Spin polarization through Floquet resonances in a driven central spin model

Pieter W. Claeys,1, 2, 3, ∗ Stijn De Baerdemacker,2 Omar El Araby,1 and Jean-Sebastien Caux1

1Institute for Theoretical Physics Amsterdam and Delta Institute for Theoretical Physics,University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands

2Department of Physics and Astronomy, Ghent University, Krijgslaan 281 S9, B-9000 Ghent, Belgium3Center for Molecular Modeling, Ghent University, Technologiepark 903, 9052 Ghent, Belgium

Adiabatically varying the driving frequency of a periodically-driven many-body quantum systemcan induce controlled transitions between resonant eigenstates of the time-averaged Hamiltonian,corresponding to adiabatic transitions in the Floquet spectrum and presenting a general tool inquantum many-body control. Using the central spin model as an application, we show how suchcontrolled driving processes can lead to a polarization-based decoupling of the central spin fromits decoherence-inducing environment at resonance. While it is generally impossible to obtain theexact Floquet Hamiltonian in driven interacting systems, we exploit the integrability of the centralspin model to show how techniques from quantum quenches can be used to explicitly construct theFloquet Hamiltonian in a restricted many-body basis and model Floquet resonances.

Introduction. – Periodically driven systems have a richhistory ranging from the simple kicked rotor to recentexperimental progress on cold atoms in optical fields1,2.The dynamics in driven systems has remarkable features,such as the absence of a well-defined adiabatic limit3,4

and the heating to infinite temperature which is expectedto occur5–9. The same physical mechanism underliesthese phenomena – in the presence of periodic driving itis possible for states to interact resonantly. States whoseenergies are separated by an integer multiple of the driv-ing frequency will interact strongly, leading to Floquetor many-body resonances10–12.

While this is generally seen as a disadvantage becauseof the experimental problems posed by heating, there ishope that in large but finite systems such many-body res-onances can be well understood and even controlled. Thisopens up ways of engineering specific many-body reso-nant quantum states by adiabatically tuning the drivingfrequency to resonance. Such “driven driving” protocols,if smartly conceived, could even lead to states with prop-erties beyond these associated with the (physical) driv-ing Hamiltonians13–18. This is illustrated on the centralspin model, which is adiabatically driven such that thecentral spin becomes completely decoupled from its envi-ronment, incompatible with the physics of the instanta-neous Hamiltonians. This model describes the (inhomo-geneous) interaction of a central spin on which a magneticfield is applied with an environment of surrounding spins,being important in the study of quantum dots, solid-statenuclear magnetic resonance, and the nitrogen-vacancydefect in diamond, a promising qubit system19. A ma-jor experimental challenge remains the decoherence dueto the presence of an environment, motivating numerousstudies20–30. Surprisingly, Floquet resonances can herebe used to construct pure spin states, seemingly at oddswith the inevitable interaction with the environment andresulting decoherence effects.

All such resonances are encoded in the spectrum ofthe Floquet Hamiltonian, governing periodic dynamics.However, due to the exponential scaling of the Hilbertspace and the inherently non-diagonal nature of time

evolution operators, it is generally impossible to obtainthis Hamiltonian in realistically-sized interacting sys-tems. In the present paper we exploit that the sys-tem is driven by periodically switching between inte-grable Hamiltonians31,32, and we show how techniquesfrom quantum quenches in integrability can be adaptedto accurately model such transitions by constructing a(numerically) exact Floquet Hamiltonian in a restrictedHilbert space spanned by the resonant (Bethe ansatz)eigenstates of the integrable time-averaged Hamiltonian.This also presents a first step toward applying the tool-box from integrability to driven systems, where inte-grability is generally expected to lose its usefulness be-cause of the general non-integrability of the FloquetHamiltonian33–35.Floquet theory. – The key result in the study of

periodically-driven systems is the Floquet theorem36, re-casting the unitary evolution operator as

U(t) = P (t)e−iHF t, (1)

with P (t) a periodic unitary operator with the same pe-riod T as the driving, satisfying P (T ) = 1, and HF theFloquet Hamiltonian (with ~ = 1). Considering time evo-lution over one full cycle leads to the Floquet operator,from which HF can be extracted as

UF ≡ U(T ) = e−iHFT . (2)

Simultaneously diagonalizing these operators leads to

HF =∑

n

εn |φn〉 〈φn| , UF =∑

n

e−iθn |φn〉 〈φn| , (3)

with quasi-energies εn = θn/T . These provide the Flo-quet equivalent of quasi-momenta in Bloch theory, simi-larly only defined up to shifts k · 2π/T, k ∈ N, and quasi-energies separated by shifts k · 2π/T, k ∈ N are said tobe quasi-degenerate. Crucially, the Floquet Hamiltonianitself also depends on the driving period T . Consider aperiodic quenching driving protocol

H(t) =

{H1 for 0 < t < ηT,

H2 for ηT < t < T,(4)

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018

2

with H(t+ T ) = H(t) and η ∈ [0, 1], leading to

UF ≡ e−iHFT = e−i(1−η)TH2e−iηTH1 . (5)

Obtaining the Floquet Hamiltonian from this expres-sion is a non-trivial task, with exact results limitedto systems where there is a clear commutator struc-ture in all involved Hamiltonians37 (e.g. non-interactingsystems38–40) or small systems for which exact diagonal-ization is feasible41.

