All-optical nonequilibrium pathway to stabilizing magnetic Weyl semimetals in pyrochlore iridates
Gabriel E. Topp, Nicolas Tancogne-Dejean, Alexander F. Kemper, Angel Rubio, Michael A. Sentef
AAll-optical nonequilibrium pathway to stabilizing magnetic Weylsemimetals in pyrochlore iridates
Gabriel E. Topp, Nicolas Tancogne-Dejean, AlexanderF. Kemper, Angel Rubio,
1, 3 and Michael A. Sentef ∗ Max Planck Institute for the Structure and Dynamics of Matter,Center for Free Electron Laser Science, 22761 Hamburg, Germany Department of Physics, North Carolina State University, Raleigh, NC, USA Center for Computational Quantum Physics (CCQ),Flatiron Institute, 162 Fifth Avenue, New York NY 10010 (Dated: September 5, 2018)
Abstract
Nonequilibrium many-body dynamics is becoming one of the central topics of modern condensedmatter physics. Floquet topological states were suggested to emerge in photodressed band struc-tures in the presence of periodic laser driving. Here we propose a viable nonequilibrium routewithout requiring coherent Floquet states to reach the elusive magnetic Weyl semimetallic phasein pyrochlore iridates by ultrafast modification of the effective electron-electron interaction withshort laser pulses. Combining ab initio calculations for a time-dependent self-consistent reducedHubbard U controlled by laser intensity and nonequilibrium magnetism simulations for quantumquenches, we find dynamically modified magnetic order giving rise to transiently emerging Weylcones that are probed by time- and angle-resolved photoemission spectroscopy. Our work offers aunique and realistic pathway for nonequilibrium materials engineering beyond Floquet physics tocreate and sustain Weyl semimetals. This may lead to ultrafast, tens-of-femtoseconds switchingprotocols for light-engineered Berry curvature in combination with ultrafast magnetism. a r X i v : . [ c ond - m a t . s t r- e l ] S e p ltrafast science offers the prospect of an all-optical design and femtosecond switching ofmagnetic and topological properties in quantum materials . Tailored laser excitations offerthe prospect of manipulating important couplings in solids, such as the magnetic exchangeinteraction or electron-phonon coupling . As a consequence light-triggered transitionsbetween ordered and non-ordered collective phases can be observed, with dramatic changesof the electronic response on ultrafast time scales. Prominent examples are the switching toa hidden phase involving charge-density wave order , possible light-induced superconduc-tivity in an organic material by phonon excitation , nonthermal melting of orbital orderin a manganite by coherent vibrational excitation , or light-induced spin-density-wave or-der in an optically stimulated pnictide material . Obviously, mutual electronic and latticecorrelations are key to understanding the physics of such phase transitions both in and outof equilibrium. By contrast, many known topological materials are well understood by theBerry curvature of Bloch electrons in an effective single-quasiparticle picture. Similarly, thenonequilibrium extension to periodically driven “Floquet states” of matter is in itssimplest form well captured by driven noninteracting models . From the point of view ofpulsed laser engineering of novel states of matter, the drawback of Floquet states in weaklyinteracting systems is that any fundamental change induced by light in the material’s bandstructure rapidly vanishes when the laser pulse is off. In that case, nonthermal distribu-tions of quasiparticles in Floquet states do not critically affect the transient band structure,though they are relevant for response functions, for instance for describing the Floquet Halleffect . This is distinctly different from the case of nonequilibrium ordered phases withan order parameter that crucially impacts the band structure itself. The slow and oftennonthermal dynamics of driven correlated ordered phases , in particular in proximity tophase transitions, present the opportunity to affect band structures on time scales that arelonger than the duration of the external perturbation. For instance, the critical slowingdown of collective oscillations and relaxation dynamics is one of the hallmark signatures ofnonthermal criticality in interacting many-body systems out of equilibrium. As anotherrecent example of ultrafast materials science in interacting systems, Floquet engineering ofstrongly correlated magnets into chiral spin states has been proposed.As a prototypical class of materials involving a striking combination of topology andmagnetism, the 227 pyrochlore iridates were theoretically suggested as possible hosts ofa variety of equilibrium phases including antiferromagnetic insulating (AFI) as well as2eyl semimetallic (WSM) states with noncollinear all-in/all-out (AIAO) spin configurations.This topologically non-trivial phase is characterized by the appearance of pairs of linearlydispersing Weyl cones of opposite chirality. A phase transition between both phases, con-trolled by the size of the ordered magnetic moment and thus depending on Hubbard U , wasconjectured to exist. However, convincing experimental evidence for the existence of theWSM phase in pyrochlore iridates has not been presented yet, with the majority of exper-iments pointing to AFI groundstates (cf. discussions in Ref. 28–30). Here we propose toinduce the elusive transition to the WSM phase by laser excitation. We use a combinationof ab initio simulations of light-reduced Hubbard U induced by an ultrashort laser pulse and time-dependent magnetization dynamics after a U quench. We restrict our study to amodel relevant for compounds with a non-magnetic R-site (R = Lu, Y, Eu), where the 5diridium electrons determine the magnetic order .Figure 1a illustrates the key idea behind our work by presenting the low-temperatureequilibrium phase diagram of a prototypical pyrochlore iridate model Hamiltonian , H = (cid:88) k (cid:88) a,b c † a ( k ) (cid:2) H NN ab ( k ) + H NNN ab ( k ) (cid:3) c b ( k ) + H U . (1)The indices 1 ≤ a, b, c ≤ H NN ab and H NNN ab describe thebare nearest-neighbour (NN) and next-nearest-neighbour (NNN) hopping plus spin-orbitcoupling, respectively. The local Hubbard repulsion H U = U (cid:80) R i n R i ↑ n R i ↓ is taken as theHartree-Fock mean-field, H U → − U (cid:88) k a (2 (cid:104) j a (cid:105) · j a ( k ) − (cid:104) j a (cid:105) ) , (2)where j a ( k ) = (cid:80) αβ = ↑ , ↓ c † aα ( k ) σ αβ c aβ ( k ) / (cid:126) ≡ m ≡ (cid:80) a |(cid:104) j a (cid:105)| / σ orbitals, t σ , and the local Coulomb repulsion, Hubbard U . In Figure 1b we show theordered magnetic moment, m , corresponding to the length of the magnetization vector inthe AIAO configuration as a function of U for our choice of t σ = − .
