Schrieffer-Wolff Transformation for Periodically Driven Systems: Strongly Correlated Systems with Artificial Gauge Fields
We generalize the Schrieffer-Wolff transformation to periodically driven systems using Floquet theory. The method is applied to the periodically driven, strongly interacting Fermi-Hubbard model, for which we identify two regimes resulting in different effective low-energy Hamiltonians. In the nonresonant regime, we realize an interacting spin model coupled to a static gauge field with a nonzero flux per plaquette. In the resonant regime, where the Hubbard interaction is a multiple of the driving frequency, we derive an effective Hamiltonian featuring doublon association and dissociation processes. The ground state of this Hamiltonian undergoes a phase transition between an ordered phase and a gapless Luttinger liquid phase. One can tune the system between different phases by changing the amplitude of the periodic drive.
The Schrieffer-Wolff transformation (SWT) Schrieffer and Wolff (1966); Zhang and Rice (1988); Bravyi et al. (2011); Barthel et al. (2009) is a generic procedure to derive effective low-energy Hamiltonians for strongly-correlated many-body systems. It allows one to eliminate high-energy degrees of freedom via a canonical transform. The SWT has proven useful for studying systems with a hugely degenerate ground-state manifold, such as the strongly-interacting limit of the Fermi-Hubbard model (FHM) Zhang and Rice (1988), without resorting to conventional perturbation theory.
Treating interactions in such a non-perturbative way is difficult in periodically-driven systems D’Alessio and Polkovnikov (2013); D’Alessio and Rigol (2014); Lazarides et al. (2014a, b); Ponte et al. (2015); Bukov et al. (2015a), which have received unprecedented attention following the realisation of dynamical localisation Dunlap and Kenkre (1986); Lignier et al. (2007); Zenesini et al. (2009); Creffield et al. (2010); Gong et al. (2009), artificial gauge fields Eckardt et al. (2010); Struck et al. (2011, 2013); Aidelsburger et al. (2013); Miyake et al. (2013); Atala et al. (2014); Kennedy et al. (2015), models with topological Oka and Aoki (2009); Kitagawa et al. (2011); Grushin et al. (2014); Jotzu et al. (2014); Aidelsburger et al. (2015); Fläschner et al. (2015) and state-dependent Jotzu et al. (2015) bands, and spin-orbit coupling Galitski and Spielman (2013); Jiménez-García et al. (2015). In this paper, we consider strongly-interacting periodically-driven systems and show how the SWT can be extended to derive effective static Hamiltonians of non-equilibrium setups. The parameter space of such models, to which we add the driving amplitude and frequency, opens up the door to new regimes. We use this to propose realisations of nontrivial Hamiltonians, including spin models in artificial gauge fields and the Fermi-Hubbard model with enhanced doublon association and dissociation processes.
SWT from the High-Frequency Expansion—Intuitively, the high-frequency expansion for periodically-driven systems (HFE) and the SWT share the same underlying concept: they allow for the elimination of virtually-populated high-energy states to provide a dressed low-energy description, as illustrated in Fig. 1. For a system driven off-resonantly (Fig. 1a), virtual absorption of a photon renormalises tunnelling. Similarly, non-driven fermions develop Heisenberg interactions via off-resonant (virtual) tunnelling processes (Fig. 1b). In this paper we combine the HFE and SWT into a single framework allowing one to treat both resonantly and non-resonantly driven systems on equal footing. Let us illustrate the connection by deriving the SWT using the HFE. Consider the non-driven FHM:
where is the bare hopping and is the fermion-fermion interaction. We are interested in the strongly-correlated regime . Going to the rotating frame w.r.t. the operator eliminates the energy in favor of fast oscillations. If , then
where and vice-versa. The first term models the hopping of doublons and holons, while the second term represents the creation and annihilation of doublon-holon pairs. Since is time-periodic with frequency , we can apply Floquet’s theorem Floquet (1883). Thus, the evolution of the system at integer multiples of the driving period [i.e. stroboscopically] is governed by the effective Floquet Hamiltonian . If we write , the HFE gives an operator expansion for Rahav et al. (2003); Goldman and Dalibard (2014); Goldman et al. (2015); Eckardt and Anisimovas (2015); Itin and Katsnelson (2015); Mikami et al. (2015). The zeroth-order term is the period-averaged Hamiltonian [here the doublon-holon hopping ], while the first-order term is proportional to the commutator , cf. Fig. 1b:
This effective Hamiltonian is in precise agreement with the one from the standard SWT
Using the HFE to perform the SWT offers a few advantages: (i) the SW generator comes naturally out of the calculation, (ii) one can systematically compute higher-order corrections Rahav et al. (2003); Goldman and Dalibard (2014); Bukov et al. (2015b); Goldman et al. (2015); Eckardt and Anisimovas (2015); Itin and Katsnelson (2015); Mikami et al. (2015), and (iii) the HFE allows for obtaining not only the effective Hamiltonian but also the kick operator, which keeps track of the mixing between orbitals and describes the intra-period dynamics Goldman and Dalibard (2014); Bukov et al. (2015b). This is important for identifying the fast timescale associated with the large frequency in dynamical measurements Trotzky et al. (2008), and expressing observables through creation and annihilation operators dressed by orbital mixing Bukov et al. (2015b).