Still, the dependence of the Floquet Hamiltonian onthe driving frequency has become well-understood in re-cent years4,5,12. At high driving frequencies, the Flo-quet Hamiltonian can be accurately approximated byan effective Hamiltonian, leading to strongly suppressedheating42–45. This effective Hamiltonian can be ob-tained from the Magnus expansion, where the time-averaged Hamiltonian HAvg = ηH1 + (1− η)H2 presentsa first-order approximation46–49. Lowering the driv-ing frequency 2π/T , many-body resonances are intro-duced where quasi-degenerate eigenstates of this effec-tive Hamiltonian interact strongly and hybridize4,5,12.Further lowering the driving frequency, these many-body resonances multiply and lead to so-called ‘infinite-temperature states’. Here, it is crucial to consider finitesystems with a bounded spectrum, since the unboundedspectrum in infinitely large systems immediately leadsto a proliferation of many-body resonances and ‘infinitetemperature’ Floquet eigenstates at all possible drivingfrequencies. However, by tuning the driving frequency infinite systems it remains possible to target specific reso-nances of non-trivial states, as will be illustrated.

The central spin model. – The model Hamiltonian is

H = BzSz0 +

L∑

j=1

Aj ~S0 · ~Sj , (6)

with Sα0 and Sαj the spin operators of the central spinand the environment respectively. These are taken to bespin-1/2 particles, and the coupling constants are com-monly chosen as Aj = exp [−(j − 1)/L], correspondingto a quantum dot in a 2D Gaussian envelope50. How-ever, the integrability of the central spin model is versa-tile enough that our proposed method holds for arbitraryspins and parametrizations. For consistency with the lit-erature on integrability, we set εj = −A−1j and ε0 = 0.The exact Bethe ansatz eigenstates

|Bz; v1 . . . vN 〉 =N∏

a=1

L∑

j=0

S+j

εj − va

|↓ . . . ↓〉 , (7)

depend on variables {v1 . . . vN} satisfying Bethe equa-tions

B−1z +1

2

L∑

j=0

1

εj − va=

N∑

b6=a

1

vb − va, ∀a = 1 . . . N,

(8)

leading to energies

E(Bz; {v1 . . . vN}) =1

2

N∑

a=1

v−1a −1

4

L∑

j=1

Aj −1

2Bz. (9)

Integrability now has two major advantages. First, theseequations can be efficiently solved in a time scaling poly-nomially with system size. This should be contrastedwith the conventional diagonalization of the Hamiltonianmatrix in an exponentially large Hilbert space, allowingfor exact results for large system sizes. Second, it al-lows for the systematic targeting of eigenstates throughthe Bethe equations. The key to our proposed ap-proach is that overlaps between eigenstates of central spinHamiltonians 〈Bz,1; v1 . . . vN |Bz,2;w1 . . . wN 〉 with differ-ent magnetic fields Bz,1 6= Bz,2 can also be efficientlycalculated numerically51.

Returning to Floquet dynamics, a protocol is consid-ered where Bz is periodically switched between valuesBz,1 and Bz,2. To fix ideas, the eigenphases of the Flo-quet operator have been given in Figure 1 for differentdriving periods T , with total spin projection 0, η = 0.5and Bz switched between 1.2 and 0.8. These calcula-tions have been performed using exact diagonalizationon a small system with L = 5 in order to avoid a visualclutter of eigenstates, but are representative for largersystem sizes. Next to the spectrum of the Floquet oper-ator, two energy measures of a Floquet state |φn〉 are

θnT

= 〈φn|HF |φn〉 ,∂θn∂T

= 〈φn|HAvg |φn〉 , (10)

with θn/T the quasi-energies and ∂T θn the dynami-cal contribution to the quasi-energies34,52. This secondquantity is convenient for the visualization of avoidedcrossings in the spectrum of the Floquet Hamiltonian.

At small driving periods, the spectrum of HF reducesto that of HAvg and both energies coincide. The onset ofmany-body resonances can be observed at Tc = 2π/W ,

with W = EAvgmax − EAvgmin the bandwidth of HAvg. Atthis critical frequency, the energy difference betweenthe ground state and the highest excited state exactlymatches the driving frequency. These states are thenquasi-degenerate and interact resonantly, which can beclearly observed in the avoided crossing between theirrespective quasi-energies in 〈HF 〉53 and the crossing be-tween their respective energies in 〈HAvg〉. Further in-creasing the driving period, more and more resonancesare introduced. Remarkably, the off-resonant parts ofthe spectrum can often be accurately approximated us-ing HAvg

12,54.Resonant transitions. – Resonances have a major in-

fluence on the concept of adiabaticity, with distinct ef-fects on the eigenstates of the Floquet Hamiltonian andthe time-averaged Hamiltonian54–59. Starting from aneigenstate of the Floquet Hamiltonian and adiabaticallychanging the driving frequency60, the initial state willadiabatically follow the eigenstate of the Floquet Hamil-tonian. Across resonance, this would result in a transi-tion from e.g. the ground state to a highly excited state

