Dynamical time-reversal symmetry breaking and photo-induced chiral spin liquids in frustrated Mott insulators

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 well-captured 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 time-reversal while preserving SU(2) spin symmetry. We fingerprint the transient phase diagram and find a stable photo-induced 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.

C ontrol 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 quasi-periodic perturbations such as wideenvelope laser pulses. Here, much of the intuition and language of equilibrium survives in a distinctly non-equilibrium 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 short-range 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 two-particle 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 low-energy physics persists in certain high-frequency regimes of time-periodic perturbations, leading for instance to enhancement of correlated hopping 8,9 , strong-field sign reversal of nearestneighbor Heisenberg exchange in a 1D magnet 10,11 , or enhancement of Cooper-pair 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 pre-thermalized 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 pre-thermalized 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 long-range 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 time-reversal 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 Fig. 1 Photo-induced Kagome chiral antiferromagnet. a Starting from a Kagome Mott antiferromagnet, pumping with circularly polarized light dynamically breaks time-reversal and parity while preserving SU(2) spin symmetry, photo-inducing scalar spin chirality S i · (S j × S k ) contributions on elementary equilateral (χ) and isosceles (χ′) triangles. Pump strength and frequency provide knobs to tune χ,χ′ as well as Heisenberg exchange J, J′, J 3 , as described in the main text. Examples of b nearest-neighbor and c three-site Floquet virtual hopping processes including absorption of photons, in the Mott-insulating regime. Boxes graphically depict the example initial, virtual intermediate and final states for second-and fourth-order virtual hopping processes, in terms of the product space of electronic degrees of freedom and the Floquet index. TRS is broken in c, inducing scalar spin chiralities χ on triangles of the lattice, whereas b solely induces nearest-neighbor Heisenberg exchange 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 three-fold: 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 steady-state description capture the many-body time evolution of an optically driven Hubbard model? In the following, we answer all three questions affirmatively.

Results
Floquet-Hubbard model. Our focus lies on Kagome antiferromagnets, which have recently garnered much attention due to candidate materials herbertsmithite, kapellasite, and others 34 with putative spin-liquid behavior at low temperatures. Experiments 54 and first-principles calculations 55,56 indicate that the ground state and low-energy excitation spectra of these materials are well-captured by antiferromagnetic Heisenberg exchange between d 9 spins localized on Cu 34 . However, as photons couple to charge, a microscopic modeling of the light-matter interaction in principle must account for the multi-orbital 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 single-orbital Hubbard model at half fillinĝ Here, t h , U, e denote nearest-neighbor 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 wide-pulse envelope A(t), coupling to electrons via Peierls substitution. Comparison of nearest-neighbor exchange J % 4t 2 h =U with first-principles predictions for herbertsmithite 56 suggests U/t h of up to 40 due to the exceedingly narrow width of Cu d-orbital derived bands.
If A(t) varies slowly with respect to the pump period, then the Hamiltonian becomes approximately periodic under a translation H t þ 2π=Ω ð Þ¼ĤðtÞ. Floquet theory then dictates that the behavior near the pump plateau is completely determined via many-body eigenstates of the form Ψ n ðtÞ j i¼ e Àiϵnt P m e imΩt Φ m j i with ϵ n the Floquet quasi-energy, where the Φ m j i conveniently follow as eigenstates of the static Floquet-Hubbard Hamiltonian where A denotes the dimensionless field strength at the pump plateau, such that A(t) ≈ Aħ/(ea 0 ) with a 0 the nearest-neighbor distance, m 2 Z is the Floquet index, and J m ðÁÞ 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 ϵ 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 red-detuned from the charge gap. However, photon-assisted 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 non-negligible contributions to longer-ranged exchange or multispin processes. Second, electrons acquire gauge-invariant 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 lowest-order necessarily involve a photo-induced 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 Kagome-Hubbard model (1), it is instructive to first consider the highfrequency limit Ω ) U; t h . Here, circularly polarized pumping induces complex nearest-neighbor analogous to a staggered magnetic flux pattern in the unit cell (Supplementary Note 1). To third order int;t 0 , a spin description then includes scalar spin chirality contributions, with χ ¼ 9 ffiffi ffi 3 p t 4 h A 2 =2U 2 Ω 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 sub-gap pumping Ω < U. Starting from Eq. (2), a microscopic derivation of the Floquet spin Hamiltonian proceeds via quasi-degenerate perturbation theory, where care must be taken to simultaneously integrate out m ≠ 0 Floquet states and many-body states with doubly occupied sites (Supplementary Note 2).  10 . While every process contributes to Heisenberg exchange, a scalar spin chirality contribution appears for multi-hop 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, Λ ð1Þ m and Λ ð2Þ m parameterize fourth-order virtual hopping processes for which the second intermediate virtual state retains a single double-occupied site or returns to local half filling (albeit with non-zero Floquet index), respectively. Furthermore, nextnearest-neighbor Heisenberg exchange and corrections to nearest-neighbor Heisenberg exchange appear at the same order, with parameterizing a twofold virtual nearest-neighbor exchange process.