Generalisation to Periodically-Driven Systems.—The HFE allows us to extend the SWT to time-periodic Hamiltonians. Related approaches have been used to study non-interacting Floquet topological insulators Nakagawa and Kawakami (2014) and ultrafast dynamical control of the spin exchange coupling Mentink et al. (2015) in fermionic Mott insulators Bermudez and Porras (2015). Let us add to the FHM an external periodic drive:
The driving protocol with frequency encompasses experimental tools such as mechanical shaking, external electromagnetic fields, and time-periodic chemical potentials, relevant for the recent realisations of novel Floquet Hamiltonians. In the following, we work in the limit and assume that the amplitude of the periodic modulation also scales with Bukov et al. (2015b).
Since both the interaction strength and the driving amplitude are large, we go to the rotating frame w.r.t. , where . The drive induces phase shifts to the hopping:
where . Notice that now there are two frequencies in the problem: and . Hence, is not strictly periodic in either. To circumvent this difficulty, we choose a common frequency by writing and where and are co-prime integers. Then becomes periodic with period , and we can proceed using the HFE. Alternatively, before going to the rotating frame, we could decompose the interaction strength as , where acts as a detuning, and can continue without including the term proportional to in .
Non-resonant Driving.— Let us first assume such that resonance effects can be ignored. We begin by Fourier-expanding the drive . If opposite spin species are driven out-of-phase, we have . Similarly, flipping the direction of the bond flips the sign of , so . We now apply the generalised SWT with frequency . At half-filling and for off-resonant driving double occupancies are suppressed, and the dominant term in the effective Hamiltonian is . Two types of commutators occur in this sum: the first comes from terms that have no oscillation with frequency , giving commutators of the form: ; all of these commutators vanish. The second type are the same commutators relevant for the SWT: , but note the presence of all higher-order harmonics induced by the drive. These involve terms rotating with , and thus will be suppressed by a –denominator. The commutators are explicitly done in the Supplemental material sup (), giving
where and .
One can Floquet-engineer the Heisenberg model with a uniform magnetic flux per plaquette , see Fig. 2. To this end, we choose the spin-dependent driving protocol (c.f. Fig. 2, inset), where , , and we denote the square-lattice position by . Such spin-sensitive drives are realised in experiments via the Zeeman effect using a periodically-modulated Jotzu et al. (2015) and static Aidelsburger et al. (2013); Miyake et al. (2013) magnetic-field gradients which couple to atomic hyperfine states. For this protocol,
where is the Bessel function of the first kind, is the dimensionless driving strength, and is the flux-modified strength
There are two physically interesting limits. For only survives and we get
where . For , we can set and sum over to obtain
The exchange strengths depend on and , but both limits give spin Hamiltonians with phases along . This phase physically appears on the flip-flop and not the Ising term because the drive is spin-dependent. Thus a phase difference only occurs if the electron virtually hops as one spin and returns as the other.
Let us discuss the regime a bit more. This spin Hamiltonian can be identified with the Heisenberg model in the presence of an artificial gauge field with flux per plaquette. Whenever the -interaction is small, the Hamiltonian reduces to the fully-frustrated XY model in 2D, in which one cannot choose a spin configuration minimizing the spin-exchange energy for all XY-couplings. In the classical limit, similarly to a type-II superconductor, the minimal energy configuration is known to be the Abrikosov vortex lattice Teitel and Jayaprakash (1983); Ryu and Stroud (1997). The realisation of the deep XY-regime with this particular driving protocol is limited, since but, at finite –interaction a semi-classical study showed that vortices persist and can be thought of as half-skyrmion configurations of the Neél field Lindner et al. (2009, 2010); Wu et al. (2004). Another interesting feature of the spin Hamiltonian is that it exhibits a Dzyaloshinskii-Moriya (DM) interaction term Cai et al. (2012); Radić et al. (2012); Cole et al. (2012); Piraud et al. (2014), . The DM coupling is spatially-dependent, polarised along the -direction , and present only along the -lattice direction.