3

−π

0

π

θ n

−2

−1

0

1

θ n/T

0 0.5 1 1.5 2 2.5

T/Tc

−2

−1

0

1

∂Tθ n

FIG. 1. Phase spectrum of the Floquet operator, quasi-energies and dynamical energies for a periodically driven cen-tral spin Hamiltonian at different driving periods T . Thedotted (blue) lines mark the edges of the Brillouin zone ±π(θn) and ±π/T (θn/T ), while the vertical dotted line denotesTc = 2π/W . The ground state and highest excited state arehighlighted red using the approximative results from integra-bility (see further).

of the time-averaged Hamiltonian, since the eigenstatesof the Floquet Hamiltonian adiabatically connect thesestates.

Focusing on the ground and highest excited state andadiabatically increasing the driving period across reso-nance, starting from the ground state of HAvg leads to

U(Tn) . . . U(T2)U(T1) |φ0(Bz)〉 , (11)

with T slowly increased from T1 to Tn and Bz = ηBz,1 +(1 − η)Bz,2. We will refer to this state as the ‘adia-batic ground state’, which is expected to adiabaticallyfollow the corresponding eigenstate of the Floquet Hamil-tonian through the frequency sweep, leading to transi-tions between resonant states. For the small system withL = 5, such transitions are shown in Figure 2 for the first(T ≈ Tc) and second-order (T ≈ 2Tc) resonance.

Slowly increasing the driving period, the system endsup in the highest excited state of HAvg in the second res-onance, while it undergoes another resonance in the firsttransition before the highest excited state can be reached.Since the initial state is not an exact eigenstate of HF ,but only a (good) approximation, oscillations are intro-duced in all expectation values corresponding to contri-butions from excited eigenstates of HF to |φ0(Bz)〉 (seeinset in Figure 2). These arise from higher-order contri-butions to the Magnus expansion and are as such control-lable (e.g. by decreasing |Bz,2 − Bz,1|). Still, it is clear

−1

0

1

〈HAvg〉

0.9 1 1.1

Tn/Tc

−0.5

0

0.5

〈Sz 0〉

−1

0

1

1.9 2 2.1

Tn/Tc

−0.5

0

0.5

FIG. 2. Expectation values of HAvg and Sz0 w.r.t. the adia-batic ground state of the Floquet Hamiltonian when the driv-ing period is slowly increased from 0.8 Tc to 1.2 Tc (firstcolumn) or from 1.8 Tc to 2.2 Tc (second column) withTi+1 − Ti = 10−4. Blue lines are exact results, while thered line is the approximation from integrability (see further).The dashed lines indicate the spectrum of HAvg.

that the ground state adiabatically follows the eigen-states of the Floquet Hamiltonian if the driving period isvaried adiabatically. In order to have a clear transitionbetween two states it is important that the resonanceis isolated, where only a single state is quasi-degeneratewith the ground state. For the ground and highest ex-cited state, the first and second resonance are guaranteedto be isolated because of the two-band nature and the lowdensity of states at the edge of the spectrum (see Figure1). Note how 〈Sz0 〉, as shown in the lower panel of Figure2, vanishes at the first resonance and nears its maximalvalue of 1/2 in the second resonance.

Modelling the resonant transition. – In general, suchcalculations require constructing the evolution operatorsfor both driving Hamiltonians at each value of the driv-ing period, and constructing and subsequently diagonal-izing the Floquet operator. Each step involves the fullHilbert space, making such calculations unfeasible for re-alistic system sizes. However, knowledge acquired fromquantum quenches (see e.g. [61] and references therein)can be transferred to the present situation under the fol-lowing key assumption. Namely, we assume that eachmany-body resonance can be modeled as a two-level sys-tem including only the corresponding quasi-degenerateeigenstates of HAvg. This assumes that quasi-degeneratestates do not interact strongly with off-resonant statesor other quasi-degenerate states with a different quasi-energy, similar in spirit to degenerate perturbation the-ory. This approximation can be validated through theFloquet-Magnus expansion and is expected to hold if thedeviations of the driving Hamiltonians are small w.r.t.

4

0.8 1 1.2

T/Tc

−3

0

3〈H

Avg〉Tc

1.98 2 2.02

T/Tc

−3

0

3

L = 5 L = 9 L = 13 L = 17

FIG. 3. Expectation value of the time-averaged Hamiltonianin the adiabatic ground state of the Floquet Hamiltonian withdriving Bz = 1± 0.2 and η = 1/2 for different system sizes L.

the time-averaged Hamiltonian43,47–49. The Floquet op-erator can then be constructed in the 2-dimensional ba-sis {|φ0(Bz)〉 , |φf (Bz)〉} spanned by the relevant quasi-degenerate eigenstates of the time-averaged Hamiltonian

UF =

[〈φ0(Bz)|UF |φ0(Bz)〉 〈φ0(Bz)|UF |φf (Bz)〉〈φf (Bz)|UF |φ0(Bz)〉 〈φf (Bz)|UF |φf (Bz)〉

].

