Equilibrium dynamics of the dissipative two-state system

4 Pages·0.18 MB·English

by  T. A. Costi

Equilibrium Dynamics of the Dissipative Two{State System
z
T. A. Costi and C. Kie(cid:11)er
Universita(cid:127)tKarlsruhe, Institutfu(cid:127)r Theorie der KondensiertenMaterie, 76128Karlsruhe, Germany
Wilson's momentum shell renormalization group method is used to solve for the dynamics of the
dissipative two{state system. We utilize themapping ofthe spin{boson model onto the anisotropic
Kondo model (AKM) and solve for the dynamics of the latter. We (cid:12)nd that the AKM captures
the physics of the dissipative two{state system for dissipation strength 0(cid:20)(cid:11)(cid:20)4 corresponding to
1 (cid:21) Jk (cid:21) (cid:0)1 in the AKM. The dynamics of the AKM shows a smooth crossover between two
strong coupling regimes corresponding to weak and strong dissipation in the spin{boson model.
!(cid:11)). In the case of anOhmicheat bath, ofinterest to us
PACS numbers: 71.27.+a,75.20.Hr,71.10.+x,72.15.Qm here, we have J(!) = 2(cid:25)(cid:11)!, for ! << !c, where !c is a
high energy cut{o(cid:11) and (cid:11) is a dimensionless parameter
6 characterizing the strength of the dissipation.
A problemwhich is of general interest in both physics
9 A great deal of work has been carried out over the
9 and chemistry is that of a quantum mechanical system
last10years inorder tounderstand the dynamicsofthis
1 tunnelingbetween twostatesandsubject toadissipative
apparently simple model [4,5]. Extensive calculations
couplingtoenvironmentaldegrees offreedom. Examples
n based on the Feynman{Vernon path integral formal-
a of this type of system abound; they include the tunnel-
ism within the \Non{Interacting Blip Approximation"
J ing of defects in metallic glasses [1], the motion of the
(NIBA) have yielded reliable informationfor weak dissi-
3 total (cid:13)ux in a SQUID between two metastable (cid:13)uxoid
pation and short times [4] and have also provided some
2 states [2]andthe di(cid:11)usionofprotons andmuonsinmet-
insight into the expected behaviour in other regimes. In
als[3]. The maintheoreticalinterest liesinadescription
addition to such direct attempts at calculating dynam-
7 of the dynamics of the generalized coordinate of the two
0 ics for the spin{boson model, it has proved fruitful to
level system subject to the in(cid:13)uence of the environment.
1 exploit analogies between this model and several other
Both the equilibriumand non{equilibriumdynamics are
1 models, including the inverse{square Ising model [6],
0 of interest for the di(cid:11)erent experimental realizations of the anisotropicKondo model(AKM) and the Vigmann{
6 two{level systems. In the case of macroscopic quantum Finkel'ste(cid:21)in (VF) model [7]. In the long time approxi-
9 coherence experiments in SQUIDs, the system can be
mation it has been shown that the partition functions,
t/ prepared inone of the twostates by applyinga bias( an
a andhence the thermodynamics,ofthese di(cid:11)erentmodels
external magnetic(cid:12)eld) fortimest<0andthen allowed
m can be put into correspondence and the parameters of
to evolve for t > 0 in zero bias. The non{equilibrium
- the models related. This correspondence has also been
correlation functions of the two state system are then of
d carried out for the respective fermionicHamiltoniansby
n primary interest. For most microscopic systems such an
applying an approximate bosonization method valid for
o initialstate preparationisnot realizableandthe interest
c then lies in the equilibrium dynamics. In this paper we low energies ! << !c [8]. In this paper we exploit such
amappingofthespin{bosonmodelontotheAKM inor-
present results for the equilibriumcase with a bias.