Finally, let us mention that spin- systems are equivalent to hard-core bosons. In this respect, and model hard-core bosons with strong nearest-neighbour interactions in the presence of a gauge field. For a flux of the non-interacting model has four topological Hofstadter bands. If we then consider the strongly-interacting model, and half-fill the lowest Hofstadter band (), the Heisenberg model supports a fractional quantum Hall ground state Wang et al. (2011); Regnault and Bernevig (2011); Hafezi et al. (2007); Grushin et al. (2014). Away from half-filling of the fermions, doublon and holon hopping terms appear in the effective Hamiltonian, cf. Suppl. sup () and it would be interesting to study the effect of such correlated hopping terms Kourtis and Castelnovo (2015) on this topological phase.
Resonant Driving.—Novel physics arises in the resonant-driving regime . To illustrate this, we choose a one-dimensional system with the driving protocol , which was realised experimentally by mechanical shaking Lignier et al. (2007, 2007); Zenesini et al. (2009); Creffield et al. (2010). Unlike off-resonant driving, resonance drastically alters the effective Hamiltonian by enabling the lowest-order term : on resonance, the doublon-holon (dh) creation/annihilation terms survive the time-averaging, and the leading-order effective Hamiltonian reads
where for , for , , and . The first term, , is familiar from the static SWT, with a renormalised coefficient . The term proportional to appears only in the presence of the resonant periodic drive and is the source of new physics in this regime. By adjusting the drive strength, one can tune and to a range of values, including zeroing out either one. Starting from a state with unpaired spins, dh pairs are created via resonant absorption of drive photons. Hence, holons and doublons become dynamical degrees of freedom governed by , with the Heisenberg model as a subleading correction. The dh production rates and further properties of the system have been investigated both experimentally and theoretically Kollath et al. (2006); Huber and Rüegg (2009); Tokuno and Giamarchi (2011, 2012); Greif et al. (2011); Sensarma et al. (2009); Strohmaier et al. (2010); Hassler and Huber (2009); Balzer and Eckstein (2014); Mentink et al. (2015); Werner and Eckstein (2016); Bello et al. (2015). A DMFT study found that the AC field can flip the band structure, switching the interaction from attractive to repulsive Tsuji et al. (2011).
Such correlated hopping models have been proposed to study high- superconductivity Arrachea and Aligia (1994); Aligia et al. (2000); Aligia and Arrachea (1999). To get an intuition about the effect of the new terms, we use the ALPS DMRG and MPS tools Bauer et al. (2011); Dolfi et al. (2014) to calculate the ground state of at half-filling. The many-body gap in the thermodynamic limit is extracted from simulations of even-length chains with open boundary conditions by extrapolation in the system size: . We numerically confirm that the model features a transition between a symmetry-broken ordered phase and a gapless Luttinger liquid phase Arrachea and Aligia (1994); Aligia et al. (2000); Aligia and Arrachea (1999) as follows
It bears mention that all regimes of the model are accessible using present-day cold atoms experiments Greif et al. (2011). We propose a loading sequence into the ground state of in the Supplemental material sup (). Moreover, by tuning the frequency away from resonance, one can write and go to the rotating frame w.r.t. the -term, keeping a finite on-site interaction in the effective Hamiltonian. This is required if one wants to capture important photon-absorption avoided crossings in the exact Floquet spectrum. Including artificial gauge fields is also straightforward in higher dimensions, see Suppl. sup () and expected to produce novel topological phases. By utilizing resonance phenomena, this scheme only requires shaking of the on-site potentials, which is easier in practice than other schemes which have suggested modulating the interaction strength to realize similar Hamiltonians Di Liberto et al. (2014); Greschner et al. (2014).