62 eV in equilibrium.Our choice of parameters is motivated by comparing with the size of the band gap from our3 b FIG. 1.
Magnetic WSM order in equilibrium a , Phase diagram as a function of Hubbard U and hopping integral, t σ , at T = 0 .
016 eV (186 K). The red dashed line indicates the chosenparameter subspace for our calculations at t σ = − .
62 eV. For small interactions, U ≤ .
22 eV,the system exhibits a paramagnetic metallic phase (PMM). In the intermediate regime, 1 .
22 eV < U ≤ .
25 eV, TRS is broken by spontaneous magnetic all-in/all-out (AIAO) order. A furtherincrease of U above 1 .
25 eV leads to a gapped antiferromagnetic insulating phase (AFI) with largermagnetization. For | t σ | ≥ .
62 eV the WSM-AFI transition is of 2nd order, indicated by blackdots. Below that value, the transition is of 1st order (bicoloured dots). The dots’ increasing sizecorresponds to the increasing region of hysteresis. The red square indicates the chosen initial state( U i = 1 .
28 eV) in parameter space. The red arrow shows the intended quenching direction. b ,Magnetization as a function of U for the choice of t σ = − .
62 eV. The dashed lines indicate theregion of hysteresis. The coloured arrows show the final values of U = U q right after the interactionquench for the non-equilibrium calculations (see FIG. 2). density functional theory calculation to fix the energy units in the model, and we fix theratio t σ /t oxy = − . m near thetransition. By contrast, the PMM-WSM transition at smaller U is continuous. In the PMMphase, m vanishes.In the following we will make a case for nonequilibrium pathways to induce the elusiveWSM state. Consistent with most of the experimental evidence, we choose an equilibrium4nitial state inside the AFI region (red square in Figure 1), corresponding to U = 1 .
28 eV.This choice is not meant to be representative of a single specific compound but rather togenerically represent to whole class of AFI pyrochlore iridates. Apparently, by controlling thestrength of U one might engineer a transition from the AFI to the WSM state. One way toeffectively control U in a correlated insulator was recently theoretically proposed , namelyby a short laser excitation, employing ab initio time-dependent density functional theoryplus dynamical U (TDDFT+U) formalism . Motivated by these results, we investigate herethe influence of a short laser pulse on the effective interaction in insulating 227 iridates. Weuse TDDFT+U to show that a reduction of U and of the magnetic order parameter m can beinduced in the pyrochlore iridates. We then take these first principles calculations to build aminimal intuitive model understanding of the nonequilibrium pathway to the light-inducedWSM state by instantaneous quenches of U . This in-depth model investigation allows us toreveal the highly nonthermal character of the nonequilibrium ordered state, and to prove itsWSM signatures by time-resolved photoemission spectroscopy calculations.In Figure 2a we show the time evolution of U eff from TDDFT+U for Y Ir O drivenby an ultrashort laser pulse, linearly polarized along the [001] direction, whose envelopeis schematically indicated by the grey shaded area. Apparently, the effective Hubbard pa-rameter is dynamically lowered by the light-matter interaction. This effect is explained bya dynamical enhancement of the electronic screening due to the delocalized nature of thepump-induced excited states . The decrease of U eff follows the squared-sinusoidal pumpenvelope. After 20 fs U eff saturates at a finite value. The relative change of the effectiveinteraction is controlled by the laser intensity. A higher intensity means a bigger and fasterdrop of the interaction. For the highest assumed realistic intensity in matter, 0 . · Wcm − , U eff changes by 40%.In a next step we investigate the real-time dynamics of the AIAO magnetic order (seeSupplementary Fig. S4) in TDDFT+U (Figure 2b). First of all, the TDDFT groundstatecalculation yields an AIAO magnetization, m = 0 . µ B , consistent with the model beforelaser excitation. Under the pump excitation, the magnetization m ( t ) is reduced from itsequilibrium value. The reduced m ( t ) saturates as a function of pump intensity more quicklythan the reduction of U eff saturates as a function of the intensity, indicating that somenonthermal effects are at play here already. We will get back to a more detailed discussionof nonthermality in the context of the model calculations below.5 b c d FIG. 2.
Optically induced nonthermal Weyl semimetal. a , Self-consistent TDDFT+Ucalculation of U eff ( t ) (Ir 5d) for Y Ir O under pump excitation (0 .
41 eV frequency, 25 . U eff decreases, initially following the pulse envelope. After 20 fs it reaches a plateau,which decreases for higher intensities. Relative changes of U eff are 1 .
2% (blue), 5 .
3% (green), 9 . .
4% (cyan). b , Time-dependent AIAO magnetization from TDDFT+U. For the small-est intensity (blue) the magnetization drops by 25%. For all higher pump intensities, there is amassive decrease of the magnetization by 88% (red) and 92% (green, brown), respectively. c , Time-dependent AIAO magnetization m ( t ) (model) for bath coupling Γ = 0 .
008 eV after instantaneousquenches at t = 0 from U i = 1 .
28 eV to U q = 1 .
24 eV, 1 .
20 eV and 1 .
16 eV, respectively (dashedcurves are without bath, Γ = 0). Thermal m values for U q are indicated by coloured arrowsfor reference. After a fast drop followed by an over-damped oscillation, the magnetization slowlyconverges towards the respective thermal value. d , Equilibrium phase diagram as a function of T and U with optically induced nonthermal states. Effective temperatures T eff (see SupplementaryFig. S3) of the nonequilibrium states at t = 123 . U q as indicated (see Supplementary Fig. S2). Pentagons (squares)indicate nonequilibrium WSM states for the open system (closed system). (see Figure 3). Theblack diamond indicates a nonequilibrium AFI state.
6n the following, we employ the dynamical reduction of U eff in TDDFT+U shown inFigure 2a as input for model calculations with dynamical U within the time-dependent self-consistent Hartree-Fock mean-field approximation. We include a dissipative coupling to aheat bath to mimick the openness of the electronic subsystem in the real material. To thisend we couple the electrons to a Markovian fermionic heat bath giving rise to a Lindblad term(see Methods). The bath is kept at fixed temperature T = 0 .
