text:
Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

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

## 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 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.

## Introduction

Control of quantum materials out of equilibrium represents one of the grand challenges of modern condensed matter physics. While an understanding of general non-equilibrium 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 wide-envelope 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 single-particle spectra1,2,3,4 and band topology or short-range entangled topological states5,6,7, a natural extension regards pumping of strongly correlated systems. Here, the essence of Floquet physics lies not merely in imbuing one-1 and two-particle2 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 hopping8, 9, strong-field sign reversal of nearest-neighbor Heisenberg exchange in a 1D magnet10, 11, or enhancement of Cooper-pair formation12,13,14. Similar ideas are being pursued in the field of ultracold atoms to simulate artificial gauge fields, to dynamically realize topological band structures15 or even propose fractional quantum Hall effects and spin liquids in optical lattices16, 17, 18. At the same time, recent advances in Floquet thermodynamics indicate that, while driven non-integrable closed systems are in principle expected to heat up to infinite temperature19, 20, heating can be exponentially slow on pre-thermalized time scales21,22,23,24,25,26 or altogether avoided via many-body localization27,28,29 or dissipation30,31,32. An ideal condensed-matter realization hence entails a charge gap to limit absorption, as well as a delicate balance of competing phases, such that time-dependent 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 magnets33 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 excitations34,35,36,37,38,39,40,41. 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 excitations42,43,44,45. While an unlikely ground state in unperturbed microscopic models, recently the CSL was found to be a competing state46,47,48,49,50,51,52,53, in particular after explicit breaking of TRS and parity46,47,48,49,50. 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 CSLs46.

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 others34 with putative spin-liquid behavior at low temperatures. Experiments54 and first-principles calculations55, 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 Cu34. 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 energies57, above the ~2 eV charge gap58. 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 filling

$$\hat H(t) = - {t_{\rm{h}}}\mathop {\sum}\limits_{\left\langle {ij} \right\rangle \sigma } {e^{i\frac{e}{\hbar }\,{{\bf{r}}_{ij}} \cdot {\bf{A}}(t)}} \hat c_{i\sigma }^\dag {\hat c_{j\sigma }} + U\mathop {\sum}\limits_i {\hat n_{i \uparrow }}{\hat n_{i \downarrow }}$$
(1)

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 $$J \approx 4t_{\rm{h}}^{\rm{2}}{\rm{/}}U$$ with first-principles predictions for herbertsmithite56 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 $$\hat H\left( {t + 2\pi {\rm{/}}\Omega } \right) = \hat H(t)$$. Floquet theory then dictates that the behavior near the pump plateau is completely determined via many-body 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 quasi-energy, where the $$\left| {{\Phi _m}} \right\rangle$$ conveniently follow as eigenstates of the static Floquet-Hubbard Hamiltonian

$$\begin{array}{*{20}{l}} {\hat H} = & { - {t_{\rm{h}}} \mathop {\sum}\limits_{{\scriptstyle \left\langle {ij} \right\rangle \sigma \atop \scriptstyle mm \prime}} {{\cal J}_{m - m \prime}}\left( A \right){e^{i\left( {m - m\prime} \right)\mathrm{arg}{{\bf{r}}_{ij}}}} \, \hat c_{i\sigma }^\dag {{\hat c}_{j\sigma }} \otimes \left| m \right\rangle \left\langle {m \prime} \right|} \\ \\ & { + U \mathop {\sum}\limits_i {{\hat n}_{i \uparrow }}{{\hat n}_{i \downarrow }} \otimes {\bf{1}} - \mathop {\sum} \limits_m m \Omega \,{\bf{1}} \otimes \left| m \right\rangle \left\langle m \right|} \end{array}$$
(2)

where A denotes the dimensionless field strength at the pump plateau, such that A(t) ≈ /(ea 0) with a 0 the nearest-neighbor 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 $${\epsilon _n}$$ +  identify with the same physical state m.

### Floquet Chiral spin model

Physically, Eq. (2) describes photon-assisted hopping in the presence of interactions, where electrons can enlist m photons to hop at a reduced energy cost U −  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 multi-spin 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:

$${\hat H_{{\rm{spin}}}} = \mathop {\sum}\limits_{ij} {J_{ij}} {{\bf{S}}_i} \cdot {{\bf{S}}_j} + \mathop {\sum}\limits_{ijk} {\chi _{ijk}} {{\bf{S}}_i} \cdot \left( {{{\bf{S}}_j} \times {{\bf{S}}_k}} \right)$$
(3)

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 high-frequency limit $$\Omega \gg U,{t_{\rm{h}}}$$. Here, circularly polarized pumping induces complex nearest-neighbor 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 purely-complex next-nearest-neighbor 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 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). Figure 1b, c depict relevant virtual processes, involving simultaneous hopping of electrons and absorption of m photons with intermediate energy cost U − . To second order in virtual hopping, two-site exchange processes (Fig. 1b) are phase-agnostic and solely renormalize the nearest-neighbor 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 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 A2g Raman response of Mott insulators59, 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

$$\chi = 3\mathop {\sum}\limits_{\bf{m}} \left[ {{\rm{sin}}\left[ {{\textstyle{{2\pi \left( {{m_1} - {m_2} + {m_3}} \right)} \over 3}}} \right]\Lambda _{\bf{m}}^{(1)} - {\rm{sin}}\left[ {{\textstyle{{2\pi {m_2}} \over 3}}} \right]\Lambda _{\bf{m}}^{(2)}} \right]$$
(4)
$$\chi ' = \mathop {\sum}\limits_{\bf{m}} \left[ {{\rm{sin}}\left[ {{\textstyle{{\pi \left( {{m_1} - {m_2} + {m_3}} \right)} \over 3}}} \right]\Lambda _{\bf{m}}^{(1)} - {\rm{sin}}\left[ {{\textstyle{{\pi {m_2}} \over 3}}} \right]\Lambda _{\bf{m}}^{(2)}} \right]$$
(5)

where m = {m 1, m 2, m 3} are Floquet indices, and

$$\Lambda _{\bf{m}}^{(1)} = \frac{{8t_{\rm{h}}^4{{\cal J}_{{m_1}}}\left( A \right){{\cal J}_{{m_2} - {m_1}}}\left( A \right){{\cal J}_{{m_2} - {m_3}}}\left( A \right){{\cal J}_{{m_3}}}\left( A \right)}}{{\mathop {\prod}\nolimits_{i = 1}^3 \left( {U - {m_i}\Omega } \right)}}$$
(6)
$$\Lambda _{\bf{m}}^{(2)} = 2\left( {1 - {\delta _{{m_2}}}} \right){\textstyle{{U - {m_2}\Omega } \over {{m_2}\Omega }}}{\left( { - 1} \right)^{{m_1} - {m_3}}}{\rm{co}}{{\rm{s}}^2}\left[ {{\textstyle{{{m_2}\pi } \over 2}}} \right]\Lambda _{\bf{m}}^{(1)}$$
(7)

Here, $$\Lambda _{\bf{m}}^{(1)}$$ and $$\Lambda _{\bf{m}}^{(2)}$$ 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, next-nearest-neighbor Heisenberg exchange

$$\begin{array}{l} J' = \mathop {\sum}\limits_{\bf{m}} \left[ {{\Gamma _{\bf{m}}} - {\rm{cos}}\left[ {{\textstyle{{\pi \left( {{m_1} - {m_2} + {m_3}} \right)} \over 3}}} \right]\frac{{\Lambda _{\bf{m}}^{(1)}}}{2} + {\rm{cos}}\left[ {{\textstyle{{\pi {m_2}} \over 3}}} \right]\frac{{\Lambda _{\bf{m}}^{(2)}}}{2}} \right]\\ {J_3} = \mathop {\sum}\limits_{\bf{m}} \left[ {{\Gamma _{\bf{m}}} - \frac{{\Lambda _{\bf{m}}^{(1)}}}{2} + \frac{{\Lambda _{\bf{m}}^{(2)}}}{2}} \right]\\ \end{array}$$
(8)

and corrections to nearest-neighbor Heisenberg exchange

$$\begin{array}{*{20}{l}} J \hfill & = \hfill & {\tilde J + \mathop {\sum}\limits_{\bf{m}} \left\{ {\left[ {2 + {\rm{cos}}\left[ {{\textstyle{{\pi \left( {{m_1} - {m_3}} \right)} \over 3}}} \right] + {\rm{cos}}\left[ {{\textstyle{{2\pi \left( {{m_1} - {m_3}} \right)} \over 3}}} \right] } \right.} \right.} \hfill \\ {} \hfill & {} \hfill & {\left. { + {\rm{cos}}\left[ {{\textstyle{{\pi \left( {{m_1} - {m_2} + {m_3}} \right)} \over 3}}} \right] + {\textstyle{1 \over 2}} \, {\rm{cos}}\left[ {{\textstyle{{2\pi \left( {{m_1} - {m_2} + {m_3}} \right)} \over 3}}} \right]} \right]\Lambda _{\bf{m}}^{(1)}} \hfill \\ {} \hfill & {} \hfill & {\left. { - \left[ {1 + 2\,{\rm{cos}}\left[ {\pi {m_2}} \right] + {\rm{cos}}\left[ {{\textstyle{{\pi {m_2}} \over 3}}} \right] + {\textstyle{1 \over 2}}{\rm{cos}}\left[ {{\textstyle{{2\pi {m_2}} \over 3}}} \right]} \right]\Lambda _{\bf{m}}^{(2)} - 7{\Gamma _{\bf{m}}}} \right\}} \hfill \\ \end{array}$$
(9)

appear at the same order, with

$${\Gamma _{\bf{m}}} = 4t_{\rm{h}}^4\left[ {\frac{{{\delta _{{m_3},0}}}}{{U - {m_1}\Omega }} + \frac{{{\delta _{{m_3},0}}}}{{U - {m_2}\Omega }}} \right]\mathop {\prod}\limits_{i = 1}^2 \frac{{{{\left[ {{{\cal J}_{{m_i}}}\left( A \right)} \right]}^2}}}{{\left( {U - {m_i}\Omega } \right)}}$$
(10)

