Scaling and Enhanced Symmetry at the Quantum Critical Point of the Sub-Ohmic Bose-Fermi Kondo Model

0.42 MB·English

by  Stefan Kirchner

Scaling and Enhanced Symmetry at the Quantum Critical Point of the Sub-Ohmic
Bose-Fermi Kondo Model
Stefan Kirchner and Qimiao Si
Department of Physics & Astronomy, Rice University, Houston, TX 77005, USA
We consider the finite temperature scaling properties of a Kondo-destroying quantum critical
point in the Ising-anisotropic Bose-Fermi Kondo model (BFKM). A cluster-updating Monte Carlo
approach is used, in order to reliably access a wide temperature range. The scaling function for
the two-point spin correlator is found to have the form dictated by a boundary conformal field
theory,eventhoughtheunderlyingHamiltonianlacksconformalinvariance. Similarconclusionsare
8
reachedforallmulti-pointcorrelatorsofthespin-isotropicBFKMinadynamicallarge-Nlimit. Our
0
results suggest that the quantum critical local properties of the sub-ohmic BFKM are those of an
0
underlyingboundaryconformal field theory.
2

n PACSnumbers: 71.10.Hf,05.70.Jk,75.20.Hr,71.27.+a
a
J

6 Quantumcriticalityiscurrentlybeingdiscussedinthe temperature counterparts. The result is the well known
1 contexts of a wide array of strongly correlated electron scaling form[15, 16]
] systems. A prototype is provided by a family of heavy π/β 2∆
l fermion metals near their antiferromagnetic quantum <Φ(τ,T)Φ(0,T)>= C , (1)
e (cid:16)sin(πτ/β)(cid:17)
critical point (QCP). The physical properties of these
-
r materials drastically deviate from the expectations of where ∆ is the scaling dimension of Φ, a conformal pri-
t
s the traditional theory of quantum criticality[1, 2, 3, 4], mary field, and C a constant.
t. so much so that the question has been raised as to We present results here which show that the scaling
a
m whether and how the Kondo effect itself becomes crit- functions of the two-point spin correlators of the sub-
icalatthe antiferromagneticquantumtransition[5,6, 7]. Ohmic BFKM have the form dictated by Eq. (1). Sim-
-
d Through the extended dynamical mean field theory, the ilar conclusions are drawn for multi-spin correlators of
n self-consistent BFKM provides one means to elucidate the model in a large-N limit. These results are surpris-
o such Kondo-destroying quantum criticality[5]. The sub- ing, since the sub-Ohmic nature [Eq. (3), with ǫ > 0]
c Ohmic BFKM is also the appropriate low-energy model of the bosonic spectrum implies that the bulk compo-
[
for single-electron transistors attached to ferromagnetic nentoftheHamiltonianitselflacksconformalinvariance.

2 leads[8]. Oneclue[9,10,11]aboutthenatureofthequan- The results imply that the symmetry is enhanced at the
v tum criticality in the sub-Ohmic BFKM is the failure boundaryQCPoftheBFKM,insuchawaythatthelocal
3
8 of the standard description[12] in terms of fluctuations propertiesarethoseofanunderlyingboundaryconformal
7 of the classical order parameter in elevated dimensions. field theory (CFT).
1 Nonetheless, a proper field theory for the QCP is not Our focus will be the Ising-anisotropic spin-1/2
6. yet available. To address this pressing open issue, it is BFKM.Inordertoaddressthefinitetemperaturescaling
0 important to identify the symmetry of the QCP. properties, it is important to access a wide temperature
7 In this letter, we study the finite temperature scal- range with sufficiently high accuracy. Here, we develop
0 a cluster-updating Monte Carlo method, and show that
ing properties of the BFKM in some detail. We have
v: been motivated by general considerations of a bound- it can reliably reach temperatures as low as 10−4 TK0,
i where T0 is the Kondo scale of the fermion-only Kondo
X ary conformal field theory[13, 14]. The latter arises in K
problem. The wide temperature range covered distin-
many quantum impurity problems whose bulk system in
r
a the continuum limit is conformally invariant. At zero guishes this method from existing ones for Kondo-type
systems[17, 18, 19].
