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. [1] H.v.L¨ohneysenetal.,Rev.Mod.Phys.79,1015(2007). whose leading behavior is 1/τǫ. Likewise, the three- [2] A. Schr¨oder et al., Nature407, 351 (2000). point correlator is 1/τ12τ13τ23 ǫ/2, and the four-point [3] S. Paschen et al.,Nature 432, 881 (2004). correlator τ−∼ǫ/3F|(x); here| the cross-ratio x = [4] P. Gegenwart et al.,Science 315, 969 (2007). i<j ij [5] Q. Si et al.,Nature413, 804 (2001). τ12τ34/τ13τ2P4, and τij ≡ τi − τj. All these are consis- [6] P. Coleman et al., J. Phys.: Conden. Matt. 13, R723 tent with the general form of a (boundary) CFT[15]. (2001). At finite temperatures we solve equations (11) on [7] T. Senthilet al.,Phys. Rev.B 69, 035111 (2004). real frequencies. The numerical parameters are as in [8] S. Kirchner et al., Proc. Natl. Acad. Sci. USA 102, Ref. [9]: We choose κ = 1/2, Q/N = 1/2, and N0(ω) = 18824 (2005); S. Kirchner and Q. Si, Physica B (2008), (1/π)exp( ω2/π) for the conduction electron density of DOI:10.1016/j.physb.2007.10.297; arXiv:0707.0062. states. T−he nominal bare Kondo scale is TK0N0(0) ≡ [1[90]] LM..ZVhoujteateatl.a,lP.,hPyhs.yRs.eRv.evL.eLtte.t9t.39,42,6702700160(420(0240)0.5). Texhpe(−bo1s/oNn0ic(0b)aJtKh)sp≈ec0tr.0a2l,fufnocrtifioxned JKδ(Nω0(0w) =) 0.ω81/−πǫ. [11] M. T. Glossop and K. Ingersent, Phys. Rev. Lett. 95, p − p ∼ 067202 (2005). is cut off smoothly at 2ωN0(0) ≈ 0P.05. The imaginary [12] J. Hertz, Phys.Rev.B 14, 1165 (1976). time correlation functions are then obtained from [13] I.AffleckandA.W.W.Ludwig,Nucl.Phys.B360,641 ∞ exp( τω) (1991). Φ(τ) = η dω − Im(Φ(ω+i0+)), (13) [14] J. L. Cardy,Nucl. Phys. B 240, 514 (1984). − Z−∞ exp(−βω)−η [15] P. Ginsparg, in Fields, Strings and Critical Phenomena (Elsevier, 1989). for 0<τ β. Here, η = for bosonic/fermionic Φ. 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Rev.B 32, 4410 (1985). In summary, we have studied the finite-temperature [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.