We study the nonequilibrium time evolution of the spin-1/2 anisotropic Heisenberg (XXZ) spin chain, with a choice of dimer product and Néel states as initial states. We investigate numerically various short-ranged spin correlators in the long-time limit and find that they deviate significantly from predictions based on the generalized Gibbs ensemble (GGE) hypotheses. By computing the asymptotic spin correlators within the recently proposed quench-action formalism [Phys. Rev. Lett. 110, 257203 (2013)], however, we find excellent agreement with the numerical data. We, therefore, conclude that the GGE cannot give a complete description even of local observables, while the quench-action formalism correctly captures the steady state in this case.

Following our previous work [PRL 113 (2014) 09020] we present here a detailed comparison of the quench action approach and the predictions of the generalized Gibbs ensemble, with the result that while the quench action formalism correctly captures the steady state, the GGE does not give a correct description of local short-distance correlation functions. We extend our studies to include another initial state, the so-called q-dimer state. We present important details of our construction, including new results concerning exact overlaps for the dimer and q-dimer states, and we also give an exact solution of the quench-action-based overlap-TBA for the q-dimer. Furthermore, we extend our computations to include the xx spin correlations besides the zz correlations treated previously, and give a detailed discussion of the underlying reasons for the failure of the GGE, especially in the light of new developments.

We revisit the question of whether and how the steady states arising after non-equilibrium time evolution in integrable models (and in particular in the XXZ spin chain) can be described by the so-called Generalized Gibbs Ensemble (GGE). It is known that the micro-canonical ensemble built on a complete set of charges correctly describes the long-time limit of local observables, and recently a canonical ensemble was built by Ilievski et. al. using particle occupation number operators. Here we provide an alternative construction by considering truncated GGE's (tGGE's) that only include a finite number of well localized conserved operators. It is shown that the tGGE's can approximate the steady states with arbitrary precision, i.e. all physical observables are exactly reproduced in the infinite truncation limit. In addition, we show that a complete canonical ensemble can in fact be built in terms of a new (discrete) set of charges built as linear combinations of the standard ones.Our general arguments are applied to concrete quench situations in the XXZ chain, where the initial states are simple two-site or four-site product states. Depending on the quench we find that numerical results for the local correlators can be obtained with remarkable precision using truncated GGE's with only 10-100 charges.

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A two dimensional disordered system of non-interacting fermions in a homogeneous magnetic field is investigated numerically. By introducing a new magnetic gauge, we explore the renormalization group (RG) flow of the longitudinal and Hall conductances with higher precision than previously studied, and find that the flow is consistent with the predictions of Pruisken and Khmelnitskii. The extracted critical exponents agree with the results obtained by using transfer matrix methods. The necessity of a second parameter is also reflected in the level curvature distribution. Near the critical point the distribution slightly differs from the prediciton of random matrix theory, in agreement with previous works. Close to the quantum Hall fixed points the distribution is lognormal since here states are strongly localized.