Steady-state phase diagram. Having established the effective steady-state physics for the duration of the pump pulse, the next question concerns whether the photo-induced 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 longer-ranged Heisenberg exchange ((c) and (d)) is comparatively weak. This immediately suggests that circularly polarized pumping grants a handle to  (Fig. 1a), whereas χ′ changes sign when Ω approaches a two-photon resonance (Ω = U/2). We analyze the steady-state 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 36-site cluster (Fig. 3a), consistent with density-matrix renormalization group simulations that find a gapped Z 2 QSL 60-63 . 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 finite-size 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 quasi-degeneracy develops continuously, with a gap to many-body 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 many-body excitation manifold above the CSL survives insertion of a flux quantum through the torus (Methods section). As shown in Fig. 3b, a robust photo-induced CSL develops already for weak A, with excited states well-separated in energy. Finally, a proper verification of the CSL necessitates characterizing its ground state topological order. We therefore fingerprint the photo-induced phase by determining a basis of minimally entangled states from combinations of the two degenerate ground states ψ 1;2 , minimizing the Rényi entropy for their reduced density matrices in two distinct bipartitions (Fig. 3d) (1)).
To this end, we consider a circularly polarized optical pump pulse with a slow sinusoidal ramp-up and a wide pump plateau (Fig. 4a), and simulate the exact many-body dynamics of driven 12-site U = 30 Kagome Hubbard clusters for long times t 1000 t À1 h . Conceptually, the transient state can then be thought of as dynamically following the instantaneous Floquet eigenstate Ψ τ; T slow À Á % e Àiϵ T slow ð Þτ P m e imΩτ Φ T slow À Á , with the time variable t "separating" into fast (τ) and slow T slow À Á moving components. Reaching the pump plateau, the timeevolved state will nevertheless retain a finite quasi-energy spread ΨðtÞ j i¼ P α ρ α e Àiϵαt P m e imΩt Φ α ðtÞ j i (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 on elementary triangles of the Kagome cluster (Fig. 4b). Vanishing in equilibrium due to TRS, the pump-period average of χ ijk (t) should saturate to its Floquet expectation value at the pump plateau. Figure 4c compares χ ijk (t), time-averaged 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 S z ð Þ 2 <1=4; additionally, the 12-site 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 Floquet-Hubbard Hamiltonian, a consequence of the photo-assisted 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 Hubbard-U.
To analyze this in detail, we focus on pumping the system across the charge resonance with the upper Hubbard band, where the photo-induced scalar spin chirality contribution is expected to be largest. Figure 5a-c show the period-averaged double occupancyn "n# 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 infinite-temperature limitn "n# ! 1=4. Importantly, this entails that thermalization at long times is independent of the pump strength A.
Conversely, in the off-resonant 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 period-shifted Floquet "fidelity measure" F ¼ Ψ t þ T ð ÞΨðtÞ j h i j j , where T = 2π/Ω is the pump period. At the pump plateau with discrete time-translation symmetry, F is time-independent and quantifies the Floquet quasi-energy spread of the transient steady state (Fig. 5d). For a pure Floquet state, 1 À F ! 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 pre-thermalized 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 period-averaged stroboscopic energy operator h b Ei (Methods section). On the pump plateau, both the double occupancy and h b Ei saturate to their pre-thermalized steady-state 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 h b EiðtÞ % E 0 þ tΔE, 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 meta-stability 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 photo-excitation for realistic materials will likely lead to a less optimistic upper bound on the time scales of interest. First, a materials-specific modeling of electron-photon coupling and multi-band effects will modify the effective photo-induced 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 magneto-elastic coupling to phonons is weak in most materials and the optical frequencies under consideration are far from resonance with infrared-active phonon modes, electron phonon coupling will nevertheless indirectly heat the lattice due to Raman-assisted 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 30-32 , 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 circularly-polarized 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 wide-pulse optical perturbations can provide an intriguing knob to tune the low-energy physics of frustrated quantum magnets, shedding light on regions of their phase diagram hitherto unexplored.

