Abstract
The nonequilibrium dynamics of matter excited by light may produce electronic phases, such as laserinduced hightransitiontemperature superconductivity, that do not exist in equilibrium. Here we simulate the dynamics of a metal driven at initial time by a spatially uniform pump that excites dipoleactive vibrational modes which couple nonlinearly to electrons. We provide evidence for rapid loss of spatial coherence, leading to emergent effective disorder in the dynamics, which arises in a system unitarily evolving under a translationinvariant Hamiltonian, and dominates the electronic behavior as the system evolves towards a correlated electronphonon longtime state, possibly explaining why transient superconductivity is not observed. Our framework provides a basis within which to understand correlation dynamics in current pumpprobe experiments of vibrationally coupled electrons, highlight the importance of the evolution of phase coherence, and demonstrate that pumped electronphonon systems provide a means of realizing dynamically induced disorder in translationinvariant systems.
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Introduction
Major efforts in condensedmatter physics are currently focused on the means to induce novel phases of matter and harness their properties for practical gain. For many years such phases were thought to robustly exist only as equilibrium, thermodynamic states. The potential outofequilibrium induction of transient phases, enabled by recent experimental advances in the creation and utilization of tailored timeresolved external fields that can excite specific degrees of freedom, opens a door to new modalities for the realization and control of new electronic states^{1,2}.
Optical, modespecific excitation of atomic vibrations^{3} serves as one broad class of outofequilibrium techniques that has been shown experimentally to lead to dramatic modifications in electronic behavior^{4,5,6}, including the possible induction of a superconducting transition at a critical temperature larger than its equilibrium counterpart in K_{3}C_{60}^{7}, YBa_{2}Cu_{3}O_{6.5}^{8}, and organic salts^{9}. In general, optically accessible phonons are longwavelength dipoleactive modes, which typically do not couple linearly to the electron density, and therefore nonlinearities are expected to govern the dynamics in centrosymmetric systems^{10,11,12,13}, stimulating many interesting theoretical proposals^{13,14,15,16,17,18,19,20}. One particular mechanism^{13} is based on the observation that since direct, local interaction between electrons and photoexcited phonons must depart from that of conventional linear Holstein^{21} and Fröhlich^{22,23} models, one must consider a quadratic coupling of driven phonons to the electron density. An approximate analysis of such a model was presented previously^{13} (see Supplementary Note 3). Here, we use exact numerical methods and an effective theory based on a loworder expansion in the electron–phonon coupling to unravel the emergent electronic behavior in this driven, nonequilibrium system. Combining a tensornetwork approach for the time evolution of an infinite onedimensional (1D) system on short timescales with propagation to long times using direct Krylov subspace methods for finitesize systems and analytical arguments, we elucidate the spatially resolved dynamics of electrons coupled to pumped phonons. Our main results are:
(1) Phononinduced disorder: we observe fast growth of local electronic correlations after the application of the pump. A dramatic flattening in the momentum dependence of charge, spin, and pairing correlations rapidly follows, pointing to the loss of electronic spatial phase coherence. We find that disorder emerges as a result of the nature of the initial lightcreated coherent phonon superposition state whose dynamics is effectively governed by a Hamiltonian that approximately conserves phonon occupations. The presence of quasiconserved phonon constants of motion implies that electronic observables selfaverage over the different disordered phonon configurations of the initial state and possess no offdiagonal coherence between different phonon sectors. This provides a realization of disorderfree localization^{24,25,26}, recently discussed in the context of lattice gauge theories^{27,28,29,30,31}. To understand this behavior, we derive an effective model whose behavior captures the qualitative features of the exact dynamics on transient timescales. Our effective theory provides a natural framework within which disorder and electron localization arise in the dynamics, and provides a perspective for the shorttime dynamics complementing analysis of the longtime behavior where the loss of phonon coherence and preservation of the Poissondistributed diagonal eigenvalues of the initial coherent state density matrix characterizes a random disorder potential, responsible for the destruction of phase coherence of the normal state electronic correlations^{13}.
