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Spins coupled to a Spin Bath: From Integrability to Chaos John Schliemann Institute for Theoretical Physics, University of Regensburg, D-93040 Regensburg, Germany (Dated: January 14, 2010) Motivated by the hyperfine interaction of electron spins with surrounding nuclei, we investigate systems of central spins coupled to a bath of noninteracting spins in the framework of random matrixtheory. WithincreasingnumberofcentralspinsatransitionfromPoissonianstatisticstothe Gaussian orthogonal ensemble occurs which can be described by a generalized Brody distribution. Theseobservationsareunaltereduponapplyinganexternalmagneticfield. Inthetransitionregion, the classical counterparts of the models studied have mixed phase space. 0 1 0 Spins coupled to a bath of other spin degrees of free- random realizations of coupling parameters. Note that 2 dom occur in a variety of nanostructures including semi- the Hamiltonian matrix represented in the usual basis of n conductor quantum dots [1–4], carbon nanotube quan- tensor-product eigentstates of Sz, Iz is always real and α i a tum dots [5], phosphorus donors in silicon [6], nitrogen symmetric. Therefore, the natural candidate for a ran- J vacencycentersindiamond[7–9],andmolecularmagnets dom matrix description of such systems is the Gaussian 4 [10]. A large portion of the presently very high both ex- orthogonal ensemble (GOE) [14]. 1 perimentalandtheoreticalinterestinsuchsystemsisdue In the important case of a single central spin, N =1, ] toproposalstoutilizesuchstructuresforquantuminfor- c l the above model has the strong mathematical property l mation processing [11–13]. Here the central spins play a of being integrable [16, 17]. Moreover, this integrability the role of the qubit whereas the surrounding bath spins h is particularly robust as it is independent of the choice s- atecrtwaes iannvedsteicgoahteerivnegryebnvasiriconpmroepnet.rtieIns otfhseupchressoe-nctallleetd- of the coupling parameters A(i1) and the length of the e spins which can even be chosen individually [16, 17]. In m central spin systems in terms of spectral statistics and fact, the model (1) for a single central spin has been random matrix theory [14]. . the basis of numerous theoretical studies on decoherence t The generic Hamiltonian is given by a properties of quantum dot spin qubits; see, for example, m - H= (cid:88)Nc S(cid:126) ·(cid:88)N A(α)I(cid:126) (1) Resetfisn.g[1q5u,e1st8i–o2n1,]b,ofothr rferovmiewasparlasoct[i2ca2l–2a4s]w. eItlliassafnroimntearn- d α i i n α=1 i=1 abstractpointofview,towhatextendtheresultsofthese o investigationsarelinkedtotheintegrabilityoftheunder- c describingNccentralspinsS(cid:126)αcoupledtoN bathspinsI(cid:126)i, lying idealized model. In particular, what changes may [ typically N (cid:29) N . Here we take all spins to be dimen- c occur if the Hamiltonian deviates from the above simple sionless quantum variables such that the coupling con- 2 case N = 1 by, e.g., involving more than one central c v stants A(α) have dimension of energy. A paradigmatic spin? Previous investigations of decoherence properties, i 3 example is given by, say, a single spin of a conduction- making strongly restrictive assumptions on the coupling 3 band electron residing in a semiconductor quantum dot constants,predictedasignificantdependenceonwhether 3 3 and being coupled via hyperfine contact interaction to the number of central spins is even or odd [25, 26]. In . the bath of surrounding nuclear spins. In a very typical thefollowingwewillinvestigateHamiltoniansofthegen- 2 material like gallium arsenide all nuclei have a spin of eral type (1) within the framework of level statistics, i.e. 0 9 I = 3/2 whereas in other systems like indium arsenide generic spectral characteristics [14]. For other studies 0 evenspinsoflengthI =9/2occur. Infact,thishyperfine of interacting quantum many-body systems using this : interaction with surrounding nuclei has been identified method see e.g. Refs. [27–29]. v i to be the limiting factor regarding coherent dynamics of X The spectra generated numerically from the Hamilto- electron spin qubits [1–3, 15]. In the above example the nian (1) clearly have a nontrivial overall structure, i.e. r hyperfinecouplingconstantsA(α) areproportionaltothe a i the locally averaged density of states is not constant as squaremodulusoftheelectronicwavefunctionatthelo- a function of energy [19, 22]. Therefore an unfolding cation of the nucleus and can therefore vary widely in of these spectra has to be performed which results in a magnitude. For the purposes of our statistical analysis transformation onto a new spectral variable s such that here we shall take an even more radical point of view the mean level density is equal to unity [14]. We have and choose the A(α) at random. To be specific, we will i compared several standard numerical unfolding proce- choose the A(α) from a uniform distribution within the dures and made sure that they yield consistent results. i interval[0,1]andnormalizethemafterwardsaccordingto Fig. 1 shows the probability distribution p(s) for the (cid:80) A(α) = 1 for each central spin. The data to be pre- nearest-neighbor level spacing for a system of a single i i sented below is obtained by averaging over typically 500 centralspinS =1/2and13bathspinsoflengthI =1/2 1 2 for several subspaces of the total angular momentum be viewed as a perturbation. This term vanishes if the J(cid:126) = S(cid:126) + (cid:80) I(cid:126) where each multiplet is counted as a coupling constants are still random but chosen to be the 1 i i single enery level. The subspaces of highest J = 5,6,7 sameforeachspin,A(1) =A(2),resultinginanintegrable i i have been discarded, and in the bottom right panel all modeloftwocentralspins,apredictionwehaveexplicitly probability distributions are joined. As to be expected verifiedinournumerics;thelattermodelwasalsostudied for an integrable model, the level statistics follow a Pois- numerically in Ref. [25]. son distribution resulting in an exponential level spacing The models studied so far have a common spin bath, distributionp(s)=e−s. Thisisincontrasttothecaseof i.e. each bath spin couples without any further restric- two central spins S1 = S2 = 1/2 shown in Fig. 2. Here tion to each central spin. Regarding the generic exam- levelrepulsiontakesclearlyplace,p(0)=0,althoughthe ple of two neighboring quantum dot spin qubits this is data considerably deviates from the Wigner surmise for not particularly realistic since in this geometry one can the GOE [14], p(s)=(π/2)sexp(−(π/4)s2). obviously identify groups of nuclear spins which couple Obviously, our numerical studies are technically re- strongly to one of the electron spins but weakly to the stricted to rather small system sizes, Nc+N ≤14. This other. The extreme case is given by two separate spin limitation, however, does not affect our results for the baths where the central spins can be coupled via an ex- level statistics as demonstrated in Fig. 3 where we have change interaction [31, 32], H(cid:48) = H+J S(cid:126) ·S(cid:126) . Here ex 1 2 plotted the same data as in the bottom right panel of we find numerically that even arbitrary small exchange Fig. 2 but for N = 10 and N = 11 bath spins. This parameters J break integrability and lead to level re- ex insensitivity to the system size seen in the figure is a pulsion. The corresponding level spacing distributions, natural consequence of the unfolding of the spectra. however, are less accurately described by the ansatz (2). Fig4showsthejointlevelspacingdistributionforJ = Ontheotherhand, forlarge|J |thesystemapproaches ex 0,...,4 and increasing number of central spins, where the integrable scenario since then the singlet and triplet p(s) approaches closer and closer the Wigner surmise. subspace of the central spins are energetically more and To quantify this observation we use the ansatz more separated. Let us now discuss the influence of an external mag- p(s)=Bsβe−Asα (2) netic field coupling to the central spins. In the case N = 1 the resulting model is known to be integrable with c [16,17], andalsofor N >1theHamiltoniancanstillbe c represented as a real and symmetric matrix. Indeed, we B = α(Γ((β+2)/α))β+1/(Γ((β+1)/α))β+2, (3) havenotseenanyqualitativedifferenceinthelevelspac- A = (Γ((β+2)/α)/Γ((β+1)/α))α (4) ing distribution with and without an external magnetic (cid:82) (cid:82) field. Inparticular,wehavenotfoundanysignforatran- such that p(s) = sp(s) = 1. Clearly, α = 1, sition between the Gaussian orthogonal to the unitary β =0correspondstoanexponentialdistributionwhereas ensemble (as appropriate for systems lacking time rever- α = 2, β = 1 reporduces the Wigner surmise. The sal symmetry [14]). In this sense, the application of an aboveansatzgeneralizestheBrodydistributiongivenby externalmagneticfieldcanbeviewedasa“falsesymme- α=β+1[14,30]. AsseeninFig.4,thenumericaldatais try breaking” which still preserves a “non-conventional verywelldescribedbytheabovedistribution,andforthe time-reversal invariance” [29, 33]. We note that recent case of 5 central spins the Wigner surmise of the GOE is theoretical works predict different time dependencies of almost reached. In particular, our level statistics do not spindynamicsindifferentmagnetic-fieldregimes[20,21]. show any odd/even-effects with respect to the number These observations are not reflected by the level statis- of central spins as predicted in Refs. [25, 26]. We at- tics. Thus, decoherence and the occurrence of integra- tribute this difference to the strongly restrictive assump- bility or chaoticity are independent phenomena in such tions made there giving rise to additional symmetries. systems, at least as far as the role of magnetic fields is Moreover, in the case of two central spins it is instruc- concerned. tive to rewrite the Hamiltonian in the following form, The data presented so far was obtained for bath spins H = (S(cid:126) +S(cid:126) )·(cid:88)1(A(1)+A(2))I(cid:126) of length I = 1/2. Motivated by the large nuclear spins 1 2 2 i i i in semiconductor materials, we have also performed sim- i ulationsforI =1whichalsodonotshowanyqualitative + (S(cid:126) −S(cid:126) )·(cid:88)1(A(1)−A(2))I(cid:126) . (5) difference to the previous case. This is indeed to be ex- 1 2 2 i i i pected since a spin bath of I = 1 can be obtained from i a bath with I =1/2 and twice the number of bath spins ThetwocentralspinsS(cid:126) =S(cid:126) +S(cid:126) cancoupletoS =0,1. bygroupingthespinsintopairsandchosingthecoupling 1 2 SincethecouplingtothesingletS =0vanishes, thefirst parameters to be the same in each pair. Similar consid- line in Eq. (5) is just the integrable Hamiltonian of a erations apply to higher bath spins. single central spin S = 1, whereas the second line can Let us come back to the case of two central spins. As 3 seeninFigs.2,4,thissystemappearstolieinbetweenthe integrablecaseandthepredictionsofrandommatrixthe- ory. Thus, in the light of the Bohigas-Giannoni-Schmitt [1] J. R. Petta et al., Science 309, 2180 (2005). conjecture [34], it is natural to speculate that the classi- [2] F. H. L. Koppens et al., Nature 442, 766 (2006). cal counterpart of this system has a mixed phase space [3] R. Hanson et al., Rev. Mod. Phys. 79, 1217 (2007). consisting of areas of regular and of chaotic dynamics. [4] P.-F. Braun et al., Phys. Rev. Lett. 94, 116601 (2005). Theclassicallimitofaquantumspinsystemisnaturally [5] H. O. H. Churchill et al., arXiv:0811.3236. obtained via spin-coherent states, and a pair of classical [6] E. Abe et al., Phys. Rev. B 70, 033204 (2004). canonicallyconjugatevariablesp,q foreachspinisgiven [7] F. Jelezko et al., Phys. Rev. Lett. 92, 076401 (2004). by p = cosϑ, q = ϕ, where ϑ, ϕ are the usual angu- [8] L. Childress et al., Science 314, 281 (2006). [9] R. Hanson et al., Science 320, 352 (2008). lar coordinates of the classical spin unit vector [35]. We [10] A. Ardavan et al., Phys. Rev. Lett. 98, 057201 (2007). have performed numerical Runge-Kutta simulations of [11] D. Loss and D. P. DiVincenzo, Phys. Rev. A 57, 120 such classical dynamics for N = 2 where one can easily c (1998). treatsystemsofseveralthousandbathspins. However,to [12] B. E. Kane, Nature 393, 133 (1998). avoidthecomplicationsofsuchahigh-dimensionalphase [13] M. Leuenberger and D. Loss, Nature 410, 789 (2001). spaceletusconcentrateonthesmallestnontrivialcaseof [14] ForareviewseeT.Guhr,A.Mu¨ller-Groeling,andH.A. just