dertomakepredictionsaboutthedynamicsoftheformer
Speci(cid:12)cally we consider the spin{boson Hamiltonian,
onthebasis ofrenormalizationgroupcalculationsonthe
1 1 X y 1 latter. It hasnot been clear [4]to whatextent the above
HSB =(cid:0) (cid:22)h(cid:1)(cid:27)x+ (cid:15)(cid:27)z+ !(cid:11)(a(cid:11)a(cid:11)+ ) mapping is valid in all parameter regimes, in particular
2 2 2
(cid:11)
for weak dissipation, and this has partly motivated this
+ 1q0(cid:27)zXp C(cid:11) (a(cid:11)+ay(cid:11)): (1) work. The AKM is given by
2 (cid:11) 2m(cid:11)!(cid:11) X y J? X y (cid:0) y +
H = (cid:15)kck(cid:27)ck(cid:27)+ (ck"ck0#S +ck#ck0"S )
Here(cid:27)i;i=x;y;zarePaulispinmatrices,thetwo{states k;(cid:27) 2 kk0
of the system correspond to (cid:27)z =" and (cid:27)z =#. (cid:1) is the Jk X y y z
bare tunneling matrix element and (cid:15) is a bias. The en- + (ck"ck0"(cid:0)ck#ck0#)S +g(cid:22)BhSz; (2)
2
vironment is represented by an in(cid:12)nite set of harmonic kk0
oscillators(labelledbythe index (cid:11))withmasses m(cid:11) and where the (cid:12)rst term represents non{interacting conduc-
frequency spectrum !(cid:11) coupling linearly to the coordi- tion electrons and the second and third terms represent
1
nate Q= 2q0(cid:27)z ofthe two{levelsystem viaa termchar- an exchange interaction between a localized spin 1=2
acterized bythecouplingsC(cid:11). Theenvironmentspectral and the conduction electrons with strength J?;Jk. For
function is given in terms of these couplings, oscillator
(cid:25) P C(cid:11)2 J? =Jk the modelreduces to the usual Kondo modelof
masses and frequencies by J(!) = 2 (cid:11)(m(cid:11)!(cid:11))(cid:14)(! (cid:0) magneticimpuritiesin metals. A localmagnetic(cid:12)eld, h,
1
coupling only to the impurity spin in the Kondo model susceptibility. The quantity we actually calculate is
00
(the last term in Eq. 2) corresponds to a (cid:12)nite bias, (cid:15), S(!) =(cid:0)1(cid:31) (!+i(cid:14)) whichisrelated tothe neutron scat-
(cid:25) !
in the spin{boson model. The correspondence between teringcross{section. Ourresultswereobtainedfor(cid:3)=2,
(cid:1)
H and HSB is then given by (cid:15) = g(cid:22)Bh, !c = (cid:26)J? and keeping the 320 lowest states at each iteration. In this
(cid:11) = (1 + 2(cid:14))2, where tan(cid:14) = (cid:0)(cid:25)(cid:26)Jk, (cid:14) is the phase paper we discuss only the T =0 results.
(cid:25) 4
shiftforscatteringofelectrons fromapotentialJk=4and The accuracy of the numerical calculations could be
(cid:26)=1=2Distheconductionelectron densityofstates per tested by (a) showing that the exactly solvable Toulouse
1
spinattheFermilevelfora(cid:13)atbandofwidth2D[4]. We limit,(cid:11)= 2, couldbe reproduced (described below) and
notethatwhereasthespin{bosonmodelcandescribedis- (b)byverifyingthegeneralizedShibarelationforthedy-
sipation with arbitrary strength, 0 (cid:20) (cid:11) (cid:20) 1, the AKM namic spin susceptibility of the spin{boson model [12].
