Ultra-fast control of magnetic relaxation in a periodically driven Hubbard model
Juan Jose Mendoza-Arenas, Fernando Javier Gomez-Ruiz, Martin Eckstein, Dieter Jaksch, Stephen R. Clark
aa r X i v : . [ c ond - m a t . s t r- e l ] O c t October 25, 2017
Ultra-fast control of magnetic relaxation in aperiodically driven Hubbard model
Juan Jose Mendoza-Arenas ∗ , Fernando Javier Gómez-Ruiz , Martin Eckstein ,Dieter Jaksch , and Stephen R. Clark Motivated by cold atom and ultra-fast pump-probe experiments we study the melting of long-range antiferromagnetic order of a perfect Néelstate in a periodically driven repulsive Hubbardmodel. The dynamics is calculated for a Bethelattice in infinite dimensions with non-equilibriumdynamical mean-field theory. In the absence ofdriving melting proceeds differently dependingon the quench of the interactions to hoppingratio U / ν from the atomic limit. For U ≫ ν decay occurs due to mobile charge-excitationstransferring energy to the spin sector, while for ν & U it is governed by the dynamics of resid-ual quasi-particles. Here we explore the rich ef-fects that strong periodic driving has on this re-laxation process spanning three frequency ω regimes: (i) high-frequency ω ≫ U , ν , (ii) res- onant l ω = U > ν with integer l , and (iii) in-gap U > ω > ν away from resonance. In case(i) we can quickly switch the decay from quasi-particle to charge-excitation mechanism throughthe suppression of ν . For (ii) the interaction canbe engineered, even allowing an effective U = regime to be reached, giving the reverse switchfrom a charge-excitation to quasi-particle decaymechanism. For (iii) the exchange interaction canbe controlled with little effect on the decay. Bycombining these regimes we show how periodicdriving could be a potential pathway for control-ling magnetism in antiferromagnetic materials.Finally, our numerical results demonstrate the ac-curacy and applicability of matrix product statetechniques to the Hamiltonian DMFT impurityproblem subjected to strong periodic driving. Relaxation of a symmetry-broken state after a quenchrepresents a class of non-equilibrium dynamics that hasbeen intensely studied both experimentally and theoret-ically. Part of the reason for this interest is the distinctdeparture of the evolution from the expected rapid ther-malization for isolated but interacting systems. Comple-mentary to quenching the application of time-periodicdriving is now emerging as a key tool for controllingmany-body systems on microscopic time scales. In coldatom systems this is achieved by directly modulatingthe optical lattice potential [1, 2], while in condensedmatter resonant THz excitation of low-energy structuraland electronic degrees of freedom is opening up simi-lar means of control [3]. As such the possibility of stabi-lizing, enhancing and switching between various formsof order like superconductivity [4, 5] and charge-densitywave [6] is a tantalizing prospect. Of particular funda- mental and technological interest is the ultrafast controlof magnetism [7–13] that may have applications in mag-netic storage devices [14–16].Motivated by these developments we examine the in-fluence of strong periodic driving on the paradigmaticcase of antiferromagnetic (AFM) Néel state relaxationwithin the repulsive Hubbard model. In the absence ∗ Corresponding author E-mail: [email protected] Departamento de Física, Universidad de los Andes, A.A. 4976,Bogotá D. C., Colombia2 Clarendon Laboratory, University of Oxford, Parks Road, Ox-ford OX1 3PU, United Kingdom3 Max Planck Institute for the Structure and Dynamics of Matter,University of Hamburg CFEL, Hamburg, Germany4 Department of Physics, University of Bath, Claverton Down,Bath BA2 7AY, United Kingdom
Copyright line will be provided by the publisher endoza-Arenas et al.: Ultra-fast control of magnetic relaxation in a periodically driven Hubbard model of driving the mechanism underpinning the melting oflong-ranged antiferromagnetic order depends on thequench of the ratio of interactions to hopping U / ν fromthe atomic limit [17]. For U ≫ ν local moments andtheir exchange coupling are retained during the evolu-tion, but the quench results in the rapid nucleation ofcharge excitations whose motion on top of the spin back-ground scrambles the staggered magnetization. For ν & U the melting is governed by the dynamics of residualquasi-particles that leads to the fast decay of both the lo-cal moments and long-range order. Suddenly introduc-ing periodic driving is expected to have a profound influ-ence on this relaxation.Our focus is on the Hubbard model for a Bethe lat-tice in infinite dimensions where non-equilibrium dy-namical mean-field theory (NE-DMFT) can solve for thedynamics. We study the relaxation process in three fre-quency ω regimes: (i) high-frequency ω ≫ U , ν , (ii) res-onant l ω = U > ν with integer l , and (iii) in-gap off-resonant U > ω > ν . The high-frequency regime (i)leads to the well-known renormalization of the hopping ν and consequently the decay process for ν > U tobe switched from a quasi-particle to charge-excitationmechanism. Resonant driving (ii) and the resulting effec-tive Hamiltonian describing the system have been thesubject of recent theory work [18]. This indicates that in-teractions can be modified significantly, eliminated all to-gether or mimic signatures of those of an opposite sign.This regime therefore allows for the reverse switch for U > ν from a charge-excitation to quasi-particle decaymechanism. For in-gap driving away from resonances(iii) we show that the increase of charge excitations iscompensated by the suppression of their hopping, leav-ing the melting mechanism unaffected. We then discusshow by combining these driving regimes we can controlthe evolution of the system to induce a particular mag-netic order parameter, and an ultrafast reversal of dynam-ics.A significant contribution of this work is to demon-strate the accuracy and applicability of matrix productstate techniques to the time-dependent Hamiltonian NE-DMFT impurity problem without increasing the num-ber of bath sites compared to the time-independent case.This adds further impetus to the use of these methodsfor more complex problems involving other symmetry-broken states like superconductivity.The plan of the