016 eV (186 K), and we choosea system-bath coupling strength Γ = 0 .
008 eV, corresponding to a characteristic time scaleΓ − ≈
80 fs. Note that the microscopic details of the system-bath coupling are unimportantfor the further discussion. The bath serves two main purposes here: (i) to thermalize theelectronic subsystem as a whole to the bath temperature, and (ii) to thermalize the electronsamong each other through the bath.The simplest and most extreme scenario to investigate is an instantaneous change of U from its initial equilibrium value to quenched values U q , which is a worst-case scenario forlight-controlled phase transitions as it produces the strongest heating effects, as will be dis-cussed below. In Figure 2c we show the resulting time evolutions after U quenches from theinitial value to three different final values, corresponding to different laser pump intensities.We observe very fast changes of the magnetization and a subsequent nonequilibrium statewith reduced but nonzero magnetization that slowly decays to its thermal value. In thefollowing we focus on the characterization of this nonthermal state at a fixed observationtime t = 123 . not indicate thermalization but is rather used as a meansof talking about nonthermality in the following. The extracted values are placed in theequilibrium temperature-versus- U phase diagram in Figure 2d. While the U q = 1 .
24 eVcase still lies within the thermal magnetic phase region, this is clearly not the case for thestronger pulses reducing U to U q = 1 .
20 eV and U q = 1 .
16 eV, respectively. For thesequenched U values, the thermal state has a vanishing magnetic order parameter in thermalequilibrium even at zero temperature, while the dynamically induced states show nonzeromagnetic order even at finite effective temperatures. Without bath coupling, Γ = 0, the7ffective nonequilibrium temperature increases quite rapidly for smaller quench values, U q ,as the increasing amount of additional energy pumped into the system is conserved. For theopen system, the increase of temperature is much slower due to dissipation induced by thelow-temperature heat bath. We take this as an indication that the observed nonequilibriumreduced but nonzero magnetic order is indeed nonthermal in character. This nonthermalitymakes Weyl states appear transiently in a much larger region of effective temperatures and U values than they would in thermal equilibrium.Importantly, such nonthermal order above the equilibrium critical temperature was ob-served for U quenches in the antiferromagnetic phase of the single-band Hubbard modelboth in static and dynamical mean-field calculations . The basic explanation for why thesystem does not thermalize rapidly is as follows: Integrability can block the system fromthermalizing due to constants of motion that hinder relaxation to thermal values for someobservables, such as the magnetization. While our mean-field model in the absence of theheat bath falls into the integrable class, thermalization is slowed down considerably even innonintegrable systems close to a nonthermal critical point, at which the thermalization timediverges. This is the case here for the second order phase transition between the magneticWSM and nonmagnetic metallic states. Therefore nonthermality on time scales that exceedthe time scale of the laser perturbation is not an artefact of the mean-field approximationemployed in the present work, but is expected to survive when true correlation effects areincluded.In order to reveal the WSM character of the pump-induced magnetically ordered statein the open system, we compute its time- and angle-resolved photoemission spectroscopy(tr-ARPES) signal . All information of the time-dependent electronic band structure isencoded in the double-time lesser Green’s function, which in our case is calculated afterthe actual time propagation in a post-processing step by unitary double-time propagationof the initial density matrix (see Methods). The double-time lesser Green’s function is theelectronic propagator of a particle removed from the system at a point in time, and addedback at a different time. Its Fourier transform from relative times to frequency gives thesingle-particle removal spectrum in equilibrium, of which the tr-ARPES signal is the time-dependent generalization for the driven case. Thus the tr-ARPES photocurrent essentiallymonitors the occupied parts of the band structure.The momentum- and frequency-dependent photocurrent at time t p has an energy resolu-8 b c d FIG. 3.
Time-resolved ARPES detection of Weyl fermions. a , Calculated photocurrentfrom tr-ARPES of the equilibrium band structure along high-symmetry path for an initial AFIstate ( U i = 1 .
28 eV, t σ = − .
62 eV) at T = 0 .
016 eV. The conduction bands are separated fromthe valence bands by an energy gap E G ≈ .
15 eV (AFI) at the L point. The yellow squarehighlights the region of interest. b - d , tr-ARPES band structure along the high-symmetry lineL − Γ at probe time t p = 123 . U q = 1 .
24 eV, 1 .
20 eV and1 .
16 eV, correspond to increasing ∆ U between initial and quenches U and thus stronger drivingfields. The black solid lines show the instantaneous band structures of the Hamiltonian at t = t p .The coloured arrows indicate the dynamically generated Weyl points. The Weyl cones are shiftedalong the high symmetry line towards Γ-point with increasing ∆ U . tion given by the inverse of the probe duration. We use a probe duration σ p = 20 . U i = 1 .
28 eV)along a high-symmetry path in the first Brillouin zone. The AFI phase is clearly identifiedby the separation of the occupied valence bands from the empty conduction bands by anenergy gap, E G ≈ .
15 eV, at the L point. The chemical potential µ = 0 lies within thatgap.In Figure 3b-d we display computed time resolved band structure of the nonequilibriumsteady-state at time t p = 32 . − L towards Γ. a bc d ef g h c d e gf h
FIG. 4.
Fate of nonthermal WSM for relaxing U ( t ) . a , Time-dependent interaction relaxingback to the initial value, U (0 ≤ t ≤ U i − ( U i − U q ) sin ( π · t/ b , Corresponding timeevolution of the magnetization. As in the instantaneous case, the magnetization is initially reducedbefore slowly relaxing back towards its initial value. The vertical solid lines indicate the probe times t p for which the tr-ARPES signals are shown in c - h , as indicated, with a probe width σ p = 20 . c - h , Tr-ARPES signals at different probe timesas indicated along L − Γ. The solid lines correspond to the effective band structures of the time-dependent Hamiltonian averaged over the FWHM of the probe pulse.