two bath spins. Here we find indeed a close vicinity Weidenmu¨ller, Phys. Rep. 299, 189 (1998). [15] A. V. Khaetskii, D. Loss, and L. Glazman, Phys. Rev. of regular and chaotic dynamics. An example is shown Lett. 88, 186802 (2002). in Fig. 5 where we have plotted in the top panel a cut [16] M. Gaudin, J. Phys. (Paris) 73, 1087 (1976). through the plane Iz = Iz = 0 as a function of p = Sz, 1 2 1 [17] M.BortzandJ.Stolze,Phys.Rev.B76,014304(2007). q =tan−1(Sy/Sx). The coupling constants are given by [18] I. A. Merkulov, A. L. Efros, and M. Rosen, Phys. Rev. 1 1 A(1) = 1−A(1) = 0.75, A(2) = 1−A(2) = 0.52, and the B 65, 205309 (2002). 1 2 1 2 initial condition is S(cid:126) = −S(cid:126) = (0,0,−1), I(cid:126) = (1,0,0), [19] J.Schliemann,A.V.Khaetskii,andD.Loss,Phys.Rev. 1 2 1 B 66, 245303 (2002). I(cid:126) = (0,1,0). This arrangement leads obviously to very 2 [20] W. A. Coish, J. Fischer, and D. Loss, Phys. Rev. B 77, regular dynamics, in stark contrast with the bottom 125329 (2008). panel where we have used the same initial condition but [21] L.Cywinski,W.M.Witzel,andandS.DasSarma,Phys. introduced a minute change in one pair of coupling con- Rev. Lett. 102, 057601 (2009). stants, A(2) = 1−A(2) = 0.5195, resulting in a clearly [22] J. Schliemann, A. V. Khaetskii, and D. Loss, J. Phys.: 1 2 Condens. Mat. 15, R1809 (2003). chaotic orbit with an inhomogeneous phase space fill- [23] W. Zhang et al., J. Phys.: Condens. Mat. 19, 083202 ing. Note that the observation that certain phase space (2007). curves are overlaid in the figure is due to the fact that [24] D. Klauser et al., arXiv:0706:1514. the remaining six phase space variables are not uniquely [25] V. V. Dobrovitski et al., Phys. Rev. Lett. 90, 210401 determined by the condition Iz = Iz = 0 and the con- (2003). 1 2 servedquantitiesH=0,J(cid:126)=(1,1,0)butoccurinseveral [26] A. Melikidze et al., Phys. Rev. B 70, 014435 (2004). [27] G. Montambaux et al., Phys. Rev. Lett. 70, 497 (1993). branches. [28] B. Georgeot and D. L. Shepelyansky, Phys. Rev. Lett. In summary, we have investigated central spin mod- 81, 5129 (1998). els via nearest-neighbor level spacing distributions. As [29] Y. Avishai, J. Richert, and P. Berkovits, Phys. Rev. B the number of central spins increases a transition from 66, 052416 (2002). Poissonian statistics to the Gaussian orthogonal ensem- [30] T. A. Brody, Lett. Nuov. Cim. 7, 482 (1973). blesetsinwhichcanbedescribedbyageneralizedBrody [31] G. Burkard, D. Loss, and D. P. DiVincenzo, Phys. Rev. B 59, 2078 (1999). distribution. These observations are not affected by the [32] J. Schliemann, D. Loss, and A. H. MacDonald, Phys. finite system size in our numerical simulations and are Rev. B 63, 085311 (2001). unaltered upon applying an external magnetic field. In [33] F. Haake, Quantum Signatures of Chaos, Springer, the transition region, the classical counterparts of the Berlin, 2000. models studied have mixed phase space. [34] O.Bohigas,M.J.Giannoni,andC.Schmitt,Phys.Rev. I thank M. Brack, K. Richter, S. Schierenberg, and T. Lett. 52, 1 (1984). Wettig for useful discussions. This work was supported [35] See, e.g., J. Schliemann and F. G. Mertens, J. Phys.: Condens. Mat.10, 1091 (1998). by DFG via SFB 631. 4 FIG. 1: (Color online) Nearest-neighbor level spacing for a system of a single central spin S =1/2 and 13 bath spins of 1 length I =1/2. The red curve is the exponential p(s)=e−s. 5 FIG. 2: (Color online) Nearest-neighbor level spacing for a system of a two central spins S = S = 1/2 and 12 bath 1 2 spins of length I = 1/2. The red curve is the GOE Wigner surmise. FIG. 3: (Color online) The same data as in the bottom right panel of Fig. 2 but for N =10 and N =11 bath spins. 6 FIG. 4: (Color online) Joint level spacing distribution for J =0,...,4 and increasing number of central spins. The red dashed lines are the exponential function (N = 1) and the c Wigner surmise (N > 1). The green solid lines are a fit to c the generalized Brody distribution (2). FIG.5: AcutthroughtheplaneIz =Iz =0oftwodifferent 1 2 phasespaceorbitsdemonstratingtheclosevicinityofregular and chaotic dynamics in a system of two central spins (see text).