(2) Correlated electron–phonon steady state: we provide evidence that the system evolves at long times to a steady state characterized by sizeable correlations between electrons and phonons. The earlytime dynamics that follow the pump already indicate rapid growth of local, negative correlations between the electron density \(\hat{n}\) and the oscillator quadratic displacement \({\hat{X}}^{2}\) at a given site and between the linear displacement \(\hat{X}\) at adjacent sites, which signals a tendency towards charge flow between neighboring sites, resulting in enhanced double occupancy. This dynamical process quenches the Friedel oscillations^{32} of the electron density profile, and manifests as a spacetimedependent feature in the densitydensity correlation function that spreads spatially outwards along a “light cone” defined by the Fermi velocity^{33}. Behind the light cone, very rapidly, the density–density correlation function becomes basically structureless, suggesting that the asymptotic state possesses a large degree of randomness. At long times, we find an overall increase in the magnitude of the expectation value of the electron–phonon interaction term, implying evolution towards a strongly correlated longtime electron–phonon state.
(3) Dynamically induced strongcoupling behavior: we compare the dynamical electronic behavior in response to a pump in the quadraticcoupling model against that in the linear (Holstein) counterpart. We observe larger double occupancy and greater largeamplitude response of momentumresolved correlation peaks in the quadratic model, indicating that in this model, in contrast to the more widely studied Holstein model, the drive pushes the system into a strongcoupling regime. This substantive dynamical response of the nonlinearly coupled system implies the existence of nonequilibrium pathways to coherent induction of electronic phases not accessible in equilibrium, and highlights the importance of the quadratic coupling in irradiated materials.
Physical setup
We consider a metal whose vibrational modes are excited at an initial time by a shortduration light pulse that creates a coherent phonon field^{34} on every site, which couples nonlinearly to the local electron density.
The Hamiltonian that governs the dynamics is given by
Here \({{{{{{{{\mathcal{H}}}}}}}}}_{{{{{{{{\rm{e}}}}}}}}}=J{\sum }_{i,\sigma }{c}_{i,\sigma }^{{{{\dagger}}} }{c}_{i+1,\sigma }+{{{{{{{\rm{h.c.}}}}}}}}\) characterizes the dynamics of electrons of spin flavor σ ∈ {↑, ↓} via the fermion creation (annihilation) operator \({c}_{i,\sigma }^{{{{\dagger}}} }\) (c_{i,σ}) and charge density \({\hat{n}}_{i}={\sum }_{\sigma }{\hat{n}}_{i,\sigma }\) at site i. The phonon Hamiltonian is \({{{{{{{{\mathcal{H}}}}}}}}}_{{{{{{{{\rm{ph}}}}}}}}}=\omega {\sum }_{i}\left({b}_{i}^{{{{\dagger}}} }{b}_{i}+\frac{1}{2}\right)\), which characterizes a local optical Einstein phonon mode with frequency ω(ℏ = 1), described by the boson creation (annihilation) operator \({b}_{i}^{{{{\dagger}}} }\) (b_{i}). The electrons of this irradiated system couple locally to the excited vibrations via the the dominant symmetryallowed interaction^{13}
This coupling also serves as a description of doublewell ^{35,36} and other^{37} systems for which the linear approximation is inadequate. This model Hamiltonian implies an equilibrium renormalization of the oscillator stiffness \(K\to K[1+4\frac{{g}_{q}}{\omega }({\langle \hat{n}\rangle }_{i}1)]\). Thus, the onsite harmonic oscillator is stable so long as \(\left{g}_{q}\right < \frac{\omega }{4}\)^{13} (see Supplementary Note 1).
Since the wavelength of the pump field extends beyond the lattice scale, we assume it produces a perfectly phasecoherent initial product state of onsite phonon coherent states \({\bigotimes }_{i}{\left\alpha \right\rangle }_{i}\), where \(\left\alpha \right\rangle ={e}^{\frac{{\left\alpha \right}^{2}}{2}}{\sum }_{\nu }\frac{{\alpha }^{\nu }}{\sqrt{\nu !}}\left\nu \right\rangle\) represents a coherent state of amplitude α written here as an appropriate superposition of phononnumber states \(\left\nu \right\rangle\). We simulate the time evolution of the initial state
where \(\left{{{{{{{\rm{FS}}}}}}}}\right\rangle ={\prod }_{k\le {k}_{{{{{{{{\rm{F}}}}}}}}}}{c}_{k,\uparrow }^{{{{\dagger}}} }{c}_{k,\downarrow }^{{{{\dagger}}} }\left0\right\rangle\) with k_{F} = π/2 describes a metal formed from a Fermi sea of spinful electrons in a halffilled (\(\langle {\hat{n}}_{i}\rangle =1\)) 1D lattice.
