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arXiv:1611.05388 · Physical review. B./Physical review. B

Filling-dependent doublon dynamics in the one-dimensional Hubbard model

Roman Rausch, Michael Potthoff

Published Jan 31, 2017• 20 citations
arXivHTMLPDF

Abstract

The fate of a local two-hole doublon excitation in the one-dimensional Fermi-Hubbard model is systematically studied for strong Hubbard interaction $U$ in the entire filling range using the density-matrix renormalization group (DMRG) and the Bethe ansatz. For strong $U$, two holes at the same site form a compound object whose decay is impeded by the lack of phase space. Still, a partial decay is possible on an extremely short time scale where phase-space arguments do not yet apply. We argue that the initial decay and the resulting intermediate state are relevant for experiments performed with ultracold atoms loaded into an optical lattice as well as for (time-resolved) CVV Auger-electron spectroscopy. The detailed discussion comprises the mixed ballistic-diffusive real-time propagation of the doublon through the lattice, its partial decay on the short time scale as a function of filling and interaction strength, as well as the analysis of the decay products, which are metastable on the intermediate time scale that is numerically accessible and which show up in the two-hole excitation (Auger) spectrum. The ambivalent role of singly occupied sites is key to understanding the doublon physics; for high fillings, ground-state configurations with single occupancies are recognized to strongly relax the kinematic constraints and to open up decay channels. For fillings close to half-filling, however, their presence actually blocks the doublon decay. Finally, the analysis of the continua in the two-hole spectrum excludes a picture where the doublon decays into unbound electron holes for generic fillings, different from the limiting case of the completely filled band. We demonstrate that the decay products as well as the doublon propagation should rather be understood in terms of Bethe ansatz eigenstates.