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 TRS-breaking 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 change the underlying many-body state. Close to the one-photon resonance (Ω = U), chiral contributions are staggered between elementary and isosceles triangles (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 QSL60,61,62,63. 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 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 $$\left| {{\psi _{1,2}}} \right\rangle$$, minimizing the Rényi entropy for their reduced density matrices in two distinct bipartitions (Fig. 3d) on a 36-site torus (Methods sections). C 6 symmetry allows extraction of both modular $${\cal U}$$, $${\cal S}$$ matrices64,65,66 that encode self- and mutual-braiding statistics of the elementary excitations. We find that the photo-induced CSL corresponds uniquely to a $$\nu = {\textstyle{1 \over 2}}$$ bosonic fractional quantum Hall state, with matching modular matrices40

$${\cal U} = {\rm e^{ - \, i\frac{{2\pi }}{{24}} \times 1.02}}\left[ {\begin{array}{*{20}{c}} 1 & 0 \\ 0 & {0.99i} \\ \end{array}} \right],\,{\cal S} = {\textstyle{1 \over {\sqrt 2 }}}\left[ {\begin{array}{*{20}{c}} {1.00} & {1.00} \\ {1.00} & { - 1.02} \\ \end{array}} \right]$$
(11)

### Time evolution

Having established a photo-induced 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 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 $$t \le 1000\,t_{\rm{h}}^{ - 1}$$. 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 time-evolved state will nevertheless retain a finite quasi-energy 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 “pre-thermalized” time scales of interest, as we show below.

First, to compare to the Floquet spin description, we focus on time-dependent 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 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 $$\left\langle {{{\left( {{S^z}} \right)}^2}} \right\rangle < 1{\rm{/}}4$$; 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 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 infinite-temperature 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 off-resonant regime, one observes a pump-strength 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” $${\cal F} = \left| {\left\langle {\Psi \left( {t + T} \right)\left| {\Psi (t)} \right.} \right\rangle } \right|$$, where T = 2π/Ω is the pump period. At the pump plateau with discrete time-translation symmetry, $${\cal F}$$ is time-independent and quantifies the Floquet quasi-energy 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 pre-thermalized time scales of interest25, 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 $$\langle {\widehat E} \rangle$$ (Methods section). On the pump plateau, both the double occupancy and $$\langle {\widehat E} \rangle$$ 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 $$\langle {\widehat E} \rangle (t) \approx {E_0} + t\Delta E$$, 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 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 follow-up 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 weakly-interacting electrons30,31,32, 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 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 invariance $$\hat H(t) = \hat H\left( {t + 2\pi {\rm{/}}\Omega } \right)$$. Instead of solving the many-body time evolution, one can reexpress the time-dependent 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 quasi-energy. Then, determination of the time-dependent eigenstates of the driven system reduces to finding the time-independent eigenstates of the Floquet Hamiltonian

$${\hat H_{{\rm{floquet}}}} = \mathop {\sum}\limits_{mm'} \left[ {{{\hat H}_{m - m'}} - m\Omega {\delta _{m,m'}}} \right]\left| m \right\rangle \left\langle {m'} \right|$$
(12)

where $${\hat H_{m - m'}}$$ are the Fourier expansion coefficients of $$\hat H(t)$$. Taking $$\hat H(t)$$ as the driven Kagome-Hubbard model of Eq. (1) in the main text straightforwardly recovers the Floquet-Hubbard 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 nearest-neighbor bond distance. While realistic estimates for materials would necessarily entail significant contributions from multi-orbital effects and local dipole transitions, all of which are not captured within the single-orbital Hubbard model, a naïve estimate from solely Peierls substitution for a single-orbital 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 near-infrared pump.

### Numerical simulations

To characterize the photo-induced CSL state, we performed exact diagonalization calculations of the Floquet chiral spin Hamiltonian, described in Eq. (3). Due to three-spin interactions and longer-ranged exchange interactions, the sparsity of the resulting Hamiltonian matrix is over an order of magnitude lower than for a nearest-neighbor Heisenberg antiferromagnet. Spin and chiral correlation functions as well as minimally entangled states were calculated for a 36-site 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 36-site cluster for a purely nearest-neighbor Heisenberg model67, 68. While density-matrix renormalization group studies indicate that such a model is likely to stabilize a Z 2 QSL in the thermodynamic limit, it is well-known that its ground state degeneracy remains inaccessible in exact diagonalization of finite-size clusters67, 68, and the 36-site cluster is likely to host a QSL close to a phase boundary with a valence bond crystal69. 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 well-stabilized already on the finite-size systems under consideration, with a robust many-body excitation gap.

The ground state degeneracy and minimal many-body excitation gap under flux insertion was calculated by imposing a spin-dependent twist of boundary 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

$${S_\alpha }(\theta ,\phi ) = - {\rm{log}}\,{\rm{tr}}\left\{ {\rho _\alpha ^2(\theta ,\phi )} \right\},\quad \alpha = A,B$$
(13)

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 quasi-degenerate 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 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 $$\left| {\Psi (\theta ,\phi )} \right\rangle$$ permit extraction of the modular matrices64, 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 12-site 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

$${\bf{A}}(t) = A(t){\left[ {{\rm{cos}}(\Omega t),\,{\rm{sin}}(\Omega t)} \right]^{\rm T}}$$
(14)

with a smooth sinusoidal pump envelope

$$A(t) = \left\{ {\begin{array}{*{20}{l}} {0, } \hfill & {t \le 0} \hfill \\ {A\,{\rm{si}}{{\rm{n}}^2}\left( {\frac{\pi }{2}\frac{t}{{{t_{{\rm{plateau}}}}}}} \right), } \hfill & {0 < \, t < \, {t_{{\rm{plateau}}}}} \hfill \\ {A,} \hfill & {t \ge {t_{{\rm{plateau}}}}} \hfill \end{array}} \right.$$
(15)

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 period-averaged energy operator

$$\begin{array}{*{20}{l}} {\hat E} \hfill & = \hfill & {\frac{1}{T}{\int}_t^{t + T} dt'\hat H\left( {t'} \right)} \hfill \\ {} \hfill & = \hfill & {U{{\hat n}_ \uparrow }{{\hat n}_ \downarrow } - {{\cal J}_0}(A){t_{\rm{h}}}\mathop {\sum}\limits_{\left\langle {ij} \right\rangle \sigma } \hat c_{i\sigma }^\dag {{\hat c}_{j\sigma }}} \hfill \end{array}$$
(16)

where $${{\cal J}_0}(A)$$ denotes the zeroth Bessel function of the first kind. Note that $$\langle {\widehat E} \rangle$$ 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.

## References

1. 1.

Wang, Y. H., Steinberg, H., Jarillo-Herrero, P. & Gedik, N. Observation of floquet-bloch states on the surface of a topological insulator. Science 342, 453–457 (2013).

2. 2.

Kim, J. et al. Ultrafast generation of pseudo-magnetic field for valley excitons in Wse2 monolayers. Science 346, 1205–1208 (2014).

3. 3.

Sie, E. J. et al. Valley-selective optical Stark effect in monolayer WS2. Nat. Mater. 14, 290–294 (2015).

4. 4.

Mahmood, F. et al. Selective scattering between Floquet-Bloch and Volkov states in a topological insulator. Nat. Phys. 12, 306–310 (2016).

5. 5.

Lindner, N. H., Refael, G. & Galitski, V. Floquet topological insulator in semiconductor quantum wells. Nat. Phys. 7, 490–495 (2011).

6. 6.

Rudner, M. S., Lindner, N. H., Berg, E. & Levin, M. Anomalous edge states and the bulk-edge correspondence for periodically-driven two dimensional systems. Phys. Rev. X 3, 031005 (2012).

7. 7.

Po, H. C., Fidkowski, L., Morimoto, T., Potter, A. C. & Vishwanath, A. Chiral floquet phases of many-body localized bosons. Phys. Rev. X 6, 041070 (2016).

8. 8.

Bukov, M., Kolodrubetz, M. & Polkovnikov, A. Schrieffer-wolff transformation for periodically driven systems: strongly correlated systems with artificial gauge fields. Phys. Rev. Lett. 116, 125301 (2016).

9. 9.

Bukov, M., Heyl, M., Huse, D. A. & Polkovnikov, A. Heating and many-body resonances in a periodically driven two-band system. Phys. Rev. B 93, 155132 (2016).

10. 10.

Mentink, J. H., Balzer, K. & Eckstein, M. Ultrafast and reversible control of the exchange interaction in Mott insulators. Nat. Commun. 6, 6708 (2015).

11. 11.

Itin, A. P. & Katsnelson, M. I. Effective hamiltonians for rapidly driven many-body lattice systems: induced exchange interactions and density-dependent hoppings. Phys. Rev. Lett. 115, 075301 (2015).

12. 12.

Knap, M., Babadi, M., Refael, G., Martin, I. & Demler, E. Dynamical Cooper pairing in non-equilibrium electron-phonon systems. Phys. Rev. B 94, 214504 (2016).