Here, we set the lattice constant a = 1 and study the physics of the model for physical parameters defined in units of J, i.e., we set J = 1. In the main text, we consider g_{q} ≤ 0.25 for ω = π/2 to study dynamics of the nonlinear model for couplings ranging from weak to strong, and use \(\alpha =\sqrt{2}\) for the pump amplitude. This choice of ω allows us to numerically resolve the quantum effects in dynamics due to a large yet amenable phonon Hilbert space.
We simulate the time evolution of \(\left{{{{{{{\mathbf{\Psi }}}}}}}}\right\rangle\) in an infinite system on transient timescales via the infinite timeevolved block decimation (iTEBD) algorithm^{38} and access its longtime behavior in finitesize systems of size L = 3–6 and local phonon Hilbert space dimension d_{ν} = 8, 10, 12 using direct Krylov subspace methods. In iTEBD, one employs a matrixproduct state (MPS) ansatz for quantum states in the thermodynamic (infinitesize) limit, which permits access to information pertaining to longrange correlations in the system. Time evolution of an MPS is however ultimately limited to finite times because of the exponential growth of entanglement. Krylov subspace methods, based on Hamiltonian matrixstate vector multiplication, are in contrast not limited to short times, but are restricted to small L due to the exponential growth of the Hilbert space with L. Combining the two approaches allows us to derive reliable conclusions about longrange correlations on finite timescales from iTEBD and local correlations at long times from Krylov propagation.
Results
Figure 1 demonstrates the energy redistribution amongst the different system subsectors in the course of the time evolution on timescales ranging from short (panel a) to long (panel b), as the system approaches its longtime limit of a correlated electron–phonon steady state. Consider the largest coupling g_{q} = 0.25 (dark lines in panel a). At early times \(t\le \frac{2\pi }{\omega }\), the electron subsystem absorbs energy from the excited phonons, and the phonon energy density oscillates about a value close to its initial value, while the electron–phonon energy density becomes more negative, see panel a of Fig. 1. At asymptotically long times, we observe an overall flow of energy from the phonon and electron–phonon subsectors to the electron subsector (panel of b of Fig. 1). Importantly, the increase in the magnitude of the (negative) electron–phonon correlation term implies a longtime correlated electron–phonon state.
Correlations between electrons and phonons already appear in the earlytime dynamics, as we demonstrate in Fig. 2. Consider the charge–phonon correlation function \(C{X}_{r}(t)=\langle {\hat{n}}_{i}{\hat{X}}_{i+r}^{2}(t)\rangle \langle {\hat{n}}_{i}(t)\rangle \langle {\hat{X}}_{i+r}^{2}(t)\rangle\) (Fig. 2, panel a), where \({\hat{X}}_{i}:= \sqrt{\frac{1}{2M\omega }}({b}_{i}^{{{{\dagger}}} }+{b}_{i})\), and we set M = 1. For r = 0, \(\hat{n}\) rapidly becomes negatively correlated with \({\hat{X}}^{2}\). Note that \(\langle {\hat{n}}_{i}(t)\rangle =1\) throughout the dynamics in the translationally invariant system under consideration and \(\langle {\hat{X}}_{i}^{2}(t)\rangle\) (dashdotted line, panel c) remains positive under time evolution. The substantial local, negative correlations in CX_{0}(t), therefore, imply a flow of electrons between neighboring sites. The same analysis applied to CX_{1}(t) reveals a positive correlation between electron density and phonons separated by a single