is only capable of describing the region 0 (cid:20) (cid:11) (cid:20) 4 with The latter relation states that, at T = 0, S(! = 0) =
0 2
(cid:11) = 0 corresponding to Jk = +1, (cid:11) = 1 corresponding 2(cid:11)(cid:31)(! =0) . Generalized with the extra factor of (cid:11), it
tothe antiferromagnetic/ferromagneticboundary ofthe is also valid for the AKM, as explicitly veri(cid:12)ed by our
AKM (Jk = 0), and 1 < (cid:11) (cid:20) 4 corresponding to the numerical results. We chose to extract the static sus-
0
ferromagnetic regime of the AKM. ceptibility via a Kramer's{Kronig relation (cid:31)(! =0) =
R+1
Therelevantdynamicalquantityforthetwo{levelsys- S(!)d!. Veri(cid:12)ed in this way,the generalized Shiba
(cid:0)1
tem is the response function (cid:31)SB(!;T) = hh(cid:27)z;(cid:27)zii relation provides a good test of the method, not just at
which translates into the local dynamic spin suscep- low frequency, but at all energy scales and for arbitrary
0
tibility, (cid:31)(!;T) = hhSz;Szii, for the Kondo model. values of (cid:11) and (cid:1). The error in (cid:31)(0) was typically 5%
In order to calculate this quantity we apply Wilson's and contributed the mainsource of error, approximately
momentum shell renormalization group method which 10%,totheShibarelation;representativecasesareshown
00
has recently been generalized to the calculation of dy- in Table I [13]. In all cases we found that (cid:31) (!)(cid:24)! for
namical quantities for a number of models (e.g. [11]). ! ! 0 showing that the spin{spin correlation function,
2
Brie(cid:13)y, the procedure, explained in detail in [9], con- h[(cid:27)z(t);(cid:27)z(0)]i,decays as1=t forlongtimesi.e. thetun-
sists of (i) linearizing the spectrum about the Fermi nelingisalwaysincoherentatlongtimes. Incontrast,the
2(1(cid:0)(cid:11))
energy (cid:15)k ! vFk, (ii) introducing a logarithmic mesh NIBAgivesanalgebraicdecay(cid:24)1=t dependingon
(cid:0)n
of k points kn = (cid:3) and (iii) performing a unitary the coupling (cid:11).
P
100.0
transformation of the ck(cid:27) such that f0(cid:27) = kck(cid:27) is
5.00
the (cid:12)rst operator in a newPbasis, fyn(cid:27);n = 0;1;:::, 4.00
w(cid:24)wi.neiht.ih!chHth(tce1rida+!biao(cid:3)gvPo(cid:0)en1(cid:22)ad)=liPis2zce1nrfse=otr0Hiz(cid:24)necnd(cid:3)=>f(cid:0)o>nr=m21k(.(cid:22)fonfy(cid:15)T+kt(cid:22)1hh(cid:22)ceekf(cid:22)knHc(cid:22)inak(cid:22)me+tiiiclhnt:oecknn:){iea;srnpgway(ict2eihs), ω) 6800..00 ωS()/S(0) 123...000000
(
S
now diagonalized by the following iterative process: (a) 40.0 0.00
0 1 2 3 4
oPnePdeN(cid:12)n(cid:0)e1s a se(cid:0)qnu=e2ncey of (cid:12)nite size HamiJl?toniyansHN(cid:0) = ω/∆*
(cid:22) n=0 (cid:24)n(cid:3) (fn+1(cid:22)fn(cid:22) + h:c:) + 2 (f0"f0#S + 20.0 r
y + Jk y y z
f0#f0"S )+ 2 (f0"f0" (cid:0)f0#f0#)S for N (cid:21) 0; (b) the
N(cid:0)1
Hamiltonians HN are rescaled by (cid:3) 2 such that the 0.0
N(cid:0)1 -0.2 0.0 0.2 0.4
energy spacing remains the same, i.e. H(cid:22)N = (cid:3) 2 HN. ω/D
This de(cid:12)nes a renormalization group transformation
H(cid:22)N+1 =(cid:3)1=2H(cid:22)N +P(cid:22)(cid:24)N(fNy+1(cid:22)fN(cid:22)+h:c:)(cid:0)EG;N+1,
FIG. 1. The response function S(!) of the AKM for
with EG;N+1 chosen so that the ground state energy of
H(cid:22)N+1 is zero. Using this recurrence relation, the se- (cid:11) = 0:1 (Jk = 4:698) and (cid:1) = 0:08 (solid line), (cid:1) = 0:1
(dotted line), (cid:1) = 0:15 (dashed line). The position of the
quenceofHamiltoniansH(cid:22)N forN =0;1;:::isiteratively (cid:3)
inelastic peaks isat (cid:1)r =0:045;0:063 and 0:087respectively.