(12)Explicitly writing out the Floquet operator (12) and ex-panding in the eigenstates of the driving Hamiltonians,each matrix element is given by

〈φi(Bz)|UF |φj(Bz)〉 =∑

m,n

e−i(1−η)Em(Bz,2)T e−iηEn(Bz,1)T

× 〈φi(Bz)|φm(Bz,2)〉× 〈φm(Bz,2)|φn(Bz,1)〉× 〈φn(Bz,1)|φj(Bz)〉 . (13)

The calculation of each matrix element generally involvesa double summation over the Hilbert space of energiesand overlaps, which in turn involve summations over theHilbert space. Integrability already provides numericallyefficient expressions for the energies and the overlaps. Asnoticed in quantum quenches, another important featureof Bethe states is that they offer a basis in which only avery small minority of eigenstates carry substantial corre-lation weight, allowing summations over the full Hilbertspace to be drastically truncated62,63. Such a truncationscheme is presented in the Supplemental Material51, andthe induced error can be checked from sum rules. Inpractice, this allows for a numerically exact constructionof the matrix elements (13) for relatively large systems.

The resulting 2×2 operator can be easily diagonalized,and integrability allows for an efficient calculation of ex-pectation values from its eigenstates64–69. So the mainapproximation in this scheme is the restriction of theHilbert space to a 2-dimensional space, but within thisspace the Floquet operator is numerically exact. Whilewe focus on the interaction between the ground and thehighest excited state only, this makes it possible to sys-tematically reconstruct part of the Floquet spectrum by

0.9 1 1.1

T/Tc

−0.5

0

0.5

〈Sz 0〉

1.99 2 2.01

T/Tc

−0.5

0

0.5

Bz = 1 Bz = 2 Bz = 5 Bz = 10

FIG. 4. Magnetization of the central spin in the adiabaticground state of the Floquet Hamiltonian at different drivingperiods with L = 25 and driving Bz = Bz ± 0.2.

including an increasing number of states in this basis.The accuracy can already be appreciated from Figures 1and 2, where the avoided crossings near the resonancesare well approximated but do not take into account theresonances involving other states. The results are ex-tended in Figures 3 and 4 to different system sizes andaverage magnetic fields. The period beyond which thetwo-level approximation fails because another state needsto be included can also be estimated51 and is marked inboth Figures. Note that this period lies outside the Fig-ure for the second-order resonance.Discussion. – While the expectation value of HAvg

varies smoothly from the initial to the final value, the be-haviour of the central spin is highly dependent on the or-der of the resonance. The magnetization 〈Sz0 〉 vanishes atthe first resonance, while it nears the maximal value 1/2at the second resonance. Such a protocol could then beused to realize a state with magnetization exceeding thatof both states, incompatible with any stationary centralspin Hamiltonian, since a maximal value of 1/2 impliesa pure state decoupled from its environment.

A simple way to understand this behavior follows fromthe structure of the ground and the highest excited state,where the environment spins tend to align either anti-parallel or parallel to the central spin. These can be ap-proximated by treating the environment as a single col-lective spin, and in this space the Hamiltonian simplifiesto

H ≈ BzSz0 +Ab~Sb · ~S0. (14)

Although not a necessary assumption51, some intuitioncan be gained by taking |Bz| � |AbSb|, where the rele-vant eigenstates can be approximated as

|φ±〉 ≈1√2

(| 12 , 12 〉0 |Sb,−

12 〉b ± |

12 ,− 1

2 〉0 |Sb,12 〉b). (15)

At resonance, the Floquet states are approximately givenby |φ〉 = 1√

2

(|φ+〉 ± eiθ |φ−〉

), where the relative phase θ

is a priori unknown. However, the magnetization of thecentral spin depends on this relative phase as 〈φ|Sz0 |φ〉 =

5

12 cos(θ). The different magnetizations hence correspondto different relative phases acquired by these states. Thisrelative phase can be deduced from second-order per-turbation theory, expanding the matrix elements of theFloquet operator (13) at resonance for small deviationsfrom the average magnetic field (Bz − Bz)51. Evolvingeither state over a full driving cycle will lead to a globalphase and introduce off-diagonal corrections on the initialstate, which are shown to either interfere constructivelyor destructively depending on the order of the resonance.This is reflected in the dependence of the off-diagonal el-ements on the order of the resonance k through termse±iηk2π, and perturbation theory leads to relative phasesπ/2 and 3π/2 in the first resonance, while it leads torelative phases 0 and π in the second resonance. Theseexplain the observed magnetization 〈Sz0 〉 = 0 or ±1/2and the decoupling of the central spin. This behaviourextends towards higher-order resonances, where the po-larization occurs at even-order resonances but vanishesat odd-order resonances. However, there is no guaranteethat such resonances will be isolated and hence observ-able.