Finally we address the question of the minimum lifetime of the nonthermal WSM state.In reality, the light-induced reduction of U will not persist indefinitely due to couplingof the electronic system to the environment, for instance to lattice vibrations, which arenot included in our TDDFT+U simulations. Electron-phonon relaxation is always presentin materials and typically has associated time scales of hundreds of femtoseconds up to10icoseconds, usually somewhat but not much slower than typical pump pulse durationsof tens to hundreds of femtoseconds . Therefore we take here a worst-case scenario andassume that U is reduced only on a time scale that is comparable to the dissipative time scaleintroduced through the heat-bath coupling. We therefore investigate the dynamics of themagnetization and the nonequilibrium Weyl cone for a time-dependent U ( t ) shown in Figure4a. Here the minimum in U ( t ) agrees with the smallest value U q used for the instantaneousquenches (see Figure 2b). Figure 4b shows the corresponding time-dependent magnetization,which closely follows U ( t ), for the same heat bath as employed before. The magnetization m ( t ) reaches a smaller minimum than in the quenched case, which reflects the fact that themagnetization follows more closely the expected thermal value corresponding to U ( t ) when U ( t ) changes more slowly than in the quench case. At the end of the U modulation, themagnetization is not fully relaxed back to its initial thermal value but slowly decays backto it due to coupling to the heat bath.Figure 4c shows the calculated ARPES spectrum of the initital state before the U mod-ulation. At t p = 20 fs (Figure 4d), the reduced magnetization leads to a reduced bandgap compared to the initial state. At t p = 50 fs (Figure 4e) the system undergoes a rapidchange in magnetization, resulting in more diffuse bands. Around t p = 100 fs (Figure 4f) m ( t ) reaches a minimum and changes more slowly than at earlier times. The Weyl coneemerging between L and Γ shows that the system has reached a WSM state. The tr-ARPESbands nicely match the instantaneous bands. Afterwards, the relaxation of the system isreflected in the shift of the Weyl cone back towards Γ at t p = 150 fs (Figure 4g). Importantlythe WSM state still persists outside the FHWM of U ( t ) indicating that interaction-inducednonthermal states of matter can live longer than the duration of the pump laser pulse, incontrast to Floquet states in quasi-noninteracting systems. This longevity is directly relatedto the fact that the dynamics of m ( t ) are slower than the dynamics of U ( t ). Figure 4h showsthe tr-ARPES signal after U ( t ) has completely relaxed. The system is again found in theAFI phase with a still slightly reduced gap compared to the initial thermal state.In summary we propose a robust and efficient novel ultrafast route towards light-inducedtopology via dynamical modulation of Hubbard U for nonequilibrium materials engineeringas an alternative to Floquet engineering in quasi-noninteracting band structures. In our sim-ulations, the laser- or quench-induced excitation leads to a dynamical reduction of magneticorder inducing an ultrafast transition to the Weyl semimetallic phase in pyrochlore iridates.11he range of effective electronic temperatures in which nonzero reduced magnetization isobtained is larger than expected for quasi-thermal states, highlighting that nonthermal or-der might allow to avoid finetuning in the quest for the light-induced magnetic Weyl phase.Importantly, the nonthermal character of transient states on long time scales has been shownto be a generic feature of ordered phases in proximity to critical points due to prethermal-ization even beyond the static mean-field approximation . Crucially, the appearance ofnonthermal order is not a prerequisite for our proposal to work at all. The time scales fornonthermal order mainly set a window of lifetimes and pump fluences under which light-induced magnetic Weyl states become observable experimentally. In practice, these windowswill be restricted mainly by the relaxation dynamics of the dynamically modulated U andmagnetization due to coupling to phonons.Overall our results imply that ultrafast pathways are a promising route to reach theelusive topological Weyl semimetallic state in pyrochlore iridates. We suggest time- andangle-resolved photoemission spectroscopy as a means of probing the dynamically generatedWeyl cones. Specifically for chiral Weyl fermions, such emergent states could also be probedall-optically in a time-domain extension of the recently demonstrated static photocurrentresponse to circularly polarized light .In a broader context our combined study of ab initio TDDFT+U under explicit laserexcitation and model dynamics under interaction quenches furthermore bridges the so-farlargely disconnected fields of laser-driven phenomena on the one hand and thermaliza-tion after quenches on the other hand. Moreover the combination of light-controlledinteractions and interaction-controlled topology adds to the variety of control knobs in thetime domain to dynamically engineer topological phases of matter in solids on ultrafast timescales.
Acknowledgment.–