paper is as follows. In Sec. (2) weintroduce the driven Hubbard model to be considered,and briefly describe the key predictions of Floquet theory.In Sec. (3) we outline the framework of NE-DMFT, withparticular emphasis on the Hamiltonian formulation ofthe impurity problem and the matrix product state tech- nique we use as an impurity solver. Details of the DMFTsetup for the Néel state melting are also described. InSec. (4) we present results for the melting without driving,extending earlier work [17], and providing a baseline forthe expected behavior. This is followed in Sec. (5) by themain results of this paper describing the effects of driv-ing in the three regimes outlined. Finally we conclude inSec. (6). We focus on the Fermi-Hubbard model at half-fillinggiven by the Hamiltonian H Hub = − J X 〈 i , j 〉 , σ c † i σ c j σ + U X i µ n i ↑ − ¶ µ n i ↓ − ¶ , (1)where c † i σ creates an electron at site i with spin σ =↑ , ↓ , n i σ = c † i σ c i σ is the corresponding number operator, J isthe hopping amplitude between nearest-neighbor sites 〈 i , j 〉 , and U is the repulsive on-site interaction. For sim-plicity we consider a Bethe lattice in the limit of infinitecoordination number Z and hopping J = ν / p Z , where ν corresponds to the unit of energy. Physically this sys-tem can be envisaged as a cycle-free rooted tree in the x - y plane with equidistant spacing a between each site.Despite this it nonetheless mimics many properties ex-pected of higher dimensional regular bipartite latticesand its Z → ∞ limit is where our numerical approach,NE-DMFT, is exact [19].We take the system as being subjected to a uniformunpolarized AC electric field propagating in the z direc-tion resulting in a time-periodic linear potential emanat-ing from the root of the tree via a driving term H drv ( t ) = X j eaE sin( ω t ) s j n j , (2)where e is the electronic charge, E is the amplitude ofthe field, ω is the angular frequency of the drive, s j isthe shell containing site j , and n j = n j ↑ + n j ↓ . Movingto the rotating-frame via the unitary transformation (we s j steps from the root of the tree to site j . Copyright line will be provided by the publisherctober 25, 2017 will take ħ = U ( t ) = exp " i φ ( t ) X j s j n j , (3)where φ ( t ) = − ( eaE / ω ) cos( ω t ), we transform H ( t ) = H Hub + H drv ( t ) into a rotating-frame Hamiltonian H rot ( t ) = i ˙ U ( t ) U † ( t ) + U ( t ) H ( t ) U † ( t ), (4)where ˙ U ( t ) is the time derivative of U ( t ). The explicit driv-ing term is therefore eliminated making H rot ( t ) equiva-lent to H Hub with a time-dependent Peierls phase on thehopping amplitude as J ( t ) = J exp £ i A ( s i − s j ) cos( ω t ) ¤ , (5)where we define a dimensionless driving amplitude A = eaE / ω . Nearest-neighbor hopping implies ( s i − s j ) = ± Since we are studying a periodically driven system wegive a brief overview of Floquet formalism applied to thedriven Hubbard lattice; see Refs. [20, 21] for recent re-views. For closed quantum systems, Floquet’s theoremestablishes that there is a complete set of solutions of thetime-dependent Schrödinger equation i dd t | ψ ( t ) 〉 = H ( t ) | ψ ( t ) 〉 , with H ( t + T ) = H ( t ), (6)and T = π / ω the period of the time-dependent Hamil-tonian H ( t ), of the form | ψ ( t ) 〉 = e − i ε α t | ψ α ( t ) 〉 , with | ψ α ( t ) 〉 = | ψ α ( t + T ) 〉 , (7)where the quasi-energies ε α lay in the “Brillouin zone" − ω /2 < ε α ≤ ω /2. This is equivalent to that of Bloch’s the-orem in time. Expanding the time-periodic function | ψ α ( t ) 〉 = X m e − i m ω t | ψ α , m 〉 , (8)in terms of the Floquet modes | ψ α , m 〉 , then Eq. (6) is re-duced to the eigenvalue problem( ε α + m ω ) | ψ α , m 〉 = X m ′ H m − m ′ | ψ α , m ′ 〉 , (9)in terms of the Fourier components of the Hamiltonian H m = T Z T d t e i m ω t H rot ( t ). (10) For the Fermi-Hubbard lattice with time-dependent hop-ping given in Eq. (5), this results in Fourier Hamiltonianterms [16] H m = − J X 〈 i j 〉 , σ ( − m J m ³ ( s i − s j ) A ´ c † i σ c j σ + δ m ,0 U X j n j ↑ n j ↓ , (11)where the Coulomb repulsion is only present for the m = m th Bessel function J m ¡ ( s i − s j ) A ¢ . Truncating thenumber of Floquet sectors, or taking some specific limits,it is possible to solve the eigenvalue problem in Eq. (9).For example, by moving to an extended Hilbert spaceand using second-order perturbation theory on the cou-plings between the m = J ex was found to be [16] J ex ( A , ω ) J ex ( A = = ∞ X n =−∞ J | n | ( A ) + n ω / U , (12)with ν / U ≪ J ex ( A =
0) can be well defined. In particular this resultshows how high-frequency driving ω ≫ U , ν will lead toa reduction of J ex . With in-gap driving ν < ω < U , onthe other hand, it is possible to increase J ex , which canbe exploited to enhance exchanging pairing below half-filling [22]. For sufficiently strong driving A Eq. (12) evenpredicts that the sign of J ex can be flipped. At half-fillingit is thus possible for periodic driving to induce an ultra-fast reversal of the magnetic dynamics of a system [16].The application of high-frequency and strong-couplingexpansions to the driven Hubbard lattice were recentlyplaced on an equal footing [18] where both can be seenas a form of generalized Schrieffer-Wolf transformation.This formalism is particularly useful for resonant drivingcase ν ≪ U = l ω , for integer l . There the effective Hamil-tonian H = H to zeroth-order in 1/ ω , equal to the time-averaged rotating-frame Hamiltonian, is H (0)eff = X 〈 i j 〉 , σ n − J eff g i j σ − K eff h ( − l η i j h † i j σ + H.c. io , (13)with the operators h † i j σ = n i ¯ σ c † i σ c j σ (1 − n j ¯ σ ), (14a) g i j σ = (1 − n i ¯ σ ) c † i σ c j σ (1 − n j ¯ σ ) + n i ¯ σ c † i σ c j σ n j ¯ σ , (14b)where ¯ ↑ =↓ and ¯ ↓ =↑ , and the parameters J eff = J J ( A ), K eff = J J l ( A ), (15)with η i j = i > j and η i j = i < j . Theterm h † i j σ corresponds to creation and annihilation of Copyright line will be provided by the publisher endoza-Arenas et al.: Ultra-fast control of magnetic relaxation in a periodically driven Hubbard model holon-doublon pairs between sites i and j , and g i j σ represents holon-doublon and projected-fermion hop-ping. The weights of these processes can be manipulatedby means of the driving amplitude A , as indicated