Conclusion. – In this work we investigated adiabatictransitions in the Floquet Hamiltonian when varying thedriving frequency, leading to a transition between theground and highest excited state away from resonance.Applying a periodically-varying magnetic field to a cen-

tral spin model, it was shown how frequency sweeps andFloquet resonances can be used to prepare the system in acoherent superposition of the targeted states, either lead-ing to a vanishing magnetization or a spin state exactlyaligned with the magnetic field, depending on the orderof the resonance. The latter effectively leads to a decou-pling of the central spin from its environment, which canbe used to purify the central spin. Integrability-basedtechniques were shown to be able to model this transi-tion, which allows for an investigation of larger systemsizes and presents a first step in applying techniques fromintegrability to interacting integrable systems subjectedto periodic driving.Acknowledgments. – We are grateful to M. Bukov, A.

Russomanno, S.E. Tapias Arze and D. Van Neck for valu-able discussions and/or comments. P.W.C. acknowledgessupport from a Ph.D. fellowship and a travel grant for along stay abroad at the University of Amsterdam fromthe Research Foundation Flanders (FWO Vlaanderen).J.-S. C. acknowledges support from the Netherlands Or-ganization for Scientific Research (NWO), and from theEuropean Research Council under ERC Advanced grant743032 DYNAMINT. This work is part of the DeltaITP consortium, a program of the Netherlands Organ-isation for Scientific Research (NWO) that is funded bythe Dutch Ministry of Education, Culture and Science(OCW).

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63 A. Faribault, P. Calabrese, and J.-S. Caux, J. Math. Phys.50, 095212 (2009).

64 N. Slavnov, Theor. Math. Phys. 79, 502 (1989).65 V. E. Korepin, N. M. Bogoliubov, and A. G. Izergin,

Quantum Inverse Scattering Method and Correlation Func-tions (Cambridge University Press, Cambridge, 1993).

66 J. Links, H.-Q. Zhou, R. H. McKenzie, and M. D. Gould,J. Phys. A: Math. Gen. 36, R63 (2003).

67 A. Faribault, P. Calabrese, and J.-S. Caux, Phys. Rev. B77, 064503 (2008).

68 G. Gorohovsky and E. Bettelheim, Phys. Rev. B 84,224503 (2011).

69 P. W. Claeys, S. De Baerdemacker, and D. Van Neck,SciPost Phys. 3, 028 (2017).

70 O. Babelon and D. Talalaev, J. Stat. Mech: Theor. Exp.2007, P06013 (2007).

71 O. El Araby, V. Gritsev, and A. Faribault, Phys. Rev. B85, 115130 (2012).

72 P. W. Claeys, S. De Baerdemacker, M. Van Raemdonck,and D. Van Neck, Phys. Rev. B 91, 155102 (2015).

Spin polarization through Floquet resonances in a driven central spin model –Supplemental Material

Pieter W. Claeys,1, 2, 3, ∗ Stijn De Baerdemacker,2 Omar El Araby,1 and Jean-Sebastien Caux1

1Institute for Theoretical Physics Amsterdam and Delta Institute for Theoretical Physics,University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands

2Department of Physics and Astronomy, Ghent University, Krijgslaan 281 S9, B-9000 Ghent, Belgium3Center for Molecular Modeling, Ghent University, Technologiepark 903, 9052 Ghent, Belgium

Appendix A: Integrability of the central spin model

The integrability of the central spin model is a crucialelement in our analysis of the transitions in the Floquetquasi-energy spectrum. In this Appendix we provide thenecessary technicalities in order to be self-contained. Fol-lowing the notation from the main text, a set of operatorscan be defined as

Qi = BzSzi +

L∑

j 6=i

~Si · ~Sjεi − εj

, (A1)

such that the central spin Hamiltonian corresponds toQ0, and these operators mutually commute [Qi, Qj ] =0,∀i, j. These are the so-called conserved charges of thecentral spin model and can be simultaneously diagonal-ized by (unnormalized) Bethe ansatz states

|Bz; v1 . . . vN 〉 =

N∏

a=1

L∑

j=0

S+j

εj − va

|↓ . . . ↓〉 , (A2)

with rapidities {v1 . . . vN} satisfying the Bethe equations

B−1z +1

2

L∑

j=0

1

εj − va=

N∑

b6=a

1

vb − va, ∀a = 1 . . . N.

(A3)Following a famous result by Slavnov1–3, overlaps be-tween Bethe states at different magnetizations can becalculated as

〈Bz,1; v1 . . . vN |Bz,2;w1 . . . wN 〉

=

∏a

∏b(va − wb)∏

b<a(wb − wa)∏a<b(vb − va)

detSN , (A4)

with SN an N ×N matrix defined as

(SN )ab =1

va − wb

[ L∑

j=0

1

(va − εj)(wb − εj)

− 2

N∑

c 6=a

1

(va − vc)(wb − vc)

]. (A5)

Taking the limit where the two sets of rapidities coincideleads to an expression for the normalization of a Bethestate as the determinant of a Gaudin matrix

〈Bz; v1 . . . vN |Bz; v1 . . . vN 〉 = detGN , (A6)

with GN an N ×N matrix defined as

(GN )ab =

{∑Lj=0

1(εj−va)2 − 2

∑Nc6=a

1(vc−va)2 if a = b

2(va−vb)2 if a 6= b

.