diagonalized within a product basis of, typically, up to
Theinset showsthatS(!)=S(0) is auniversal function ofthe
1200 states for each iteration. This gives the excitations rescaled energy !=(cid:1)(cid:3)r (S(0)(cid:24) (cid:1)1(cid:3)r2). Energies are in units of
aerngdymscaanlyesbo!dNydeieg(cid:12)ennesdtatbeys !atNa=cor(cid:3)re(cid:0)spNo2(cid:0)n1dainngdsaetlloowf sena- D= !2c =1.
direct calculationofresponse functions. Thus, for exam- The case of zero dissipation, (cid:11) = 0, corresponds to
0 00
pZ1lNe,P(cid:31)(m!;;nTjM) imNs;gnijv2e!en+(cid:0)b(cid:12)i0(cid:15)ym(cid:0)(cid:31)((cid:0)(cid:15)(me!(cid:0)(cid:0);(cid:12)(cid:15)(cid:15)Tnn)),=w(cid:31)he(r!e;(cid:15)Tm);+(cid:15)ni(cid:31)are(!m;Tan)y={ J41!k[=(cid:14)(!1(cid:0)Jin?)th(cid:0)e(cid:14)A(!K+MJ.?T)]hiissaissuexmacotflytwsooldvealbtale;fuSn(c!ti)on=s
body excitations of HN, ZN(T) the corresponding par- at ! = (cid:6)J?. In the spin{boson model the response is
N
tition function of HN, and Mm;n =< mjSzjn >N the also a sum of two delta functions at ! = (cid:6)(cid:1). We can
(cid:1)
relevant many{body matrix elements for the dynamic thusidentifythecut{o(cid:11)!c appearingin !c =(cid:26)J? asthe
2
bandwidth, !c = 2D. In this case coherent oscillations (cid:1)b((cid:15)), is well describedpby the weak coupling theory ex-
withfrequency (cid:1)arerealized. On increasing(cid:11),Fig.(1), pression [5], (cid:1)b((cid:15)) = (cid:15)2+(cid:1)(cid:3)r2. The e(cid:11)ect of a bias,
the above peaks survive but are broadened. The sys- however, is not simply to renormalize the tunneling fre-
temexhibits oscillationsfor short timest<(cid:24)1=(cid:1)r,where quency, since we couldnotscale S(!) for di(cid:11)erent (cid:15)onto
(cid:1)r is the renormalized tunneling frequency (see below). a single curve, as is the case for di(cid:11)erent tunneling am-
As discussed above, the long time behaviour is always plitudes and zero bias. Thus a (cid:12)nite bias for weak dis-
incoherent for any (cid:12)nite (cid:11) and (cid:1). The position of the sipation does not change the qualitative aspects of the
peaks, which is a measure of the renormalized tunnel- dynamics. Beyond (cid:11)(cid:25) 0:2 the inelastic peaks are negli-
ing frequency, is reduced i.e. as expected, dissipation gible,even forvery smallvaluesof(cid:1),andthe incoherent
hinders tunneling. Fig. (1) illustrates these features for part dominates (see also Fig. (4)).
1
(cid:11) = 0:1. For (cid:11) << 1 and small (cid:1), scaling arguments For the exactly solvable Toulouse limit, (cid:11) = , the
(cid:11) 2
givearenormalizedtunnelingfrequency (cid:1)r =(cid:1)(!(cid:1)c)1(cid:0)(cid:11) numericalresults forS(!) (cid:12)t very wellontothe resonant
[4]. Numerical results for (cid:11)=0:01 gave for the inelastic levelmodelresult forallvaluesof(cid:1)asshowninFig.(3).
