/9701022; J. T. Stock-
bandwidthmodel. The scalingdiscussedhereis validfor
burgerandC.H.Mak,J.Chem.Phys.,105,8126(1996);
0 < α < 1 (0 < J < ) as long as ∆/ω remains the
k ∞ c B. Golding et al., Phys. Rev.Lett. 68, 998 (1992).
smallest bare energy scale.
[4] S.ChakravartyandJ.Rudnick,Phys.Rev.Lett.75,501
The scaling and universality discussed above can be
(1995).
useful in interpreting experiments on dissipative two– [5] T. A. Costi and C. Kieffer, Phys. Rev. Lett. 76, 1683
state systems. The dissipation strength α can be deter- (1996); T. A. Costi, Phys.Rev. B55, 3003 (1997).
minedbyfittingthedataforsomequantity,suchasS(ω), [6] F. Lesage, H.Saleur and S.Skorik,Phys. Rev.Lett. 76,
tothe appropriateuniversalscalingfunctionS (ω). The 3388 (1996).
α
low energy scale, ∆ , can be extracted from the low en- [7] S. P. Strong, Phys.Rev. E55, 6636 (1997).
r
ergy/temperature behaviour (e.g. from S (ω) by using [8] P.B.VigmannandA.M.Finkel’ste˘in,Sov.Phys.JETP;
α
the generalized Shiba relation [19] S (0) = 2α[χ (T = 48,102 (1978)
0)]2 and χ (T = 0)= 1/2∆ ). In prαactice, this msbay be [9] F.Guinea,V.HakimandA.Muramatsu,Phys.Rev.B32
sb r 4410, (1985); F. Guinea, Phys.Rev.B32 4486, (1985).
difficult for strong dissipation α 1/3 since the scaling
≥ [10] L. N. Oliveira and J. W. Wilkins, Phys. Rev. 23, 1553
functions for different α will differ appreciably only for
(1981).
kBT,ω > ∆r. In this case an alternative is to extract α [11] A. J. Bray and M. A. Moore, Phys. Rev Lett. 49, 1546
and ∆r from γ = π2kB2α/3∆r and χsb = 1/2∆r. α and (1982); S.Chakravarty,Phys.RevLett.50,1811(1982).
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systems α<1/3,for which the scaling functions depend (1969); Phys.Rev.B1, 4464 (1970).
sensitively on α, even for k T,ω ∆ . [13] K. G. Wilson, Rev. Mod. Phys.47, 773 (1975); H. B.
B r
≪
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Ohmic spin–boson model to the AKM in order to study Rev. B21, 1044 (1980).
4
[14] A.M.TsvelickandP.B.Wiegmann,Adv.Phys.32,453
(1983).
[15] T. A. Costi et al., in preparation.
[16] This is not true for fixed ∆/ω ≪ α and different α,
c
except in the special case of theKondolimit α→1−.
[17] This is consistent with the Bethe–Ansatz for the AKM
[14] where the scaling trajectories are labelled by a pa-
rameter which we can relate to α for 0 < α < 1 and
∆/ω ≪α[15].Renormalizationsofα,enteringthescal-
c
ingfunctions,aretypicallyafraction ofapercent(being
O((∆/ω )2)) and unimportant in the regime of interest
c
for theOhmic two-state system studied in thispaper.
[18] B. Jones and J. W. Wilkins, Phys. Rev. Lett. 61, 125
(1988).
[19] M. Sassetti and U. Weiss, Phys. Rev. Lett. 65, 2262
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mat/970413.
5