site with a dynamical profile somewhat similar (albeit of opposite sign) to CX_{0}(t). With a slightly delayed onset, much weaker positive correlations buildup at larger r in CX_{r}(t). The interplay between onsite and nearestneighbor correlations in CX_{r}(t) reflects the tendency of charge to flow from a site to its neighbors, implying that doublons (doubly occupied sites) and holons (empty sites) emerge in the dynamics on such timescales. Indeed, in panel b, we observe a rapid enhancement of local electron density–density correlations \({{{{{{{{\mathcal{D}}}}}}}}}_{0}(t)=\langle {\hat{n}}_{i}{\hat{n}}_{i}(t)\rangle =\langle {\hat{n}}_{i}\rangle +2\langle {\hat{n}}_{i,\uparrow }{\hat{n}}_{i,\downarrow }(t)\rangle\), accompanied by the suppression of \({{{{{{{{\mathcal{D}}}}}}}}}_{1}(t)=\langle {\hat{n}}_{i}{\hat{n}}_{i+1}(t)\rangle\) due to doublon creation, as expected if there is a tendency towards formation of an alternating pattern of doubly and singly occupied sites. For times greater than \(t\approx 0.175[\frac{2\pi }{\omega }]\), \({{{{{{{{\mathcal{D}}}}}}}}}_{1}(t)\) begins to grow and becomes positive, whilst \({{{{{{{{\mathcal{D}}}}}}}}}_{2}(t)=\langle {\hat{n}}_{i}{\hat{n}}_{i+2}(t)\rangle\) diminishes, and a wavefront behavior in r appears to arise. In fact, when normalized against the t = 0 metal Friedel density profile, a density–density correlation lightcone C_{r}(t)/C_{0}(t) (\({C}_{r}(t)=\langle {\hat{n}}_{i}{\hat{n}}_{j}(t)\rangle \langle {\hat{n}}_{i}(t)\rangle \langle {\hat{n}}_{j}(t)\rangle\))^{33,39} propagating outwards in r can be clearly seen (Fig. 2, panel d). A characteristic feature that emerges for larger r at later time delays closely trails the secondintime maximum. Thus, to sharply characterize the light cone, we track the inflection point preceding that maximum (diamond symbols). A line of best fit through these data points (Fig. 2, panel d; inset) reveals linear charge propagation with a velocity v_{c} ≈ 3.5J, slightly larger than the free metal Fermi velocity 2k_{F}J = πJ. On the timescales accessed by iTEBD, we find no evidence for a wavefront propagating in either of CX_{r}(t) or \(\langle {\hat{X}}_{i}(t){\hat{X}}_{i+r}(t)\rangle\), reflecting the resistance to the propagation of the dispersionless Einstein oscillator modes of the initialtime (g_{q} = 0) state. The behavior exhibited by CX_{r}(t) and C_{r}(t) implies nonequilibrium induction of enhanced double occupancy 〈n_{i,↑}n_{i,↓}(t)〉, which we have directly verified.
Turning to the dynamics of longrange electronic correlations, in Fig. 3 we study the evolution with time of the momentumresolved charge C_{k}(t), spin S_{k}(t), and pairing P_{k}(t) correlation functions to fully characterize the electronic features. Apart from the fast initial growth of C_{π}(t) for t ⪅ \(0.2[\frac{2\pi }{\omega }]\) due to the enhanced double occupancy, we observe rapid flattening in momentum space of these correlations, marking the loss of spatial coherence, despite the persistent growth of local density–density and charge–phonon correlations, indicating that the pattern of doubly and singly occupied sites is becoming random. This remarkable behavior implies an effective disordered state forms on transient timescales, and a more subtle role played by phonons in the dynamics, as we show next.
Effective model for the disorder
To understand the mechanism behind the appearance of disorder, we derive an effective theoretical picture for the dynamics to leading order in g_{q}/ω.