inEq. (15), leading to novel physics that we will illustratein Sec. (5). The first-order correction in 1/ ω is H (1)eff = (1/ ω ) P ∞ m = [ H m , H − m ]/ m and gives rise to driving modi-fied exchange terms. To study the dynamics of the driven high-dimensionalFermi-Hubbard lattice described in Section 2.1, we usethe non-equilibrium extension [19] of DMFT [23]. Inbrief, this approach consists of replacing the correlatedlattice by an impurity site, which retains the strictly lo-cal interactions of the lattice, and a coupling to an effec-tive mean-field Λ σ ( t ) representing the rest of the system.This mean-field must be determined self-consistentlyand is time-dependent in order to capture dynamicalfluctuations arising from the exchange of particles withthe impurity separated by a time t . For non-equilibriumconfigurations, which in general do not possess time-translational invariance, the dynamical mean-field de-pends on two times, and is denoted as Λ σ ( t , t ′ ). Althoughthis mapping is exact only in the limit of infinite di-mensions (equivalent to Z → ∞ ), it has been success-fully used as the starting point of different approachesfor finite-dimensional lattices including nonlocal spatialcorrelations [24–28]. A variety of methods have been developed to solve theeffective impurity problem of strongly correlated non-equilibrium high dimensional systems; see Ref. [19] fordetailed descriptions. These include weak- and strong-coupling perturbation theories, which cannot deal withintermediate interaction regimes, and continuous-timequantum Monte-Carlo algorithms, that are limited bysign problems. To overcome these difficulties the use of aso-called
Hamiltonian-based impurity solver , which hasbeen successful for equilibrium DMFT [23, 29], has beenrecently proposed for the non-equilibrium case [30]. Theapproach consists of discretizing the mean-field environ-ment Λ σ ( t , t ′ ) by mapping the effective impurity prob-lem to a time-dependent single-impurity Anderson model(SIAM). This new problem corresponds to the impuritysite coupled to an in-principle infinite, but in practise finite number of noninteracting bath orbitals L bath , de-scribed by the Hamiltonian H SIAM ( t ) = H imp + H bath ( t ) + H hyb ( t ), (16)with H imp = U µ n ↑ − ¶ µ n ↓ − ¶ − µ X σ n σ , (17a) H hyb = X p > σ ³ V σ p ( t ) c †0 σ c p σ + H.c. ´ , (17b) H bath = X p > σ £ ǫ p σ ( t ) − µ ¤ c † p σ c p σ . (17c)Here we denote with 0 the impurity site and bath orbitalsby the index p > µ is the chemical potential, V σ p ( t )is the time-dependent hopping amplitude of a fermionwith spin σ from orbital p to the impurity, and ǫ p σ ( t ) isthe time-dependent on-site energy of bath orbital p forspin σ . Equation (17a) corresponds to the local impurityHamiltonian possessing the repulsive interaction of theoriginal lattice problem. Equation (17b) describes the hy-bridization between the impurity and bath orbitals, andEq. (17c) gives the on-site bath Hamiltonian. In this sys-tem there are only direct couplings between the bath or-bitals and the impurity, therefore it is sketched in Fig. 1(a)as the impurity surrounded by bath orbitals in a star ge-ometry.The dynamics of the impurity site of the SIAM corre-sponds to that of each site of the original high-dimensionallattice once self-consistency conditions are met. Thusthe original problem is solved by performing the timeevolution of the SIAM, given the time-dependent pa-rameters of Eq. (17a)- Eq. (17c), as obtained by the self-consistency to be outlined in Sec. (3.3). While simplerthan the original lattice system, solving the impurityproblem is still non-trivial. To aid this the use of state-of-the-art experimental quantum technologies has recentlybeen proposed based on quantum simulating the dy-namics of the SIAM with trapped ions [31] and supercon-ducting qubits [32]. Otherwise popular approaches ona classical computer include exact diagonalization [30],multi-configurational time-dependent Hartree Fock [33]and matrix product state (MPS) calculations [17, 34]. Inthis work we exploit a slightly different version of the lat-ter as we now describe. The MPS tensor network has a one-dimensional chaingeometry and as such this has made it an extremelysuccessful ansatz for describing short-ranged interacting Copyright line will be provided by the publisherctober 25, 2017
Figure 1 (a) Star geometry of the SIAM, with the impuritysite (blue big circle) surrounded by noninteracting bath orbitals(small red circles). The green lines indicate the hybridizationbetween both. (b) Linear chain geometry of the SIAM, with theimpurity on the left edge and long-range hopping to the bathorbitals. (c) td-DMRG sweep for a single time step, where theimpurity moves across the lattice by the action of swap gates. one-dimensional quantum systems. While the star geom-etry of the SIAM is straightforward with exact diagonal-ization, it does pose an issue for MPS approaches. To ap-ply MPS we therefore reshape the system as a chain withthe impurity site on an edge and long-range hopping, asdepicted in Fig. 1(b). It is not a priori clear whether aMPS simulation can be performed efficiently for this sys-tem, since entanglement is generally expected to growvery fast in the presence of long-range coupling. How-ever it has been shown that entanglement grows slowlydue to the inhomogeneous distribution of the couplingsacross the lattice, making the problem suitable for ma-trix product calculations [34]. For non-equilibrium prob-lems, Refs. [17, 34] successfully applied MPS simulationmethods to the SIAM with long-range hopping using aKrylov-based time evolution algorithm [35].In our work, we show that a conventional time-dependentdensity matrix renormalization group (td-DMRG) ap-proach based on Trotterized two-site unitaries [36, 37]works similarly well, not only for time-independent evo-lution but also for periodic driving. The codes used herewere implemented with the open-source Tensor NetworkTheory (TNT) library [38, 39]. The long-range interactionis dealt with as shown in Fig. 1(c). We perform the evolu-tion during a single time step of length δ t by means of asecond-order Trotter expansionexp ³ − i H SIAM δ t ´ ≈ Ã L bath Y p = W (0, p ) !Ã Y p = L bath W ( p ,0) ! , (18) which consists of a left-to-right sweep of local two-sitegates W (0, p ) followed by a right-to-left sweep of gates W ( p ,0), defined by W (0, p ) = S exp ³ − i h (0, p ) δ t ´ , (19a) W ( p ,0) = S exp ³ − i h ( p ,0) δ t ´ . (19b)Here S is the fermionic swap gate that exchanges the po-sition of the impurity with the nearest neighbor in the di-rection of the sweep, h (0, p ) is the SIAM Hamiltonian ofimpurity site 0 and bath site p for the left-to-right sweepwhere the impurity is to the left of the bath site, and simi-larly for h ( p ,0) in the right-to-left sweep where the impu-rity is to the right of the bath site.In this scheme the impurity is moved across the chain,becoming nearest neighbor of every bath orbital at somepoint in the sweep, and thus allowing for a standard im-plementation of each two-site gate. Our td-DMRG calcu-lations were performed using particle-number conserva-tion for both spins σ =↑ , ↓ , bath size L bath =