Discussion/Outlook.—It becomes clear from the discussion above how to generalise the SWT to arbitrary strongly-interacting periodically-driven models: First, we identify the large energy scale denoted by (e.g., ) and write the Hamiltonian as . Second, we go to the rotating frame using the transformation to get a new time-dependent Hamiltonian with frequencies
Although isolated interacting Floquet systems are generally expected to heat up to infinite temperature at infinite time D’Alessio and Polkovnikov (2013); D’Alessio and Rigol (2014); Lazarides et al. (2014a, b); Ponte et al. (2015); Roy and Das (2015), the physics of such systems at experimentally-relevant timescales is well-captured by the above effective Hamiltonians; indeed, it was recently argued that typical heating rates at high frequencies are suppressed exponentially Abanin et al. (2015a); Kuwahara et al. (2016); Mori et al. (2015); Abanin et al. (2015b), and long-lived pre-thermal Floquet steady states have been predicted Canovi et al. (2016); Bukov et al. (2015c); Kuwahara et al. (2016); Abanin et al. (2015b). In particular, rigorous mathematical proofs Kuwahara et al. (2016); Mori et al. (2015); Abanin et al. (2015b) supported by numerical studies Bukov et al. (2015a) showed that the mistake in the dynamics due to the approximative character of the HFE is under control for the large frequencies and the experimentally-relevant times considered. Our work paves the way for studying such strongly-driven, strongly-correlated systems. Both the resonant and non-resonant regimes that we analyse for the FHM yield systems directly relevant to the study of high-temperature superconductivity. More generally, we show that by using the generalised SWT, one can Floquet-engineer additional knobs controlling the model parameters of strongly-correlated systems, such as the spin-exchange coupling. Our methods are readily extensible to strongly-interacting bosonic systems, as well as many other systems under active research.
Acknowledgements.We thank L. D’Alessio, E. Altman, W. Bakr, E. Demler, M. Eckstein, A. Grushin, M. Heyl, D. Huse, A. Iaizzi, G. Jotzu, R. Kaul, S. Kourtis, M.Piraud, A. Sandvik and R. Singh for insightful and interesting discussions. We are especially grateful to M. Dolfi and all contributors to the ALPS project Bauer et al. (2011); Dolfi et al. (2014) for developing the ALPS MPS and DMRG tools used in this work. We thank A. Rosch for pointing out to us the potential connection between the HFE and the SWT. This work was supported by AFOSR FA9550-13-1-0039, NSF DMR-1506340, and ARO W911NF1410540. M. K. was supported by Laboratory Directed Research and Development (LDRD) funding from Berkeley Lab, provided by the Director, Office of Science, of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231.
Appendix A Supplemental Material
Appendix B The High-Frequency Expansion.
We open up the discussion by briefly reviewing the basic tool used in the main text – the (van Vleck) High-Frequency Expansion (HFE). For a more-detailed description, consult Refs. Rahav et al. (2003); Goldman and Dalibard (2014); Bukov et al. (2015b); Goldman et al. (2015); Eckardt and Anisimovas (2015); Itin and Katsnelson (2015); Mikami et al. (2015). Consider a time-periodic Hamiltonian . According to Floquet’s theorem, the evolution operator , where denotes time ordering, can be cast in the form
with the time-independent effective Hamiltonian governing the slow dynamics and the -periodic kick operator describing the micromotion, i.e. the fast dynamics within a period. In the high-frequency limit, one can calculate perturbatively the kick operator and the effective Hamiltonian as follows:
where we Fourier-decomposed the Hamiltonian as with operator-valued coefficients and the functions and , in the integrands are understood periodic with period Eckardt and Anisimovas (2015). Since we are interested in the low-energy spectrum of the Floquet Hamiltonian, it suffices to calculate only. However, we remark that the effective kick operator is crucial for the correct description of the dynamics – both stroboscopic and non-stroboscopic Bukov and Polkovnikov (2014); Bukov et al. (2015b).
Appendix C Applying the Schrieffer-Wolff Transformation to the driven Fermi-Hubbard Model.
In this section, we give the details of the calculation of the effective Hamiltonian in the periodically-driven Fermi-Hubbard model (FHM). The starting point is the Hamiltonian:
Going to the rotating frame is equivalent to a re-summation of two infinite lab-frame inverse-frequency subseries Bukov et al. (2015b). The first subseries leads to a non-perturbative renormalisation of the hopping amplitude by resumming single-particle terms, while the second subseries contains the many-body nn-interaction-dependent hopping terms. Using the change-of-reference-frame transformation , we arrive at the Hamiltonian in the rotating frame:
where is the anti-derivative of and . It is convenient to cast this expression in the following form
The first term in gives rise to the hopping of holons, while the second one yields hopping of doublons. The term in is, in turn, responsible for creation of doublons and holes. We draw the reader’s attention to the fact that the overall sign of the function in the Hamiltonian above depends on the direction of hopping. For instance, for a one-dimensional chain with drive the Hamiltonian (9), when fully written out, reads
Note also that while , ; in other words destroying a doublon to the left is different from creating a doublon to the left .
Non-driven case. Let us pause for a moment and check the non-driven case, i.e. . Then the terms proportional to vanish in after time-averaging over one period , cf. Eq. (7). On the other hand, the -terms do not have a time-dependent pre-factor and hence they give rise to the leading-order Hamiltonian.
where the above expression is understood as the defining relation for the projector which projects out the subspace of doubly-occupied states. The -correction as given by Eq. (7) is proportional to the commutator , and results in the familiar Heisenberg spin exchange. Notice that already at this level the calculation for the static model reduces exactly to the standard SW calculation.