We find it convenient to consider a rotating frame in which the offdiagonal phonon terms (in the occupationnumber basis) of Eq. (2) are eliminated via a Bogoliubovtype squeezing transformation^{13}: \({{{{{{{\mathcal{H}}}}}}}}\to \tilde{{{{{{{{\mathcal{H}}}}}}}}}=U{{{{{{{\mathcal{H}}}}}}}}{U}^{{{{\dagger}}} }\), where U = e^{S}, \(S={\sum }_{j}\frac{1}{2}{\zeta }_{j}({b}_{j}^{{{{\dagger}}} }{b}_{j}^{{{{\dagger}}} }{b}_{j}{b}_{j})\), and \({\zeta }_{i}=\frac{1}{4}{{{{{{\mathrm{ln}}}}}}}\,[1+4\frac{{g}_{q}}{\omega }({\hat{n}}_{i}1)]\), the squeezing parameter, is chosen so that the \({({b}_{i}^{{{{\dagger}}} })}^{2}\) and \({({b}_{i})}^{2}\) terms vanish. This yields, in the squeezed frame, \(\tilde{{{{{{{{\mathcal{H}}}}}}}}}={e}^{S}{{{{{{{{\mathcal{H}}}}}}}}}_{{{{{{{{\rm{e}}}}}}}}}{e}^{S}+{\sum }_{i}\omega \sqrt{1+4\frac{{g}_{q}}{\omega }({\hat{n}}_{i}1)}({\beta }_{i}^{{{{\dagger}}} }{\beta }_{i}+\frac{1}{2})\), where β^{†} creates a squeezedphonon state. Perturbatively expanding the transformed coupling term in orders of g_{q}/ω, we find
Here, \({J}^{* }=J{e}^{\frac{1}{2}{(\frac{{g}_{q}}{\omega })}^{2}({\langle {\hat{n}}_{{{{{{{{\rm{B}}}}}}}}}\rangle }^{2}+2\langle {\hat{n}}_{{{{{{{{\rm{B}}}}}}}}}\rangle +1)}\) (\(\langle {\hat{n}}_{{{{{{{{\rm{B}}}}}}}}}\rangle\) is the average number of excited bosons in the dynamics) and \({\omega }^{* }=\omega {g}_{q}^{2}/\omega\). Aside from renormalization of the electron and phonon energy scales, we see that the electron density, at \({{{{{{{\mathcal{O}}}}}}}}\{{g}_{q}/\omega \}\), and double occupancy, at \({{{{{{{\mathcal{O}}}}}}}}\{{({g}_{q}/\omega )}^{2}\}\), couple to the squeezedphonon density. This Hamiltonian is exact to \({{{{{{{\mathcal{O}}}}}}}}\{{g}_{q}/\omega \}\), and approximate to \({{{{{{{\mathcal{O}}}}}}}}\{{({g}_{q}/\omega )}^{2}\}\) (and higher orders). See Supplementary Note 2 for details of the derivation and approximations employed. For the time dependence of electronic operators \({\hat{O}}_{{{{{{{{\rm{e}}}}}}}}}\) measured in the original frame, we derive in a similar approximation (details in Supplementary Note 2) a theory in the squeezed frame in which \({\hat{O}}_{{{{{{{{\rm{e}}}}}}}}}\) transforms as \({\hat{O}}_{{{{{{{{\rm{e}}}}}}}}}\to {e}^{S}{\hat{O}}_{{{{{{{{\rm{e}}}}}}}}}{e}^{S}\), the initial state as \(\left0\right\rangle \equiv \left{{{{{{{\mathbf{\Psi }}}}}}}}\right\rangle \to {e}^{S}\left0\right\rangle\), and \({{{{{{{{\mathcal{U}}}}}}}}}_{{{{{{{{\rm{eff.}}}}}}}}}={e}^{i{{{{{{{\mathcal{{H}}}}}}}_{{{{{{{{\rm{eff.}}}}}}}}}}}t}\) governs the time evolution. Within this scheme in which terms larger than \({{{{{{{\mathcal{O}}}}}}}}\{{g}_{q}/\omega \}\) are neglected, the equaltime expectation value of \({\hat{O}}_{{{{{{{{\rm{e}}}}}}}}}\) in the squeezed frame becomes
with \({{{\Gamma }}}_{{\hat{O}}_{{{{{{{{\rm{e}}}}}}}}}}=\left[S,{\hat{O}}_{{{{{{{{\rm{e}}}}}}}}}\right]\). In Fig. 4, we test the predictions of Eq. (5) against the exact results. Not only does the effective theory reproduce the flattening in momentumresolved C_{k}(t) and S_{k}(t) (Fig. 4, panels a and b) observed in the exact simulations (Fig. 3), it also provides an overall qualitative proxy for the exact raw and timeaveraged quantities 〈n_{i,↑}n_{i,↓}(t)〉, C_{π}(t) and S_{π}(t) (Fig. 4, panels c–h), even for relatively large g_{q}/ω for which the approximations we employ are less justified (we discuss limitations of the effective model in Supplementary Note 2).
