The interaction between electrons, phonons, and spins in solids becomes manifest on the macroscopic scale trough the very rich equilibrium phase diagrams of correlated materials. One goal of our theoretical effort is to explore the additional manifold of metastable phases that can only be addressed along non-thermal pathways, e.g., by means of photo-excitation. Understanding those non-thermal photo-excited states is an important step towards controlling material properties on ultra-fast timescales.
One way to prepare a metastable state has been described recently in a simple one-band Hubbard model [work with Philipp Werner at ETH Zurich, Phys. Rev. B 84, 035122 (2011)]: We start at interaction U>>W (W is the bandwidth), where the system is a half-filled Mott insulator (see illustration, 1). The photoemission spectrum (upper left plot) reveals a filled lower Hubbard band and an almost unoccupied upper Hubbard band. A strong laser pulse almost instantaneously creates doublons and holes in the system. Because there is no effective scattering process to release their large energy, these excitations cannot simply recombine. The resulting stability of doublons and holes provides the necessary relaxation bottleneck which allows the system to stabilize into a quasi-stationary photo-excited state on the timescale of the inverse bandwidth (2). This state can be clearly distinguished by its photoemission spectrum from a state that is thermally excited to the same energy (3). The thermalization time of the photo-excited state is exponentially long in U/W, in accordance with the lifetime of doublons that has been measured in experiments with ultra-cold atoms in an optical lattice.
[R. Sensarma, D. Pekker, E. Altman, E. Demler, N. Strohmaier, D. Greif, R. Jördens, L. Tarruell, H. Moritz, and T. Esslinger, Phys. Rev. B 82, 224302 (2010)].
Our goal is to build on those investigations and study long-lived excited states in more complex system, including antiferromagnetic order and coupling to lattice degrees of freedom. In the simple Mott insulator described above, the question is still open which process would yield a photo-excited state which is metallic and has a proper Drude response.
A dynamical transition is an abrupt change in the dynamical behavior of a system, depending on parameters such as the interaction strength or on external influences such as the excitation energy. In a collaboration with Marcus Kollar and Philipp Werner we found evidence that such a transition occurs in the single-band Hubbard model, in a parameter regime where the equilibrium phase diagram displays only a smooth crossover between metallic and insulating behavior [Eckstein, Kollar, and Werner, Phys. Rev. Lett. 103, 056403 (2009)]. The transition is between phases in which the model resembles the non-ergodic behavior of simple models on short timescales. The three panels in the illustration show the time evolution of the momentum distribution n(k) in the Hubbard model along a cut through the Brillouin zone, after the repulsive interaction is suddenly switched on in the free Fermi sea. For small interactions and large interactions, the systems is trapped in long-lived non-thermal states. At U/J=3.3, the relaxation behavior changes from a monotonous decay to oscillations, and rapid thermalization is observed on the timescale of the inverse bandwidth. In future, we would like to understand whether similar transitions are a generic feature in the thermalization of correlated systems.
The true bottleneck for our microscopic calculations is the solution of impurity models in real time, i.e., the dynamics of a single interacting site or a small cluster which is coupled to a continuous manifold of noninteracting bath states. Such models can provide a realistic description of a quantum dot - on the other hand, they are the effective model onto which lattice models are mapped within DMFT and its extensions.
Continuos-time Monte Carlo yields the numerically exact solution of a single impurity model. The approach has given useful insights both into the dynamics of the Hubbard model within DMFT [Eckstein, Kollar, and Werner, Phys. Rev. Lett. 103, 056403 (2009)], and to the steady state transport through a quantum dot [Werner, Oka, Eckstein, and Millis, Phys. Rev. B 81, 035108 (2010)]. However, it is applicable only to short times due to the dynamical sign problem.
Together with Philipp Werner we have implemented a systematic perturbation theory around the limit of a decoupled impurity [M. Eckstein and Ph. Werner, Phys. Rev. B 82, 115115 (2010)]. This allows us to simulate considerably larger times than with Monte Carlo, while we can still keep track of the systematic error by comparing results from various orders of the approximation. The method is the prerequisite for a number of investigations which start from an excitation of the Mott insulating state.
Exact diagonalization (ED) is another promising approach: The impurity model with a continuous bath degrees of freedom is truncated to finitely many bath sites, and the resulting model is then solved exactly. The method is non-perturbative and can yield large times, but the dynamics of a small system will suffer from severe finite size effects.
To tackle the problem of the real-time dynamics in impurity models one will ultimately have to make use of all three of those approaches, and combine them in a suitable way. For example, the strong-coupling expansions can take an ED solution of a small cluster as its zeroth order.