In this paper recent substantial progress in applying the density-matrix renormalizationgroup (DMRG) to the simulation of the time-evolution of strongly correlated quantum systems in one dimension is reviewed. Various approaches to generating a time-evolution numerically are considered. The key focus of this review is on current strategies to circumvent the limitations of the quite small subspace well approximated by DMRG, by either enlarging or changing it as time evolves. All these approaches can be extended to the simulation of mixed, i.e. finitetemperature states. While this paper is phrased almost entirely in standard DMRG language, I finish by considering the alternative formulation of time-evolutions in the language of matrix product states which is less well-known but conceptually more powerful.KEYWORDS: density-matrix renormalization-group, strongly correlated systems, quantum information theory IntroductionMore than ten years after the invention of the Density-Matrix Renormalization Group (DMRG) by Steve White, 1, 2 this method has become the method of choice for the numerical simulation of the equilibrium properties of strongly correlated one-dimensional quantum systems.3, 4 Ground-state properties of both bosonic and fermionic systems have been calculated often at almost machine precision and comparatively low computational cost. While these results have been of prime importance in understanding the details of gapped Haldane systems or critical Luttinger liquids, to name but a few applications, for a long time hardly any attention had been paid to the time-evolution of strongly correlated quantum systems, both due to the comparative lack of experimental input in the past and to the inherent difficulties of actually calculating such time-evolutions. The last years have seen an increasing number of experimental results on non-trivial time-evolutions. Perhaps most spectacular was the recent mastery of storing ultracold bosonic atoms in a magnetic trap superimposed by an optical lattice: This has allowed to drive these atoms, at will, by time-dependent variations of the optical lattice strength, from the superfluid (metallic) to the Mott insulating regime. These regimes are linked by one of the key phase transitions in strongly correlated systems.5 Quite generally, progress in the fields of nanoelectronics and spintronics raises the question how (strongly correlated) quantum many-body systems react to external timedependent perturbations and how transport can be calculated quantitatively also far from the linear-response regime. On the computational physicists' side, the last twelve months or so have seen an extraordinary surge of activity in developing DMRG variants applicable to timeevolutions, such that by now we have various very powerful and conceptually innovative DMRG algorithms for pure as well as mixed quantum states, for non-dissipative as well as dissipative dynamics.The relationship between these various algorithms is

Heavy electron superconductors mostly have anisotropic gap function which vanishes (has nodes) for certain directions in the momentum space. Since the nodal structure is closely related to the pairing mechanism, its experimental determination is very important. In anisotropic superconductors, low energy spectrum of the quasiparticles in the vortex state much depends on the nodal structure. In particular, the electronic specific heat has been demonstrated to exhibit the characteristic dependence on the angle between the field and the nodal direction, thereby the nodal structure can be probed experimentally. In this contribution, we present our recent experimental results on the field-orientation dependence of the specific heat on f electron superconductors CeRu 2 , CeCoIn 5 , PrOs 4 Sb 12 , and URu 2 Si 2 , and discuss their gap structures.

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Inelastic neutron scattering measurements of magnetic excitations in the charge-ordered state of Yb4As3 have been performed under magnetic field up to about 6 T. By applying a magnetic field, the spectrum at the one-dimensional wave vector q = 1 [ pi/d] changes drastically from a broad one corresponding to the spinon excitation continuum of the one-dimensional S = 1 / 2 spin system to a sharp one at a finite energy, indicating the opening of an energy gap in the system. The magnetic field dependence of the gap is well fitted by the power law H2/3. The experimental result gives strong evidence for the existence of a staggered field alternating along Yb3+ chains induced by the Dzyaloshinsky-Moriya interaction.

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