We applied the functional renormalization group to elastic systems such as interfaces or lattices pinned by correlated quenched disorder considering two different types of correlations: columnar disorder and quenched defects correlated as $sim x^{-a} $ for large separation $x$. We computed the critical exponents and the response to a transverse field $h$ to two-loop order. The correlated disorder violates the statistical tilt symmetry resulting in nonlinear response to a tilt. Elastic systems with columnar disorder exhibit a transverse Meissner effect: disorder generates the critical field $h_c$ below which there is no response to a tilt, and above which the tilt angle behaves as $varthetasim(h-h_c)^{phi}$ with a universal exponent $phi 1$. The obtained results is applied to the Kardar-Parisi-Zhang equation with temporally correlated noise. We also studied the long-distance properties of $O(N)$ spin systems with long-range correlated random fields and random anisotropies. Below the lower critical dimension, there exist two different types of quasi-long-range-order with zero order-parameter but infinite correlation length.

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