temperature, the s-wave component of the bulk degrees
BFKM with Ising Anisotropy: In a BFKM, a quan-
of freedom can be thought of as living on a half-plane,
tum spin is simultaneously coupled to a fermionic bath
which is composed of the imaginary time (τ) dimen-
and a bosonic one. For the Ising-anisotropic case, the
sion and the radial spatial (r) dimension; the quan-
Hamiltonian is
tum impurity is located at the boundary of the half-
plane[13]. At finite temperature (T), the extent along = J S s + E c† c
Hbfkm K · c p pσ pσ
the imaginary time direction becomes finite, of length X
pσ
β = 1/T, and periodic boundary condition (along τ)
+g˜ Sz φ +φ† + w φ†φ ,(2)
turns it into a half-cylinder of circumference β. A con- p −p p p p
X (cid:16) (cid:17) X
formal mapping between the half-plane and the half- p p
cylinder[13, 15], say z = tan(πw/β), can then be used where S is a spin-1/2 local moment, c† describes
pσ
to obtain finite-temperature correlators from their zero- a fermionic bath with a constant density of states,
2
Ptrupmδ(iωs−suEb-pO)h=mNic0(,ǫa>nd0φ):p† a bosonic bath whose spec- hdaersttoo bhaevteakthene steumchptehraattuarefirneigteioTnK0ofisinptreerseesrtv,eTd,<inTo0r-;
K
in the limit τ0 0 at finite β we approach the high-
[δ(ω ω ) δ(ω+ω )] ω 1−ǫsgn(ω). (3) temperaturefixe→dpoint. Forthenumericalvalueschosen
p p
− − ∼| |
Xp inthemanuscriptwearealwaysabletofocusonthescal-
ingpropertiesinthelowtemperaturerangeofT <<T0.
K
We adopt bosonization and a canonical
We study this model using a cluster-updating Monte
transformation[19] to map to
Hbfkm Carlo (MC) scheme. The long-range nature of the in-
teraction is most conveniently incorporated using the
′ = ΓSx+Γ Szsz +
Hbfkm z c H0 method of Ref. [22]. We specifically use a Wolff
+g˜ Sz φ +φ† + w φ†φ ,(4) algorithm[23]. The improved estimator for the spin-
p −p p p p
Xp (cid:16) (cid:17) Xp spin correlation function implies that the susceptibility
in Matsubara frequency is given by
whereH0 andszc describethe(bosonized)fermionicbath
andlocalconductionelectronspin. ThequantitiesΓand χ(iω ) =< eiωnτj 2/n >/TL, (8)
n C
Γz are,respectively,determinedby the spin-flip andlon- |XjεC |
gitudinalcomponentsoftheKondocoupling. Integrating
out both the fermionic and bosonic baths, we arrive at where the sum runs over all spins in the cluster, <>
the partition function Z′ Tr exp[ ′ ] with indicatestheaverageoverallMonteCarlorunsandn is
bfkm ∼ −Simp C
thenumberofspinsinagivencluster. Thesusceptibility
β 1 β as a function of imaginary time τ is given by χ(τn) =
Si′mp =Z0 dτ[ΓSx(τ)− 2Z0 dτ′Sz(τ)Sz(τ′) 41 Smz,Smz+nεC <Sz(τm+τn)Sz(τm)>, where τn =nτ0.
(χ−1(τ τ′) (τ τ′))], (5) WPe measure χ(iωn) and χ(τn) directly.
0 c
× − −K − Thismethodallowsustoreachconsiderablylowertem-
where the trace is taken over spin degrees of freedom. peratures than approaches using local updates[18, 19].
cχa−0m1e=frg˜o2mPinptGegφr,0a,tianngdoKutc(tihωen)fe=rmκioc|nωicn|b(aκtch.∼TΓr2zoNtt02e)r WabeoutytpNicraulnly=bui1ld0620M0C0 MclCusctelurssteartshaisgha wteamrmpe-uraptuarneds
decomposing the effective action and re-expressing the andincreasethe number of clusters built to about 50000
leadingorder(in1/L,Lbeingthenumberoftime slices) warm-ups and about Nrun = 1010 clusters at β = 512
through the transfer matrix for a one-dimensional Ising for all τ0. While every cluster built contributes to
model[20], one finally obtains, χ(τ = 0) only the subset of clusters with spins sepa-
rated by τ β/2 will contribute to χ(τ β/2). One
≥ ≈
Tr exp KNNSiSi+1+ KLR(i j)SiSj . (6) mightthereforeexpectthatvarianceandautocorrelation
Z ∼ −
(cid:2)Xi Xi,j (cid:3) effects strongly depend on τ, but this turned out not to
be the case. For an error estimate, we performed a bin-
The mapping procedureis essentially equivalent to what ninganalysisofourdatainordertoobtaintheintegrated
was done for the pure Kondo model[21]. The effective autocorrelationtime τ and variance [30]. The relative
int
action at inverse temperature β is equivalent to that of error of our results is (∆χ)/χ 10−2 and below (de-
a one-dimensional chain of L Ising spins, with a peri- pending on τ and β) and the int≈egrated autocorrelation
odic boundary condition. The nearest neighbor inter- time is τ <100 N for all τ and β.
int run
action is KNN = ln(τ0Γ/2)/2, where τ0 = β/L; it is For concreteness,≪we will now present the results for
−
singular in the limit τ0 0. KLR(i j) is the sum of ǫ=0.4.