(A7)This is sufficient for the evaluation of all terms in the Flo-quet operator. In order to calculate expectation valuesfrom the resulting wave function one more result needs tobe used, namely that for the matrix elements of the spinprojection Sz0 (see e.g. Refs. [4–6]). From the Hellmann-Feynman theorem, it follows that

〈Bz; v1 . . . vN |Sz0 |Bz; v1 . . . vN 〉〈Bz; v1 . . . vN |Bz; v1 . . . vN 〉

=∂

∂BzE(Bz; {v1 . . . vN}), (A8)

while the off-diagonal elements can be calculated as

〈Bz; v1 . . . vN |Sz0 |Bz;w1 . . . wN 〉

=

∏c(vc − ε0)∏c(wc − ε0)

det (TN +QN )∏b<a(wb − wa)

∏a<b(vb − va)

, (A9)

with TN following from SN as

(TN )ab = (SN )ab∏

c

(vc − wb), (A10)

and QN an N ×N matrix defined as

(QN )ab =

∏c6=b(wc − wb)(ε0 − va)2

. (A11)

Given a set of Bethe states, it is now possible to calculateall necessary overlaps and expecation values. Further-more, the Bethe ansatz approach reduces the problemof obtaining eigenvalues and eigenstates to solving a setof coupled nonlinear equations, instead of diagonalizingthe Hamiltonian in the full (exponentially large) Hilbertspace. Many techniques have been proposed for solvingthese equations, where we have implemented the so-calledeigenvalue-based approach7–9.

The other necessary result for our approach was an ef-ficient expansion of a Bethe state at given magnetic fieldBz in a set of Bethe states at slightly different magneticfield. In Refs. 10 and 11, it was shown how such an ex-pansion can be efficiently implemented by targeting rele-vant eigenstates in a systematic way. This targeting can

arX

iv:1

712.

0311

7v2

[co

nd-m

at.s

tr-e

l] 2

3 A

ug 2

018

2

FIG. 1. Graphical representation of relevant states for L = 5and N = 3 starting from the ground state and the highestexcited state.

be represented in a graphical way by making the connec-tion with the Bz →∞ limit. In this limit, the rapiditiesbehave as va = εi(a) + O(B−1z ), where i(a) associates aspin index with each rapidity such that the eigenstatesreduce to

|Bz →∞; v1 . . . vN 〉 ∝N∏

a=1

S+i(a) |↓ . . . ↓〉 . (A12)

In this limit, all states reduce to simple product states de-fined in terms of a set occupied spin levels {i(1) . . . i(N)}.For the given parametrization, the ground state reducesto the state where the central spin is unoccupied and thestates interacting most strongly with the central spin areoccupied, which can be represented graphically (for e.g.L = 5 and N = 3) as

|φ0(Bz →∞)〉 ∝ S+1 S

+2 S

+3 |↓ . . . ↓〉

= |◦| • • • | ◦ ◦〉 , (A13)

|φf (Bz →∞)〉 ∝ S+0 S

+1 S

+2 |↓ . . . ↓〉

= |• • •| ◦ ◦◦〉 . (A14)

When expanding an eigenstate at fixed Bz into eigen-states of a model at different Bz, it is often sufficient torestrict the expansion to states which can be related tothe initial state by simple spin-flip excitations in the limitBz → ∞. This results in a set of N(L + 1 − N) statesfor which the overlap needs to be calculated, which areagain represented graphically in Figure 1.

When expanding an initial state |φi(Bz)〉 in such a re-stricted basis, a measure for the error is easily calculatedas 1 −∑n | 〈φi(Bz)|φn〉 |2, which reduces to zero for acomplete basis. If this initial truncation would prove tobe insufficient and the error exceeds a certain thresh-old (when e.g. there are large differences between thetime-averaged Hamiltonian and the driving Hamiltoni-ans), this summation can be extended in a systematicway by including higher-order spin-flip excitations.

Appendix B: Perturbation expansion of the Floquetoperator

In order to better understand the behaviour of the Flo-quet eigenstates near resonance, we perform a perturba-tive expansion of the Floquet operator when the model

is being driven in such a way that there are only smalldeviations from the average magnetic field. Then withineach matrix element all non-diagonal overlaps in the sum-mation will be of order Bz,i − Bz ≡ O(∆), allowing thesummation to be severely restricted. For corrections upto O(∆2), only the initial and final state are relevantas intermediate states in the summation. The diagonalelements can easily be found as

〈φ0(Bz)|UF |φ0(Bz)〉 = e−ikE0(Bz)Tc +O(∆2), (B1)

〈φf (Bz)|UF |φf (Bz)〉 = e−ikEf (Bz)Tc +O(∆2), (B2)

which holds for arbitrary values of the driving period. Inthe first element, the summation has been restricted to(m,n) = (0, 0), while it has been restricted to (m,n) =(f, f) in the second element. Here 0 and f label the reso-nant states, which we will take to be the ground (initial)state and highest excited (final) state. All other terms in-volve at least two off-diagonal overlaps and are as such ofO(∆2). Due to the expansion around Bz, the first-ordercorrections on the phases can also be shown to vanish.Note that we discard a summation of O(∆2) over an ex-ponentially large Hilbert space, but we can assume thatthe phases do not add coherently and as such this sum-mation can still be neglected. If this would not be thecase then the resonant states have a strong interactionwith off-resonant states and our 2-level approximationwould similarly prove to be insufficient.