Driven case. Now let us turn on the periodic drive again. Pay attention how the zeroth order Hamiltonian changes, since the terms proportional to , which average to zero in the non-driven case, now remain finite after averaging over one period. These are precisely the doublon association and dissociation processes in the resonant limit whose physics we discuss in the main text.
Appendix D The Strongly-Interacting Periodically-Driven Fermi-Hubbard-Model Away from Half-Filling.
In this section, we give the details of the calculations for the resonant and non-resonant driving regimes, for which we derive the low-energy effective Hamiltonian. In the non-resonant case, we use two consecutive SW transformations, applied in the limits and [these two limits are reconciled in the next section]. We label the lattice sites by .
(i) Non-resonant Driving Limit. In this regime, we choose spin-dependent periodic driving of the type used to engineer the Harper-Hofstadter Hamiltonian Aidelsburger et al. (2013); Miyake et al. (2013):
where is the fermion spin, and . In this section, we choose , which results in a quarter flux quantum per plaquette. From the definition of the drive, it becomes clear that opposite spin species are subject to opposite gradient potentials. Notice that spin-exchange processes along the -direction are enabled by a resonant absorption of two photons, leading to an effective gauge field for the Heisenberg model at half-filling. We denote by the dimensionless interaction strength.
Let us first focus on the regime and show the derivation of the effective Hamiltonian comprising the Heisenberg model in an artificial gauge field. We can identify the largest frequency in the problem to be the interaction strength , followed by the driving frequency . Time-scale separation allows us to first perform a SW transformation to the Hamiltonian in Eq. (8) w.r.t. the fast period . In doing so we treat the time-fluctuations in the Hamiltonian due to the driving protocol at frequency as slow variables, and apply the HFE expansion with the fast period only. This allows us to effectively take the -oscillating terms out of the integrals in the HFE, which results in the familiar model in a presence of a -periodic drive. The remaining effective dynamics induced by the drive happens at time-scales and, in the rotating frame, it is governed by the following intermediate Hamiltonian:
where, again we drop the holon hopping term to order , as it will be a minor correction to the order- hopping above Keeling (). If we consider the system away from half-filling, double occupancies are not suppressed and the spin part of the Hamiltonian (13) is merely a correction. The leading effective Hamiltonian away from half-filling after applying the HFE once again with period reads
Notice the presence of a gauge field in the hopping of doublons and holons.
We now switch to half filling. Then one can safely neglect the terms in Eq. (13) containing the projectors , as well as the terms proportional to , similarly to the case for the static SW transformation. Now we apply the HFE again with the slow frequency . Since the leading correction term scales as we can safely neglect it to obtain
We thus see that in the regime , applying the SW transformation at half filling leads to the Heisenberg model in an artificial gauge field. We stress that the effective dynamics of the system is best governed by the above effective Hamiltonian for times , set by the magnitude of the next-order correction term. Furthermore, choosing and to be incommensurate will lead to suppression of resonant effects, thus enhancing the time interval for which time-scale separation holds. This is possible because the spectra of both and are discrete and commensurate.
Let us also briefly discuss the other non-resonant case . This time the fastest frequency in the problem is the driving frequency , followed by the interaction strength . Thus, we go to the rotating frame w.r.t. the driving term first:
Once again we make use of time-scale separation; applying the HFE with period results in the intermediate Hamiltonian to order :
To complete the derivation, all one has to do is to apply the static SW transformation with frequency . This mimics the static SW transformation and directly leads to the following Heisenberg model at any filling
with the effective exchange interactions and . Notice that since the leading -correction succumbs to the leading -Heisenberg model, so our assumption to drop the former is justified.
(ii) Resonant Driving Limit. Last, let us focus on the commensurate case . Unlike in the main text, we choose the same driving protocol as in Eq. (12) which allows us to show how to engineer doublon-holon physics in the presence of a gauge field. In this regime, the Hamiltonian in Eq. (8) is indeed periodic with the single frequency . Locking the driving frequency to the interaction strength leads to resonances which drastically change the behaviour of the system. Here, we show that they are captured by the HFE, beyond linear response theory. Moreover, this procedure does not suffer from vanishing denominators as is the case in conventional perturbation theory. To this end, we average Eq. (8) over one period which is equivalent to keeping only the leading order term in the effective Hamiltonian:
with , and . If the resonant periodic drive couples to the interaction strength instead, one can realise homogeneous doublon-holon creation amplitudes along the -direction