Methods
Floquet theory. Consider a generic time-dependent many-body Hamiltonian with discrete time-translation invarianceĤðtÞ ¼Ĥ t þ 2π=Ω ð Þ . Instead of solving the many-body time evolution, one can reexpress the time-dependent Schrödinger equation in a Floquet eigenbasis Φ α ðtÞ j i¼e Àiϵα t P m e imΩt u α;m , where α indexes the basis wave function and ϵ α is its respective Floquet quasi-energy. Then, determination of the time-dependent eigenstates of the driven system reduces to finding the time-independent eigenstates of the Floquet Hamiltonian  Lines are guides to the eye. On resonance (gray region), the system heats rapidly, withn "n# thermalizating independent of the pump strength A and approaching its infinite-temperature expectation value 1/4. Below resonance (white region), the system transiently realizes the TRS-breaking chiral quantum magnet, with a tunable Hubbard interaction U (as well as correspondingly tunable magnetic J, χ, see Fig. 4) as a function of pump strength. d Floquet fidelity FðTÞ ¼ Ψ t þ T ð ÞΨðtÞ j h i j j on the pump plateau-below charge resonance, (1 − F) → 0, indicating the controlled preparation of a Floquet eigenstate. e Extracted stroboscopic heating rates per pump cycle and below resonance. Remarkably, heating is strongly suppressed close to the charge resonance, with the driven system requiring many thousands of pump cycles to finally absorb energy on the order of the Heisenberg exchange J NATURE COMMUNICATIONS | DOI: 10.1038/s41467-017-00876-y ARTICLE conditions along one direction and tracking the many-body 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 30-site clusters, spanned by vectors R 1 = 2a 1 + a 2 , R 2 = −2a 1 + 4a 2 , and checked against the 36-site cluster. The winding of the quasi-degenerate ground states upon flux insertion is a signature of CSLs, with the two quasi-degenerate ground states exchanging once under flux insertion, or remaining separated, depending on whether they lie in different (30-site cluster) or the same (36-site cluster) momentum sectors.
We furthermore consider two bipartitions A, B of the 36-site cluster with periodic boundary conditions, as depicted in the inset in Fig. 3c, and calculate the Rényi entropies S α ðθ; ϕÞ ¼ Àlog tr ρ 2 α ðθ; ϕÞ where ρ α ðθ; ϕÞ ¼ tr α Ψðθ; ϕÞ j iΨðθ; ϕÞ h j f g is the reduced density matrix on bipartition α = A, B for superpositions Ψðθ; ϕÞ j i¼ cosðθÞ ψ 1 j iþ sinðθÞe iϕ ψ 2 j i of the quasi-degenerate ground states ψ 1 j i; ψ 2 j i, as a function of θ, ϕ. Figure 3c depicts S A and S B calculated from twofold quasi-degenerate 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 Ψðθ; ϕÞ j i permit extraction of the modular matrices 64 , which match expectations for a Kalmeyer-Laughlin CSL and are quoted in the main text.
Time evolution. The electronic many-body time evolution was simulated for a 12site Kagome-Hubbard 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 AðtÞ ¼ AðtÞ cosðΩtÞ; sinðΩtÞ with a smooth sinusoidal pump envelope where t plateau ¼ 700t À1 h 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 period-averaged energy operator where J 0 ðAÞ denotes the zeroth Bessel function of the first kind. Note that h b Ei is time-independent for a pure Floquet state, in theory. Instead, the finiteness of the pump envelope entails a residual quasi-energy 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 infinite-temperature state at infinite times, the results of the main text demonstrate a long-lived and remarkably stable pre-thermalized regime with negligible absorption.
Data availability. The data that support the results presented in this study are available from the corresponding authors on request.