Abstract
The search for quantum spin liquids in frustrated quantum magnets recently has enjoyed a surge of interest, with various candidate materials under intense scrutiny. However, an experimental confirmation of a gapped topological spin liquid remains an open question. Here, we show that circularly polarized light can provide a knob to drive frustrated Mott insulators into a chiral spin liquid, realizing an elusive quantum spin liquid with topological order. We find that the dynamics of a driven Kagome Mott insulator is wellcaptured by an effective Floquet spin model, with heating strongly suppressed, inducing a scalar spin chirality S _{ i } · (S _{ j } × S _{ k }) term which dynamically breaks timereversal while preserving SU(2) spin symmetry. We fingerprint the transient phase diagram and find a stable photoinduced chiral spin liquid near the equilibrium state. The results presented suggest employing dynamical symmetry breaking to engineer quantum spin liquids and access elusive phase transitions that are not readily accessible in equilibrium.
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Introduction
Control of quantum materials out of equilibrium represents one of the grand challenges of modern condensed matter physics. While an understanding of general nonequilibrium settings beyond heating and thermalization is still in its infancy, a loophole concerns considering instead the transient quantum states of quasiperiodic perturbations such as wideenvelope laser pulses. Here, much of the intuition and language of equilibrium survives in a distinctly nonequilibrium setting within the framework of Floquet theory. While recently enjoying much attention and experimental success in the manipulation of singleparticle spectra^{1,2,3,4} and band topology or shortrange entangled topological states^{5,6,7}, a natural extension regards pumping of strongly correlated systems. Here, the essence of Floquet physics lies not merely in imbuing one^{1} and twoparticle^{2} responses with the pump frequency as an additional energy scale, but in reshaping the underlying Hamiltonian to stabilize phases of matter that might be inaccessible in equilibrium.
Indeed, initial investigations suggest that the notion of effective lowenergy physics persists in certain highfrequency regimes of timeperiodic perturbations, leading for instance to enhancement of correlated hopping^{8, 9}, strongfield sign reversal of nearestneighbor Heisenberg exchange in a 1D magnet^{10, 11}, or enhancement of Cooperpair formation^{12,13,14}. Similar ideas are being pursued in the field of ultracold atoms to simulate artificial gauge fields, to dynamically realize topological band structures^{15} or even propose fractional quantum Hall effects and spin liquids in optical lattices^{16, 17, 18}. At the same time, recent advances in Floquet thermodynamics indicate that, while driven nonintegrable closed systems are in principle expected to heat up to infinite temperature^{19, 20}, heating can be exponentially slow on prethermalized time scales^{21,22,23,24,25,26} or altogether avoided via manybody localization^{27,28,29} or dissipation^{30,31,32}. An ideal condensedmatter realization hence entails a charge gap to limit absorption, as well as a delicate balance of competing phases, such that timedependent perturbations and dynamical symmetry breaking can be expected to have an outsized effect and phase boundaries can be reached on prethermalized time scales with moderate effort.
Frustrated quantum magnets^{33} are prime candidates for such ideas. Strong local Coulomb repulsion between electrons freezes out the charge degrees of freedom, whereas the spin degrees of freedom are geometrically obstructed from ordering, hosting a delicate competition of conventionally ordered phases as well as quantum spin liquids (QSLs) with longrange entanglement and exotic excitations^{34,35,36,37,38,39,40,41}. The chiral spin liquid (CSL) constitutes one of the earliest proposals of a topologically ordered QSL; it breaks timereversal symmetry (TRS) and parity, while preserving SU(2) spin symmetry, and can be regarded as a bosonic ν = 1/2 fractional quantum Hall state of spins with zero net magnetization and gapped semion excitations^{42,43,44,45}. While an unlikely ground state in unperturbed microscopic models, recently the CSL was found to be a competing state^{46,47,48,49,50,51,52,53}, in particular after explicit breaking of TRS and parity^{46,47,48,49,50}. However, TRS breaking in experiment is realized canonically via external magnetic fields, necessarily entailing a Zeeman shift as the dominant contribution, which breaks SU(2) symmetry and disfavors CSLs^{46}.
