Spontaneous versus explicit replica symmetry breaking in the theory of disordered systems

Bose-glass phase of a one-dimensional disordered Bose fluid: Metastable states, quantum tunneling, and droplets

We study a one-dimensional disordered Bose fluid using bosonization, the replica method, and a nonperturbative functional renormalization-group approach. We find that the Bose-glass phase is described by a fully attractive strong-disorder fixed point characterized by a singular disorder correlator whose functional dependence assumes a cuspy form that is related to the existence of metastable states. At nonzero momentum scale k, quantum tunneling between the ground state and low-lying metastable states leads to a rounding of the cusp singularity into a quantum boundary layer (QBL). The width of the QBL depends on an effective Luttinger parameter K_{k}∼k^{θ} that vanishes with an exponent θ=z-1 related to the dynamical critical exponent z. The QBL encodes the existence of rare "superfluid" regions, controls the low-energy dynamics, and yields a (dissipative) conductivity vanishing as ω^{2} in the low-frequency limit. These results reveal the glassy properties (pinning, "shocks," or static avalanches) of the Bose-glass phase and can be understood within the "droplet" picture put forward for the description of glassy (classical) systems.

Disorder-Driven Quantum Transition in Relativistic Semimetals: Functional Renormalization via the Porous Medium Equation

In the presence of randomness, a relativistic semimetal undergoes a quantum transition towards a diffusive phase. A standard approach relates this transition to the U(N) Gross-Neveu model in the limit of N→0. We show that the corresponding fixed point is infinitely unstable, demonstrating the necessity to include fluctuations beyond the usual Gaussian approximation. We develop a functional renormalization group method amenable to include these effects and show that the disorder distribution renormalizes following the so-called porous medium equation. We find that the transition is controlled by a nonanalytic fixed point drastically different from that of the U(N) Gross-Neveu model. Our approach provides a unique mechanism of spontaneous generation of a finite density of states and also characterizes the scaling behavior of the broad distribution of fluctuations close to the transition. It can be applied to other problems where nonanalytic effects may play a role, such as the Anderson localization transition.

Glassy phase in quenched disordered crystalline membranes

We investigate the flat phase of D-dimensional crystalline membranes embedded in a d-dimensional space and submitted to both metric and curvature quenched disorders using a nonperturbative renormalization group approach. We identify a second-order phase transition controlled by a finite-temperature, finite-disorder fixed point unreachable within the leading order of ε=4-D and 1/d expansions. This critical point divides the flow diagram into two basins of attraction: that associated with the finite-temperature fixed point controlling the long-distance behavior of disorder-free membranes and that associated with the zero-temperature, finite-disorder fixed point. Our work thus strongly suggests the existence of a whole low-temperature glassy phase for quenched disordered crystalline membranes and, possibly, for graphene and graphene-like compounds.