20, and U -dependent time steps 5 × − ≤ δ t ν ≤ × − . The deci-mation process during the time evolution was controlledby fixing a maximal truncation error per time step, witha maximal internal dimension of χ = χ during the evolution is determined by the Hamilto-nian parameters, being fast for systems with strong cor-relations. This increase limits the final times that can beaccessed within the algorithm, as the states of each stepmust be saved to perform the DMFT self consistency (seeSec. 3.3) for which a huge amount of memory is required.However the timescales reached in our simulations, simi-lar to those of the previous NE-DMFT+MPS works [17,34]and obtained with essentially the same computationaleffort, allow us to provide a complete discussion of therelevant physical processes underlying the dynamics. Ap-proximate schemes to reach even longer times have beenpreviously proposed [34]. Now we summarize the main steps to implement the NE-DMFT for an initial state that is a classical antiferromag-netic Néel state | Ψ Néel 〉 = Y i ∈ A c † i ↑ Y j ∈ B c † j ↓ | 〉 . (20)A detailed description can be found in Refs. [17, 30, 34].The system consists of two homogeneous interpenetrat-ing sub-lattices, A for σ =↑ and B for σ =↓ . Since both Copyright line will be provided by the publisher endoza-Arenas et al.: Ultra-fast control of magnetic relaxation in a periodically driven Hubbard model lattices are identical except for their opposite magnetiza-tion, only the dynamics of one of them needs to be calcu-lated, with that of the other sub-lattice following imme-diately.The central quantity of the NE-DMFT method andits self-consistency condition is the local single-particleGreen function G k σ ( t , t ′ ) = − i 〈 T c k σ ( t ) c † k σ ( t ′ ) 〉 , (21)where T indicates time ordering along a Keldysh con-tour on which t , t ′ are placed, and c k σ ( c † k σ ) is the anni-hilation (creation) operator of a fermion at a particularsite of sub-lattice k = A , B ; since the sub-lattices are ho-mogeneous, the site index is not included.For a Bethe lattice its semi-elliptical density of states D ( ǫ ) = q ν − ǫ /(2 πν ) results in the lattice Green func-tions being related to the mean-fields Λ k σ ( t , t ′ ) by a sim-ple self-consistency condition, Λ A ( B ), σ ( t , t ′ ) = ν ( t ) G B ( A ), σ ( t , t ′ ) ν ∗ ( t ′ ), (22)where J ( t ) = ν ( t )/ p Z is the time-dependent hopping inthe driven Hubbard model. Since the Green functions ofthe two sub-lattices are related by the symmetry condi-tion G A , σ = G B , − σ , we may drop the k index. ThereforeEq. (22) becomes Λ σ ( t , t ′ ) = ν ( t ) G − σ ( t , t ′ ) ν ∗ ( t ′ ). (23)Additionally, the mean-field obtained from the SIAMwith a star geometry is Λ SIAM σ ( t , t ′ ) = X p V σ p ( t ) g p σ ( t , t ′ ) V σ p ( t ′ ), (24)with g p σ ( t , t ′ ) the noninteracting Green function for theisolated bath orbital p , and whose exact analytical formis well known [19]. The time-dependent parameters ofthe SIAM, namely V σ p ( t ) and ǫ p σ ( t ), must be chosen so Λ σ ( t , t ′ ) = Λ SIAM σ ( t , t ′ ). (25)If this condition is satisfied, the SIAM correctly capturesthe physics of the original high-dimensional lattice.Due to the particle-hole symmetry of the problem,and the possibility to freely choose ǫ p σ ( t = k = A sub-lattice, the former corresponds to a single fermionwith spin up. So the initial state of the SIAM is | ψ SIAM ( t = 〉 = c †0 ↑ L bath Y p = ( L bath /2) + c † p ↑ c † p ↓ | 〉 , (26)with | 〉 the SIAM vacuum. The full implementation ofthe NE-DMFT requires several technical details and sub-tleties not mentioned here [17, 30, 34]. Instead we sketchthe main ingredients required for the algorithm. The ba-sic steps for one of the sub-lattices are the following.1. Start with a guess of the Green function G σ ( t , t ′ ).2. Obtain mean field Λ σ ( t , t ′ ) from self consistency Eq. (23).3. Calculate the SIAM hopping parameters V σ p ( t ) fromDMFT conditions Eq. (24) and Eq. (25). This can beperformed, for example, by means of a Cholesky de-composition [30]. For simplicity, the on-site bath or-bital energies are taken to be ǫ p σ ( t > =
0, so all timedependence is assigned to the hopping.4. Obtain a new Green function G σ ( t , t ′ ) by calculatingEq. (21) for the impurity site of the SIAM. The re-quired time evolution of the initial state under Hamil-tonian Eq. (16) is calculated with td-DMRG as de-scribed in Sec. (3.1).5. Repeat steps 2-4 until convergence is reached.Once convergence has been obtained, single-site ex-pectation values can be calculated. In particular we focuson the double occupation d ( t ) = 〈 n ↑ ( t ) n ↓ ( t ) 〉 , (27)and the staggered magnetic order parameter M stagg ( t ) =〈 n ↑ ( t ) − n ↓ ( t ) 〉 , along with M ( t ) = M stagg ( t )1 − d ( t ) , (28)which is normalized by the probability of the site beingsingly occupied [17]. The magnetization M ( t ) signals theexistence of antiferromagnetic order in the system, de-scribing spin dynamics, and d ( t ) indicates the formationof charge excitations during the dynamics. To determine the AFM melting mechanism during the dy-namics, the following setup was proposed in Ref. [17]. Ona single probe site o of sub-lattice A, the spin of the ini-tially located fermion is flipped in the x direction. Thisaction leads to a change of O (1/ Z ) on Λ σ , and thus has anegligible back-action on the rest of the lattice [17]. The Copyright line will be provided by the publisherctober 25, 2017