The off-diagonal elements contain at least one off-diagonal overlap, restricting the summation to (m,n) =(0, 0), (f, f) and (0, f) or (f, 0), leading to

〈φ0(Bz)|UF |φf (Bz)〉= e−i(1−η)E0(Bz)T e−iηE0(Bz)T (Bz,1 −Bz) 〈∂Bzφ0|φf 〉+ e−i(1−η)Ef (Bz)T e−iηEf (Bz)T (Bz,2 −Bz) 〈φ0|∂Bzφf 〉+ e−i(1−η)E0(Bz)T e−iηEf (Bz)T

×[(Bz,2 −Bz) 〈∂Bzφ0|φf 〉+ (Bz,1 −Bz) 〈φ0|∂Bzφf 〉

]

+O(∆2), (B3)

and similar for 〈φ0(Bz)|UF |φf (Bz)〉. Here the innerproducts have been expanded as e.g. |φ0(Bz,i)〉 =

|φ0(Bz)〉+ (Bz,i − Bz) |∂Bzφ0(Bz)〉+O(∆2), where the

dependence on Bz has been made implicit in the finalexpressions. Evaluating these at T = k · Tc, k ∈ N andexplicitly setting Ef = E0 + 2π/Tc, the matrix elementscan (up to O(∆2)) be rewritten as

〈φ0(Bz)|UF |φ0(Bz)〉 = e−ikE0(Bz)Tc ,

〈φf (Bz)|UF |φf (Bz)〉 = e−ikEF (Bz)Tc = e−ikE0(Bz)Tc ,

〈φ0(Bz)|UF |φf (Bz)〉 = e−ikE0(Bz)Tc(Bz,1 −Bz,2)

× 〈∂Bzφ0|φf 〉 (1− e−iηk2π),

〈φf (Bz)|UF |φ0(Bz)〉 = e−ikE0(Bz)Tc(Bz,1 −Bz,2)

× 〈∂Bzφf |φ0〉 (1− e+iηk2π).

3

−3

0

3

〈HAvg〉Tc

−3

0

3

2.8 3 3.2

T/Tc

−0.5

0

0.5

〈Sz 0〉

3.9 4 4.1

T/Tc

4.8 5 5.2

T/Tc

5.9 6 6.1

T/Tc

−0.5

0

0.5

L = 5 L = 9 L = 13 L = 17

FIG. 2. Expectation value of the time-averaged Hamiltonian 〈HAvg〉 and magnetization of the central spin 〈Sz0 〉 in the adiabatic

ground (and highest excited) state of the Floquet Hamiltonian at higher-order resonances with driving Bz = 1±0.2 and η = 1/2for different system sizes L.

For the central spin model, it is known that all overlapsare purely real4 and hence 〈∂Bz

φ0|φf 〉 = −〈∂Bzφf |φ0〉.

Taking k = 1 and η = 1/2 then results in

UF = e−iE0(Bz)Tc

×(1 + 2(Bz,1 −Bz,2) 〈∂Bzφ0|φf 〉

[0 1−1 0

])+O(∆2).

(B4)

Diagonalizing the 2 × 2 perturbation matrix leads toeigenstates

|φ0(Bz)〉 ± i |φf (Bz)〉 , (B5)

corresponding to relative phases of π/2 or 3π/2, as men-tioned in the main text.

For the second resonance (1 − e±iηk2π) = 0 andthe first-order correction vanishes, so higher-order termsneed to be included. It is no longer possible to explicitlyobtain the corrections without performing the summa-tion over the Hilbert space, but symmetry arguments canbe used to predict the relative phase. If the first-ordercorrection vanishes, the Floquet operator can be writtenas

UF = e−iE0(Bz)T(1 + (Bz,1 −Bz,2)2V (Bz) +O(∆3)

),

(B6)with V (Bz) a 2 × 2 matrix containing the second-ordercorrections on the matrix elements. Demanding UF tobe unitary then results in the constraint that V (Bz) isanti-hermitian V †(Bz) = −V (Bz). Hence, taking thetranspose of UF with η = 1/2 and exchanging Bz,1 and

Bz,2 leaves UF invariant, since

(UF (Bz,1, Bz,2))T

=(e−iTH(Bz,2)/2e−iηTH(Bz,1)/2

)T

= e−iTH(Bz,1)T /2e−iηTH(Bz,2)

T /2

= e−iTH(Bz,1)/2e−iηTH(Bz,2)/2

= UF (Bz,2, Bz,1). (B7)

This restricts V (Bz) to be symmetric, since exchang-ing Bz,1 and Bz,2 leaves the second-order contribution

(Bz,1 −Bz,2)2V (Bz) invariant. Combining these restric-

tions allows this matrix to be rewritten as V (Bz) =iW (Bz), with W (Bz) a purely real and symmetric ma-trix. Diagonalizing this perturbation matrix, the eigen-states will be purely real and result in relative phases of0 or π.

