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.

We propose to observe many-body localization in cold atomic gases by realizing a Bose-Hubbard chain with binary disorder and studying its nonequilibrium dynamics. In particular, we show that measuring the difference in occupation between even and odd sites, starting from a prepared density-wave state, provides clear signatures of localization. Furthermore, we confirm as hallmarks of the many-body localized phase a logarithmic increase of the entanglement entropy in time and Poissonian level statistics. Our numerical density-matrix renormalization group calculations for infinite system size are based on a purification approach; this allows us to perform the disorder average exactly, thus producing data without any statistical noise and with maximal simulation times of up to a factor 10 longer than in the clean case.

We investigate the phase transition between an ergodic and a many-body localized phase in infinite anisotropic spin-1/2 Heisenberg chains with binary disorder. Starting from the Néel state, we analyze the decay of antiferromagnetic order ms(t) and the growth of entanglement entropy Sent(t) during unitary time evolution. Near the phase transition we find that ms(t) decays exponentially to its asymptotic value ms(∞) = 0 in the localized phase while the data are consistent with a power-law decay at long times in the ergodic phase. In the localized phase, ms(∞) shows an exponential sensitivity on disorder with a critical exponent ν ∼ 0.9. The entanglement entropy in the ergodic phase grows sub-ballistically, Sent(t) ∼ t α , α ≤ 1, with α varying continuously as a function of disorder. Exact diagonalizations for small systems, on the other hand, do not show a clear scaling with system size and attempts to determine the phase boundary from these data seem to overestimate the extent of the ergodic phase.