We thank C. Timm for suggesting the pyrochlore iridates as a candi-date class of materials to light-induce nontrivial topology beyond Floquet states. Discussionswith A. de la Torre and D. Kennes are gratefully acknowledged. G.E.T. and M.A.S. acknowl-edge financial support by the DFG through the Emmy Noether programme (SE 2558/2-1).A.R. and N.T.-D. acknowledge financial support by the European Research Council (ERC-2015-AdG-694097), and European Union’s H2020 program under GA no. 676580 (NOMAD).12 . BIBLIOGRAPHY ∗ [email protected] Kimel, A. V., Kirilyuk, A., Hansteen, F., Pisarev, R. V. & Rasing, T. Nonthermal optical controlof magnetism and ultrafast laser-induced spin dynamics in solids. J. Phys.: Condens. Matter , 043201 (2007). URL http://stacks.iop.org/0953-8984/19/i=4/a=043201 . Kirilyuk, A., Kimel, A. V. & Rasing, T. Ultrafast optical manipulation of magnetic or-der. Rev. Mod. Phys. , 2731–2784 (2010). URL https://link.aps.org/doi/10.1103/RevModPhys.82.2731 . Kampfrath, T. et al. Coherent terahertz control of antiferromagnetic spin waves. Nat Photon ,31–34 (2011). URL . Nova, T. F. et al. An effective magnetic field from optically driven phonons. Nat Phys , 132–136 (2017). URL . Basov, D. N., Averitt, R. D. & Hsieh, D. Towards properties on demand in quantum materials.Nature Materials , 1077 (2017). URL . Shin, D. et al. Phonon-driven spin-Floquet magneto-valleytronics in MoS 2.Nature Communications , 638 (2018). URL . Mentink, J. H., Balzer, K. & Eckstein, M. Ultrafast and reversible control of the exchangeinteraction in Mott insulators. Nature Communications , 6708 (2015). URL http://arxiv.org/abs/1407.4761 . ArXiv: 1407.4761. Pomarico, E. et al. Enhanced electron-phonon coupling in graphene with periodically distortedlattice. Phys. Rev. B , 024304 (2017). URL http://link.aps.org/doi/10.1103/PhysRevB.95.024304 . Kennes, D. M., Wilner, E. Y., Reichman, D. R. & Millis, A. J. Transient superconductivity fromelectronic squeezing of optically pumped phonons. Nat Phys , 479–483 (2017). URL . Sentef, M. A. Light-enhanced electron-phonon coupling from nonlinear electron-phonon cou-pling. Phys. Rev. B , 205111 (2017). URL https://link.aps.org/doi/10.1103/PhysRevB.95.205111 . Sentef, M. A., Ruggenthaler, M. & Rubio, A. Cavity quantum-electrodynamical polaritoni-cally enhanced superconductivity. arXiv:1802.09437 [cond-mat, physics:quant-ph] (2018). URL http://arxiv.org/abs/1802.09437 . ArXiv: 1802.09437. Stojchevska, L. et al. Ultrafast Switching to a Stable Hidden Quantum State in an ElectronicCrystal. Science , 177–180 (2014). URL http://science.sciencemag.org/content/344/6180/177 . Mitrano, M. et al. Possible light-induced superconductivity in K3c60 at high temperature.Nature , 461–464 (2016). URL . Tobey, R. I., Prabhakaran, D., Boothroyd, A. T. & Cavalleri, A. Ultrafast electronic phasetransition in la / sr / mno by coherent vibrational excitation: Evidence for nonthermal meltingof orbital order. Phys. Rev. Lett. , 197404– (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.197404 . Kim, K. W. et al. Ultrafast transient generation of spin-density-wave order in the normal stateof bafe2as2 driven by coherent lattice vibrations. Nature Materials , 497– (2012). URL http://dx.doi.org/10.1038/nmat3294 . Oka, T. & Aoki, H. Photovoltaic hall effect in graphene. Phys. Rev. B , 081406 (2009). URL https://link.aps.org/doi/10.1103/PhysRevB.79.081406 . Lindner, N. H., Refael, G. & Galitski, V. Floquet topological insulator in semiconductor quan-tum wells. Nature Physics , 490– (2011). URL http://dx.doi.org/10.1038/nphys1926 . Sentef, M. A. et al. Theory of Floquet band formation and local pseudospin textures in pump-probe photoemission of graphene. Nat Commun , 7047 (2015). URL . H¨ubener, H., Sentef, M. A., Giovannini, U. D., Kemper, A. F. & Rubio, A. Creating sta-ble Floquet–Weyl semimetals by laser-driving of 3d Dirac materials. Nature Communications , 13940 (2017). URL . Dehghani, H., Oka, T. & Mitra, A. Out-of-equilibrium electrons and the Hall conductance of aFloquet topological insulator. Phys. Rev. B , 155422 (2015). URL https://link.aps.org/doi/10.1103/PhysRevB.91.155422 . Sentef, M. A., Kemper, A. F., Georges, A. & Kollath, C. Theory of light-enhanced phonon- ediated superconductivity. Phys. Rev. B , 144506 (2016). URL http://link.aps.org/doi/10.1103/PhysRevB.93.144506 . Sentef, M. A., Tokuno, A., Georges, A. & Kollath, C. Theory of Laser-Controlled CompetingSuperconducting and Charge Orders. Phys. Rev. Lett. , 087002 (2017). URL http://link.aps.org/doi/10.1103/PhysRevLett.118.087002 . Tsuji, N., Eckstein, M. & Werner, P. Nonthermal antiferromagnetic order and nonequilibriumcriticality in the hubbard model. Phys. Rev. Lett. , 136404– (2013). URL https://link.aps.org/doi/10.1103/PhysRevLett.110.136404 . Claassen, M., Jiang, H.-C., Moritz, B. & Devereaux, T. P. Dynamical time-reversalsymmetry breaking and photo-induced chiral spin liquids in frustrated Mott insula-tors. Nature Communications , 1192 (2017). URL . Kitamura, S., Oka, T. & Aoki, H. Probing and controlling spin chirality in Mott insulators bycircularly polarized laser. Phys. Rev. B , 014406 (2017). URL https://link.aps.org/doi/10.1103/PhysRevB.96.014406 . Pesin, D. & Balents, L. Mott physics and band topology in materials with strong spin–orbitinteraction. Nature Physics , 376–381 (2010). URL . Wan, X., Turner, A. M., Vishwanath, A. & Savrasov, S. Y. Topological semimetal and fermi-arc surface states in the electronic structure of pyrochlore iridates. Phys. Rev. B , 205101–(2011). URL https://link.aps.org/doi/10.1103/PhysRevB.83.205101 . Nakayama, M. et al. Slater to Mott Crossover in the Metal to Insulator Transition of$ {\ mathrm { Nd }} { }{\ mathrm { Ir }} { }{\ mathrm { O }} { } $. Phys. Rev. Lett. , 056403(2016). URL https://link.aps.org/doi/10.1103/PhysRevLett.117.056403 . Wang, R., Go, A. & Millis, A. J. Electron interactions, spin-orbit coupling, and intersitecorrelations in pyrochlore iridates. Phys. Rev. B , 045133 (2017). URL https://link.aps.org/doi/10.1103/PhysRevB.95.045133 . Wang, Y., Weng, H., Fu, L. & Dai, X. Noncollinear Magnetic Structure and Multipolar Order inEu2ir2o7. Phys. Rev. Lett. , 187203 (2017). URL https://link.aps.org/doi/10.1103/PhysRevLett.119.187203 . Tancogne-Dejean, N., Sentef, M. A. & Rubio, A. Ultrafast Modification of Hubbard $U$ in a trongly Correlated Material: Ab initio High-Harmonic Generation in NiO. Phys. Rev. Lett. , 097402 (2018). URL https://link.aps.org/doi/10.1103/PhysRevLett.121.097402 . Witczak-Krempa, W., Chen, G., Kim, Y. B. & Balents, L. Correlated Quantum Phenomenain the Strong Spin-Orbit Regime. Annu. Rev. Condens. Matter Phys. , 57–82 (2014). URL . Witczak-Krempa, W., Go, A. & Kim, Y. B. Pyrochlore electrons under pressure, heat, andfield: Shedding light on the iridates. Phys. Rev. B , 155101– (2013). URL https://link.aps.org/doi/10.1103/PhysRevB.87.155101 . Tancogne-Dejean, N., Oliveira, M. J. T. & Rubio, A. Self-consistent DFT + u method for real-space time-dependent density functional theory calculations. Phys. Rev. B , 245133 (2017).URL https://link.aps.org/doi/10.1103/PhysRevB.96.245133 . Freericks, J. K., Krishnamurthy, H. R. & Pruschke, T. Theoretical description of time-resolvedphotoemission spectroscopy: Application to pump-probe experiments. Phys. Rev. Lett. ,136401 (2009). URL http://link.aps.org/doi/10.1103/PhysRevLett.102.136401 . Rameau, J. D. et al. Energy dissipation from a correlated system driven out of equilib-rium. Nature Communications , 13761 (2016). URL . Ma, Q. et al. Direct optical detection of Weyl fermion chirality in a topological semimetal.Nature Physics , nphys4146 (2017). URL . Zhang, J. & Averitt, R. Dynamics and Control in Complex Transition Metal Oxides.Annual Review of Materials Research , 19–43 (2014). URL http://dx.doi.org/10.1146/annurev-matsci-070813-113258 . Giannetti, C. et al. Ultrafast optical spectroscopy of strongly correlated materials and high-temperature superconductors: a non-equilibrium approach. Advances in Physics , 58–238(2016). URL . Calabrese, P. & Cardy, J. Time Dependence of Correlation Functions Following a QuantumQuench. Phys. Rev. Lett. , 136801 (2006). URL https://link.aps.org/doi/10.1103/PhysRevLett.96.136801 . Rigol, M., Dunjko, V. & Olshanii, M. Thermalization and its mechanism for generic isolatedquantum systems. Nature , 854–858 (2008). URL . Eckstein, M., Kollar, M. & Werner, P. Thermalization after an Interaction Quench in theHubbard Model. Phys. Rev. Lett. , 056403 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.103.056403 . Canovi, E., Rossini, D., Fazio, R., Santoro, G. E. & Silva, A. Quantum quenches, thermalization,and many-body localization. Phys. Rev. B , 094431 (2011). URL https://link.aps.org/doi/10.1103/PhysRevB.83.094431 . Yamaguchi, M., Yuge, T. & Ogawa, T. Markovian quantum master equation beyond adia-batic regime. Phys. Rev. E , 012136 (2017). URL https://link.aps.org/doi/10.1103/PhysRevE.95.012136 . Andrade, X. et al. Real-space grids and the octopus code as tools for the development ofnew simulation approaches for electronic systems. Phys. Chem. Chem. Phys. , 31371–31396(2015). URL http://dx.doi.org/10.1039/C5CP00351B . Agapito, L. A., Curtarolo, S. & Buongiorno Nardelli, M. Reformulation of DFT + u as apseudohybrid hubbard density functional for accelerated materials discovery. Phys. Rev. X ,011006 (2015). URL http://link.aps.org/doi/10.1103/PhysRevX.5.011006 . Hartwigsen, C., Goedecker, S. & Hutter, J. Relativistic separable dual-space gaussian pseu-dopotentials from h to rn. Phys. Rev. B , 3641–3662 (1998). URL https://link.aps.org/doi/10.1103/PhysRevB.58.3641 . Taira, N., Wakeshima, M. & Hinatsu, Y. Magnetic properties of iridium pyrochlores R 2 Ir2 O 7 (R = Y, Sm, Eu and Lu). J. Phys.: Condens. Matter , 5527 (2001). URL http://stacks.iop.org/0953-8984/13/i=23/a=312 . ETHODS a. Pyrochlore Hamiltonian
The Pyrochlore structure is given by an fcc Bravais latticewith a four-atomic basis. We define the lattice vectors b = (0 , , , b = (0 , , , b = (1 , , , b = (1 , , , (3)with a lattice constant a = 4. We work with the time-reversal invariant Hamiltonian intro-duced in Ref. 33, which, by assuming a single Kramers doublet at each Ir site, results in aneight-band model including splitting of the partially filled 5 d electron shell due to spin-orbitcoupling. In momentum space, the Hamiltonian takes the form H = (cid:88) k (cid:88) a,b c † a ( k ) (cid:2) H NN ab ( k ) + H NNN ab ( k ) (cid:3) c b ( k ) + H U , with H NN ab ( k ) = 2( t + t i σ · d ab ) cos [ k · b ab ] , H NNN ab ( k ) = (cid:88) c (cid:54) = a,b { t (cid:48) (1 − δ ab ) + i σ · [ t (cid:48) ( b ac × b cb )+ t (cid:48) ( d ac × d cb )] } cos [ k · ( − b ac + b cb )] . (4)Here, the indices 1 ≤ a, b, c ≤ c a ( k ) † = ( c a ↑ ( k ) † , c a ↓ ( k ) † )is a spinor, where the second lower index refers to a global pseudospin-1/2 degree offreedom. The matrices H NN ab and H NNN ab describe the nearest-neighbour (NN) and next-nearest-neighbour (NNN) hopping including spin-orbit coupling, respectively. The vector σ = ( σ x , σ y , σ z ) contains the Pauli matrices. The following real-space vectors appear in theHamiltonian: d ij = 2 a ij × b ij , (5) a ij = 12 ( b i + b j ) − (1 , , / , (6) b ij = b j − b i . (7)Starting from a localized Hubbard repulsion of the form H U = U (cid:80) R i n R i ↑ n R i ↓ , we use theHartree-Fock mean-field decoupling, H U → − U (cid:88) k i (2 (cid:104) j i (cid:105) · j i ( k ) − (cid:104) j i (cid:105) ) , (8)18here j i ( k ) = (cid:80) αβ = ↑ , ↓ c † iα ( k ) σ αβ c iβ ( k ) / i = 1 , . . . , (cid:104) j i (cid:105) = V (cid:80) k j i ( k ) its mean expectation value. Here, k runs over the k-space volume of the1st Brillouin zone V . We sample the BZ with a grid of 30 × ×
30 points for the full BZin k -space. For our calculations we use a half-volume reduced BZ by exploiting inversionsymmetry. b. Microscopic parameters Following Ref. 33 we use t = 130 t oxy
243 + 17 t σ − t π , t (cid:48) = 233 t (cid:48) σ − t (cid:48) π ,t = 28 t oxy
243 + 15 t σ − t π , t (cid:48) = t (cid:48) σ − t (cid:48) π ,t (cid:48) = 25 t (cid:48) σ t (cid:48) π , (9)where t oxy and t σ , t π denote the oxygen-mediated and direct-overlap NN hopping, respec-tively. The dashed parameters refer to the NNN hopping. We choose t π = − t σ / t (cid:48) σ /t σ = t (cid:48) π /t π = 0 .