Perturbation theory thus explains the relative phasesin both resonances, where it is important to note that thepresented arguments can be extended to systems wherethe condition |Bz| � |AbSb| does not hold. The orthogo-nal eigenstates ofHAvg can be approximately constructedas

|φ0〉 = cos(ϕ) | 12 , 12 〉0 |Sb,−12 〉b + sin(ϕ) | 12 ,− 1

2 〉0 |Sb,12 〉b ,

|φf 〉 = sin(ϕ) | 12 , 12 〉0 |Sb,−12 〉b − cos(ϕ) | 12 ,− 1

2 〉0 |Sb,12 〉b .(B8)

In the first-order resonance, it can easily be checked thata coherent superposition of these states with a relativephase of π/2 and 3π/2 (|φ0〉 ± i |φf 〉) always has a van-ishing magnetization. In the second-order resonance, the

4

relative phases 0 or π result in Floquet eigenstates withreal coefficients interpolating between these two states.This then necessitates a change of sign in one of the twocomponents (either when going from + cos(ϕ) to + sin(ϕ)or from + sin(ϕ) to − cos(ϕ)), resulting in an intermedi-ate state where only a pure state remains and the centralspin is decoupled. While this might not always be exactlyat the point of resonance where T = 2Tc, this will gener-ally occur close to resonance because of the second-ordernature of the perturbation.

Perturbation theory then generally predicts odd-orderresonances where the magnetization is approximatelyzero and even-order resonances near which the magne-tization is either maximal or minimal, as supported bynumerical results from the integrability-based approxi-mation. This corresponds to the behaviour at first- andsecond-order resonances as discussed in the main text,and is shown in Figure 2 for the first four higher-order res-onances. It should be kept in mind that there is no guar-antee that these higher-order resonances will be isolated,since other states might be quasi-degenerate with the res-onant states and the two-level approximation would fail.However, if the resonance only involves the two targetedstates, the described dependence of the magnetization onthe order of the resonance is expected to hold.

Appendix C: Range of validity of the approximation

As shown in the main text, the two-level approxima-tion fails when other resonances occur. Starting from theground state of HAvg and increasing the driving periodacross the first-order resonance with the highest excited

state, the range of periods for which no other resonancesare expected to occur can be estimated. Specifically, thenext relevant resonance occurs when the highest excitedstate is quasi-degenerate with the first excited state. This

state has energy EAvgmin + EAvggap , with EAvggap the ground-state energy gap in the time-averaged Hamiltonian, and a

resonance is expected at T = 2π/(EAvgmax−EAvgmin−EAvggap ).This is the relevant period discussed in the main text,leading to

T − TcTc

=EAvggap

EAvgmax − EAvgmin − EAvggap

≈ EAvggap

W. (C1)

The relevant state can again be efficiently targeted fromthe Bz → ∞ limit and is related to the ground statethrough a simple spin-flip excitation (e.g. for the previ-ously mentioned example |◦| • • ◦ | ◦ •〉). This allows theenergy gap to be easily calculated, returning the resultsmentioned in the main text. For the chosen parametriza-tion the bandwidth W scales linearly with system sizeL, whereas the energy gap EAvggap quickly converges to aconstant (non-zero) value with increasing system size L.The relevant range of periods for which no additional res-onances are expected to occur then scales inversely withsystem size since EAvggap /W ∝ L−1. For the second-orderresonance, this region can similarly be shown to be ap-proximately 2EAvggap /W .

Two effects hence combine to lead to a second-orderresonance which is more isolated compared to the first-order one – it is a second-order perturbative effect andhence more narrow, and the subsequent region where noadditional resonances are expected to occur is larger.

∗ PieterW.Claeys@UGent.be1 N. Slavnov, Theor. Math. Phys. 79, 502 (1989).2 V. E. Korepin, N. M. Bogoliubov, and A. G. Izergin,Quantum Inverse Scattering Method and Correlation Func-tions (Cambridge University Press, Cambridge, 1993).

3 P. W. Claeys, S. De Baerdemacker, and D. Van Neck,SciPost Phys. 3, 028 (2017).

4 J. Links, H.-Q. Zhou, R. H. McKenzie, and M. D. Gould,J. Phys. A: Math. Gen. 36, R63 (2003).

5 A. Faribault, P. Calabrese, and J.-S. Caux, Phys. Rev. B77, 064503 (2008).

6 G. Gorohovsky and E. Bettelheim, Phys. Rev. B 84,

224503 (2011).7 O. Babelon and D. Talalaev, J. Stat. Mech: Theor. Exp.2007, P06013 (2007).

8 O. El Araby, V. Gritsev, and A. Faribault, Phys. Rev. B85, 115130 (2012).

9 P. W. Claeys, S. De Baerdemacker, M. Van Raemdonck,and D. Van Neck, Phys. Rev. B 91, 155102 (2015).

10 A. Faribault, P. Calabrese, and J.-S. Caux, J. Stat. Mech.2009, P03018 (2009).

11 A. Faribault, P. Calabrese, and J.-S. Caux, J. Math. Phys.50, 095212 (2009).