→ −
two ferromagnetic long-ranged interactions proportional ConsiderfirsttheKondolimit(g =0). InFig.1(a)we
to 1/i j 2 and 1/i j 2−ǫ; it results from discretizing show the static spin susceptibility versus temperature.
(χ−01(|τ− τ|′) c(τ| −τ′|)): It correctly captures the Pauli behavior at temperatures
− −K −
below T0. Because we have placed the Kondo couplings
τ2 2α(π/β)2 g(π/β˜)2−ǫ K
K (i j )= 0 + . (7) at the Toulouse point, we can compare our results with
LR | − | 4 hsin(πτ0|i−j|)2 sin(πτ0|i−j|)2−ǫi the exact expression: Fig. 1(a) demonstrates the agree-
β β˜
ment for more than 4 decades of temperature! In addi-
The coupling constant α is related to the electron scat- tion, our results for the dynamical susceptibility[24] are
tering phase shift. We choose α = 1/2, so that g = 0 consistentwith the standardexpectations for the Kondo
corresponds to the Toulouse limit of the Kondo prob- problem, including the asymptotic long time (low fre-
lem. For the most part, the parameter β˜ β is taken quency) Fermi-liquid power-law behavior and the exact
≫
to be 20000τ0. (In the cases we have checked, we found limit at short time, χ(τ τ0) 1/4.
→ →
identical results when the second term in the brackets is Fig. 1(a) also shows the static susceptibility at the
simply replaced by g/τ2−ǫ.) The thermodynamic limit QCP, g /T0 = 0.821. We find χ (T,g ) 1/T0.608,
c K stat c ∼
3
10 (a) f(x)=0.333x−0.608
·0χTT()statK01.1 Tf(oxu)lo=us0e.3l3im3ixt−0.608 ·0χTT()statK150 ττττττττ00000000−−−−−−−−11111111========4681123626024
cKroitnicdaolbcoehuapvliinogr((ggc=/T0K0),=τ0−01.8=211)6,τ0−1=64 (b)
10−4 10−3 10−2 10−1 1 010−3 10−2 10−1 1
T/TK0 T/TK0
FIG. 1: (a) Static local spin susceptibility in the Kondocase
(g =0, τ0 =1/16, black circles) and for the critical coupling
(ǫ = 0.4,Γ = 0.75,gc = 0.821TK0,τ0 = 1/64, red diamonds).
The dashed blue line is the fit to the Toulose limit (see e.g.
Ref. [29]), and the dashed dotted line a fit to the critical
behavior. Defining TK0 ≡ 1/χstat(T = 0) we obtain TK0τ0 ≈ FIG. 2: Scaling of the local spin susceptibility (at the
0.688; (b) Static susceptibility at the critical coupling, for QCP), which is plotted as a function of πTτ0/sin(πτT).
various values of thecutoff parameter, τ0. The parameters are ǫ = 0.4,Γ = 0.75,g = 0.821T0 ≈
K
gc,τ0 = 1/64. Note that πTτ0/sin(πτT) becomes small (or-
der πτ0/β) as τ approaches the long-time limit, τ → β/2.
for over two decades of temperature. Since χ(ω,gc) The power-law collapse occurs over two decades of the pa-
1/ω1−ǫ is expected[25, 26], the temperature exponent i∼s, rameterπTτ0/sin(πτT). Thedeviationatthelow-leftcorner
withinabout1%accuracy,the sameasthefrequencyex- is attributed to finite-size effects.
ponent; this is consistent with the NRG result[11]. The
dependence ofthecriticalsusceptibilityonthe cutoffpa- contains N2 1 components. This dynamical large-N
rameter τ0 is illustrated in Fig. 1(b). Compared to the limit[27, 28] i−s expressed in terms of pseudo-fermions f
σ
KHoowndevoerc,asweit(hnionttshheowmne)a,suthreisd dteempepnedreantucereisrasntgroe,ngtehre. andabosonicdecouplingfieldBα,whereSσ,σ′ =fσ†fσ′−
δσ,σ′Q/N, where Q is related to the chosen irreducible
resultdoesnotchangesignificantlyforthesmallestthree
representation of SU(N)[27]. The large-N equations are
τ0 values. We now turn to the τ-dependence of the dy-
namical spin susceptibility, χ(τ,T), near the quantum ΣB(τ) = 0(τ)Gf( τ);
−G −
cstrritaitceasltchoautpχli(nτg,Tg)≈isgac.fuTnhcetisocnaolifnπgTp/losti,nF(πigτ.T2),odnelmy.oInn- G−1Σ(ifω(τ)) == κ1/GJ0(τ)GΣB(τ(i)ω+)g;2Gf(τ)GΦ(τ);
the long-timelimit (lower-leftcorner),the dependence is B n K − B n
G−1(iω ) = iω λ Σ (iω ); (11)
asimplepower-lawforovertwodecadesofπT/sin(πτT). f n n− − f n
We therefore reach one of our key results, namely together with a constraint G (τ 0−) = Q/N.