(cid:3)
peak positions, (cid:1)r, values within 5% of this result for 10.0
(cid:1) = 0:05;0:07and 0:1 [13]. Although, the amplitude of
the dampedoscillationsdecreases with increasing(cid:1), the 8.0
rescaled spin response S(cid:11)(!)=S(cid:11)(0) is a universal func-
tion of !=(cid:1)(cid:3)r (depending on (cid:11)). This is shown in the 00 6.0
insettoFig.(1)for(cid:11)=0:1. Hence, forweakdissipation, 0
1
increasing (cid:1), for (cid:1) < !c, does not destroy the damped )/
ω 4.0
oscillations. As we show below, only increasing (cid:11) has (
S
this e(cid:11)ect.
15.0 2.0
ε/∆*=0
r
0.0
ε/∆*=1/2 -0.02 -0.01 0.00 0.01 0.02
r
10.0
ε/∆*=1 ω/D
r
)
ωS( ε/∆r*=2
2∆ FIG.3. S(!)fortheexactlysolvable Toulousecase,(cid:11)~ = 1,
2
5.0 1
(J~k = 1:312). (cid:1) = 0:1 (circles), (cid:1) = 240:1 (diamonds),
1
(cid:1) = 220:1 (squares). The solid lines are the exact analytic
results for the resonant level model.
0.0
-0.6 -0.4 -0.2 0.0 0.2 0.4 Theresponseconsistsofasinglepeakat! =0ofwidth
ω/∆ (cid:1)~RL = (cid:25)(cid:26)VR2L where VRL!R(cid:0)L1 = J2?!K(cid:0)1Mcos2((cid:14)) and
!RL;!KM are the respective high{energy cut{o(cid:11)s ofthe
FIG. 2. S(!) for (cid:11) = 0:2 (Jk = 3:01), (cid:1) = 0:001 resonantlevelandKondomodels[7]. Since!KM =2D =
(cid:3) (cid:3)
and di(cid:11)erent bias (cid:15)=(cid:1)r. (cid:1)r = 0:079(cid:1) is the renormal- !c,we canrelatethe variousmodelsbydetermining!RL
ized tunneling amplitude for the zero bias case. For the (cid:12)- from the exact resonant level model result S(! = 0) =
nite bias cases, (cid:15)=(cid:1)(cid:3)r =(cid:3)21;1(cid:3);2, we (cid:12)nd for the renormalized (cid:25)21(cid:1)~2. We found that !!KRML = 1:125 in all cases with a
tunneling amplitude (cid:1)b=(cid:1)r = 1:05;1:33;1:93, respectively,
variationof less than 0.1%[13].
whichpis within 6% agreement of the weak coupling result [5] The regime 1 < (cid:11) < 1 corresponds to the antiferro-
(cid:1)b= (cid:1)r(cid:3)2+(cid:15)2. Energies are in units of D= !2c =1. magnetic Kond2o regime, for which there have been few
reliableresultsforthedynamicsusceptibility. Therenor-
Fig. (1) for (cid:11) = 0:1 and the zero bias case of Fig. (2)
malizationgroupresultspresented here provideanessen-
for (cid:11) = 0:2 show that the damped oscillations become
tially exact solution in this region, as can be seen from
strongly suppressed with increasing (cid:11), even for (cid:1)<<1.
Table I. The single quasielastic peak in S(!) narrows
The oscillations disappear completely at approxi-
exponentially for (cid:11) ! 1, i.e. J?;k !0, (Fig. (4)). Tun-
mately (cid:11) = 0:33 [14], well below the Toulouse point
neling is incoherent for all values of (cid:1). Fig. (4) shows
1
(cid:11) = (see below). A small bias, (cid:15) << (cid:1), i.e. small
2 the crossover from damped oscillationsfor weak dissipa-
relativetothebaretunnelingamplitude,butcomparable
tion to incoherent relaxation for strong dissipation. The
(cid:3)
to the renormalized tunneling amplitude, (cid:1)r, leads to a crossover occurs at approximately (cid:11) = 0:33. From the
suppression ofthe inelasticpeaks, and anincrease in the
scaling properties of S(cid:11)(!) for di(cid:11)erent (cid:1) this value is
tunneling frequency relative to the zero bias value, Fig.
