Three-loop order approach to flat polymerized membranes.

We derive the three-loop order renormalization group equations that describe the flat phase of polymerized membranes within the modified minimal subtraction scheme, following the pioneering one-loop order computation of Aronovitz and Lubensky [Phys. Rev. Lett. 60, 2634 (1988)10.1103/PhysRevLett.60.2634] and the recent two-loop order one of Coquand, Mouhanna, and Teber [Phys. Rev. E 101, 062104 (2020)2470-004510.1103/PhysRevE.101.062104]. We analyze the fixed points of these equations and compute the associated field anomalous dimension η at three-loop order. Our results display a marked proximity with those obtained using nonperturbative techniques and reexpanded in powers of ε=4-D. Moreover, the three-loop order value that we get for η at the stable fixed point, η=0.8872, in D=2, is compatible with known theoretical results and within the range of accepted numerical values.

Flat phase of quenched disordered membranes at three-loop order.

We study quenched disordered polymerized membranes in their flat phase by means of a three-loop perturbative analysis performed in dimension D=4-ε. We derive the renormalization group equations at this order and solve them up to order ε^{3}. Our results confirm those obtained by Coquand et al. within a nonperturbative approach [Phys. Rev. E 97, 030102(R) (2018)PREHBM2470-004510.1103/PhysRevE.97.030102] predicting a finite-temperature, finite-disorder wrinkling transition and those obtained by Coquand and Mouhanna within a recent two-loop order approach [Phys. Rev. E 103, L031001 (2021)PREHBM2470-004510.1103/PhysRevE.103.L031001], while correcting some of the results obtained in this last reference. We compute the anomalous dimensions that characterize the scaling behavior at the various fixed points of the renormalization group flow diagram. They appear to be in strong agreement with those predicted within the nonperturbative context.

Wrinkling transition in quenched disordered membranes at two loops.

We investigate the flat phase of quenched disordered polymerized membranes by means of a two-loop, weak-coupling computation performed near their upper critical dimension D_{uc}=4, generalizing the one-loop computation of Morse et al. [D. C. Morse et al., Phys. Rev. A 45, R2151 (1992)PLRAAN1050-294710.1103/PhysRevA.45.R2151; D. C. Morse and T. C. Lubensky, Phys. Rev. A 46, 1751 (1992)PLRAAN1050-294710.1103/PhysRevA.46.1751]. Our work confirms the existence of the finite-temperature, finite-disorder wrinkling transition, which has been recently identified by Coquand et al. [O. Coquand et al., Phys. Rev. E 97, 030102(R) (2018)2470-004510.1103/PhysRevE.97.030102] using a nonperturbative renormalization group approach. We also point out ambiguities in the two-loop computation that prevent the exact identification of the properties of the novel fixed point associated with the wrinkling transition, which very likely requires a three-loop order approach.

Flat phase of polymerized membranes at two-loop order.

We investigate two complementary field-theoretical models describing the flat phase of polymerized-phantom-membranes by means of a two-loop, weak-coupling, perturbative approach performed near the upper critical dimension D_{uc}=4, extending the one-loop computation of Aronovitz and Lubensky [Phys. Rev. Lett. 60, 2634 (1988)PRLTAO0031-900710.1103/PhysRevLett.60.2634]. We derive the renormalization group equations within the modified minimal substraction scheme, then analyze the corrections coming from two-loop with a particular attention paid to the anomalous dimension and the asymptotic infrared properties of the renormalization group flow. We finally compare our results to those provided by nonperturbative techniques used to investigate these two models.

Universal behaviors in the wrinkling transition of disordered membranes.