Here, we show that pumping a Mott insulator with circularly polarized light below the Mott gap can dynamically break TRS without breaking of SU(2) or translation symmetry, providing a knob to drive a frustrated quantum magnet into a CSL. Starting from a prototypical Hubbard model, the key questions posed by this work are threefold: First, how does optically induced TRS breaking manifest itself in a Mott insulator; second, can the ensuing effective Floquet spin model support a transient CSL and what are its signatures; and finally, does such an effective Floquet steadystate description capture the manybody time evolution of an optically driven Hubbard model? In the following, we answer all three questions affirmatively.
Results
FloquetHubbard model
Our focus lies on Kagome antiferromagnets, which have recently garnered much attention due to candidate materials herbertsmithite, kapellasite, and others^{34} with putative spinliquid behavior at low temperatures. Experiments^{54} and firstprinciples calculations^{55, 56} indicate that the ground state and lowenergy excitation spectra of these materials are wellcaptured by antiferromagnetic Heisenberg exchange between d ^{9} spins localized on Cu^{34}. However, as photons couple to charge, a microscopic modeling of the lightmatter interaction in principle must account for the multiorbital structure at higher energies^{57}, above the ~2 eV charge gap^{58}. Here, we take a phenomenological approach, and, as an effective starting point that captures the essential physics but without pretense of a direct materials connection, start from a driven singleorbital Hubbard model at half filling
Here, t _{h}, U, e denote nearestneighbor hopping, Coulomb interaction, and electron charge, r _{ ij } denotes vectors between sites i,j, and A(t) = A(t)[cos(Ωt), sin(Ωt)]^{T} models a circularly polarized pump beam with widepulse envelope A(t), coupling to electrons via Peierls substitution. Comparison of nearestneighbor exchange \(J \approx 4t_{\rm{h}}^{\rm{2}}{\rm{/}}U\) with firstprinciples predictions for herbertsmithite^{56} suggests U/t _{h} of up to 40 due to the exceedingly narrow width of Cu dorbital derived bands.
If A(t) varies slowly with respect to the pump period, then the Hamiltonian becomes approximately periodic under a translation \(\hat H\left( {t + 2\pi {\rm{/}}\Omega } \right) = \hat H(t)\). Floquet theory then dictates that the behavior near the pump plateau is completely determined via manybody eigenstates of the form \(\left {{\Psi _n}(t)} \right\rangle = {e^{  i{\epsilon _n}t}}\mathop {\sum}\nolimits_m {e^{im\Omega t}}\left {{\Phi _m}} \right\rangle\) with \({\epsilon _n}\) the Floquet quasienergy, where the \(\left {{\Phi _m}} \right\rangle\) conveniently follow as eigenstates of the static FloquetHubbard Hamiltonian
where A denotes the dimensionless field strength at the pump plateau, such that A(t) ≈ Aħ/(ea _{0}) with a _{0} the nearestneighbor distance, \(m \in {\Bbb Z}\) is the Floquet index, and \({{\cal J}_m}( \cdot )\) denotes the Bessel function of the first kind (Methods section). Note that the apparent Hilbert space expansion is merely a gauge redundancy of Floquet theory, as eigenstates with energy \({\epsilon _n}\) + mΩ identify with the same physical state ∀m.
Floquet Chiral spin model
Physically, Eq. (2) describes photonassisted hopping in the presence of interactions, where electrons can enlist m photons to hop at a reduced energy cost U − mΩ of doubly occupying a site. Deep in the Mott phase the formation of local moments persists out of equilibrium as long as the pump remains off resonance and reddetuned from the charge gap. However, photonassisted hopping reduces the energy cost of virtual exchange, pushing the system closer to the Mott transition and enlarging the range of virtual hopping paths that provide nonnegligible contributions to longerranged exchange or multispin processes. Second, electrons acquire gaugeinvariant phases when hopping around loops on the lattice for circular polarization. Crucially, and in contrast to an external magnetic field, an optical pump precludes a Zeeman shift, retaining the SU(2) spin rotation symmetry that is shared by CSL ground states. Symmetry considerations dictate that a manifestation of TRS breaking must to lowestorder necessarily involve a photoinduced scalar spin chirality χ _{ ijk } term, with:
This Floquet Chiral Spin Hamiltonian is the central focus of the paper; to derive it microscopically from the driven KagomeHubbard model (1), it is instructive to first consider the highfrequency limit \(\Omega \gg U,{t_{\rm{h}}}\). Here, circularly polarized pumping induces complex nearestneighbor hoppings \(\tilde t = {t_{\rm{h}}}\left( {1  {A^2}{\rm{/}}4} \right) + i\left( {\sqrt 3 {\rm{/}}4} \right)t_{\rm{h}}^{\rm{2}}{A^2}{\rm{/}}\Omega\) as well as purelycomplex nextnearestneighbor hoppings \(\tilde t' =  i\left( {\sqrt 3 {\rm{/}}4} \right)t_{\rm{h}}^2{A^2}{\rm{/}}\Omega\), analogous to a staggered magnetic flux pattern in the unit cell (Supplementary Note 1). To third order in \(\tilde t,\tilde t'\), a spin description then includes scalar spin chirality contributions, with \(\chi = 9\sqrt 3 t_{\rm{h}}^4{A^2}{\rm{/}}2{U^2}\Omega\) of equal handedness for both equilateral triangles per unit cell, as depicted in Fig. 1a, and six isosceles triangles of opposite handedness with χ′ = χ/3, such that the total chiral couplings in the unit cell sum to zero.