13. 13.

Sentef, M. A., Kemper, A. F., Georges, A. & Kollath, C. Theory of light-enhanced phonon-mediated superconductivity. Phys. Rev. B 93, 144506 (2016).

14. 14.

Coulthard, J., Clark, S. R., Al-Assam, S., Cavalleri, A. & Jaksch, D. Enhancement of superexchange pairing in the periodically driven Hubbard model. Phys. Rev. B 96, 085104 (2017).

15. 15.

Jotzu, G. et al. Experimental realization of the topological Haldane model with ultracold fermions. Nature 515, 237–240 (2014).

16. 16.

Yao, N. Y. et al. Realizing fractional chern insulators with dipolar spins. Phys. Rev. Lett. 110, 185302 (2012).

17. 17.

Cooper, N. R. & Dalibard, J. Reaching fractional quantum hall states with optical flux lattices. Phys. Rev. Lett. 110, 185301 (2013).

18. 18

Yao, N. Y., Zaletel, M. P., Stamper-Kurn, D. M. & Vishwanath, A., A quantum dipolar spin liquid. Preprint at http://arxiv.org/abs/1510.06403 (2015).

19. 19.

D’Alessio, L. & Rigol, M. Long-time behavior of isolated periodically driven interacting lattice systems. Phys. Rev. X 4, 041048 (2014).

20. 20.

Lazarides, A., Das, A. & Moessner, R. Equilibrium states of generic quantum systems subject to periodic driving. Phys. Rev. E 90, 012110 (2014).

21. 21.

Canovi, E., Kollar, M. & Eckstein, M. Stroboscopic prethermalization in weakly interacting periodically driven system. Phys. Rev. E 93, 012130 (2016).

22. 22.

Kuwahara, T., Mori, T. & Saito, K. Floquet-magnus theory and generic transient dynamics in periodically driven many-body quantum systems. Ann. Phys. 367, 96–124 (2016).

23. 23.

Mori, T., Kuwahara, T. & Saito, K. Rigorous bound on energy absorption and generic relaxation in periodically driven quantum systems. Phys. Rev. Lett. 116, 120401 (2016).

24. 24.

Bukov, M., Gopalakrishnan, S., Knap, M. & Demler, E. Prethermal floxquet steady states and instabilities in the periodically driven, weakly interacting bose-hubbard model. Phys. Rev. Lett. 115, 205301 (2015).

25. 25.

Abanin, D. A., De Roeck, W. & Huveneers, F. Exponentially slow heating in periodically driven many-body systems. Phys. Rev. Lett. 115, 256803 (2015).

26. 26.

Ho, W. W. & Abanin, D. A. Quasi-adiabatic dynamics and state preparation in Floquet many-body systems. Preprint at http://arxiv.org/abs/1611.05024 (2016).

27. 27.

D’Alessio, L. & Polkovnikov, A. Many-body energy localization transition in periodically driven systems. Ann. Phys. 333, 19–33 (2013).

28. 28.

Ponte, P., Papić, Z., Huveneers, F. & Abanin, D. A. Many-body localization in periodically driven systems. Phys. Rev.Lett. 114, 140401 (2015).

29. 29.

Lazarides, A., Das, A. & Moessner, R. Fate of many-body localization under periodic driving. Phys. Rev. Lett. 115, 030402 (2015).

30. 30.

Dehghani, H., Oka, T. & Mitra, A. Dissipative Floquet topological systems. Phys. Rev. B 90, 195429 (2014).

31. 31.

Iadecola, T., Neupert, T. & Chamon, C. Occupation of topological Floquet bands in open systems. Phys. Rev. B 91, 235133 (2015).

32. 32.

Seetharam, K. I., Bardyn, C.-E., Lindner, N.-H., Rudner, M.-S. & Refael, G. Controlled population of Floquet-Bloch states via coupling to BOSE and Fermi Baths. Phys. Rev. X 5, 041050 (2015).

33. 33.

Balents, L. Spin liquids in frustrated magnets. Nature 464, 199–208 (2010).

34. 34.

Norman, M. R. Herbertsmithite and the search for the quantum spin liquid. Rev. Mod. Phys. 88, 041002 (2016).

35. 35.

Zhou, Y., Kanoda, K. & Ng, T.-K. Quantum spin liquid states. Rev. Mod. Phys. 89, 025003 (2017).

36. 36.

Yamashita, M. et al. Highly mobile gapless excitations in a two-dimensional candidate quantum spin liquid. Science 328, 1246–1248 (2010).

37. 37.

Han, T.-H. et al. Fractionalized excitations in the spin-liquid state of a kagome-lattice antiferromagnet. Nature 492, 406–410 (2012).

38. 38.

Banerjee, A. et al. Proximate Kitaev quantum spin liquid behaviour in a honeycomb magnet. Nat. Mater. 15, 733–740 (2016).

39. 39.

Fu, M., Imai, T., Han, T.-H. & Lee, Y. S. Evidence for a gapped spin-liquid ground state in a kagome Heisenberg antiferromagnet. Science 350, 655–658 (2016).

40. 40.

Wen, X.-G. Topological orders in rigid states. Int. J. Mod. Phys. B 4, 239–271 (1990).

41. 41.

Wen, X.-G. Quantum orders and symmetric spin liquids. Phys. Rev. B 65, 165113 (2002).

42. 42.

Kalmeyer, V. & Laughlin, R. B. Equivalence of the resonating-valence-bond and fractional quantum hall states. Phys. Rev. Lett. 59, 2095–2098 (1987).

43. 43.

Kalmeyer, V. & Laughlin, R. B. Theory of the spin liquid state of the Heisenberg antiferromagnet. Phys. Rev. B 39, 11879–11899 (1989).

44. 44.

Schroeter, D. F., Kapit, E., Thomale, R. & Greiter, M. Spin Hamiltonian for which the chiral spin liquid is the exact ground state. Phys. Rev. Lett. 99, 097202 (2007).

45. 45.

Thomale, R., Kapit, E., Schroeter, D. F. & Greiter, M. Parent Hamiltonian for the chiral spin liquid. Phys. Rev. B 80, 104406 (2009).

46. 46.

Bauer, B. et al. Chiral spin liquid and emergent anyons in a Kagome lattice Mott insulator. Nat. Commun. 5, 5137 (2014).

47. 47.

Wietek, A., Sterdyniak, A. & Läuchli, A. M. Nature of chiral spin liquids on the kagome lattice. Phys. Rev. B 92, 125122 (2015).

48. 48.

Kumar, K., Sun, K. & Fradkin, E. Chiral spin liquids on the kagome lattice. Phys. Rev. B 92, 094443 (2015).

49. 49.

Hickey, C., Cincio, L., Papić, Z. & Paramekanti, A. Haldane-Hubbard mott insulator: from tetrahedral spin crystal to chiral spin liquid. Phys. Rev. Lett. 116, 137202 (2016).

50. 50.

Wietek, A. & Läuchli, A. M. Chiral spin liquid and quantum criticality in extended S = 1/2 Heisenberg models on the triangular lattice. Phys. Rev. B 95, 035141 (2016).

51. 51.

He, Y.-C., Sheng, D. N. & Chen, Y. Chiral spin liquid in a frustrated anisotropic Kagome Heisenberg model. Phys. Rev. Lett. 112, 137202 (2014).

52. 52.

Gong, S.-S., Zhu, W. & Sheng, D. N. Emergent chiral spin liquid: fractional quantum Hall effect in a Kagome Heisenberg model. Sci. Rep. 4, 6317 (2014).

53. 53.

Gong, S.-S., Zhu, W., Balents, L. & Sheng, D. N. Global phase diagram of competing ordered and quantum spin-liquid phases on the kagome lattice. Phys. Rev. B 91, 075112 (2015).

54. 54.

Han, T., Chu, S. & Lee, Y. S. Refining the spin Hamiltonian in the spin-1/2 Kagome lattice antiferromagnet ZnCu3(OH)6Cl2 using single crystals. Phys. Rev. Lett. 108, 157202 (2012).

55. 55.

Janson, O., Richter, J. & Rosner, H. Modified Kagome physics in the natural spin-1 = 2 Kagome lattice systems: kapellasite Cu3n(OH)6Cl2 and haydeeite Cu3Mg(OH)6Cl2. Phys. Rev. Lett. 101, 106403 (2008).

56. 56.

Jeschke, H. O., Salvat-Pujol, F. & Valentí, R. First-principles determination of Heisenberg Hamiltonian parameters for the spin-1/2 Kagome antiferromagnet ZnCu3(OH)6Cl2. Phys. Rev. B 88, 075106 (2013).

57. 57.

Claassen, M., Jia, C., Moritz, B. & Devereaux, T. P. All-optical materials design of chiral edge modes in transition-metal dichalcogenides. Nat. Commun. 7, 13074 (2016).

58. 58.

Helton, J. S. et al. Spin dynamics of the spin-1/2 kagome lattice antiferromagnet ZnCu3(OH)6Cl2. Phys. Rev. Lett. 98, 107204 (2007).

59. 59.

Ko, W.-H., Liu, Z.-X., Ng, T.-K. & Lee, P. A. Raman signature of the U(1) Dirac spin-liquid state in the spin-1/2 Kagome system. Phys. Rev. B 81, 024414 (2010).

60. 60.

Jiang, H. C., Weng, Z. Y. & Sheng, D. N. Density matrix renormalization group numerical study of the kagome antiferromagnet. Phys. Rev. Lett. 101, 117203 (2008).