Figure 2 (a) Initial state of the SIAM. The impurity site ( )is coupled to the bath orbitals, with ǫ p σ = , through time-dependent hopping rates V σ p ( t ) . At time t = the impurityis isolated, so V σ p (0) = . Initially half of the bath orbitals areempty and half are doubly occupied. The initial state of theimpurity site, represented by a blue circle, depends on the par-ticular problem of interest. (b) Scheme of spin precession dy-namics. A single magnetic moment S ( t ) in the Bethe lattice isflipped to the x direction, which precesses in the x - y planeunder the effective magnetic field B eff (green arrows). Fromthe precession angle φ ( t ) the exchange interaction J ex canbe obtained as described in Sec. (3.4). resulting dynamics corresponds to the local magneticmoment precessing in the effective mean field Λ σ ( t , t ′ )obtained in the DMFT simulations. So a new SIAM simu-lation is performed in which the spin at site o (the impu-rity) starts in x direction. The initial state of the SIAM isthus | ψ SIAM ( t = 〉 = p c †0 ↑ + c †0 ↓ ) L bath Y p = ( L bath /2) + c † p ↑ c † p ↓ | 〉 , (29)and the dynamics takes place with the same hopping pa-rameters obtained during the DMFT. As the neighboringsites of the probe o are oriented along the z direction, wecan assume that they lead to an effective parallel mag-netic field B eff = B eff ˆ z . Under this approximation the dy-namics is easily shown to correspond to a harmonic pre-cession of the magnetic moment in the x - y plane, with frequency Ω = ˙ φ ( t ) = B eff ; this is depicted in Fig. 2(b).Thus the components of the magnetic moment S ( t ) atthe impurity site S x = 〈 c †0 ↑ c ↓ + c †0 ↓ c ↑ 〉 , (30a) S y = − i 〈 c †0 ↑ c ↓ − c †0 ↓ c ↑ 〉 , (30b) S z = 〈 n ↑ − n ↓ 〉 , (30c)with initial conditions S x (0) = S y (0) =
0, are de-scribed by the functions S x ( t ) = P x cos( Ω x t ), S y ( t ) = P y sin( Ω y t ), (31)with amplitudes P x = P y = Ω x = Ω y = Ω . From these components an effective exchange inter-action between the impurity and the mean field can beobtained, J ex = ˙ φ ( t )/ | M stagg | . Here φ ( t ) = tan − · S y ( t ) S x ( t ) ¸ (32)is the angle of the spin in the x - y plane [17] and M stagg is the staggered magnetization of the environment (ob-tained from the magnetic melting simulation), which de-fines the exchange interaction through B eff = J ex M stagg .With this it is possible to find out whether the initial AForder is melted by (a) the decay of magnetic moments intime, (b) the decay of the effective exchange interactionwhile the moments persist, or (c) the energy exchange be-tween charge and spin sectors (if (a) and (b) don’t occur). We start by discussing the physics of the undriven FermiHubbard model in a Bethe lattice ( A = ν = ν =
1) toprovide a baseline of expected effects. Even though thisscenario has been studied in detail in Ref. [17] (see alsoRef. [40] for a study of the 1D case), we outline the melt-ing process here focusing on the two limits of U ≫ ν and ν > U . In Sec. (5) we will describe how this physics is af-fected when the system is periodically driven. We first consider the case of strong Coulomb interactions U ≫ ν . In Fig. 3(a) we show the decay in the magnetic or-der parameter M ( t ), and correspondingly in Fig. 3(b) thedouble occupation over the time 0 ≤ ν t . U . The behavior for both quantities divides up Copyright line will be provided by the publisher endoza-Arenas et al.: Ultra-fast control of magnetic relaxation in a periodically driven Hubbard model Figure 3
Dynamics of a system with no driving and U ≫ . (a) Decay of order parameter M ( t ) . The symbols corre-spond to the results of the simulations, and the dashed linesto linear fits M ( t ) = − m ν t / U . For U = wehave m = respectively. Inset. Early-time dynamics of the staggered magnetization M stagg ( t ) , withscaled axes. (b) Double occupation d ( t ) . Inset. Early-time dy-namics, with scaled axes. The times for both insets correspondto the shaded areas of the main panels. We note that the os-cillations at early times of both M stagg ( t ) and d ( t ) have U -dependent decays, not captured by the perturbative results ofEqs. (33) and (34) . However we verified numerically that theseare well described by a function ∼ exp( − qU t ) with constant q . distinctly into times before and after ν t ≈
1. For ν t > M ( t ) with a U dependentgradient, while d ( t ) converges to a U dependent steady-state value. Thus M ( t ) and d ( t ) relax at a different rate, the charge dynamics being faster than the spin dynam-ics. For ν t < M ( t )and d ( t ) that are well captured by a second-order time-dependent perturbation theory [40] as M stagg ( t ) = − ν U sin µ U t ¶ , (33)and d ( t ) = ν U sin µ U t ¶ . (34)This is made clearer in the insets of Fig. 3(a) and (b)where oscillations in M ( t ) and d ( t ) for different U ’s col-lapse on top of one another and damp away within ν t ≈
1. These results are consistent with the charge excita-tion decay mechanism outlined in Ref. [17]. The earlytime oscillations, resulting from the Néel state not beingan eigenstate of the Hubbard model at finite U , nucle-ate a finite amount of charge excitations d ss ≈ ν / U in the system, as highlighted in Fig. 3(b). The motion ofthese excitations, with a hopping amplitude ν , on top ofa spin background then scrambles the AFM order. Owingto strong spin-charge coupling via the exchange interac-tion J ex the kinetic energy of these charge carrier is trans-ferred to the spin sector. This process is well captured bya so-called t − J z model [17, 41], where it is found that for ν t > f of spins flipped by a single carriergrows linear as f ( t ) ∼ ν t before saturation occurs. Thedecay of magnetization is therefore M ( t ) ≈ − d ss f ( t ) = − ν U ( ν t ). (35)In Fig. 3(a) we indeed find a linear decay with a gradientscaling as 1/ U and coefficient close to 6 for times ν t >
1. In this regime the spin precession dynamics, describedin Section 3.4, gives rise to Fig. 4. At early times ν t < J ex = Ud ( t ),strongly oscillating around its static perturbative value J pertex = ν / U . (36)For longer times ν t > M ( t ) is not due to the suppressionof local moments. For U / ν .