Now consider subgap pumping Ω < U. Starting from Eq. (2), a microscopic derivation of the Floquet spin Hamiltonian proceeds via quasidegenerate perturbation theory, where care must be taken to simultaneously integrate out m ≠ 0 Floquet states and manybody states with doubly occupied sites (Supplementary Note 2). Figure 1b, c depict relevant virtual processes, involving simultaneous hopping of electrons and absorption of m photons with intermediate energy cost U − mΩ. To second order in virtual hopping, twosite exchange processes (Fig. 1b) are phaseagnostic and solely renormalize the nearestneighbor Heisenberg exchange \(\tilde J = 4\mathop {\sum}\nolimits_m {\left {{{\cal J}_m}\left( A \right)} \right^2}t_{\rm{h}}^2{\rm{/}}\left( {U  m\Omega } \right)\) ^{8, 10}. While every process contributes to Heisenberg exchange, a scalar spin chirality contribution appears for multihop processes that enclose an area. Naïvely, to third order, an electron could simply circumnavigate the elementary triangles of the Kagome lattice; however, these processes interfere destructively and cancel exactly to all orders in A even though TRS is broken, and in contrast to an external magnetic field (Supplementary Note 2). This is consistent with results on the resonant A_{2g } Raman response of Mott insulators^{59}, that connect to the A → 0, m = 1 limit. Instead, TRS breaking first manifests itself to fourth order in virtual hopping. Here, processes (Fig. 1c) can either encompass an elementary triangle, or virtually move an electron back and forth two legs of a hexagon, inducing scalar spin chirality contributions as shown in Fig. 1a, with
where m = {m _{1}, m _{2}, m _{3}} are Floquet indices, and
Here, \(\Lambda _{\bf{m}}^{(1)}\) and \(\Lambda _{\bf{m}}^{(2)}\) parameterize fourthorder virtual hopping processes for which the second intermediate virtual state retains a single doubleoccupied site or returns to local half filling (albeit with nonzero Floquet index), respectively. Furthermore, nextnearestneighbor Heisenberg exchange
and corrections to nearestneighbor Heisenberg exchange
appear at the same order, with
parameterizing a twofold virtual nearestneighbor exchange process.
Steadystate phase diagram
Having established the effective steadystate physics for the duration of the pump pulse, the next question concerns whether the photoinduced Floquet spin model (Eq. (3)) can indeed stabilize a CSL. Consider its parameter space as a function of A, Ω, depicted in Fig. 2 for U = 20t _{h}. Adiabatic ramping up of the circularly polarized pump then corresponds to horizontal trajectories with fixed Ω. Figure 2a, b show that TRSbreaking scalar spin chiralities develop with increasing field strength, whereas the effect on longerranged Heisenberg exchange ((c) and (d)) is comparatively weak. This immediately suggests that circularly polarized pumping grants a handle to change the underlying manybody state. Close to the onephoton resonance (Ω = U), chiral contributions are staggered between elementary and isosceles triangles (Fig. 1a), whereas χ′ changes sign when Ω approaches a twophoton resonance (Ω = U/2).