DOI
10.1103/physrevb.95.045152

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The fate of a local two-hole doublon excitation in the one-dimensional Fermi-Hubbard model is systematically studied for strong Hubbard interaction UUU in the entire filling range using the density-matrix renormalization group (DMRG) and the Bethe ansatz. For strong UUU, two holes at the same site form a compound object whose decay is impeded by the lack of phase space. Still, a partial decay is possible on an extremely short time scale where phase-space arguments do not yet apply. We argue that the initial decay and the resulting intermediate state are relevant for experiments performed with ultracold atoms loaded into an optical lattice as well as for (time-resolved) CVV Auger-electron spectroscopy. The detailed discussion comprises the mixed ballistic-diffusive real-time propagation of the doublon through the lattice, its partial decay on the short time scale as a function of filling and interaction strength, as well as the analysis of the decay products, which are metastable on the intermediate time scale that is numerically accessible and which show up in the two-hole excitation (Auger) spectrum. The ambivalent role of singly occupied sites is key to understanding the doublon physics; for high fillings, ground-state configurations with single occupancies are recognized to strongly relax the kinematic constraints and to open up decay channels. For fillings close to half-filling, however, their presence actually blocks the doublon decay. Finally, the analysis of the continua in the two-hole spectrum excludes a picture where the doublon decays into unbound electron holes for generic fillings, different from the limiting case of the completely filled band. We demonstrate that the decay products as well as the doublon propagation should rather be understood in terms of Bethe ansatz eigenstates.
Two fermions occupying the same site of a lattice model with strongly repulsive Hubbard-type interaction UUU form a doublon, a long-living excitation, the decay of which is suppressed because of energy conservation. By means of an exact-diagonalization approach based on the Krylov-space technique, we study the dynamics of a single doublon, of two doublons, and of a doublon in the presence of two additional fermions prepared locally in the initial state of the extended Hubbard model. The time dependence of the expectation value of the double occupancy at the different sites of a large one-dimensional lattice is analyzed by perturbative arguments. In this way the spatiotemporal evolution of the doublon can be understood. The initial decay takes place on a short time scale 1/U1/U1/U, and the long-time average of the decayed fraction of the total double occupancy scales as 1/U21/{U}^{2}1/U2. We demonstrate how the dynamics of a doublon in the initial state is related to the spectrum of two-fermion excitations obtained from linear-response theory, we work out the difference between doublons composed of fermions vs doublons composed of bosons, and we show that despite the increase of phase space for inelastic decay processes, the stability of a doublon is enhanced by the presence of additional fermions on an intermediate time scale.
In this work, the appearance of bound states in spectroscopies and their effects on real-time dynamics of strongly correlated quantum systems is investigated with the help of the density matrix renormalization group (DMRG) and exact diagonalization (ED). The main focus surrounds the behaviour of the so-called doublons in the Hubbard model, which are repulsively bound states of two electrons. The doublon manifests itself in high-resolution CVV Auger electron spectroscopy (AES) in the form of a satellite in the spectrum. Using DMRG, its behaviour is traced and explained through all fillings in the Hubbard model. It is furthermore demonstrated that there appears a second correlation satellite corresponding to bound states of four electrons (or two doublons) which are termed quadruplons. The quadruplon can be further stabilized with nearest-neighbour Coulomb interaction which creates even more bound states: triplons (three electrons) and dimerons (nearest-neighbour electrons). The spectra are also analysed with the help of the Bethe ansatz and decay channels of the doublon into spinons and holons in one spatial dimension are revealed. The real-time dynamics and decay of a local doublon excitation is investigated and it is shown that the doublon becomes more stable at intermediate fillings and that scattering channels add a diffusive component to its propagation, while a ballistic wavefront remains. Finally, a formalism for pump-probe Auger spectroscopy is developed and possible setups are discussed. It is shown that due to the high stability of the doublon at the intermediate time scale, the spectra show no dependence on the pump-probe delay and that either a modification of the setup or of the model is necessary. In the latter approach, spectra are calculated for the dynamic Hubbard model, where the double occupancy couples to a bosonic degree of freedom and it is shown how the parameters of the coupling are relayed to shifts and intensities in the spectra as a function of the delay time. The secondary focus surrounds the dynamics of a quantum or classical impurity spin coupled to a free or strongly interacting substrate chain. It is demonstrated that for a large enough interaction at half filling, the spin cannot fully relax by aligning itself completely to an external magnetic field because the excess energy cannot be taken up by the substrate. This behaviour is very similar to the reason why the doublon is bound despite repulsive interactions and very stable at an intermediate time scale. Die vorliegende Arbeit untersucht das Auftreten von gebundenen Zustanden in Spektroskopien und ihre Auswirkungen auf die Realzeitdynamik von stark korrelierten Quantensystemen mit Hilfe der Dichtematrix-Renormierungsgruppe (DMRG) und exakter Diagonalisierung (ED). Der Hauptfokus betrifft das Verhalten der sogenannten Dublonen im Hubbard-Modell, bei denen es sich um gebundene Zustande zweier Elektronen handelt. Das Dublon manifistiert sich in hochaufgeloster CVV-Auger-Elektronenspektroskopie (AES) als ein Satellit im Spektrum. Mit Hilfe der DMRG wird sein Verhalten uber alle Fullungen des Hubbard-Modells hinweg nachverfolgt und erklart. Daruberhinaus wird demonstriert dass ein