independent of (cid:1) (for (cid:1) << !c). The crossover point
(2). This increase in the e(cid:11)ective tunneling frequency,
separates twostrong{coupling regimes,corresponding to
3
(a)dampedshorttimeoscillationsforsmall(cid:11),and(b)in- [1] B.Golding, N.M.Zimmermann and S.N.Coppersmith,
coherent relaxationof the spin and Kondo behaviour for Phys. Rev.Lett.68, 998 (1992).
large(cid:11). Thedetailsofthe(cid:12)xedpointsandthebehaviour [2] S.ChakravartyandS.Kivelson,Phys.Rev.Lett.50,1811
(1983).
in the ferromagnetic regime Jk <0, (cid:0)Jk > J?, ((cid:11)> 1),
[3] H. Wipf, D. Steinbinder, K. Neumaier, P. Gutsmiedl,
whereweobtaintheexpected localizationoftheparticle,
A. Magerl and A. J. Dianoux, Europhys. Lett. 4, 1379
will be discussed elsewhere. The inset to Fig. (4) shows
(1989).
thatthelowenergy scaleisgivencorrectly bythe scaling
[4] A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. A.
result, (cid:1)r, for weak dissipationand by the Bethe ansatz Fisher, A. Garg and W. Zwerger, Rev. Mod. Phys. 59,1
expression for the Kondo temperature of the AKM [10] (1987); ibid.,67, 725 (1995).
for (cid:11)!1. [5] U. Weiss, \Quantum Dissipative Dynamics", Series in
Modern Condensed Matter Physics, Vol.2,World Scien-
4.0
ti(cid:12)c, Singapore (1993).
α=0.1
α=0.2 [6] S.Chakravarty and J.Rudnick, Phys.Rev.Lett.75,501
3.0 α=0.3 10-4 (1995).
α=0.4 [7] P. B. Vigmann and A.M. Finkel'ste(cid:21)in,Sov. Phys.JETP
(0) αα==00..69 10-9 NScRaGling 48,102 (1978). The VF model, as used here, contains a
S 2.0 Bethe Ansatz Coulomb term. When this is set to zero the VF model
/
ω) 10-14 reduces tothe exactly solvable resonant level model. See
S( α 1 also P.Schlottmann, Phys.Rev.B25,4805(1982); L.N.
1.0 Oliveira and J. W. Wilkins, Phys.Rev.23, 1553 (1981);
L. N.Oliveira, PhD Thesis, Cornell University (1985).
[8] F.Guinea,V.HakimandA.Muramatsu,Phys.Rev.B32
0.0
4410, (1985).
1 10 100
[9] K. G. Wilson, Rev. Mod. Phys. 47, 773 (1975); H. B.
ω/∆NRG Krishnamurthy, J. W. Wilkins & K. G. Wilson, Phys.
r
Rev. B21, 1044 (1980).
[10] A.M.Tsvelick andP.B.Wiegmann, Adv.Phys.32,453
FIG. 4. S(!) for (cid:1) = 0:1 and di(cid:11)erent values of (cid:11). The
(1983).
inelastic peakdisappears atapproximately (cid:11)=0:33. Thelow
NRG [11] T.A.Costi,A.C.HewsonandV.Zlati(cid:19)c J.Phys.: Cond.
energy scale, (cid:1)r , for (cid:1)=0:1 and for (cid:11)=0:1;0:2;:::;0:9
Matt. 6, 2251 (1994); T. A. Costi, A. C. Hewson, J.
is shown in the inset. It is de(cid:12)ned to be the position of
(cid:3) Phys.: Cond. Matt. 5, 361 (1993); T. A. Costi, A. C.