We analyze the steadystate phase diagram using exact diagonalization of the Floquet spin model (Eq. (3)), parameterized by pump strength and frequency. In equilibrium (A = 0), the ground state of Eq. (3) is gapped and TRS invariant. Absence of conventional spin order is evidenced by a rapid decay of spin–spin and chiral–chiral correlation functions on a 36site cluster (Fig. 3a), consistent with densitymatrix renormalization group simulations that find a gapped Z _{2} QSL^{60,61,62,63}. We adopt this view for the thermodynamic limit, but note that the ground state degeneracy of a Z _{2} QSL remains inaccessible in exact diagonalization of finitesize clusters (Methods section). Upon pumping (A ≠ 0), the spin correlator displays no propensity for ordering; however, chiral correlations develop smoothly (Fig. 3a). Importantly, a twofold ground state quasidegeneracy develops continuously, with a gap to manybody excitations, indicative of a CSL. To track the phase boundary as a function of A, Ω, we determine the parameter space region within which the ground state degeneracy as well as the gap Δ _{CSL} to the manybody excitation manifold above the CSL survives insertion of a flux quantum through the torus (Methods section). As shown in Fig. 3b, a robust photoinduced CSL develops already for weak A, with excited states wellseparated in energy. Finally, a proper verification of the CSL necessitates characterizing its ground state topological order. We therefore fingerprint the photoinduced phase by determining a basis of minimally entangled states from combinations of the two degenerate ground states \(\left {{\psi _{1,2}}} \right\rangle\), minimizing the Rényi entropy for their reduced density matrices in two distinct bipartitions (Fig. 3d) on a 36site torus (Methods sections). C _{6} symmetry allows extraction of both modular \({\cal U}\), \({\cal S}\) matrices^{64,65,66} that encode self and mutualbraiding statistics of the elementary excitations. We find that the photoinduced CSL corresponds uniquely to a \(\nu = {\textstyle{1 \over 2}}\) bosonic fractional quantum Hall state, with matching modular matrices^{40}
Time evolution
Having established a photoinduced CSL for the Floquet spin model, the final question concerns whether this effective spin model qualitatively captures the time evolution of the driven Hubbard model (Eq. (1)).
To this end, we consider a circularly polarized optical pump pulse with a slow sinusoidal rampup and a wide pump plateau (Fig. 4a), and simulate the exact manybody dynamics of driven 12site U = 30 Kagome Hubbard clusters for long times \(t \le 1000\,t_{\rm{h}}^{  1}\). Conceptually, the transient state can then be thought of as dynamically following the instantaneous Floquet eigenstate \(\left {\Psi \left( {\tau ,{{\bar T}_{{\rm{slow}}}}} \right)} \right\rangle \approx {e^{  i\epsilon \left( {{{\bar T}_{{\rm{slow}}}}} \right)\tau }}\mathop {\sum}\nolimits_m {e^{im\Omega \tau }}\left {\Phi \left( {{{\bar T}_{{\rm{slow}}}}} \right)} \right\rangle\), with the time variable t “separating” into fast (τ) and slow \(\left( {{{\bar T}_{{\rm{slow}}}}} \right)\) moving components. Reaching the pump plateau, the timeevolved state will nevertheless retain a finite quasienergy spread \(\left {\Psi (t)} \right\rangle = \mathop {\sum}\nolimits_\alpha \rho_\alpha {e^{  i{\epsilon _\alpha }t}}\mathop {\sum}\nolimits_m {e^{im\Omega t}}\left {{\Phi _\alpha }(t)} \right\rangle\) (with α indexing the Floquet eigenstates). While dephasing of these constituent Floquet eigenstates should ultimately thermalize the system to infinite temperature, the system nevertheless matches the effective spin dynamics described by Eq. (3) and barely absorbs energy on the broad “prethermalized” time scales of interest, as we show below.
First, to compare to the Floquet spin description, we focus on timedependent scalar spin chirality expectation values \({\chi _{ijk}}(t) = \left\langle {{{\bf{S}}_i} \cdot \left( {{{\bf{S}}_j} \times {{\bf{S}}_k}} \right)} \right\rangle\) \(\left( {{{\bf{S}}_i} = \hat c_{i\sigma }^\dag {{\overrightarrow {\bf{s}} }_{\sigma \sigma '}}{{\hat c}_{i\sigma '}}} \right)\) on elementary triangles of the Kagome cluster (Fig. 4b). Vanishing in equilibrium due to TRS, the pumpperiod average of χ _{ ijk }(t) should saturate to its Floquet expectation value at the pump plateau. Figure 4c compares χ _{ ijk }(t), timeaveraged over the pump plateau, to corresponding static χ _{ ijk } expectation values of the Floquet spin model (3) ground state. The latter follows from choosing χ, χ′, J, J′, J _{3} via Eqs. (4)–(10), with A, Ω the pump parameters of the Hubbard time evolution. Intriguingly, the electronic time evolution is in excellent qualitative agreement with predictions for the Floquet spin model, even when driven close to the Mott transition.