08. If not denoted otherwise, we work in units of fs (time) and eV(energy). A comparison bewtween the groundstate band structures obtained from DFTcalculations and model calculations yields an oxygen-mediated hopping of approximately t oxy = 0 . c. Self-consistent thermal state In the mean-field decoupled Hubbard term Eq. (8) the (cid:104) j i (cid:105) are proportional to the spontaneous magnetic moments of the 5 d electrons and thusdetermine the overall magnetic configuration. We define the magnetic order parameter asthe average length of the pseudospin vector, m ≡ (cid:88) a |(cid:104) j a (cid:105)| = 14 (cid:88) a (cid:112) (cid:104) j xa (cid:105) + (cid:104) j ya (cid:105) + (cid:104) j za (cid:105) , (10)which is proportional to the magnetic moment per unit cell, carried by the 5d iridium elec-trons (in units of (cid:126) ≡ n target = 4, constant. In order to determine the thermal initial state configurationfor a fixed temperature, β − , we calculate the initial values of the magnetic moments and thechemical potential, µ , in a self-consistent procedure. To this end we adjust the chemical po-tential accordingly. This procedure is repeated until a converged magnetic order parameter | m n +1 − m n | < − is reached in consecutive iterations of the self-consistency loop. d. Unitary time propagation For a closed system the dynamics of the density opera-tor and thus the time-dependent magnetization are governed by the von-Neumann equa-tion, ˙ ρ ( t ) = − i [ H ( t ) , ρ ( t )]. We use the two-step Adams predictor-corrector method, a linear19ulti-step scheme, for its numerical solution. First, the explicit two-step Adams-Bashforthmethod to calculate a prediction of the value at the next time step is used. This takes theform, ρ ( t ) = ρ ( t − ∆ t ) − i∆ t [ H ( t − ∆ t ) , ρ ( t − ∆ t )] ,ρ pred ( t + ∆ t ) = ρ ( t ) − i 3∆ t H ( t ) , ρ ( t )]+i ∆ t H ( t − ∆ t ) , ρ ( t − ∆ t )] . (11)The first line’s Euler step is only needed to calculate the very first time step. Afterwards,the implicit Adams-Moulton corrector is applied, ρ corr ( t + ∆ t ) = ρ ( t ) − i ∆ t H ( t + ∆ t ) , ρ pred ( t + ∆ t )]+ [ H ( t ) , ρ ( t )]) . (12)We typically use 200 . − .
000 time steps for the propagation. This corresponds to asmallest step size of ∆ t = 0 . e. Non-unitary time evolution We introduce relaxation processes by coupling the elec-tronic pyrochlore system to thermal fermionic reservoirs. The non-unitary dynamics of thereduced system’s degrees of freedom are described in the framework of system-bath theoryby a Lindblad master equation . We choose the following form for the Lindblad dissipatorfor a time-independent Hamiltonian D ( ρ S ) = (cid:88) ω (cid:88) αβ γ αβ ( ω ) (cid:18) A β ( ω ) ρ S A † α − (cid:8) A † α ( ω ) A β ( ω ) , ρ S (cid:9)(cid:19) . (13)Here, A α ( ω ) describe system coupling operators of a general interaction Hamiltonian, H I = (cid:80) α A α ⊗ B α , expanded in the energy eigenbasis of the system Hamiltonian. As the system un-der consideration and thus its Hamiltonian is time-dependent, the instantaneous eigenbasisapproximation is carried out. Here, the coupling operators take the form A α ( ω ) = (cid:88) (cid:15) b − (cid:15) a = ω | (cid:15) a (cid:105) (cid:104) (cid:15) a | A α | (cid:15) b (cid:105) (cid:104) (cid:15) b |→ A α ( ω ( t )) = (cid:88) (cid:15) b ( t ) − (cid:15) a ( t )= ω ( t ) | (cid:15) a ( t ) (cid:105) (cid:104) (cid:15) a ( t ) | A α | (cid:15) b ( t ) (cid:105) (cid:104) (cid:15) b ( t ) | , (14)where H S ( t ) | (cid:15) a ( t ) (cid:105) = (cid:15) a ( t ) | (cid:15) a ( t ) (cid:105) denotes the instantaneous energy eigenbasis of the systemHamiltonian. The Hermitian coefficient matrix γ αβ is defined by the Fourier transform ofthe bath correlation functions + ∞ ´ −∞ d se i ω ( t ) s (cid:104) B † α ( s ) B β (0) (cid:105) .20or every band a and momentum k we define two system coupling operators A a,k ≡ d a,k ( t )and A a,k ≡ d † a,k ( t ), which annihilate and create a quasi particle with the instantaneous bandenergy (cid:15) a,k ( t ), respectively. Accordingly, we define two bath coupling operators B a,k ≡ (cid:80) n t a,kn ( b a,kn ) † and B a,k ≡ (cid:80) n ( t a,kn ) ∗ b n , where t a,kn denotes the transition matrix element and n the mode index. Assuming the bath to be in a thermal steady state and momentum-independent transition rates Γ a,k ( ω ) ≡ (cid:80) n | t a,kn | δ ( ω − w a,kn ) → Γ in the wide-band limit,this introduces relaxation dynamics of the formdd t (cid:104) d † a,k ( t ) d b,k ( t ) (cid:105) t = unitary evolution (15) − (cid:104) (cid:104) d † a,k ( t ) d b,k ( t ) (cid:105) t − n F ( (cid:15) a,k ( t ) , µ ( t )) (cid:105) δ ab − (cid:104) d † a,k ( t ) d b,k ( t ) (cid:105) t (1 − δ ab ) . In the above equation n F ( (cid:15), µ ) ≡ (1 + exp( β ( (cid:15) − µ ))) − denotes the Fermi-Dirac distributionat inverse temperature β = ( k B T ) − . Throughout the paper we use eV units for temperature,setting k B ≡ µ → µ ( t ), in order to keep the particle number inthe system constant. The first non-unitary term on the right hand side of the above equationleads to thermalization of the occupations, the second term induces an exponential decayof the interband transitions. The relaxation time scale is governed by the inverse couplingΓ − ≈
80 fs. We use a bath temperature T = 0 .