1, close to the non-interacting integrablelimit U =
0, approximately conserved quasi-particles are Copyright line will be provided by the publisherctober 25, 2017
Figure 4
Spin precession dynamics for non-driven lattices.(a) Trajectory in the x - y plane of the precessing magnetic mo-ment. We checked numerically that this is approximately welldescribed by Eq. (31) . (b) Exchange interaction J ex extractedfrom the dynamics of simulations with U ≫ . The perturbativeexchange values (36) are indicated by arrows ( ← ). Matchingcolors among both panels correspond to equal values of U . expected to govern the dynamics leading to oscillatorybehavior of M ( t ) and prethermalization [17]. In Fig. 5 weshow the results for M ( t ) and d ( t ) for different low val-ues of the Coulomb repulsion . The magnetic order pa-rameter oscillates and decays very fast to zero, and thedouble occupancy saturates to high values, approach- U due to adifferent growth rate of χ , as described in Sec. 3.2. This is alsoobserved in results to be presented in subsequent Sections. Figure 5
Dynamics of a system with no driving and U . .Main panel. Order parameter M ( t ) . Inset. Double occupation d ( t ) . Matching colors correspond to equal values of U . ing the non-interacting limit d ( t → ∞ ) = U de-creases. The spin precession dynamics, shown in Fig. 4,displays a very small decaying component in the y direc-tion and a rapidly decaying x component. This indicatesthat the magnetic moments are short lived, and that aneffective exchange interaction cannot be well defined. Inthis regime magnetic and charge dynamics occur on thesame time scale, namely that of the hopping ν . The melt-ing of AFM order is therefore via a different mechanismto that seen at U / ν ≫
1, corresponding to the destruc-tion of magnetic moments.
Having reviewed the behavior for the static case we nowconsider the impact of the driving, represented by thetime-dependent hopping (5). To do so we explore in turnthree different regimes for the driving frequency ω : (i)high-frequency ω ≫ U , ν , (ii) resonant l ω = U > ν withinteger l , and (iii) in-gap U > ω > ν . The high-frequency limit, ω ≫ U , ν , where the driving ismuch faster than all microscopic time scales of the sys-tem, is the conventional regime for Floquet theory. InFig. 6 we show the behavior of M ( t ) and d ( t ) for ν = U = ω =
15. As expected for the weakly interacting
Copyright line will be provided by the publisher endoza-Arenas et al.: Ultra-fast control of magnetic relaxation in a periodically driven Hubbard model Figure 6
Time evolution of a lattice with high-frequency driv-ing. (a) Magnetic order parameter. (b) Double occupation. Thesolid lines correspond to systems with ν = U = , ω = and different amplitudes A . The dashed lines correspond tosystems with no driving, U = and renormalized hopping ν = ν J ( A ) , with A the driving amplitude of the solid lineof the same color. regime, the static system displays magnetic order that de-cays very fast and features oscillations around M ( t ) = d ( t ) = A = A = M ( t ) and a significant suppression of d ( t ). Thisbehavior is typical of static systems with a larger value of U / ν . Figure 7
Spin precession dynamics for high-frequency driv-ing. (a) Trajectory in the x - y plane of the precessing magneticmoment. The solid lines correspond to ν = U = , ω = and different amplitudes A . The dashed lines correspond tosystems with no driving, U = and hopping ν = ν J ( A ) .(b) Exchange interaction J ex extracted from the precession dy-namics of the system with renormalized hopping (solid lines),and perturbative values J ( A ) ν ) / U (dashed lines). These results are well known from Floquet theory. Inthe high-frequency limit, different Floquet sectors arelargely separated in energy, as suggested by Eq. (9). Sothe system can be effectively described by keeping onlysector m =
0. From Eq. (11), this corresponds to a Fermi-Hubbard model with the hopping being renormalized bya Bessel function J ( A ). Since | J ( A ) | <
1, this corre-sponds to a suppression of the hopping, or equivalentlyto an effective enhancement of U / ν . Copyright line will be provided by the publisherctober 25, 2017
As depicted in Fig. 6 for both M ( t ) and d ( t ), thedynamics of the driven system is (after averaging outsmall modulations at frequency ω ) very well captured bya static system with a hopping renormalized by J ( A ).This picture is confirmed by the spin precession dy-namics; the results are shown in Fig. 7. For weak driv-ing amplitudes the precessing magnetic moment decaysquickly, while for large amplitudes it persists for longtimes. The average dynamics is again well described bya non-driven system with a renormalized hopping alongwith an exchange interaction (obtained as described inSection 3.4) close to the renormalized perturbative value4( J ( A ) ν ) / U .Our simulations thus corroborate the prediction fromFloquet theory of effective hopping renormalization athigh frequency driving. Choosing amplitudes A so | J ( A ) | ≪ U / ν < A so J ( A ) =
0, thehopping is completely suppressed (effectively correspond-ing to infinite Coulomb interaction) and the dynamics isfrozen . This phenomenon of dynamical localization bycoherent driving was predicted long ago in a noninteract-ing system with the same type of driving used here [42]. Ithas also been studied theoretically in spin chains [43,44],and was observed experimentally in a system of coldatoms where a potential of the form of Eq. (2) was createdby shaking an underlying optical lattice [45]. We now decrease ω to the point that it is directly reso-nant as ω = U ≫ ν ( l = ω = U =
15. In the non-driven case,the magnetic order decays very slowly and the double oc-cupation remains low; the Coulomb repulsion is so largethat fermions are prevented from hopping across the lat-tice. This picture is strongly modified when driving at res-onance, even at weak amplitudes. The magnetic order de-cays very fast, and the double occupation increases tolarge values. Importantly, for A = M ( t ) has been suppressed at short times while d ( t ) hasnot saturated. For strong driving A > Figure 8
Time evolution of a lattice with resonant driving. (a)Magnetic order parameter. (b) Double occupation. The solidlines correspond to ν = , U = ω = and various ampli-tudes A . The dashed line corresponds to the non-driven case. onant driving it is possible to control the speed of chargedynamics compared to spin dynamics up to the point ofmaking the former slower, a behavior not seen in the non-driven case.The magnetic melting mechanism is elucidated fromthe spin precession dynamics shown in the main panelof Fig. 9. In the static case the magnetic moment is pre-served for a long time, while for the resonantly drivencase the magnetic moments decay very fast. Resonantabsorption of energy from the drive thus destroys the lo-cal moments so an exchange interaction cannot be de-fined, and suppresses magnetic order. The behavior is ex-tremely reminiscent of the static case with U / ν <
1, andsuggests that resonant driving can be used to switch from
Copyright line will be provided by the publisher endoza-Arenas et al.: Ultra-fast control of magnetic relaxation in a periodically driven Hubbard model Figure 9