Quantitative discrepancies predominantly originate from deviations of the local moment \(\left\langle {{{\left( {{S^z}} \right)}^2}} \right\rangle < 1{\rm{/}}4\); additionally, the 12site cluster with periodic boundary conditions permits weak ring exchange contributions from loops of virtual hopping around the cluster. Importantly, the transient increase in double occupancies is not an indication of heating—instead, this follows from a reduction of the effective U of the transient FloquetHubbard Hamiltonian, a consequence of the photoassisted hopping processes depicted in Fig. 1b, c. Quantitative differences between spin and fermionic observables are therefore analogous to differences between canonical spin and fermionic descriptions of equilibrium quantum magnets for a finite HubbardU.
To analyze this in detail, we focus on pumping the system across the charge resonance with the upper Hubbard band, where the photoinduced scalar spin chirality contribution is expected to be largest. Figure 5a–c show the periodaveraged double occupancy \(\left\langle {{{\hat n}_ \uparrow }{{\hat n}_ \downarrow }} \right\rangle\) as a function of pump strength and detuning from the charge resonance ≈U − 5.5t _{h}. Upon resonant charge excitation, the system heats up rapidly and the double occupancy approaches its infinitetemperature limit \(\left\langle {{{\hat n}_ \uparrow }{{\hat n}_ \downarrow }} \right\rangle \to 1{\rm{/}}4\). Importantly, this entails that thermalization at long times is independent of the pump strength A.
Conversely, in the offresonant regime, one observes a pumpstrength dependent saturation of the double occupancy. Here, proper heating is strongly suppressed and the system instead realizes the effective Floquet chiral quantum magnet with a transient reduction of the Hubbard interaction U. To verify that the driven steady state indeed follows the ground state of the effective Floquet Hamiltonian adiabatically, consider a periodshifted Floquet “fidelity measure” \({\cal F} = \left {\left\langle {\Psi \left( {t + T} \right)\left {\Psi (t)} \right.} \right\rangle } \right\), where T = 2π/Ω is the pump period. At the pump plateau with discrete timetranslation symmetry, \({\cal F}\) is timeindependent and quantifies the Floquet quasienergy spread of the transient steady state (Fig. 5d). For a pure Floquet state, \(1  {\cal F} \to 0\), suggesting that the driven state below resonance adiabatically follows a Floquet eigenstate, whereas adiabaticity is lost when crossing the absorption edge.
To distinguish residual heating on these prethermalized time scales of interest^{25, 26} from a transient increase in energy in the chiral quantum magnet due to modulation of the Hamiltonian, consider the periodaveraged stroboscopic energy operator \(\langle {\widehat E} \rangle\) (Methods section). On the pump plateau, both the double occupancy and \(\langle {\widehat E} \rangle\) saturate to their prethermalized steadystate expectation values; however, a minuscule residual gradient over thousands of pump cycles remains. To good approximation for the time scales considered here, we can linearize the energy on the pump plateau \(\langle {\widehat E} \rangle (t) \approx {E_0} + t\Delta E\), and extract the heating rate ΔE from simulations. Figure 5e depicts the absorbed energy per pump cycle on the pump plateau, as a function of pump strength and detuning from the absorption edge. Remarkably, residual heating is largely suppressed close to resonance, with an absorbed energy on the order of 10^{−6} t _{h} per pump cycle. Naïvely, this extraordinary metastability suggests that it could take on the order of tens of thousands of pump cycles for heating to dominate the dynamics, absorbing a total energy ~J the exchange coupling.