The wrinkling transition experimentally identified by Mutz et al. [Phys. Rev. Lett. 67, 923 (1991)PRLTAO0031-900710.1103/PhysRevLett.67.923] and then thoroughly studied by Chaieb et al. [Phys. Rev. Lett. 96, 078101 (2006)]PRLTAO0031-900710.1103/PhysRevLett.96.078101 in partially polymerized lipid membranes is reconsidered. One shows that the features associated with this transition, notably the various scaling behaviors of the height-height correlation functions that have been observed, are qualitatively and quantitatively well described by a recent nonperturbative renormalization group approach to quenched disordered membranes by Coquand et al. [Phys. Rev E 97, 030102(R) (2018)]2470-004510.1103/PhysRevE.97.030102. As these behaviors are associated with fixed points of renormalization group transformations they are universal and should also be observed in, e.g., defective graphene and graphene-like materials.

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.

Flat phase of quantum polymerized membranes.

We investigate the flat phase of quantum polymerized phantom membranes by means of a nonperturbative renormalization group approach. We first implement this formalism for general quantum polymerized membranes and derive the flow equations that encompass both quantum and thermal fluctuations. We then deduce and analyze the flow equations relevant to study the flat phase and discuss their salient features: quantum to classical crossover and, in each of these regimes, strong to weak coupling crossover. We finally illustrate these features in the context of free-standing graphene physics.

First-order phase transitions in polymerized phantom membranes.

The crumpled-to-flat phase transition that occurs in D-dimensional polymerized phantom membranes embedded in a d-dimensional space is investigated nonperturbatively using a field expansion up to order 8 in powers of the order parameter. We get the critical dimension dcr(D) that separates a second-order region from a first-order one everywhere between D=4 and 2. Our approach strongly suggests that the phase transitions that take place in physical membranes are of first order in agreement with most recent numerical simulations.

Crumpled-to-tubule transition in anisotropic polymerized membranes: beyond the ϵ expansion.

Anisotropic D-dimensional polymerized phantom membranes are investigated within a nonperturbative renormalization group framework. One focuses on the transition between a high-temperature, crumpled phase and a low-temperature, tubular phase where the membrane is flat along one direction and crumpled along the other ones. While the upper critical dimension--D(uc)=5/2--is close to D=2, the weak-coupling perturbative approach is qualitatively and quantitatively wrong. We show that our approach is free of the problems encountered within the perturbative framework and provides physically meaningful critical quantities.

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

We investigate the relation between spontaneous and explicit replica symmetry breaking in the theory of disordered systems. On general ground, we prove the equivalence between the replicon operator associated with the stability of the replica-symmetric solution in the standard replica scheme and the operator signaling a breakdown of the solution with analytic field dependence in a scheme in which replica symmetry is explicitly broken by applied sources. This opens the possibility to study, via the recently developed functional renormalization group, unresolved questions related to spontaneous replica symmetry breaking and spin-glass behavior in finite-dimensional disordered systems.

Crumpling transition and flat phase of polymerized phantom membranes.

Polymerized phantom membranes are revisited using a nonperturbative renormalization-group approach. This allows one to investigate both the crumpling transition and the low-temperature flat phase in any internal dimension D and embedding dimension d and to determine the lower critical dimension. The crumpling phase transition for physical membranes is found to be of second order within our approximation. A weak first-order behavior, as observed in recent Monte Carlo simulations, is however not excluded.

Short-time dynamics of a family of XY noncollinear magnets.

Critical scaling and universality in the short-time dynamics for antiferromagnetic models on a three-dimensional stacked triangular lattice are investigated using Monte Carlo simulation. We have determined the critical point by searching for the best power law for the order parameter as a function of time and measured the critical exponents. Our results indicate that it is possible to distinguish weak first-order from second-order phase transitions and confirm that XY antiferromagnetic systems undergo a (weak) first-order phase transition accompanied by pseudocritical scaling.

Frustrated heisenberg magnets: A nonperturbative approach

Frustrated magnets are a notorious example of where usual perturbative methods fail. Using a nonperturbative Wilson-like approach, we get a coherent picture of the physics of frustrated Heisenberg magnets everywhere between d = 2 and d = 4. We recover all known perturbative results in a single framework and find the transition to be weakly of first order in d = 3. We compute effective exponents that are in good agreement with numerical and experimental data.