61. 61.

Yan, S., Huse, D. A. & White, S. R. Spin-liquid ground state of the S = 1/2 Kagome Heisenberg antiferromagnet. Science 332, 1173 (2011).

62. 62.

Jiang, H. C., Wang, Z. & Balents, L. Identifying topological order by entanglement entropy. Nat. Phys. 8, 902–905 (2012).

63. 63.

Depenbrock, S., McCullough, I. P. & Schollwöck, U. Nature of the spin-liquid ground state of the S = 1/2 Heisenberg model on the Kagome lattice. Phys. Rev. Lett. 109, 067201 (2012).

64. 64.

Zhang, Y., Grover, T., Turner, A., Oshikawa, M. & Vishwanath, A. Quasiparticle statistics and braiding from ground-state entanglement. Phys. Rev. B 85, 235151 (2012).

65. 65.

Cincio, L. & Vidal, G. Characterizing topological order by studying the ground states on an infinite cylinder. Phys. Rev. Lett. 110, 067208 (2013).

66. 66.

Zhu, W., Sheng, D. N. & Haldane, F. D. M. Minimal entangled states and modular matrix for fractional quantum Hall effect in topological flat bands. Phys. Rev. B 88, 035122 (2013).

67. 67.

Läuchli, A. M., Sudan, J. & Sørensen, E. S. Ground-state energy and spin gap of spin-1/2 Kagomé-Heisenberg antiferromagnetic clusters: large-scale exact diagonalization results. Phys. Rev. B 83, 212401 (2011).

68. 68.

Nakano, H. & Sakai, T. Numerical-diagonalization study of spin gap issue of the Kagome lattice Heisenberg antiferromagnet. J. Phys. Soc. Jpn 80, 053704 (2011).

69. 69.

Li, T. The spin-1/2 Heisenberg model on Kagome lattice as a quantum critical system. Preprint at http://arxiv.org/abs/1106.6134 (2011).

## 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. DE-AC02-76SF00515. 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- AC02-05CH11231.

## Author information

Authors

### Contributions

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.

### Corresponding authors

Correspondence to Martin Claassen or Thomas P. Devereaux.

## Ethics declarations

### Competing interests

The authors declare no competing financial interests.

Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Rights and permissions

Reprints and Permissions

Claassen, M., Jiang, HC., Moritz, B. et al. Dynamical time-reversal symmetry breaking and photo-induced chiral spin liquids in frustrated Mott insulators. Nat Commun 8, 1192 (2017). https://doi.org/10.1038/s41467-017-00876-y

• Accepted:

• Published:

• ### Edge states of Floquet–Dirac semimetal in a laser-driven semiconductor quantum-well

• Boyuan Zhang
• Ken-ichi Hino

Scientific Reports (2021)

• ### Band structure engineering and non-equilibrium dynamics in Floquet topological insulators

• Mark S. Rudner
• Netanel H. Lindner

Nature Reviews Physics (2020)

• ### Ultrastrong coupling between light and matter

• Anton Frisk Kockum
• Franco Nori

Nature Reviews Physics (2019)

• ### Light-induced evaporative cooling of holes in the Hubbard model

• Philipp Werner
• Martin Eckstein
• Gil Refael

Nature Communications (2019)

• ### Theory of photoinduced ultrafast switching to a spin-orbital ordered hidden phase

• Jiajun Li
• Hugo U. R. Strand
• Martin Eckstein

Nature Communications (2018)


text:
# Tag Info

## Hot answers tagged express.js

18

You don't have to inject express into each modules. requiring express in each file will return the same instance. http://nodejs.org/docs/latest/api/modules.html#modules_caching Also, ask yourself who should be the one registering the router into the app. The app it self or the router ? For code reuse and clean separation you should let the app handle it. ...

13

The power of the aggregation framework is it's ability to iterate over the dataset in various useful ways without incurring extra round trips between the database and the app. Your code uses one stage for aggregation (grouping by id1), and then jumps out of aggregation to iterate over the entire FUNDS collection for every user. That is very expensive, ...

12

You can create a file db.js var mysql = require('mysql'); var settings = require('./settings.json'); var db; function connectDatabase() { if (!db) { db = mysql.createConnection(settings); db.connect(function(err){ if(!err) { console.log('Database is connected!'); } else { ...

8

How about a ternary operator? var redirect = req.from.redirect ? url.parse(req.from.redirect) : { protocol: 'http', hostname: config.domain, pathname: 'thanks' }; Looks cleaner in my opinion.

7

Okay I'm going to be a bit more comprehensive here since I've already read your code quite a few times, and I'm gonna assume that you are on node v6 The setup for the TodoControllerSpec feels kind of verbose. Can it be simplified? If you mean all the require calls you have to do...unfortunately not as far as I know. As a side note, I really like jest for ...

5

Interesting, I actually really like your constructor, it is neat and shows in one place what the class does. I am not sure I would call it SignupDoctorPostRequest, since it is not a request at all but processes a request. I would consider DoctorSignupModel or since it seems to leverage several models DoctorSignupMetaModel. As for your eager validation, I ...

5

There is a better way. If you wish to serve static files with express - it comes bundled with static middleware. To use it, you simply add this line before your routes: app.use(express.static(__dirname + '/public')); This may require slight adjustment to your paths but it replaces all those manual routes and is more efficient (caching, expiration etc). ...

5

I know you are generating data for the sake of completion but it does complicate things on what aspect of the code you'd like to focus on. But here are a couple thoughts. With Express, I often like to treat the routes as controllers in an MVC pattern. So any fetching of data and mutating data, I would create at least a model per resource to encapsulate that ...

5

Don't use callbacks, it's hard to read and maintain, the fs.readdir function will return a promise if you didn't provide a callback function. Use arrow functions when you could. Respond with an error instead of throwing (unless you have a catch wrapper middleware). Use property shorthands. You forgot req and res. Try this: const express = require('express')...

4

That does not look very DRY ( Dont repeat yourself ), not to mention that it does not look good that every function must be aware of the connection string. For updates, you could do something like this: exports.setPassword = function (id, password, callback) { genericUpdate('UPDATE users SET password=$1 WHERE id=$2', [password, id], callback ); }; ...

4

First, read about how .get() can chain multiple callbacks, then read how the sample code of Passport uses ensureAuthenticated as a callback. That should give you enough inspiration. Furthermore: English only in source code, especially for variables, even for comments Comments, your code could use more of them, and less, this comment is pretty useles: //...

4

What you're looking for is JSON.parse. It's not in the Node docs, because it's "lower" than that: Every modern javascript runtime has it (see MDN). Here's a simple function to read a file, parse it as JSON, and send it to a callback (all in Node) function readJSONFile(filename, callback) { require("fs").readFile(filename, function (err, data) { if(...

4

To answer your question: yes, I think it could be interesting as a stand-alone module. There's can be a lot of repetitive boilerplate involved in connecting a REST api to a CRUD backend, reducing that is certainly worth a module, in my opinion. Now to dive into the code... index.js if(!config.collection || !config.type){ //Raise an error here... ...

4

I would just name the variable for Bluebird promises as Promise as it mostly acts like the standard Promise object in ES6. That way, if you happen to run an ES6-compatible Node.js, you can just remove the import, and you'll be set. In the promise constructor, you need not do a return. The code will also feel awkward to a new developer, thinking Promise needs ...

4

There's a big problem here. getUserByEmail() can either return a promise or throw synchronously which is a bad design and very unfriendly for the caller. Instead, you want any synchronous errors to be returned as a rejected promise. The beginning of your code should be changed from this: // SERVICE / MODEL LOGIC getUserByEmail() { if (!this._model....

4

If you use 1 promise, you would encounter a long chain of then method and all of async operations would be executed in sequence style which increases response time. If you create 3 promises, all of that can be performed in parallel. So the answer is yes, you should implement with 3 promises. By the way, with help of bluebird library, You can write more ...

4

Utilize the query object, this type of querying is more suitable there than as params. Use object destructuring. Dry your code, see how we're mapping over [countryId, stateId] to check the validity of the input. Ideally, you would have been using a module like Joi for that. Maintain a happy path and break out of the function early when erroring. Use async/...

4

Ways of improving/optimizations: What fixApostrophes and fixDashes functions actually try to do is replacing a specific punctuation chars with respective HTML entities. Instead of those numerous horrifying while loops - a more optimized, concise and extendable approach would be to: compose a predefined replacement map (where keys are search patterns and ...

3

Heyo! This is pretty good, but doesn't have any password hashing (storing your passwords in plain text is bad). You might want to instead consider using an authentication library like either passportjs or stormpath. If you're really set on rolling your own auth stuff, you could use this project I wrote as an example (it's using the same tools you are): ...

3

Get rid of all those commas: i18n.init detectLngQS: "lang" ,ns: { namespaces: ['ns.common', 'ns.layout'], defaultNs: 'ns.common'} ,resSetPath: "./locales/__lng__/new.__ns__.json" ,... CoffeScript doesn't need them: i18n.init detectLngQS: "lang" ns: { namespaces: ['ns.common', 'ns.layout'], defaultNs: 'ns.common'} resSetPath: "./locales/...

3

Sorry to tell you, but unfortunately it is near impossible to dry html in itself if you want to keep the elements in the same way. As far as I can see, you already use only the minimum amount of elements. Move on, your code is fine

3

Use map/reduce Whenever you can, you should consider using map functions to transform data. or example, you transform a JSON items array, to a sites array. This is currently done like: var sites = []; for (var i = 0; i < json.items.length; i++) { var site = json.items[i]; sites.push({ title: site.name, icon: site.icon_url, ...