the inelastic peak, (cid:1)r, for small (cid:11) and the half{width of the
Hewson, Phil. Mag.B65, 1165 (1992).
quasielastic peakforlarge(cid:11). Itagreeswiththescalingresult,
[12] M. Sassetti and U. Weiss, Phys. Rev. Lett. 65, 2262
(cid:1)r, for small (cid:11), and with the Bethe ansatz Kondo tempera-
(1990). This relation is remarkable, since it generalizes
ture for the AKM [10], TK(J?;Jk), for (cid:11) ! 1. At (cid:11) (cid:25) 0:3,
the original Shiba relation for the Anderson impurity
close to the crossover, there is an ambiguity in de(cid:12)ning the
model (H. Shiba Prog. Theor. Phys. 54, 967 (1975)) to
low energy scale and some discrepancy is seen. Similarly,
NRG the AKM for arbitrary coupling.
TK(Jk;J?)deviates fromthetrue energyscale, (cid:1)r ,when
[13] Inverifying thegeneralized Shiba relation, andtheexact
(cid:11)<0:5(TK(Jk;J?)in[10]isderivedforsmallJki.e. (cid:11)!1). 1
results in the Toulouse limit (cid:11)= , it was important to
2
take into account a renormalization in (cid:11)!(cid:11)~((cid:3)) due to
To summarize, we have shown, that at low energies,
the logarithmic discretization. Our prescription follows
0 (cid:20) ! <(cid:24) (cid:1) << !c, the equilibrium dynamics of the dis- Oliveira in [7] and replaces the bare Jk in the expres-
sipative two{state system is very well described by the sion for (cid:11) by J~k((cid:3))=Jk=A(cid:3), where A(cid:3) = 12ln(cid:3)11(cid:0)+(cid:3)(cid:3)(cid:0)(cid:0)11.
AKM for dissipation 0 (cid:20) (cid:11) (cid:20) 1. Agreement with exact
1 Similarly, J? is renormalized to J~?((cid:3))=J?=A(cid:3).
results was obtained in the limiting cases (cid:11) ! 0; ;1.
2 [14] F. Lesage, H. Saleur and S. Skorik, preprint cond{
The validityof the generalized Shiba relation, within er-
mat/9512087,1995, (cid:12)nd the same value of (cid:11).
rors consistent with the numerical data, was veri(cid:12)ed in
the range 0<(cid:11)<1 and (cid:1)<<!c.
0 2
We acknowledge useful discussions with A. Rosch, P. TABLE I. The Shiba relation, S(0) = 2(cid:11)~(cid:31)(0) , for the
Wo(cid:127)l(cid:13)e and T. Kopp. This work was supported by E.U. dynamic spin susceptibility; (cid:11)~ is de(cid:12)ned in [13].
grants CT92-0068(TAC), CT93-0115 (TAC,CK). 0 2 (cid:31)00(!)
(cid:11) (cid:1)=J? Jk (cid:11)~ 2(cid:11)~[(cid:31)(0)] [(cid:0) (cid:25)! ]!!0 %error
3 3
0.01 0:001 16:078 0:0108 6:536(cid:2)10 6:450(cid:2)10 1.3%
3 3
0.1 0:01 4:698 0:1068 2:181(cid:2)10 2:420(cid:2)10 10.9%
2 1
0.2 0:1 3:008 0:2111 7:095(cid:2)10 7:638(cid:2)10 7.4%
10 10
0.4 0:001 1:659 0:4144 1:372(cid:2)10 1:291(cid:2)10 6.4%
6 6
0.5 0:025 1:262 0:5139 3:748(cid:2)10 3:965(cid:2)10 5.5%
8 8
0.8 0:1 0:4262 0:8071 8:568(cid:2)10 7:919(cid:2)10 8.2%
z
Present address: Institut Laue-Langevin, B.P.156 38042 13 13
0.9 0:1 0:2057 0:9037 2:262(cid:2)10 2:437(cid:2)10 7.7%
Grenoble, Cedex 9, France.
4