The origin of disorder becomes manifest in the effective model. Noting that in Eq. (5) \({{{\Gamma }}}_{{C}_{k}},{{{\Gamma }}}_{{S}_{k}}=0\) because the charge and spin correlations are conserved under the squeezing transformation implies that the exact dynamics is approximately captured by an effective theory that conserves the squeezedphonon occupations. This effective theory thus encodes dynamics of the electrons within independent trajectories of different squeezedphonon configurations in an ensemble given by an initial Poissondistributed linear combination that describes the t = 0 state (now in the squeezed basis) and is thus formally equivalent to the disorderaveraged dynamics of an electronic system quenched in a random, static Poissondistributed potential determined by the initial state occupations. Charge and spin correlations, by construction, possess no coherence between different squeezedphonon sectors, and thus very quickly flatten in the course of the dynamics. The exact electronic behavior on transient timescales is therefore dominated by a large degree of effective disorder despite that the initial state and the Hamiltonian in both squeezed and unsqueezed frames are disorder free.
Note that, however, while this effective model remains valid on intermediate timescales, higherorder terms in g_{q}/ω, neglected in our treatment, will eventually become important, possibly leading at long times to deviations from the above behavior. Nonetheless, our numerics seem to suggest evolution towards a state with a large disorder that remains robust for extended timescales. We provide in Supplementary Note 3 a complementary treatment of electronic disorder at later times in the unsqueezed frame based on the dynamics induced by phonon decoherence. This disordered behavior persists despite the attractive electron density–density interaction term of \({{{{{{{\mathcal{{H}}}}}}}_{{{{{{{{\rm{eff.}}}}}}}}}}}\), which, at least in 1D, implies that the system lies within a regime far from the superfluid transition^{40}.
The picture we obtain here indicates that a translationally uniform system excited by a spatially uniform field governed by electron–phonon nonlinearity will flow towards a state characterized by a high level of randomness in absence of quenched disorder. This behavior was noted in ref. ^{13} based on an analysis of phonon decoherence (see also Supplementary Note 3) and has become a subject of major theoretical interest within the field of disorderfree localization^{24,25,26,27} (see also refs. ^{41,42,43}). In this regard, our effective theory reveals a mechanism operative in experiments for dynamically induced disorder reminiscent of that found in the context of the particular lattice gauge theory models studied in refs. ^{27,28,29,30,31}. These models describe the coupling of fermions to background gauge fields modeled as spin degrees of freedom, in which a duality transformation^{44,45} maps the Hamiltonian onto one with conserved gauge charge configurations and the gauge charge couples directly onsite to the fermion occupation. Time evolution with this manifestly translationally invariant Hamiltonian of an initial product state of fermions and gauge spins, equivalent to a linear superposition over different superselection gauge charge configurations, exhibits disorderfree localization due to selfaveraging of observables over the different initial gauge configurations. In contrast to these models, our theory reveals that an approximate effective model governed by similar behavior dominates the exact dynamics of the quenched electron–phonon system on extended timescales. Thus, our work paves a way towards the physical realization of disorderfree localization in current pumpprobe experiments. Furthermore, the emergence of an attractive Hubbard interaction in the effective model presents an unexplored avenue within the context of disorderfree localization to study the competition between disorder and attractive interactions in the dynamics of spinful fermionic systems.
Comparison with a linearly coupled electron–phonon model: dynamically induced strongcoupling behavior from nonlinear electron–phonon coupling
We contrast the dynamics of our nonlinear model to that of the (linear) Holstein model (which cannot be driven by a light pulse in an inversion symmetric system). We use two methods to choose an appropriate coupling strength in the Holstein model corresponding to a given coupling strength of the quadratic model against which we perform a comparison, see Supplementary Note 4 for details. In one approach, we choose the Holstein coupling that yields the same equilibrium ground state double occupancy as in the quadratic model. In the other, the Holstein coupling is chosen to produce the same double occupancy as that obtained analytically from a disentangling transformation that serves as a lowenergy description of the dynamics (Eq. (4)). Both methods of comparison show that even a relatively large nonlinear coupling such as g_{q} = 0.25, proximate to the oscillator instability threshold, gives rise in equilibrium to weakcoupling behavior. In contrast, Fig. 5 shows that the quadratic model exhibits a much stronger dynamical response to the pump, displaying both a large enhancement of double occupancy (panel a) and largeamplitude dynamics in momentumresolved electronic correlations (panels b–d) including flattening of P_{k}(t) (panel d). This is in sharp contrast to the behavior of the Holstein model and implies that dynamics of nonlinear coupled electron–phonon systems operative in pumpprobe experiments afford nonequilibrium pathways to correlated physics unavailable in the static limit and which lies outside the frame of conventional theoretical models.