Precession dynamics in the x - y plane for ν = ,resonant driving l = U = ω = and various amplitudes A . a charge-excitation mechanism to quasi-particle melt-ing.Resonant driving with l even can in fact map the in-teracting system exactly to the U = J eff = K eff in Hamiltonian (13), the driving precisely induces holon-doublon creation terms that would exist if U = l = ω = U /2) and A = ν J ( A ) confirming this novel effectfor an infinite-dimensional system. Therefore at this spe-cial driving strength and frequency ω = U /2 we have in-duced an effective suppression of the Coulomb repulsion U , which corresponds to the opposite effect of the effec-tive interaction enhancement at high-frequency driving. For larger driving strengths A = l = l = d ( t ) ≈ d ss = A sothat J ( A ) <
0, flipping the sign of the hopping. Therethe appearance of d ( t ) > attractive in- Figure 10
Resonant l = dynamics for U = , ω = andseveral driving amplitudes A . (a) Magnetic order parameter.(b) Double occupation. We go from repulsive interactions ( A = ) to an effective non-interacting case ( A = ,well described by a non-interacting lattice with renormalizedhopping ν = ν J ( A ) = ), to positive effective interac-tions ( A = ). After creation of holon-doublon pairs is maxi-mized ( A = ), the interactions start decreasing whilestill being positive ( A = ). teraction. Here similar signatures are seen, but cruciallythey occur on a much faster time scale and emerge froman initial state with zero double occupancies.This can be qualitatively understood from the res-onant effective Hamiltonian (13) with l =
1. For A = J eff > K eff , so the dynamics is dom-inated by holon and doublon hopping processes. Onthe other hand, for A = | K eff | > | J eff | , so holon-doublon creation processes dominate, explaining the Copyright line will be provided by the publisherctober 25, 2017
Figure 11
Time evolution of a lattice with in-gap driving. Thesolid lines correspond to ν = , A = , ω = and differentvalues of U > ω . The dashed lines of the same color corre-spond to equal values of U with no driving. (a) Magnetic orderparameter. Inset. Collapse of M ( t ) = (1 − M ( t ))/ J ( A ) 〈 d 〉 as a function of time for the values of U of the main panel;results for U = and different amplitudes A have also beenincluded, to further test Eq. (37) . (b) Double occupation. Theresults for non-driven systems are shown in the inset for clarity. greater propensity for double occupancies. For l = A = J eff =
0, so the dynamics isthen entirely governed by creation and annihilation ofdoublon-holon pairs. This temporarily maximizes d ( t ),as depicted in Fig. 10(b). However in either case the effec-tive Hamiltonian is far from simple, owing to frustrationeffects from overlapping holon-doublon creation terms.We therefore postpone a more detailed analysis of poten-tial pairing [22] to future work where it maybe of rele- vance to observations of light-induced superconductiv-ity [4, 5]. Now we consider the regime of in-gap driving ν < ω < U .For our analysis we fix ω and take different values of U , staying away from resonant points U = l ω . In thisform the divergences of J ex ( A , ω ) predicted by Eq. (12)are avoided. The magnetic order parameter M ( t ) andthe double occupation d ( t ) are depicted in Fig. 11, forboth the driven and the non-driven cases. Although eachtime evolution shows a general decay of M ( t ) with strongfluctuations, it closely follows the melting of the corre-sponding non-driven case. The double occupation, onthe other hand, notably increases on average due to thedriving, although still remains low.These results suggest that the melting mechanismof the off-resonant in-gap driving discussed here is thesame of the strongly-interacting non-driven case, de-scribed in Sec. 4.1. In fact, as depicted in the inset ofFig. 11(a), besides oscillations the magnetic order param-eter satisfies M ( t ) ∼ − 〈 d 〉 ˜ f ( t ), (37)with ˜ f ( t ) = J ( A ) ν t and 〈 d 〉 the time-average of d ( t ).This behavior is completely analogous to that of Eq. (35)for the non-driven case, with renormalized hopping ν J ( A )as expected for in-gap off-resonant driving [22]. So themagnetic melting emerges from the strong spin-chargecoupling, being mostly dependent on the hopping andthe number of charge excitations nucleated at early times,and weakly dependent on the value of the exchange inter-action. The fact that the melting of the order parameterclosely follows that of the non-driven case indicates thatthe effective suppression of the hopping by J ( A ) is ap-proximately compensated by the increase of the doubleoccupation.This picture is reinforced by the spin precession dy-namics. In Fig. 12(a) we show fitted trajectories of the pre-cessing spin in the x - y plane, which for each value of U are very similar to those of the non-driven case. The mag-netic moments thus remain long-lived under in-gap off-resonant driving. The resulting effective exchange inter-actions of the driven and non-driven cases are also sim-ilar, as shown in Fig. 12(b). The relation between bothis very well captured by the Floquet result Eq. (12). Forthe different U we obtain that 1.02 < J ex ( A , ω )/ J ex ( A = < Copyright line will be provided by the publisher endoza-Arenas et al.: Ultra-fast control of magnetic relaxation in a periodically driven Hubbard model Figure 12 (a) Trajectories in the x - y plane of precessing mag-netic moments in systems with ν = , A = , ω = anddifferent values of U > ω . The solid lines are the resulting tra-jectories when fitting S x and S y to the functions (31) , with theoriginal trajectories being shown in the inset (with the samescale for both S x and S y ). The dashed lines of matching col-ors are for systems with equal U and no driving. (b) Exchangeinteraction J ex for the driven dynamics extracted from the fit-ting functions (31) (solid lines; extracting J ex from the trajec-tories depicted in the inset of (a) is complicated due to theirmany loops), and for systems with equal U and no driving(dashed lines). The perturbative exchange (36) for the non-driven cases are indicated by arrows ( ← ), and those from Flo-quet theory for in-gap off-resonant driving (12) by solid circles( • ). change are long-lived, the magnetic melting occurs dueto energy transfer from charge to spin sectors.In summary the results of this Section show that forin-gap driving away from resonances, a similar magnetic dynamics to that of the non-driven case is induced. Thenotably larger double occupancies created by the driv-ing are compensated by the suppression of the hopping.In addition the exchange interaction is weakly modified.This approximately leads to the same magnetic meltingrate in both the driven and non-driven scenarios, leavingthe underlying melting mechanism unaffected. Finally we show two examples of how combining the dif-ferent driving schemes previously described, it is possi-ble to engineer a ultra-fast protocol to manipulate themagnetic order and pairing of a fermionic interacting sys-tem. The first corresponds to rapidly stabilizing a partic-ular magnetization value from an initial Néel state withlarge U / ν . An immediate form to do it is following a driv-ing scheme like that depicted in Fig. 13(a). This consistsof two stages. First the desired magnetization is quicklyreached by resonant driving. Then the dynamics is frozenby high-frequency driving, maintaining the magnetiza-tion for the required time. This protocol could also beused for stabilizing a state with enhanced pairing, target-ing a particular value of the double occupation instead ofmagnetization (inset of Fig. 13(a)).The second example illustrates the induction of dy-namics reversal by an appropriate choice of the driving.In Ref. [16] a scheme for achieving this effect in longtime scales ν t ≈
