Quantum typicality: A novel numerical approach to dynamics and thermalization

Robin Steinigeweg, TU Braunschweig

03.07.2015 at 09:00 

The concept of typicality states that a single pure state can have the same properties as the full statistical ensemble. This concept is not restricted to specific states and applies to the overwhelming majority of all possible states, drawn at random from a high-dimensional Hilbert space. In the cleanest realization, even a  single eigenstate of the Hamiltonian may feature the properties of the full equilibrium density matrix, assumed in the well-known eigenstate thermalization hypothesis. The notion of property is manifold in this context and also refers to the expectation values of observables. Remarkably, typicality is not only a static concept and includes the dynamics of expectation values. Recently, it has become clear that typicality even provides the basis for powerful numerical approaches to the dynamics [1] and thermalization [2] of quantum many-particle systems at nonzero temperatures. These approaches are in the center of my talk.

In my talk, I demonstrate that typicality allows for significant progress in the study of real-time spin and energy dynamics of low-dimensional quantum magnets. To this end, I present a numerical analysis of current  autocorrelation fuctions of the integrable XXZ spin-1/2 chain [1] and nonintegrable modifications with staggered magnetic fields [3] and inter-chain couplings [4]. This analysis includes a comprehensive comparison with state-of-the-art methods, including exact and Lanczos diagonalization, time-dependent density-matrix renormalization group, and perturbation theory. This comparison unveils that typicality is satisfied in finite systems over a wide range of temperature and is fulfilled in both, integrable and nonintegrable systems. For the integrable case, I calculate the long-time dynamics of the spin current and extract the spin Drude weight for large systems outside the range of exact diagonalization. I particularly provide strong evidence that the high-temperature Drude weight vanishes at the isotropic point. For the nonintegrable cases, I obtain the full relaxation curve of the  energy current and determine the heat conductivity as a function of model parameters and temperature.

[1]R. Steinigeweg, J. Gemmer, and W. Brenig, Phys. Rev. Lett. 112, 120601 (2014).
[2]R. Steinigeweg et al., Phys. Rev. Lett. 112, 130403 (2014).
[3]R. Steinigeweg, J. Gemmer, and W. Brenig, Phys. Rev. B 91, 104404 (2015).
[4]R. Steinigeweg, J. Herbrych, X. Zotos, and W. Brenig, arXiv:1503.03871 (2015).