It has been conjectured that transport in integrable one-dimensional systems is necessarily ballistic. The large diffusive response seen experimentally in nearly ideal realizations of the S=1/2 1D Heisenberg model is therefore puzzling and has not been explained so far. Here, we show that, contrary to common belief, diffusion is universally present in interacting 1D systems subject to a periodic lattice potential. We present a parameter-free formula for the spin-lattice relaxation rate which is in excellent agreement with experiment. Furthermore, we calculate the current decay directly in the thermodynamic limit using a time-dependent density matrix renormalization group algorithm and show that an anomalously large time scale exists even at high temperatures.

In integrable one-dimensional quantum systems an infinite set of local conserved quantities exists which can prevent a current from decaying completely. For cases like the spin current in the XXZ model at zero magnetic field or the charge current in the attractive Hubbard model at half filling, however, the current operator does not have overlap with any of the local conserved quantities. We show that in these situations transport at finite temperatures is dominated by a diffusive contribution with the Drude weight being either small or even zero. For the XXZ model we discuss in detail the relation between our results, the phenomenological theory of spin diffusion, and measurements of the spin-lattice relaxation rate in spin chain compounds. Furthermore, we study the Haldane-Shastry model where the current operator is also orthogonal to the set of conserved quantities associated with integrability but becomes itself conserved in the thermodynamic limit.

A boundary transfer matrix formulation allows to calculate the Loschmidt echo for onedimensional quantum systems in the thermodynamic limit. We show that nonanalyticities in the Loschmidt echo and zeros for the Loschmidt amplitude in the complex plane (Fisher zeros) are caused by a crossing of eigenvalues in the spectrum of the transfer matrix. Using a density-matrix renormalization group algorithm applied to these transfer matrices we numerically investigate the Loschmidt echo and the Fisher zeros for quantum quenches in the XXZ model with a uniform and a staggered magnetic field. We give examples-both in the integrable and the nonintegrable cases-where the Loschmidt echo does not show nonanalyticities although the quench leads across an equilibrium phase transition, and examples where nonanalyticities appear for quenches within the same phase. For a quench to the free fermion point, we analytically show that the Fisher zeros sensitively depend on the initial state and can lie exactly on the real axis already for finite system size. Furthermore, we use bosonization to analyze our numerical results for quenches within the Luttinger liquid phase.