Discussion
Prior studies of nonlinear electron–phonon dynamics have relied on approximate lowenergy treatments. Our exact numerical approach to spatially resolved dynamics of a pumped nonlinear electron–phonon systems fills an urgent need. We use iTEBD to provide a detailed exact analysis of shorttime (up to \(t \sim \frac{2\pi }{\omega }\)) dynamics of an infinite nonlinear electron–phonon coupled metal upon coherent excitation of vibrational modes by light. We supplement this by direct Krylov propagation of small systems to asymptotically long times. We explicitly describe the flow towards a correlated electron–phonon steady state at long times, the indication of which already manifests on short timescales. Remarkably, although we consider a spatially uniform system evolving after application of a spatially uniform pump field, the key feature of the longtime state is the appearance of properties consistent with a high degree of effective disorder that dominates the physical behavior, unveiling an intriguing connection to the scenario of disorderfree localization^{27,28}. These properties are a consequence of the quasiconserved squeezedphonon constants of motion that effectively govern the time evolution of the initial linear superposition state and the very rapid loss of coherence of the phonons, which we found to be directly tied to the buildup of disorder, implying that the intermediate and longtime state is an incoherent superposition of different oscillator configurations on different sites. These incoherent phonon configurations result in a dynamic effective disorder potential for the electrons, which leads to the suppression of the (powerlaw) quasilongrange charge, spin and pairing correlations. Analysis of the energy redistribution amongst the different system subsectors and of electron and phonon distribution functions of the longtime state obtained in finitesize systems, presented in Supplementary Note 3, suggests that the terminal state obtained in finitesize simulations may not be thermal. Determining the fate of the established longtime entangled electron–phonon state in which the phonons in effect provide strong onsite potential fluctuations that substantially broaden all momentum–space distribution functions and fully disentangling the contributions of electron heating from localization due to the transient phononinduced disorder to this entangled electron–phonon state are beyond the scope of this paper, and are left to future work. However, the results of this paper establish that the pumpactivated transient phononinduced disorder in electron dynamics presents an opportunity to explore the interplay between correlations and randomness in outofequilibrium electronic matter.
A crucial question relates to the possibility of pumpinduced superconductivity as predicted in ref. ^{13}. In our calculations no evidence for superconductivity is found and we only find weak evidence for charge density wave correlations for very short time delays; the results are more consistent with the system falling within the disorderdominated Anderson insulating regime of the phase diagram presented in ref. ^{13}. One possibility would be that superconducting and density wave regimes either do not exist or are not accessible with the current pump protocol (perhaps because the pump transfers too much energy to the electronic subsystem). A second possibility would be that the 1D model considered here disfavors superconductivity. In fact, it has been shown that quantum fluctuations can destroy superconductivity in dirty superconductors below a mobility threshold^{46}. In one dimension, all singleparticle states are localized in presence of a static disorder potential. Despite that in 1D systems superconductivity can overcome the localizing tendency of disorder to some extent^{47}, the effects of the disorder are stronger than in higher dimensions. The accurate simulation of pumpinduced dynamics in higherdimensional systems in the thermodynamic limit faces challenges, but is urgently needed.
The quadratic model reacts more strongly to a pump than the linear Holstein model, highlighting the importance of this mechanism in pumpprobe experiments, e.g., ref. ^{48}. Questions such as the consideration of additional electron–vibration interactions consistent with inversion symmetry^{20,49}, which may aid in the stabilization of a transient superconducting state, as well as how the electron–phonon steady state exposed in this work manifests experimentally are also important open challenges and call for the development of new tools for the study of outofequilibrium nonlinear electron–phonon problems. An intriguing possibility is to use the information obtained here about the properties of the longtime state to motivate a variational ansatz in order to simulate the dynamics.