3

if(typeof req.body.data !== 'undefined' && req.body.data ... // to if(req.body.data && req.query.hash && typeof req.body.id) Assuming that false, 0, null and an empty string has no special meaning in this program, you can simply use loose comparison. Anything falsy (null, undefined, false, 0, NaN, empty string) will just fall off. ...

3

I only looked at the tests briefly, but they look good and I like your approach The following changes will help the readability of your main code: Prefer early returning guard clauses over if/else. It's easier to follow, uses less code, and avoids a level of indentation. Name concepts clearly and move them to helper functions, so that your main body ...

3

It's often a good idea to separate routes into their own files to get a better overview of the project. Try out express-generator and see how the example project is organized. Use a logging framework to log requests. I've used morgan which is really nice. Callbacks are like any other parameters and should be given a descriptive name. For instance, query ...

3

One of the biggest things you can do to clean up you code is to simply remove the unnecessary if-else blocks. In many cases you have a return inside the first conditional, which means there is no reason for an else block at all. For example, you code rewrite like this: Student.findOne({parent_ids:parent_id},function(err,parentF){ if(err){ ...

3

As for the jQuery/function usage, there is definitely room for improvement but it would be helpful to see an example HTML element that contains the article img, title, and url to understand the page structure a bit. I think your variable names are clear, but could be simplified. Also, it is common practice to use Camel Case with the first letter lowercase ...

3

A short review; Your database related functions should be separated, in my mind your socket methods should only do this: Normalize parameters your receive like with isCheckedNum Call a method to Create/Read/Update/Delete data (with a promise mechanism) Emit to rooms if succesful and needed Send back the appropriate HTTP response code with extra info if ...

3

This is missing the point. It is not about promises. You are right that if done correctly, you'll get a nice flat structure, but not if you inline all your code. This would be true if you weren't using promises as well. What you should do is split your code into smaller building blocks, and the nice structure will follow. Also, be wary of your variable ...

3

Not a node expert, but it looks like you cover the basics. CSRF protection via token is standard practice and very reliable. You are generating your token via a cryptographically secure random number generator. You are leaving it in the session and making sure it is present on all post requests. The only way an attacker can break it is via XSS attack, ...

Only top voted, non community-wiki answers of a minimum length are eligible

format: plain


text:
# Why have brown dwarf classes been dubbed L, T and Y?

The classes used to categorize stars (O, B, A, F, G, K, M) are in a bizarre order for historical reasons. Stars were labeled based on the spectral lines that were visible, then the categories were put in a different order to account for their temperature. When discovering the temperature of stars, the categories were not renamed, so that the old catalogs could still be used, and everyone just learned "Oh be a fine guy/gal kiss me".

The first seven classes were updated as knowledge improved, but brown dwarfs were discovered more recently, so why have the letters L, T and Y been chosen, rather than the easier to remember P, Q and R (or some other set of letters), for instance?

• Have you tried looking at the Stellar Classification Wikipedia article? It explains, for example, that R and N have already been used for carbon stars (though these are now generally classed as "C" stars), and that P was used for the central stars of planetary nebulae. Jan 16 '20 at 9:09

The choice of L and T is explained in Kirkpatrick et al. (1999) 'Dwarfs Cooler than "M": The Definition of Spectral Type "L" Using Discoveries from the 2 Micron All-Sky Survey (2MASS)'. The principles behind the choice are given at the start of section 5.1:

In choosing a letter designation for the new spectral class, three important points must be considered : (1) The letter must be unambiguous, having not been used for any currently recognized spectral type. For example, though "N" follows "M" in the alphabet, it would be a poor choice of letter since it is used for a class of carbon stars. (2) The letter must create a taxonomy that is clearly distinguished from other types of astronomical objects. In this case, the letter must be accepted by the entire community, both by researchers involved in low-mass star and brown dwarf science and by astronomers in general. Although stellar spectroscopists might find "E0," "E1," "E2," etc., perfectly acceptable as new spectral subclasses, extragalactic morphologists already recognize these designations as elliptical galaxy types. (3) The letter must stand the test of time. For example, choosing "D" to mean that these objects are "degenerate" brown dwarfs would be flawed reasoning. Some of these dwarfs certainly are substellar (see § 7), but such a designation cannot be tied uniquely to any particular spectroscopic trait. The designation should apply to spectral features alone and be free of physical interpretation. Our understanding of the underlying physics may change with time ; our choice of letter should be impervious to such changes.

Applying these, they find the letters H, L, T and Y to be ok: the rationale for excluding the other letters is given in their Table 5:

Letter Status Notes
A In use Standard spectral class
B In use Standard spectral class
C In use Standard carbon-star class
D Ambiguous Confusion with white dwarf classes DA, DB, DC, etc.
E Ambiguous Confusion with elliptical galaxy morphological types E0–E7
F In use Standard spectral class
G In use Standard spectral class
H OK
I Problematic Transcription problems with I0 (10, Io) and I1 (11, II, Il)
J In use Standard carbon-star class
K In use Standard spectral class
L OK
M In use Standard spectral class
N In use Standard carbon-star class
O In use Standard spectral class
P Problematic? Incorrect association with planets?
Q Problematic? Incorrect association with QSOs?
R In use Standard carbon-star class
S In use Standard spectral class for ZrO-rich stars
T OK
U Problematic? Incorrect association with ultraviolet sources?
V Problematic Confusion with vanadium oxide (V0 vs VO)
W Ambiguous Confusion with Wolf-Rayet WN and WR classes
X Problematic Incorrect association with X-ray sources
Y OK
Z Problematic? Incorrect implication that we have reached "the end"?

They go with "L" as it is the closest letter to "M" that is still available. They prefer it to "H" because of the occasional use of the term "hydride dwarf" to refer to M subdwarfs which are dominated by CaH and MgH bands.

After that, the sequence is going alphabetically through the remaining available letters, so they put Gl 229B, which has a notably different spectrum to the other (L-class) brown dwarfs known at the time, into spectral class T. Spectral type Y took a few more years to show up in observations, and was chosen by the same principle.

If observations get to the point where it is worth erecting a new spectral class beyond Y then things will get interesting.


text:
## Chemistry and Chemical Reactivity (9th Edition)

$He\rightarrow He^++e^-$
From the description, these ions are formed by the removal of one electron from Helium: $$He\rightarrow He^++e^-$$


text:
# Snooker shot - does margin of error increase or decrease as the target angle increases?

There is a perception (widely held) in snooker that a straight shot is more difficult than an angled shot. There are many forum discussion about this, and the reasons are usually accepted to be psychological.

But I was wondering, is there a mathematical reason for it. Is the margin of error greater when the shot being taken is at an angle?

Example - if the white ball was 1 degree off target on a straight shot, and one degree off target on an angled shot (same target for both shots), and each shot hit at the same speed, would the red ball travel off line to the same extent?

• Interesting. I guess from experience one could "pinch the pocket" for angled shots. Apr 25 '17 at 9:50
• I should have been clearer. The target (red) ball would be in the same position - in line with the pocket (so that would not be a changing variable), the only difference would be the white ball position. In one shot it's straight in line with the red and pocket, in the other it would be at an angle Apr 25 '17 at 9:54
• Exactly what I had in mind, and I still stand by my comment (from experience). I hope someone has a nice mathematical answer to this question ! Apr 25 '17 at 9:57
• Could it simply be that inaccurate slower balls are more likely to drop into the pocket than inaccurate faster balls? At a larger angle, the cue ball imparts less momentum to the target ball. Apr 25 '17 at 13:43
• I updated the question - with Example of what I mean Apr 25 '17 at 14:16

Let the ball be radius $r$ and distance between balls $d$. Let $\theta$ be angle white is struck from line between centre of balls and $\phi$ direction red moves from line between red ball and white before struck. Then get $2r sin(\theta + \phi) = d sin(\theta)$. Call $a=\frac{d}{2r}$. This gives $\frac{d\phi}{d\theta}=-1+ \frac{a cos(\theta)}{\sqrt{1-a^2 sin^2(\theta)}}$. This represents the ratio of the error in the direction of red to the error in hitting the white. It increases monotonically from straight shot to a fine cut. If $d \approx r$ then this will not hold true as the target spot on the red is further away for a cut compared to a straight shot.

• What does this mean in a real world scenario - e.g. I don't understand the answer:-) Apr 26 '17 at 15:05
• For a given amount of error in the direction of the cue ball the error in the direction of the red will be bigger the greater the cut. So a straight shot ($\theta=0$) is easiest. It turns out that it is only the fine cuts where the increase in error is significant. See graph here wolframalpha.com/input/… Apr 26 '17 at 19:15
• PS for those who argue that straight shots seem hard, I agree :-) Apr 26 '17 at 19:17

For straight shots, one needs to hit very close to Center of mass of the ball.Agreed? Now consider a standard sphere. I will take projected area as $2\pi r^2$ ($r$ is radius)

To hit at the center of mass. Let's consider that hitting anywhere within $\frac{r}{2}$ of the projected area. Making it's favorable area will be $\frac{\pi r^2}{4}$ which happens to be $0.125$ of the total area. What I want to conclude by this is that the probability of a direct hit near center of mass is a tougher job Iff the probability of hitting anywhere on the ball is uniform.