100 was discussed. There a 1D systemwas allowed to evolve freely for some time, after which itwas in-gap and non-resonantly driven in such a way that,from Eq. (12), J ex ( A , ω )/ J ex ( A = ≈ −
1. By slowly ramp-ing the driving field electronic excitations were stronglyimpeded, leading to spin-dominated dynamics whichcould be reversed by changing the sign of the exchangeinteraction.Here we use a similar idea to induce a dynamics re-versal on much shorter time scales ν t <
8, but with asudden quench of the amplitude of a l = A tilltime t , leading to a fast melting of the magnetic orderand large charge excitations. For the second stage thedriving amplitude A is chosen so the sign of both J eff and K eff is inverted, while keeping their ratio to the val-ues of the first stage approximately constant. The ampli-tudes used in Fig. 13(b) are such that J eff ( A )/ J eff ( A ) ≈ Copyright line will be provided by the publisherctober 25, 2017
Figure 13 (a) Simple protocol to stabilize a strongly-interacting system ( ν = , U = ) with a particular mag-netization (main panel) or double occupation (inset). From t = to t = t resonant driving is applied ( A = , ω = )to quickly decrease the magnetization. Then for t > t high-frequency driving is applied ( A = so J ( A ) = , ω = ), which freezes the system in the final state of the first driv-ing scheme. (b) Protocol to induce dynamics reversal on astrongly-interacting system at l = resonance ( U = ω = ),indicated by both the magnetic order parameter (main panel)and the double occupation (inset). From t = to t = (red dashed lines) the system in driven with A = , andfrom t = onwards it is driven with A = . In this form J ( A )/ J ( A ) = − and J ( A )/ J ( A ) = − .Reversal of dynamics is observed at time ν t ≈ (greendashed lines). K eff ( A )/ K eff ( A ) ≈ −
2. The dynamics for t > t is thusslower than that for t < t by a factor of 2, in addition to the change of sign, resulting in an almost-complete rever-sal of both the magnetic order parameter and the doubleoccupation after an evolution of length 2 t . The reversalis not perfect due to the higher-order terms in Hamilto-nian (13). However our results show that it is possibleto almost entirely re-magnetize a system whose orderparameter has been strongly suppressed on a ultra-fasttime scale. Notably this re-magnetization occurs eventhough the double occupation reaches large values, by re-versing the dynamics of the latter as well. In the present work we have discussed several formsin which the melting of an initial perfect Néel state ina high-dimensional Fermi-Hubbard lattice can be con-trolled by external periodic driving. Using the recently-introduced non-equilibrium dynamical mean-field the-ory with a Hamiltonian-based matrix product impuritysolver, we have performed the time evolution of drivensystems with a computational effort similar to that ofquenched Hamiltonians. In addition, insights from Flo-quet theory have allowed us to understand the underly-ing mechanisms responsible for the melting of the mag-netic order in each scenario considered.We focused on three different driving regimes. Firstwe observed how high-frequency driving suppresses thehopping and the exchange interaction, slowing down themelting of the magnetic order. For weak Coulomb repul-sion this corresponds to switching from quasiparticle-governed magnetic melting to the scrambling of mag-netic order due to mobile charge carriers. Second we con-sidered resonant driving, and observed how for stronginteractions it leads to a very fast magnetization decayand a large enhancement of double occupations. Thiscorresponds to switching from melting due to chargecarriers to quasiparticle-governed dynamics. In addi-tion, we showed how resonant driving can effectivelyinduce zero or attractive interactions on an ultra-fasttime scale, where the latter might be relevant for non-equilibrium superconductivity. Third we considered in-gap off-resonant driving, where for large interactions thedriving-induced double occupation is compensated bythe suppression of the hopping, while the exchange in-teraction is weakly affected. Thus the underlying mag-netic melting mechanism is the same of the non-drivensystem, governed by a strong spin-charge coupling. It isimportant to stress that these results illustrate how driv-ing with the same amplitude A , in particular those forwhich J ( A ) =
0, can lead to completely different conse-
Copyright line will be provided by the publisher endoza-Arenas et al.: Ultra-fast control of magnetic relaxation in a periodically driven Hubbard model quences for distinct frequency regimes. In this case hop-ping processes are suppressed, but while it leads to dy-namical localization at high frequency, it maximizes thecreation of charge excitations at resonance.Gathering ideas from the different driving regimeswe presented combined schemes of magnetic control,namely the stabilization of states with determined mag-netic order parameter or double occupation, and the in-duction of dynamics reversal in a short time scale. Ourwork thus indicates that ultra-fast periodic modulationmight provide a viable mechanism to control antifer-romagnetic order in strongly interacting systems. Thiscould be observed experimentally in condensed mattersystems by excitation with THz light pulses [3], or in coldatomic gases [47] in an oscillating underlying optical lat-tice [1, 2].Finally we emphasize that our work manifests thepower of the NE-DMFT algorithm to successfully de-scribe strongly-driven many-body systems in differentinteracting regimes, without resorting to perturbationtheory. It also shows that conventional MPS time evo-lution based on Trotterized two-site unitary operationsworks as well as the previously-used Krylov-space algo-rithms [17,34], since with both methods similar timescalesare reached with comparable computational effort. Thismotivates the use of our approach for analyzing how tomanipulate and enhance, by external periodic driving,different types of ordered states in high-dimensional lat-tices such as charge-density waves and superconductingphases, for repulsive [22] and attractive [48,49] fermionicmodels. Acknowledgements
The authors would like to acknowledge the use of the Uni-versity of Oxford Advanced Research Computing (ARC)facility in carrying out this work. http://dx.doi.org/10.5281/zenodo.22558.This research is partially funded by the European Re-search Council under the European Union’s SeventhFramework Programme (FP7/2007-2013)/ERC Grant Agree-ment no. 319286 Q-MAC. This work was also supportedby the EPSRC National Quantum Technology Hub in Net-worked Quantum Information Processing (NQIT) EP/M013243/1,and the EPSRC projects EP/P009565/1 and EP/K038311/1.J.J.M.A. and F.J.G.R. acknowledge financial support fromFacultad de Ciencias at UniAndes-2015 project "Quan-tum control of non-equilibrium hybrid systems-Part II",and F. Rodríguez, L. Quiroga and J. Coulthard for interest-ing discussions. J.J.M.A. also acknowledges S. Al-Assam and J. Kreula for their help during the development of theDMFT code.
Key words.
Ultrafast control, antiferromagnetism, Floquettheory, non-equilibrium dynamical mean-field theory