Methods
We study pumpinduced dynamics via exact numerical simulations of the nonlinear model coupled with an effective theory derived within a treatment formally similar to a linear response theory in a loworder expansion in powers of g_{q}/ω.
Details of exact numerical simulations of the nonlinear electron–phonon system
We simulate the time evolution of \(\left{{{{{{{\mathbf{\Psi }}}}}}}}\right\rangle\) representing the metal on an infinite chain irradiated at initial time t = 0 by a pump via the iTEBD algorithm^{38} utilizing the TeNPy Library^{50}. We use d_{ν} = 12 phonon states to represent the local phonon Hilbert space. We allow the bond dimension χ to grow without saturation in the iTEBD time evolution, and converge our results with respect to the truncation error ϵ_{TEBD}. This allows access to time t ~ 5J for which we find ϵ_{TEBD} = 10^{−3.5} achieves satisfactory convergence. We refer the reader to Supplementary Note 5 for more information. To shed light on the longtime behavior, we also propagate the initial state using direct Krylov subspace methods for finite system sizes L = 3 − 6 with d_{ν} = 8, 10, 12 and twisted boundary conditions, see Supplementary Note 5 for more details.
Details of the effective model obtained within a loworder expansion in g _{q}/ω
We derive an effective model within a framework similar to linear response, consistently incorporating contributions of \({{{{{{{\mathcal{O}}}}}}}}\{{g}_{q}/\omega \}\), with judiciously selected \({{{{{{{\mathcal{O}}}}}}}}\{{({g}_{q}/\omega )}^{2}\}\) corrections (e.g., the effective electron density–density interaction term). This theory, strictly valid to \({{{{{{{\mathcal{O}}}}}}}}\{{g}_{q}/\omega \}\), qualitatively captures the exact behavior of the timeevolved initial state in infinite systems obtained using iTEBD. We simulate the dynamics governed by the effective model by timeevolving \(\left0\right\rangle \equiv \left{{{{{{{\mathbf{\Psi }}}}}}}}\right\rangle\) under the action of \({{{{{{{{\mathcal{U}}}}}}}}}_{{{{{{{{\rm{eff.}}}}}}}}}(t)\) in Eq. (5) using iTEBD, employing d_{ν} = 12 local squeezedphonon states and allowing χ to grow without saturation in the time evolution, while converging results with respect to ϵ_{TEBD}. This allows access to time t ~ 5J for which we find ϵ_{TEBD} = 10^{−3.5} achieves satisfactory convergence. Details of the derivation of the effective model and additional discussion of the dynamics are presented in Supplementary Note 2 and Supplementary Note 3.
Data availability
All data are available upon reasonable request.
Code availability
All the custom codes used in this work are available upon reasonable request.
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Acknowledgements
J.S., D.R.R. and A.J.M. acknowledge support from the National Science Foundation (NSF) Materials Research Science and Engineering Centers (MRSEC) program through Columbia University in the Center for Precision Assembly of Superstratic and Superatomic Solids under Grant No. DMR1420634. B.K. acknowledges support from NSF Grant No. CHE1954791. D.M.K. acknowledges support from Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy—Cluster of Excellence Matter and Light for Quantum Computing (ML4Q) EXC 2004/1—390534769 and from the Max PlanckNew York City Center for NonEquilibrium Quantum Phenomena. This work used the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by NSF grant number ACI1548562, through allocation TGDMR190074. J.S. also acknowledges the hospitality of the Center for Computational Quantum Physics (CCQ) at the Flatiron Institute. The Flatiron Institute is a division of the Simons Foundation.
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J.S. and B.K. designed computer codes and performed numerical calculations. J.S., D.R.R. and A.J.M. performed analytical calculations. J.S., B.K., D.M.K., D.R.R. and A.J.M. contributed to the conception, development, analysis, and writing of this work.
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Sous, J., Kloss, B., Kennes, D.M. et al. Phononinduced disorder in dynamics of optically pumped metals from nonlinear electronphonon coupling. Nat Commun 12, 5803 (2021). https://doi.org/10.1038/s41467021260303
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DOI: https://doi.org/10.1038/s41467021260303
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