One more Physics related stuff.Hitting very close to the ball also does not guarantee you a head on collision. We simply can't ignore the Angular Impulse factor and friction offered by the table. That angular impulse will surely provide it a rotational motion along a axis which ain't passing through center of mass. This will surely affect it's trajectory (I can't calculate the error margin though).

Your question was interesting. +1 for it.

• I was only thinking about the angles - I'm sure there is less margin for error the greater the angle of the shot - but the force applied is much less as there is a lot less mass behind the impact. This is more complicated than I thought. Apr 25 '17 at 11:03
• I think You need to see that the linear momentum is conserved along the Common Normal of impact. Moreover. You can't ignore the imparted angular momentum upon the impact. After all no collision has coefficient of restitution=1. It ain't wonderland . I think that angular momentum imparted is the criminal you are looking for error in trajectory. @Brian Apr 25 '17 at 11:07
• I'm not trained in any of those areas - math via computer science only Apr 25 '17 at 13:22

A practical reason: Experience. Naturally, angled shots arise more frequently than straight shots, so we have more practice shooting the former.

• That's the sort of answer I was getting in snooker forums.... so I came here for a mathematical approach Apr 25 '17 at 14:14


text:
# Magnetization of ferromagnetic material

lorenz0
Homework Statement:
Consider a ferromagnetic material and suppose we can approximate a portion of its hysteresis curve as a segment as shown in the figure. With this material we make a permanent magnet in the shape of a toroid with an overall length equal to ##D = 80 cm## including an air gap with a thickness of ##h = 5 mm##. Calculate the magnitude of the magnetization vector ##\vec{M}## of the magnet.
Relevant Equations:
##\vec{H}=\frac{\vec{B}}{\mu_0}-\vec{M}##
I have thought about the following
##\oint \vec{H}\cdot d\vec{l}=0\Leftrightarrow H_{int}(D-h)+H_{ext}h=0\Leftrightarrow (\frac{B}{\mu_0}-M)(D-h)+\frac{B}{\mu_0}h=0\Leftrightarrow M=\frac{D}{D-h}\frac{B}{\mu_0}## but (supposing what I have done is correct) I don't understand which value of ##B## I should take: perhaps the one at ##H=0\frac{kA}{m}##, namely ##B=0.5T##?

Thanks

#### Attachments

• ferromagnet.png
2 KB · Views: 34

## Answers and Replies

Homework Helper
Gold Member
This one is not simple, but you got a good start, with ## \oint H \cdot dl=0 ##, so that ## H_{inside} \approx -H_{gap} h/D ##. To a good approximation, ## H_{gap}=M ##, from the pole model of magnetostatics, (compare to the computation of ## E ## with two planes of surface charge density ## \pm \sigma ##).
From the hysteresis curve, you need to find the linear relationship between ## M ## and ## H ##. Let me write it out for you: Using ## B=\mu_o H+\mu_o M ##, we have one point at ## M=.5/\mu_o=M_o ## and ## H=0 ##. The other point is at ## H=-2000 ## and ## M=+2000 ##. We can write ## (M-M_o)/(H-0)=(2000-M_o)/(-2000-0) ##, and solve for ## M ## in terms of ## H ##.
Combine this with ## H=-M h/D ## to solve for ## M ##.

Edit: alternatively to see that ## H_{gap}=M ##, in the material ## B \approx \mu_o M ##. The ## B ## must be continuous, so that ## B=\mu_o M=\mu_o H_{gap} ##.

Last edited:
lorenz0 and Delta2
Homework Helper
Gold Member
for additional reading on the pole model vs. the surface current model of magnetostatics, especially post 1 of this link.
See https://www.physicsforums.com/insig...tostatics-and-solving-with-the-curl-operator/
for including the contribution to ## H ## from currents in conductors.
(Note: ## H ## in general comes from two contributions: magnetic poles and currents in conductors. For the above problem, to find ## H_{gap} ##, we just had magnetic poles to consider, as there were no currents in conductors).

Last edited:
lorenz0
lorenz0
This one is not simple, but you got a good start, with ## \oint H \cdot dl=0 ##, so that ## H_{inside} \approx -H_{gap} h/D ##. To a good approximation, ## H_{gap}=M ##, from the pole model of magnetostatics, (compare to the computation of ## E ## with two planes of surface charge density ## \pm \sigma ##).
From the hysteresis curve, you need to find the linear relationship between ## M ## and ## H ##. Let me write it out for you: Using ## B=\mu_o H+\mu_o M ##, we have one point at ## M=.5/\mu_o=M_o ## and ## H=0 ##. The other point is at ## H=-2000 ## and ## M=+2000 ##. We can write ## (M-M_o)/(H-0)=(2000-M_o)/(-2000-0) ##, and solve for ## M ## in terms of ## H ##.
Combine this with ## H=-M h/D ## to solve for ## M ##.

Edit: alternatively to see that ## H_{gap}=M ##, in the material ## B \approx \mu_o M ##. The ## B ## must be continuous, so that ## B=\mu_o M=\mu_o H_{gap} ##.
You have been very helpful, thank you very much!

Homework Helper
Gold Member
and a follow-on: More useful than the value of ## M ## is the value of ## \mu_o M ##. I get ## \mu_o M=.22 ## T. Does this agree with what you obtained?

lorenz0
Homework Helper
Gold Member
Perhaps a couple additional comments about the solution of this would be worthwhile. We are given a hysteresis curve that is linear, relating ## B ## to ## H ##. We can thereby write ## B=aH+b ## for constants ## a ## and ## b ##. With ## B=\mu_o H+\mu_o M ##, it should be apparent, without a rigorous proof, that ## M=a'H+b' ##, for constants ## a' ## and ## b' ##, i.e. there is a linear relationship between ## M ## and ## H ##. We made use of this in solving the problem above.

lorenz0
Homework Helper
Gold Member
It may be of interest that changing the geometry from that of a toroid to a toroid with a gap made for the solution where we had the line ## M=-160 \, H ## intersecting the hysteresis curve, rather than having ## H=0 ##,
(## \mu_o M=.5 ## T), for the complete toroid where there aren't any poles. (Magnetic pole density ## \rho_m=-\mu_o \nabla \cdot M ## gives rise to a non-zero ## H ## in the magnetic pole model of doing the calculations).

Another case of interest is a spherical shape for the permanent magnet, where some advanced calculations show that ## H=-M/3 ##. The factor (1/3) is known as the demagnetizing factor ## D ## for the spherical geometry. Using the same hysteresis curve, the result would be that ## \mu_o M ## would be considerably less for the permanent spherical magnet than the toroid with the small gap. If I get a little extra time, I may compute this value. See https://encyclopedia2.thefreedictionary.com/Demagnetizing+Factor

Last edited:
lorenz0
Homework Helper
Gold Member
For the spherical geometry, using the same hysteresis data, I get that ## \mu_o M=.0074 ## T. It would be interesting to see some experimental data for this type of thing. They could also try gaps of different thicknesses with the toroid, and see how it affects the strength of the permanent magnet. The larger the gap, the smaller the magnetization ## \mu_o M ## of the permanent magnet.

lorenz0
Homework Helper
Gold Member
One other item of interest worth mentioning in regards to these permanent magnets is what the ## H ## represents. The ## H ##, computed from the currents in conductors along with a contribution from magnetic poles in general does not represent an actual magnetic field, but is a useful mathematical construction that always obeys the pole model formula ## B=\mu_o H +\mu_o M ##.

In a permanent magnet, surprisingly, the ## H ## is found to point opposite the magnetization ## M ##. For the permanent magnet, it is the ## B ## that is the actual magnetic field, and it is found to point in the direction of ## M ##, as expected.

For linear materials such as the iron in transformers, where we can write ## B=\mu H ##, the ## H ## can be treated like an actual field, but that is not the case with the permanent magnet.

lorenz0


text:
# Subspace of Complete Metric Space is Closed iff Complete

## Theorem

Let $\struct {M, d}$ be a complete metric space.

Let $\struct {S, d}$ be a subspace of $\struct {M, d}$.

Then $S$ is closed if and only if $S$ is complete.

## Proof

This will be proved by demonstrating the contrapositive:

$S$ is not complete if and only if $S$ is not closed.

### Necessary Condition

Suppose that $S$ is not complete.

Then there exists a Cauchy sequence $\sequence {x_n}$ in $S$ such that the limit $\ds x = \lim_{n \mathop \to \infty} x_n$, which exists in the complete metric space $M$, is not a member of $S$.

For all $\epsilon > 0$, there exists an $N \in \N$ such that for all $n \ge N$:

$\map d {x, x_n} < \epsilon$

Hence $M \setminus S$ is not open.

Therefore, $S$ is not closed.

$\Box$

### Sufficient Condition

Suppose that $S$ is not closed.

Then $M \setminus S$ is not open.

Therefore, there exists a $x \in M \setminus S$ such that for all $\epsilon > 0$, there exists a $y \in S$ such that $\map d {x, y} < \epsilon$.

So there exists a sequence $\sequence {y_n}$ in $S$ such that for all $n \in \N$:

$\map d {x, y_n} < \dfrac 1 n$

Now, we show that $\sequence {y_n}$ is a Cauchy sequence.

Let $N \in \N$ be such that for all $n \ge N$:

$\map d {x, y_n} < \dfrac \epsilon 2$

Let $m, n \ge N$.

Then, by the triangle inequality:

$\map d {y_m, y_n} \le \map d {x, y_m} + \map d {x, y_n} < \epsilon$

Hence $\sequence {y_n}$ is a Cauchy sequence.

Because $\struct {M, d}$ is a complete metric space by assumption, the limit $\ds \lim_{n \mathop \to \infty} y_n$ exists and is in $M$.