A more microscopic analysis of photoexcitation for realistic materials will likely lead to a less optimistic upper bound on the time scales of interest. First, a materialsspecific modeling of electron–photon coupling and multiband effects will modify the effective photoinduced spin Hamiltonian, albeit necessarily retaining the salient symmetry properties and scalar spin chirality contributions that stabilize the CSL. Second, an intriguing followup question regards the role of coupling to—and heating of—the lattice. While magnetoelastic coupling to phonons is weak in most materials and the optical frequencies under consideration are far from resonance with infraredactive phonon modes, electron phonon coupling will nevertheless indirectly heat the lattice due to Ramanassisted hopping processes. Conversely, the separation of time scales for electrons and phonons suggests that the phonon bath could similarly play the role of a dissipative channel, effectively “cooling” the electronic system. While initial investigations have already studied the case of free or weaklyinteracting electrons^{30,31,32}, a proper understanding of the confluence of strong interactions, external drive and dissipation remains an interesting topic for future study.
Conclusions
In summary, we have shown that pumping a frustrated Mott insulator with circularlypolarized light can dynamically break TRS while preserving SU(2) symmetry of the underlying spin system, by augmenting its effective dynamics with a transient scalar spin chirality term. Remarkably, on the Kagome lattice this effective Floquet spin model was found to stabilize a transient CSL in a broad parameter regime. Our results suggest that widepulse optical perturbations can provide an intriguing knob to tune the lowenergy physics of frustrated quantum magnets, shedding light on regions of their phase diagram hitherto unexplored.
Methods
Floquet theory
Consider a generic timedependent manybody Hamiltonian with discrete timetranslation invariance \(\hat H(t) = \hat H\left( {t + 2\pi {\rm{/}}\Omega } \right)\). Instead of solving the manybody time evolution, one can reexpress the timedependent Schrödinger equation in a Floquet eigenbasis \(\left {{\Phi _\alpha }(t)} \right\rangle = {e^{  i{\epsilon _\alpha }t}}\mathop {\sum}\nolimits_m {e^{im\Omega t}}\left {{u_{\alpha ,m}}} \right\rangle\), where α indexes the basis wave function and \({\epsilon _\alpha }\) is its respective Floquet quasienergy. Then, determination of the timedependent eigenstates of the driven system reduces to finding the timeindependent eigenstates of the Floquet Hamiltonian
where \({\hat H_{m  m'}}\) are the Fourier expansion coefficients of \(\hat H(t)\). Taking \(\hat H(t)\) as the driven KagomeHubbard model of Eq. (1) in the main text straightforwardly recovers the FloquetHubbard Hamiltonian (Eq. (2)). Here, the dimensionless pump strength A that enters via Peierls substitution relates to the electric field \({\cal E}\) as \({\cal E} = \Omega {\rm{/}}\left( {e{a_0}} \right)\), with e the electron charge and a _{0} the nearestneighbor bond distance. While realistic estimates for materials would necessarily entail significant contributions from multiorbital effects and local dipole transitions, all of which are not captured within the singleorbital Hubbard model, a naïve estimate from solely Peierls substitution for a singleorbital approximation of herbertsmithite yields A = 0.25…0.5 for \({\cal E} \ldots 100 \ldots 200\,{\rm{meV}}\,{{\rm{{\AA}}}^{  1}}\) for a 900 nm nearinfrared pump.
Numerical simulations
To characterize the photoinduced CSL state, we performed exact diagonalization calculations of the Floquet chiral spin Hamiltonian, described in Eq. (3). Due to threespin interactions and longerranged exchange interactions, the sparsity of the resulting Hamiltonian matrix is over an order of magnitude lower than for a nearestneighbor Heisenberg antiferromagnet. Spin and chiral correlation functions as well as minimally entangled states were calculated for a 36site cluster with periodic boundary conditions, spanned by vectors R _{1} = 4a _{1} − 2a _{2}, R _{2} = −2a _{1} + 4a _{2}, where a _{1}, a _{2} are the lattice vectors. This choice retains the rotational symmetry of the Kagome lattice, facilitating extraction of the modular matrices. Previous exact diagonalization studies have simulated the 36site cluster for a purely nearestneighbor Heisenberg model^{67, 68}. While densitymatrix renormalization group studies indicate that such a model is likely to stabilize a Z _{2} QSL in the thermodynamic limit, it is wellknown that its ground state degeneracy remains inaccessible in exact diagonalization of finitesize clusters^{67, 68}, and the 36site cluster is likely to host a QSL close to a phase boundary with a valence bond crystal^{69}. However, the nature of the equilibrium ground state and its extrapolation to the thermodynamic limit does not affect the conclusions regarding transient state; crucially, the CSL state is wellstabilized already on the finitesize systems under consideration, with a robust manybody excitation gap.