zweiter Satellit auftaucht, der zu gebundenen Zustanden von vier Elektronen (bzw. zwei Dublonen) korrespondiert, fur den der Begriff eingefuhrt wird. Das Quadruplon kann durch Nachste-Nachbarn-Coulomb-Wechselwirkung zusazlich stablisiert werden, die zudem weitere gebunde Zustande erzeugt: Triplonen (drei Elektronen) und Dimeronen (Nachste-Nachbarn-Elektronen). Die Spektren werden auch mit Hilfe des Bethe-Ansatzes analysiert und es werden die Zerfallskanale des Dublons in Spinonen und Holonen in einer Raumdimension aufgedeckt. Die Echtzeit-Dynamik und der Zerfall einer lokalen Dublon-Anregung wird untersucht und es wird gezeigt, dass das Dublon bei mittleren Fullungen stabiler wird. Zudem wird gezeigt, dass Zerfallskanale eine diffusive Komponente zu der Propagation hinzufugen, wahrend eine ballistische Wellenfront erhalten bleibt. Schlieslich wird ein Formalismus fur Pump-Probe-Augerspektroskopie entwickelt und mogliche Aufbauten werden werden diskutiert. Es wird gezeigt, dass die hohe Stabilitat des Dublons auf mittleren Zeitskalen bedingt, dass die Spektren keine Abhangigkeit vom Pump-Probe-Delay zeigen und dass entweder eine Modifikation des Aufbaus oder des Modells vonnoten ist. Der letztere Pfad wird verfolgt und es werden Spektren fur das dynamische Hubbard-Modell ausgerechnet, wo die Doppelbesetzung an einen bosonischen Freiheitsgrad koppelt. Es wird gezeigt, wie die Parameter der Kopplung zu den Verschiebungen und Intensitaten als Funktion des Delays in den Spektren in Beziehung stehen. Der sekundare Fokus liegt in der Untersuchung eines quantenmechanischen oder klassischen Spins als Storstelle auf einem freien oder wechselwirkenden Substrat. Unter anderem wird demonstriert, dass wenn die Wechselwirkung bei Halbfullung stark genug ist, der Spin nicht vollstandig relaxieren kann, d.h. er stellt sich nicht komplett parallel zu einem auseren Magnetfeld, da die uberschussige Energie vom Substrat nicht aufgenommen werden kann. Dieses Verhalten ist ahnlich zu dem Grund warum das Dublon trotz repulsiver Wechselwirkung gebunden und sehr stabil auf einer mittleren Zeitskala ist.
Using the density-matrix renormalization group in combination with the Chebyshev polynomial expansion technique, we study the two-hole excitation spectrum of the one-dimensional Hubbard model in the entire filling range from the completely occupied band ( n = 2) down to half-filling ( n = 1). For strong interactions, the spectra reveal multiplon physics, i.e., relevant final states are characterized by two (doublon), three (triplon), four (quadruplon) and more holes, potentially forming stable compound objects or resonances with finite lifetime. These give rise to several satellites in the spectra with largely different spectral weights as well as to different two-hole, doublon–hole, two-doublon etc continua. The complex multiplon phenomenology is analyzed by interpreting not only local and k -resolved two-hole spectra but also three- and four-hole spectra for the Hubbard model and by referring to effective low-energy models. In addition, a filter-operator technique is presented and applied which allows to extract specific information on the final states at a given excitation energy. While multiplons composed of an odd number of holes do neither form stable compounds nor well-defined resonances unless a nearest-neighbor density interaction V is added to the Hamiltonian, the doublon and the quadruplon are well-defined resonances. The k -resolved four-hole spectrum at n = 2 represents an interesting special case where a completely stable quadruplon turns into a resonance by merging with the doublon–doublon continuum at a critical wave vector. For all fillings with , the doublon lifetime is strongly k -dependent and is even infinite at the Brillouin zone edges as demonstrated by k -resolved two-hole spectra. This can be traced back to the 'hidden' charge-SU(2) symmetry of the model which is explicitly broken off half-filling and gives rise to a massive collective excitation, even for arbitrary higher-dimensional but bipartite lattices.
We study the real-time evolution of a charge-excited state in a one-dimensional ege_{\rm g}eg​-orbital degenerate Hubbard model, by a time-dependent density-matrix renormalization group method. Considering a chain along the zzz direction, electrons hop between adjacent 3z2−r23z^2-r^23z2−r2 orbitals, while x2−y2x^2-y^2x2−y2 orbitals are localized. For the charge-excited state, a holon-doublon pair is introduced into the ground state at quarter filling. At initial time, there is no electron in a holon site, while a pair of electrons occupies 3z2−r23z^2-r^23z2−r2 orbital in a doublon site. As the time evolves, the holon motion is governed by the nearest-neighbor hopping, but the electron pair can transfer between 3z2−r23z^2-r^23z2−r2 orbital and x2−y2x^2-y^2x2−y2 orbital through the pair hopping in addition to the nearest-neighbor hopping. Thus holon and doublon propagate at different speed due to the pair hopping that is characteristic of multi-orbital systems.
We study non-equilibrium properties of one-dimensional Hubbard model by the density-matrix renormalization-group method. First, we demonstrate stability of "doublon", which characterized by double occupation on a site due to the integrability of the model. Next, we present a kind of anomalous transport caused by the doublons created under strong non-equilibrium conditions in an optical lattice system regarded as an ideal testbed to investigate fundamental properties of the Hubbard model. Finally, we give a result on development of the pair correlation function in a strong non-equilibrium condition. This can be understood as a development of coherence among many excited doublons.
We study the real-time dynamics of a hole and doubly occupied site pair, namely, a holon and a doublon, in a 1D Hubbard insulator with on-site and nearest-neighbor Coulomb repulsion. Our analysis shows that the pair is long-lived and the expected decay mechanism to underlying spin excitations is actually inefficient. For a nonzero intersite Coulomb repulsion, we observe that part of the wave function remains in a bound state. Our study also provides insight on the holon-doublon propagation in real space. Because of the one-dimensional nature of the problem, these particles move in opposite directions even in the absence of an applied electric field. The potential relevance of our results to solar cell applications is discussed.