Denote this limit by $y$.

By the definition of $\sequence {y_n}$:

$\ds \lim_{n \mathop \to \infty} \map d {x, y_n} = 0$
$\map d {x, y} = 0$

By definition of a metric, this implies that $x = y$.

Since $y \notin S$, $S$ is not complete.

$\blacksquare$


text:
# Another edge partitioning problem on cubic graphs

This question was motivated by a closely related problem An edge partitioning problem on cubic graphs

Input: at most cubic graph ( maximum node degree is 3) $G=(V,E)$, a natural number $k$

Question: Is there a partition of $E$ into subgraphs isomorphic to $K_{1,2}$ and $K_{1,1}$ such that the sum of the orders of the corresponding subgraphs is exactly $k$ ?

Is this problem polynomial time solvable or is it $NP$-complete?

Edit: Changed input graph to at most cubic graphs to avoid triviality.

• Replacing K_1,2 by two K_1,1s increases the total order by one, so the problem is just finding the maximum, which you can do by Edmond's algorithm on the line graph. – Colin McQuillan Oct 28 '10 at 17:39
• I'm looking for a partition with exact sum of orders? I do not see how finding the maximum matching in the line graph would help. Notice that for a given $k$, the edge partition is not necessarily a maximum matching? – Mohammad Al-Turkistany Oct 28 '10 at 18:22
• Could you describe what you are looking for, in the line graph? A partition of the vertices such that... – Colin McQuillan Oct 28 '10 at 18:39
• @Colin, The problem is not restricted to line graphs. I'm looking for an Edge partition with sum of orders exactly equal to $k$. – Mohammad Al-Turkistany Oct 28 '10 at 18:54

Perhaps I don't understand the question, but the way I interpret it, Colin's answer seems correct. It is enough to find the minimum possible k for which such partition exists, say k_0. (This is a partition with maximum number of K_21's.) There is a partition for any k between k_0 and 3n/2, because splitting K_21 into two K_11's increases the sum of cardinalities by 1. The K_21's in your partition induce a matching in the line graph of G, so this k_0 corresponds to the maximum matching in the line graph, and this matching can be found in polynomial time.

• Hmm. What am I missing? This algorithm does not use regularity of G. Finding max matching in the line graph G will work for any graph G, as far as I can tell. – Marek Chrobak Oct 29 '10 at 19:22


text:
# Derivát e ^ x-1

Derivative of (x+1)/(x-1). Simple step by step solution, to learn. Simple, and easy to understand, so dont hesitate to use it as a solution of your homework. Below you can find the full step by step solution for you problem. We hope it will be very helpful for you and it will help you to understand the solving process.

\left[e^x\right]' \ = \ \lim_{. Per un ripasso sulle derivate - click! Risposta di   The product of ex and eX is exCX. Connection we need : The derivative of ex is ex. Calculus is about pairs of functions. Function 1 (the distance we travel or the

Type in any function derivative to get the solution, steps and graph Jul 10, 2007 An online derivative calculator that differentiates a given function with respect to a given variable by using analytical differentiation. A useful mathematical differentiation calculator to simplify the functions. From the table below, you can notice that sech is not supported, but you can still enter it using the identity sech(x)=1/cosh(x)`. If you get an error, double-check your expression, add parentheses and multiplication signs where needed, and consult the table below. All suggestions and improvements are welcome. Please leave them in comments. Derivative of (x+1)/(x-1).

## 1. Demonstrate how the moments of a random variable x may be obtained from its moment generating function by showing that the rth derivative of. E(ext) with

The derivative of the logarithmic function is given by: f ' (x) = 1 / (x ln(b) ) x is the function argument. Sigmoid function is defined as $$\\frac{1}{1+e^{-x}}$$ I tried to calculate the derivative and got $$\\frac{e^{-x}}{(e^{-x}+1)^2}$$ Wolfram|Alpha however give me the same function but with exponents Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … Teorema fundamentală a calculului integral specifică relația dintre cele două operații de bază ale calculului integral, derivarea și integrarea..

### Sep 21, 2017

We can now apply that to calculate the derivative of other functions involving the exponential. Example 1: f An online derivative calculator that differentiates a given function with respect to a given variable by using analytical differentiation. A useful mathematical differentiation calculator to simplify the functions. Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals. For math, science, nutrition, history Derivative of (x+1)/(x-1). Simple step by step solution, to learn.

Mar 30, 2020 Example 17 Calculate the derivative of the function $y = \left( {x + 1} \right)\cos x + \left( {x + 2} \right)\sin x$ at $$x = 0.$$ Derivat ive E x a mp le s of C o nt inu it y Comparing Limits and Continuity At x = −1, the function has the value f(−1) = 1 The function is not continuous nor does a limit exist at this point At x = 0, the function is not defined There is a vertical asymptote At x = 1, the function has the value f(1) = 4 (5pts) Suppose f(x) is a function whose derivative is f'(x) = (x + 1)e". If f(1) = e, find an equation of the tangent line to the curve y = f(x) at the point (1,e).

For every x value in this graph, the function is changing at a rate that is proportional to 2x. General Rules For Computing Derivatives Free secondorder derivative calculator - second order differentiation solver step-by-step f x x x e Am aplicat proprietatile 1 si 2, adica se deriveaza fiecare separate,fiind adunare si scadere. Iar numarul din fata functiei nu pateste nimic, se deriveaza doar functia . 11c 0, pt ca 11 e constanta, fiindca n-are X. c c x x x 1 2 ln 2 ln 2 Proving this formula by induction, https://youtu.be/gpjcCPEN-tc🔑 If you enjoy my videos, then you can click here to subscribe https://www.youtube.com/blackp How to differentiate the natural exponential function using chain rule.

K-Meleon is free (open source) software released under … Apr 27, 2007 · if tha is just x-1 then the derrivative is the same because the derivative of that inside function is just 1. if you mean an inverse x then the answer would be - (e^x^-.1)/x^2 Jul 11, 2007 · b) The derivative of e^x is e^x itself, which means the slope of the graph e^x i at any point x is the value of e^x at x The proof follows Picard–LindelÃ¶f theorem which can be seen at the link While graphing, singularities (e. g. poles) are detected and treated specially. The gesture control is implemented using Hammer.js. If you have any questions or ideas for improvements to the Derivative Calculator, don't hesitate to write me an e-mail. Find the Derivative - d/dx e^(1/x) Differentiate using the chain rule, which states that is where and .

Free math lessons and math homework help from basic math to algebra, geometry and beyond. Students, teachers, parents, and everyone can find solutions to their math problems instantly. Mar 30, 2020 Example 17 Calculate the derivative of the function $y = \left( {x + 1} \right)\cos x + \left( {x + 2} \right)\sin x$ at $$x = 0.$$ Derivat ive E x a mp le s of C o nt inu it y Comparing Limits and Continuity At x = −1, the function has the value f(−1) = 1 The function is not continuous nor does a limit exist at this point At x = 0, the function is not defined There is a vertical asymptote At x = 1, the function has the value f(1) = 4 (5pts) Suppose f(x) is a function whose derivative is f'(x) = (x + 1)e". If f(1) = e, find an equation of the tangent line to the curve y = f(x) at the point (1,e). Write your answer in slope intercept form.

+···+ x8. 8!

banka new yorku 1 swift kód ulice
aké sú požiadavky na americký pas
3 000 gbp na usd
1,5 milióna dolárov v indických rupiách
londýn dobrá dodávka zlatá tehlička
história miery inflácie v usa bls

### Derivative Calculator This simple and convenient derivative calculator will help you solve any problem, just enter the value of the function and you will immediately get a solution with a detailed step-by-step description.

We only needed it here to prove the result above. We can now apply that to calculate the derivative of other functions involving the exponential. Example 1: f An online derivative calculator that differentiates a given function with respect to a given variable by using analytical differentiation. A useful mathematical differentiation calculator to simplify the functions. Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals.


text:
Find all School-related info fast with the new School-Specific MBA Forum

 It is currently 26 Jun 2016, 21:28

### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

# Events & Promotions

###### Events & Promotions in June
Open Detailed Calendar

# A manufacturing company employs staff.work full or part time

Author Message
TAGS:

### Hide Tags

Intern
Joined: 27 Sep 2010
Posts: 27
Followers: 1

Kudos [?]: 50 [1] , given: 3

A manufacturing company employs staff.work full or part time [#permalink]

### Show Tags

23 Jan 2011, 02:57
1
KUDOS
00:00

Difficulty:

(N/A)

Question Stats:

100% (02:45) correct 0% (00:00) wrong based on 2 sessions

### HideShow timer Statistics

Q) A manufacturing company employs staff who work either on full time basis or on part time basis. The ratio of number of males who work on part time basis to the number of males who work on full time basis is 4 to 5.And the ratio of number of females who work on part time basis to the number of males who work on part time basis is 3 to 2.If the ratio of the number of females who work on full time basis to the number of males who work on full time basis is 3:1, what is the ratio of number of females who work on part time basis to the number of females who work on full time basis ?

Math Expert
Joined: 02 Sep 2009
Posts: 33510
Followers: 5933

Kudos [?]: 73554 [0], given: 9902

Re: A manufacturing company employs staff.work full or part time [#permalink]

### Show Tags

23 Jan 2011, 04:19
Expert's post
MichelleSavina wrote:
Q) A manufacturing company employs staff who work either on full time basis or on part time basis. The ratio of number of males who work on part time basis to the number of males who work on full time basis is 4 to 5. And the ratio of number of females who work on part time basis to the number of males who work on part time basis is 3 to 2. If the ratio of the number of females who work on full time basis to the number of males who work on full time basis is 3:1, what is the ratio of number of females who work on part time basis to the number of females who work on full time basis ?

