Cusps and shocks in the renormalized potential of glassy random manifolds: How Functional Renormalization Group and Replica Symmetry Breaking fit together

Abstract

We compute the Functional Renormalization Group (FRG) disorder- correlator function R(v) for d-dimensional elastic manifolds pinned by a random potential in the limit of infinite embedding space dimension N. It measures the equilibrium response of the manifold in a quadratic potential well as the center of the well is varied from 0 to v. We find two distinct scaling regimes: (i) a "single shock" regime, v^2 ~ 1/L^d where L^d is the system volume and (ii) a "thermodynamic" regime, v^2 ~ N. In regime (i) all the equivalent replica symmetry breaking (RSB) saddle points within the Gaussian variational approximation contribute, while in regime (ii) the effect of RSB enters only through a single anomaly. When the RSB is continuous (e.g., for short-range disorder, in dimension 2 <= d <= 4), we prove that regime (ii) yields the large-N FRG function obtained previously. In that case, the disorder correlator exhibits a cusp in both regimes, though with different amplitudes and of different physical origin. When the RSB solution is 1-step and non- marginal (e.g., d < 2 for SR disorder), the correlator R(v) in regime (ii) is considerably reduced, and exhibits no cusp. Solutions of the FRG flow corresponding to non-equilibrium states are discussed as well. In all cases the regime (i) exhibits a cusp non-analyticity at T=0, whose form and thermal rounding at finite T is obtained exactly and interpreted in terms of shocks. The results are compared with previous work, and consequences for manifolds at finite N, as well as extensions to spin glasses and related models are discussed.

arXiv commentv2: Note added in proof

References in corpus (51)

• Analytical Solution of the Off-Equilibrium Dynamics of a Long Range Spin-Glass Model
• Elastic theory of flux lattices in presence of weak disorder
• Creep and depinning in disordered media
• Elastic Theory of pinned flux lattices
• The structural glass transition and the entropy of the metastable states
• Vortex-glass phases in type-II superconductors
• Phase Transitions in the Coloring of Random Graphs
• Renormalization of pinned elastic systems: how does it work beyond one loop ?
• 2-loop Functional Renormalization Group Theory of the Depinning Transition
• Two-Loop Renormalization Group Analysis of the Burgers-Kardar-Parisi-Zhang Equation
• SCALING AND INTERMITTENCY IN BURGERS' TURBULENCE
• Glass models on Bethe lattices
• Instability of one-step replica-symmetry-broken phase in satisfiability problems
• Functional Renormalization Group and the Field Theory of Disordered Elastic Systems
• Large N Expansion for 4-Epsilon Dimensional Oriented Manifolds in Random Media
• The glass transition and the Coulomb gap in electron glasses
• Large times off-equilibrium dynamics of a particle in a random potential
• On the Perturbation Expansion of the KPZ-Equation
• Variational theory of elastic manifolds with correlated disorder and localization of interacting quantum particles
• Nonperturbative functional renormalization group for random field models: the way out of dimensional reduction
• On the Renormalization of the Kardar-Parisi-Zhang equation
• The Large Scale Energy Landscapes of Randomly Pinned Objects
• A unified picture of ferromagnetism, quasi-long range order and criticality in random field models
• Numerical Calculation of the Functional renormalization group fixed-point functions at the depinning transition
• Mean field theory for the three-dimensional Coulomb glass
• Creep via dynamical functional renormalization group
• Quasi-long range order in glass states of impure liquid crystals, magnets, and superconductors
• Threshold values, stability analysis and high-q asymptotics for the coloring problem on random graphs
• Measuring functional renormalization group fixed-point functions for pinned manifolds
• Amorphous-amorphous transition and the two-step replica symmetry breaking phase
• Quasi-long-range order in the random anisotropy Heisenberg model: functional renormalization group in 4-εdimensions
• Random field spin models beyond one loop: a mechanism for decreasing the lower critical dimension
• Stability of Random-Field and Random-Anisotropy Fixed Points at large N
• Functional renormalization group at large N for random manifolds
• Low temperature solution of the Sherrington-Kirkpatrick model
• Higher correlations, universal distributions and finite size scaling in the field theory of depinning
• Can non-linear elasticity explain contact-line roughness at depinning?
• 2-loop Functional Renormalization for elastic manifolds pinned by disorder in N dimensions
• Two-loop Functional Renormalization Group of the Random Field and Random Anisotropy O(N) Models
• Functional renormalization group for anisotropic depinning and relation to branching processes
• Thermal fluctuations in pinned elastic systems: field theory of rare events and droplets
• Critical Discussion of the 2-Loop Calculations for the KPZ-Equation
• Exact renormalization group and applications to disordered problems: part I
• Statics and dynamics of elastic manifolds in media with long-range correlated disorder
• Universal distribution of threshold forces at the depinning transition
• Broad relaxation spectrum and the field theory of glassy dynamics for pinned elastic systems
• Derivation of the Functional Renormalization Group Beta-Function at order 1/N for Manifolds Pinned by Disorder
• Structural glass on a lattice in the limit of infinite dimensions
• Dynamic Compressibility and aging in Wigner crystals and quantum glasses
• Interface fluctuations in disordered systems: Universality and non-Gaussian statistics
• Low Temperature Specific Heat of some Quantum Mean Field glassy phases

Cited by in corpus (23)

• Avalanches in mean-field models and the Barkhausen noise in spin-glasses
• Exponential number of equilibria and depinning threshold for a directed polymer in a random potential
• Elastic systems with correlated disorder: Response to tilt and application to surface growth
• Hessian spectrum at the global minimum of high-dimensional random landscapes
• Temperature-induced crossovers in the static roughness of a one-dimensional interface
• Freezing Transition in Decaying Burgers Turbulence and Random Matrix Dualities
• Step-wise responses in mesoscopic glassy systems: a mean field approach
• Bose-glass phase of a one-dimensional disordered Bose fluid: metastable states, quantum tunneling and droplets
• Spherical spin glass model with external field
• Universality class of replica symmetry breaking, scaling behavior, and the low-temperature fixed-point order function of the Sherrington-Kirkpatrick model
• Manifolds in high dimensional random landscape: complexity of stationary points and depinning
• Field theory of disordered systems -- Avalanches of an elastic interface in a random medium
• Spontaneous versus explicit replica symmetry breaking in the theory of disordered systems
• Field Theory of Disordered Elastic Interfaces at 3-Loop Order: Critical Exponents and Scaling Functions
• Functional renormalization-group approach to decaying turbulence
• Field Theory of Disordered Elastic Interfaces at 3-Loop Order: The $\beta$-Function
• One step replica symmetry breaking and extreme order statistics of logarithmic REMs
• The random anisotropy model revisited
• Non-differentiability of the effective potential and the replica symmetry breaking in the random energy model
• Manifolds pinned by a high-dimensional random landscape: Hessian at the global energy minimum
• Theory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles
• Disordered statistical physics in low dimensions: extremes, glass transition, and localization
• The quantum $p$-spin renormalization group in the large $N$ limit as a benchmark for functional renormalization group