The ground state degeneracy and minimal manybody excitation gap under flux insertion was calculated by imposing a spindependent twist of boundary conditions along one direction and tracking the manybody spectral flow as a function of twist angle, as depicted in the inset of Fig. 3b. A fine sampling of flux insertion, as depicted in Fig. 3b, was performed for 30site clusters, spanned by vectors R _{1} = 2a _{1} + a _{2}, R _{2} = −2a _{1} + 4a _{2}, and checked against the 36site cluster. The winding of the quasidegenerate ground states upon flux insertion is a signature of CSLs, with the two quasidegenerate ground states exchanging once under flux insertion, or remaining separated, depending on whether they lie in different (30site cluster) or the same (36site cluster) momentum sectors.
We furthermore consider two bipartitions A, B of the 36site cluster with periodic boundary conditions, as depicted in the inset in Fig. 3c, and calculate the Rényi entropies
where \({\rho _\alpha }(\theta ,\phi ) = {\rm{t}}{{\rm{r}}_\alpha }\left\{ {\left {\Psi (\theta ,\phi )} \right\rangle \left\langle {\Psi (\theta ,\phi )} \right} \right\}\) is the reduced density matrix on bipartition α = A, B for superpositions \(\left {\Psi (\theta ,\phi )} \right\rangle = {\rm{cos}}(\theta )\left {{\psi _1}} \right\rangle + {\rm{sin}}(\theta ){e^{i\phi }}\left {{\psi _2}} \right\rangle\) of the quasidegenerate ground states \(\left {{\psi _1}} \right\rangle ,\left {{\psi _2}} \right\rangle\), as a function of θ, ϕ. Figure 3c depicts S _{ A } and S _{ B } calculated from twofold quasidegenerate ground states of the Floquet chiral spin model. For a CSL, S _{ α }(θ, ϕ) is expected to display two entanglement minima; the two corresponding minimally entangled states \(\left {\Psi (\theta ,\phi )} \right\rangle\) permit extraction of the modular matrices^{64}, which match expectations for a KalmeyerLaughlin CSL and are quoted in the main text.
Time evolution
The electronic manybody time evolution was simulated for a 12site KagomeHubbard cluster (4 unit cells) with periodic boundary conditions, spanned by vectors R _{1} = 2a _{1}, R _{2} = 2a _{2}, and with the time propagation employing adaptive step size control. We note this is the minimum cluster size to faithfully host all permutations of virtual hopping processes that give rise to the effective Floquet chiral spin Hamiltonian discussed in the main text (Eq. (3)).
To model broad circularly polarized pump pulses, we consider a pulsed field
with a smooth sinusoidal pump envelope
where \({t_{{\rm{plateau}}}} = 700t_{\rm{h}}^{  1}\) for the results of the main text. Details on pump envelope dependence can be found in Supplementary Note 3.
Finally, to quantify energy absorption in the driven system, we compute the periodaveraged energy operator
where \({{\cal J}_0}(A)\) denotes the zeroth Bessel function of the first kind. Note that \(\langle {\widehat E} \rangle\) is timeindependent for a pure Floquet state, in theory. Instead, the finiteness of the pump envelope entails a residual quasienergy spread, with the resulting dephasing of the driven state leading to residual heating on the pump plateau. While the driven state is ultimately expected to thermalize to an infinitetemperature state at infinite times, the results of the main text demonstrate a longlived and remarkably stable prethermalized regime with negligible absorption.
Data availability
The data that support the results presented in this study are available from the corresponding authors on request.
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Acknowledgements
We acknowledge support from the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under Contract No. DEAC0276SF00515. Computational resources were provided by the National Energy Research Scientific Computing Center supported by the Department of Energy, Office of Science, under Contract No. DE AC0205CH11231.
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M.C. conceived the project, developed the theoretical description, and performed the numerical simulations. M.C., H.C.J., and B.M. analyzed the numerical results. The manuscript was written by M.C. with input from all authors. T.P.D. supervised the project.
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Claassen, M., Jiang, HC., Moritz, B. et al. Dynamical timereversal symmetry breaking and photoinduced chiral spin liquids in frustrated Mott insulators. Nat Commun 8, 1192 (2017). https://doi.org/10.1038/s4146701700876y
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DOI: https://doi.org/10.1038/s4146701700876y
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