The ratio of number of males who work on part time basis to the number of males who work on full time basis is 4 to 5: $$\frac{male_p}{male_f}=\frac{4}{5}$$;

The ratio of number of females who work on part time basis to the number of males who work on part time basis is 3 to 2: $$\frac{female_p}{male_p}=\frac{3}{2}=\frac{6}{4}$$, so from above: $$\frac{female_p}{male_f}=\frac{6}{5}$$ --> $$male_f=\frac{5*female_p}{6}$$;

The ratio of the number of females who work on full time basis to the number of males who work on full time basis is 3:1: $$\frac{female_f}{male_f}=\frac{3}{1}$$ --> $$\frac{female_f}{\frac{5*female_p}{6}}=\frac{3}{1}$$ --> $$\frac{female_p}{female_f}=\frac{2}{5}$$.
_________________
Intern
Joined: 13 Oct 2010
Posts: 21
Location: Milpitas, CA
Followers: 2

Kudos [?]: 2 [0], given: 0

Re: A manufacturing company employs staff.work full or part time [#permalink]

### Show Tags

23 Jan 2011, 04:38
MichelleSavina wrote:
A manufacturing company employs staff who work either on full time basis or on part time basis. The ratio of number of males who work on part time basis to the number of males who work on full time basis is 4 to 5.And the ratio of number of females who work on part time basis to the number of males who work on part time basis is 3 to 2.If the ratio of the number of females who work on full time basis to the number of males who work on full time basis is 3:1, what is the ratio of number of females who work on part time basis to the number of females who work on full time basis ?

Say,
Number of part-time male = m
Number of full-time male = M
Number of part-time female = f
Number of full-time female = F

Hence,
m:M = 4:5 => m = (4M/5) => M = (5m/4)
f:m = 3:2 => f = (3m/2)
F:M = 3:1 => F = 3M = (15m/4)

Therefore, $$\frac{f}{F} = (\frac{3m}{2})/(\frac{15m}{4}) = \frac{3}{2}*\frac{4}{15} = \frac{2}{5}$$
_________________

Anurag Mairal, Ph.D., MBA
GMAT Expert, Admissions and Career Guidance
Gurome, Inc.
1-800-566-4043 (USA)
+91-99201 32411 (India)

Manager
Joined: 24 May 2013
Posts: 86
Followers: 0

Kudos [?]: 13 [0], given: 99

Re: A manufacturing company employs staff.work full or part time [#permalink]

### Show Tags

20 Mar 2016, 07:39
A manufacturing company employs staff who work either on full time basis or on part time basis. The ratio of number of males who work on part time basis to the number of males who work on full time basis is 4 to 5.And the ratio of number of females who work on part time basis to the number of males who work on part time basis is 3 to 2.If the ratio of the number of females who work on full time basis to the number of males who work on full time basis is 3:1, what is the ratio of number of females who work on part time basis to the number of females who work on full time basis ?

Mp =Male part time, Fp= female part time, Mf= male full time, Ff= female full time
Given,
Mp/Mf=4/5---------------(1)
Fp/Mp=3/2-----------------(2)
Ff/Mf=3/1-----------------(3)
Fp/Ff=? = (2)*(1)/(3) = 2:5
Re: A manufacturing company employs staff.work full or part time   [#permalink] 20 Mar 2016, 07:39
Similar topics Replies Last post
Similar
Topics:
1 Of the 150 employees at company X, 80 are full-time, and 100 have work 3 27 Dec 2015, 09:54
15 An automated manufacturing unit employs N experts such that the range 13 07 Apr 2015, 06:34
3 In the Excel Manufacturing Company, 46 percent of the employ 5 08 Oct 2013, 07:46
17 In a company with 48 employees, some part-time and some full 10 12 May 2012, 06:26
1 A company has 540 employees, 40% of whom are employed part time. If it 8 19 Oct 2010, 08:06
Display posts from previous: Sort by


text:
## Filters

Sort by :
Q

The total angle turned through during this interval is

3rd option is correct

$\omega _{0}=0$[initial]

$\omega$ = 24 rad/s[final]

t=8sec

$\alpha = \frac{\omega }{t} = \frac{24}{8} = 3$[angular acceleration]

$\theta = \frac{1}{2} \alpha t^{2} = \frac{1}{2}\times 3 \times 8^{2} = 96$ is the answer..!

View More

water column of height  7 cm. If refractive index of water is

1. 1.269

2. 1.286

3. 1.259

4. 1.310

It is given that wave number in 6cm of glass is the same as the wave number in water with height 7 cm. Let  is wavelength of light in glass and   is wavelength of light in water. According to question,  Also we know that relation between refractive index and wavelength,  Putting values,  Option (2) is correct

The true power consumed in a.c. circuit is

1. 400 W

2. 600 W

3. 220 W

4. 800 W

Given, R = 18 ohm Z = 33 ohm Vrms = 220 V For calculating Power,  option (4) is correct

3. In which photo current is independent of intensity of incident radiation

4. Which may be operated in forward or reverse bias

Option (1) is correct Photo Diode is operated only in reverse bias

already i register and paid fee but i am unable to locate course

Hello, Go to www.entrance360.com then then login to your account. When you logged in to your account there is a dashboard option, click on it you'll find your dashboard of MHT-CET product option. When you click on it, your dashboard will open.


text:
# LHCb pins down the Bs mixing phase

23 September 2011

The LHCb collaboration’s presentation at Lepton Photon 2011 included one of the most eagerly awaited measurements in flavour physics: the CP violation phase in BsB mixing. This is the counterpart of sin 2β in the B0 system, which was measured by the B-factory experiments BaBar and BELLE using the channel B0→ J/Ψ Ks. They provided the first measurement of CP violation in B0 mixing, which is both large and now well measured, with sin 2β = 0.69 ± 0.02. In contrast, the Standard Model prediction for φs, the corresponding phase for the Bs meson, is extremely small and precise: φs = 0.036 ± 0.002 rad (Charles et al. 2005). It is therefore an interesting place to search for new physics beyond the Standard Model, which may enhance the value. Time-dependent analyses of Bs mesons were not accessible at the B-factories, so this remained a key measurement for hadronic machines, first at the Tevatron and now at the LHC.

The golden mode for this study is Bs → J/Ψ φs where the J/Ψ decays to μ+μ and the φ decays to K+K . The measurement is very challenging: the final state is not a pure CP eigenstate, so an angular analysis has to be made to separate the CP-even and CP-odd components. In addition, the fast BsBs oscillation necessitates precise vertex reconstruction, and tagging of the production state (whether it was a Bs or Bs ) is also important. The result for φs is correlated to another quantity in the fit, ΔΓs, the difference in width of the two Bs mass eigenstates. (It is the mass difference of these two states that determines the oscillation frequency.) ΔΓs can be positive or negative, but in the Standard Model is predicted to be 0.087 ± 0.021 ps–1 (Lenz and Nierste 2011). The uncertainties on φs and ΔΓs are correlated, and furthermore the fit turns out to be insensitive to the replacement φs → π – φs when ΔΓs → – ΔΓs so there are two ambiguous solutions. As a result, the measurements are usually plotted as contours in the φs vs ΔΓs plane.

The CDF and DØ experiments at the Tevatron made the first measurements. Their early results agreed with each other and appeared, when combined, to indicate a large value for φs, about 3σ away from the Standard Model expectation. More recent updates have moved their preferred values somewhat closer to the Standard Model, but a hint of a possible discrepancy remained, as shown by the red and green contours in figure 1 (Burdin and DØ 2011, CDF 2010).

LHCb has now accumulated the largest sample of Bs → J/Ψ φ decays in the world, over 8000 signal candidates with very high purity (figure 2). The resulting constraint is shown as the blue contour in figure 1 (LHCb 2011a). It is much more precise than the preceding measurements, with one of the two solutions being in good agreement with the Standard Model expectation – the hint of a discrepancy is not confirmed. This result also gives the first significant direct measurement of ΔΓs, 0.123 ± 0.029 ± 0.008 ps–1, where the first uncertainty is statistical and the second systematic.

Another related analysis presented by LHCb uses a different decay mode, Bs → J/Ψ f0, which should measure the same phase. Although the statistics are lower, the final state is CP-odd in this case, so the analysis is simpler (LHCb 2011b). It gives a consistent result to B0s → J/Ψ φ, and the preliminary combined result from LHCb is φs = 0.03 ± 0.16 ± 0.07 rad (LHCb 2011c). This result is statistically limited, but as data continue to pour in from the LHC there are good prospects for substantial further improvement. So, although LHCb has now ruled out a gross effect from new physics, the experiment should be able to measure the true value even if it is as small as predicted in the Standard Model – and test any subtle effects from new physics.

S Burdin, DØ collaboration 2011 EPS-HEP 2011, Grenoble.
CDF collaboration 2010 CDF/ANAL/BOTTOM/PUBLIC/10206.
J Charles et al. (CKMfitter group) 2005 Eur. Phys. J. C 41 1.
A Lenz and U Nierste 2011 arXiv: 1102.4274v1.
LHCb collaboration 2011a LHCb-CONF-2011-049.
LHCb collaboration 2011b LHCb-CONF-2011-051.
LHCb collaboration 2011c LHCb-CONF-2011-056.