f
→
χcrit(τ,T) = Φ(cid:0)sinπ(τπ0τTT)(cid:1) T−≪→TK0 c·(cid:0)sinπ(τπ0τTT)(cid:1)ǫ, (9) H−ehTreτ,cσκα(=τ)cM†σα/(N0),i0λ, aisndaGLΦag=ranhgTiτaΦn(mτ)uΦlt†i(p0l)iie0r., GN0ot=e
that, when g = 0, the Kondo Hamiltonian contains a
for τ−1 T0; here c is a constant ( 0.89). In conformally-invariant bulk and the corresponding corre-
≪ K ≈
otherwords,inthelong-time(low-energy)limit,thetwo- lation functions naturally have the form of a boundary
point spin correlator has precisely the form dictated by CFT[27,28]. Hereweaddresswhathappens attheQCP
a boundary CFT, inspite of the lack of conformal in- of the model with finite g [9], for which the bulk lacks
variance in the Hamiltonian. This scaling form implies conformal invariance.
ω/T-scaling. Consider first the zero-temperature case. The quan-
We now turn to complementary results on the multi- tum critical properties of the model have been deter-
pointcorrelators,whichareprovidedbythelarge-Nlimit mined in Ref. [9]. At g = g , both the pseudo-fermion
c
of a spin-isotropic BFKM. propagator G (τ) and the auxiliary boson propagator
f
Spin-isotropic BFKM in a large-N limit: The limit is G (τ) are critical; their leading terms are G (τ) =
B f
taken for the Hamiltonian of the SU(N) S(M) BFKM, A/τ ǫ/2sgn(τ) and G (τ) = B/τ 1−ǫ/2, respectively.
× | | B | |
Here, we observe that the local two-spin correlator, as
HMBFK = (JK/N) S·sα+ Ep c†pασcpασ well as all the local higher-multiple-spin correlators,fac-
X X
α p,α,σ torize in terms of G (τ) according to Wick’s theorem.
f
+ (g/√N)S Φ+ w Φ† Φ , (10) This immediately implies that the scaling functions for
· p p · p
Xp all these correlators have the form of a boundary CFT.
The dynamical spin susceptibility, e.g., is
where the spin andchannel indices areσ =1,...,N and
α = 1,...,M, respectively, and Φ ≡ p(Φp + Φ−†p) χ(τ)≡hTτSσ6=σ′(τ)Sσ′σ(0)i=−Gf(τ)Gf(−τ), (12)
P
4
1 1 an enhanced symmetry. This insight is expected to be
Gτ,T()B1100−−42 TTTT////TTTTKKKK,,,,0000====1111....66667777····111100000−−−123 Gτ,T()f10−1 TTTTTT//////TTTTTTKKKKKK,,,,,,000000======111111......666666777777······111111000000−−−−−−654321 itcmhiaeWploloyrer,ytatAohnf.attnWhfkoisr.CKtWh.oe.nJduL.onu-Bdddeowerlsisetgtcrahofn,yodirnHingu.gsqeGouffua.nlthEtduevimseucrnutczdsrsietiaroilcnnyadsiln.,pgeoTfisinpehtleid.s-
10−6 (a) fTTT(x///TTT)KKK=,,,0000.===16118...x6630773.8···5111000−−−456 10−2 (b) fTT(x//TT)KK=,,000.==12189..63x730.··1115008−−78 wDoMrkR-h0a7s06b6e2e5n,thsuepRpoobrteerdtAin.WpaelrcthbFyouNnSdFatiGonr,atnhteNWo..
10−8 10−6 10−4 10−2 1 10−8 10−6 10−4 10−2 1 M. Keck Foundation, the Rice Computational Research
πT πT
sin(πτT) sin(πτT) Cluster funded by NSF and a partnership between Rice
University, AMD and Cray, and (for S.K.) DOE Grant
FIG. 3: Scaling of the propagators for the auxiliary boson, No. DE-FG-02-06ER46308.
GB(τ) [panel (a)] and for the pseudo-fermion, Gf(τ) [panel
(b)],forǫ=0.3andthenumericalparametersspecifiedinthe
main text,at thecritical coupling gc =25.5TK0.
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[30] The integrated autocorrelation time of χ is defined as
quantum critical properties of the BFKM. Our results τint = 12Pi(<χkχk+i >−<χ>2)/(<χ2 >−<χ>2
suggestthatthequantumcriticalpointoftheBFKMhas ),whereχk andχk+1 aresuccessivemeasurementsofχ.