016 eV (186 K). f. Time-resolved ARPES
In order to monitor the nonequilibrium changes in the elec-tronic band structure in a time-resolved fashion, we use a theoretical implementation of atime-resolved angle-resolved photoemission spectroscopy (tr-ARPES) probe measurement.The central mathematical object, in order to calculate the observable photocurrent, is thetwo-times lesser Green’s function G 01 fs.The photocurrent can be computed by , I ( k , ω, ∆ t ) = Im ˆ d t ˆ d t (cid:48) s σ p ( t p − t ) s σ p ( t p − t (cid:48) ) exp [i ω ( t − t (cid:48) )]Tr { G < ( k , t, t (cid:48) ) } . (19)Here, t p refers to the center time of the probe pulse, ω to its frequency. Here s σ p ( t ) definesthe probe pulse duration via the width of the Gaussian-shaped probe pulse envelope in bothdirections in time. g. TDDFT+U simulations The evolution of the spinor states and the evaluation ofthe time-dependent Hubbard U and magnetization are computed by propagating the gen-eralized Kohn-Sham equations within time-dependent density functional theory includingmean-field interactions (TDDFT+U), as provided by the Octopus package , using theACBN0 functional together with the local-density approximation (LDA) functional fordescribing the semilocal DFT part. We compute ab initio the Hubbard U and Hund’s J for the 5 d orbitals of iridium. We employ norm-conserving HGH pseudopotentials , aFermi-Dirac smearing of 25 meV, a real-space grid spacing of 0.3 atomic units, and an8 × × U . We find that the inclusion of semi-corestates of yttrium and iridium are important for obtaining accurate band structures; thusvalence electrons explicitly included are Y: 4s , 4p , 4d and 5s ; and Ir: 5s , 5p , and 6s .All TDDFT calculations for bulk Y Ir O (spacegroup F d m , number 227) were per-formed using the primitive cell containing 22 atoms, without a priori assuming symmetriesin order to obtain the correct magnetic ground-state. We employ the experimental val-ues for the lattice parameter and atomic positions . Starting from a random magneticconfiguration as an initial guess, we find the all-in all-out configuration at the end of theself-consistent ground-state calculation (see Supplementary Fig. S4), in agreement with themodel results and previous studies . Very similar values for the magnetic moments for22he Ir atoms are obtained from the spherical averaging of the total electronic density orfrom the density matrix of the localized orbitals (entering in the evaluation of the ACBN0functional), demonstrating that the magnetic properties arise from the localized orbitals.We obtain the groundstate ab initio values of U = 2 . 440 eV and J = 0 . 74 eV, leading to aneffective U eff = U − J = 1 . 69 eV.For the time-dependent simulations, the laser is coupled to the electronic degrees offreedom via the standard minimal coupling prescription using a time-dependent, spatiallyhomogeneous vector potential A ( t ), with electric field E ( t ) = − c ∂ A ∂t . We consider a laserpulse of 12.7-fs duration at full-width half-maximum (FWHM) with a sin-square envelopecorresponding to a total width of 25.4 fs, and the carrier wavelength λ is 3000 nm, corre-sponding to ω = 0 . 41 eV. We choose the driving field to be linearly polarized along the [001]direction (c-axis in Fig. S4). In all our calculations, we used a carrier-envelope phase of φ = 0. h. Data availability All data generated and analysed during this study are availablefrom the corresponding author upon reasonable request.23 UPPLEMENTARY INFORMATION FIG. S1. Computed ground state magnetization. a , Self-consistently calculated zero-temperature magnetic order parameter as a function of Hubbard U for different hopping values t σ ,varied in steps of 0 . 02 eV from − . 56 eV to − . 70 eV. Starting from t σ = − . 62 eV, with increasinghopping an increasing region of hysteresis is found, indicating a first-order phase transition. b c d FIG. S2. Nonequilibrium magnetization and energy. a , Closed system (Γ = 0) timeevolution of the magnetic order parameter for different quench values U q , varied in steps of 0 . . , . 26] eV. b , Nonequilibrium energy per particle per unit cell for thecorresponding interaction interval. The system energy only changes at t=0, at which the (closed)system is quenched. The amount of energy pumped into the system scales linearly with ∆ U . Itremains constant before and after the quench. The final energy values are used to calculate theeffective nonequilibrium temperatures T eff , displayed in Figure 2(c). c , Open system (Γ = 0 . d , Time-dependent totalenergy of the open system. After the quench at t = 0, the total energy relaxes towards its thermalsteady-state value on a time scale Γ − ≈ 80 fs. The dashed vertical line denotes the probe time, t p = 123 . b FIG. S3. Temperature dependence of total energy and magnetization. a , Temperature-dependent equilibrium mean energy per particle per unit cell for different values U , varied insteps of 0 . 02 eV within the interval [1 . , . 28] eV. From these curves, the effective temperatures T eff , assigned to the nonequilibrium states in Figure S2(b, d), can be read off by attributing eachenergy value at the probe time, t p = 123 . b , ∆ ≡ U · m independence of the equilibrium temperature. The WSM-PMM phase boundary can be read off atthat point where ∆( T ) reaches zero. ∆ crit ≈ . 24 eV denotes the critical value at which the AFI-WSM transition takes place. The dip appearing for intermediate interactions is associated withthe semi-metallic band structure and change in density of states in the WSM phase as opposed tothe insulating AFI phase. IG. S4. Ab initio magnetization configuration. All-in all-out magnetic configuration (bluearrows) obtained for Y Ir O , computed from the density matrix of the localized 5 d orbitals ofiridium atoms (green spheres).orbitals ofiridium atoms (green spheres).