Self-consistent Screening Approximation Research Articles

Lessons from [formula omitted] models in one dimension

Various topics related to the O(N) model in one spacetime dimension (i.e. ordinary quantum mechanics) are considered. The focus is on a pedagogical presentation of quantum field theory methods in a simpler context where many exact results are available, but certain subtleties are discussed which may be of interest to active researchers in higher dimensional field theories as well.Large N methods are introduced in the context of the zero-dimensional path integral and the connection to Stirling’s series is shown. The entire spectrum of the O(N) model, which includes the familiar l(l+1) eigenvalues of the quantum rotor as a special case, is found both diagrammatically through large N methods and by using Ward identities. The large N methods are already exact at ON−1 and the ON−2 corrections are explicitly shown to vanish. Peculiarities of gauge theories in d=1 are discussed in the context of the CPN−1 sigma model, and the spectrum of a more general squashed sphere sigma model is found. The precise connection between the O(N) model and the linear sigma model with a ϕ4 interaction is discussed. A valid form of the self-consistent screening approximation (SCSA) applicable to O(N) models with a hard constraint is presented. The point is made that at least in d=1 the SCSA may do worse than simply truncating the large N expansion to ON−1 even for small N. In both the supersymmetric and non-supersymmetric versions of the O(N) model, naive equations of motion relating vacuum expectation values are shown to be corrected by regularization-dependent finite corrections arising from contact terms associated to the equation of constraint.

Thermal ripples in bilayer graphene

We study thermal fluctuations of freestanding bilayer graphene subject to vanishing external tension. Within a phenomenological theory, the system is described as a stack of two continuum crystalline membranes, characterized by finite elastic moduli and a nonzero bending rigidity. A nonlinear rotationally invariant model guided by elasticity theory is developed to describe interlayer interactions. After neglection of in-plane phonon nonlinearities and anharmonic interactions involving interlayer shear and compression modes, an effective theory for soft flexural fluctuations of the bilayer is constructed. The resulting model, neglecting anisotropic interactions, has the same form of a well-known effective theory for out-of-plane fluctuations in a single-layer membrane, but with a strongly wave-vector-dependent bare bending rigidity. Focusing on AB-stacked bilayer graphene, parameters governing interlayer interactions in the theory are derived by first-principles calculations. Statistical-mechanical properties of interacting flexural fluctuations are then calculated by a numerical iterative solution of field-theory integral equations within the self-consistent screening approximation. The bare bending rigidity in the considered model exhibits a crossover between a long-wavelength regime governed by in-plane elastic stress and a short wavelength region controlled by monolayer curvature stiffness. Interactions between flexural fluctuations drive a further crossover between a harmonic and a strong-coupling regime, characterized by anomalous scale invariance. The overlap and interplay between these two crossover behaviors is analyzed at varying temperatures.

Scaling behavior of crystalline membranes: An ε-expansion approach

We study the scaling behavior of two-dimensional (2D) crystalline membranes in the flat phase by a renormalization group (RG) method and an ε-expansion. Generalization of the problem to non-integer dimensions, necessary to control the ε-expansion, is achieved by dimensional continuation of a well-known effective theory describing out-of-plane fluctuations coupled to phonon-mediated interactions via a scalar composite field, equivalent for small deformations to the local Gaussian curvature. The effective theory, which will be referred to as Gaussian curvature interaction (GCI) model, is equivalent to theories of elastic D-dimensional manifolds fluctuating in a (D+dc)-dimensional embedding space in the physical case D=2 for arbitrary dc. For D≠2, instead, the GCI model is not equivalent to a direct dimensional continuation of elastic membrane theory and it defines an alternative generalization to generic internal dimension D. After decoupling interactions through a Hubbard-Stratonovich transformation, we study the GCI model by perturbative field-theoretic RG within the framework of an expansion in ε=(4−D). We calculate explicitly RG functions at two-loop order and determine the exponent η characterizing the long-wavelength scaling of correlation functions to order ε2. The value of η at this order is shown to be insensitive to Feynman diagrams involving vertex corrections. As a consequence, the self-consistent screening approximation for the GCI model is shown to be exact to O(ε2). In the physical case of a single out-of-plane displacement field, dc=1, the O(ε2) correction is suppressed by a small numerical prefactor. As a result, despite the large value of ε=2, extrapolation of the first and second order results to D=2 leads to very close numbers, η=0.8 and η≃0.795. The calculated exponent values are close to earlier reference results obtained by non-perturbative renormalization group, the self-consistent screening approximation and numerical simulations. These indications suggest that a perturbative analysis of the GCI model could provide an useful framework for accurate quantitative predictions of the scaling exponent even at D=2.

Absolute Poisson’s ratio and the bending rigidity exponent of a crystalline two-dimensional membrane

We compute the absolute Poisson’s ratio ν and the bending rigidity exponent η of a free-standing two-dimensional crystalline membrane embedded into a space of large dimensionality d=2+dc, dc≫1. We demonstrate that, in the regime of anomalous Hooke’s law, the absolute Poisson’s ratio approaches material independent value determined solely by the spatial dimensionality dc: ν=−1+2∕dc−a∕dc2+… where a≈1.76±0.02. Also, we find the following expression for the exponent of the bending rigidity: η=2∕dc+(73−68ζ(3))∕(27dc2)+…. These results cannot be captured by self-consistent screening approximation.

Anomalous elasticity, fluctuations and disorder in elastic membranes

Motivated by freely suspended graphene and polymerized membranes in soft and biological matter we present a detailed study of a tensionless elastic sheet in the presence of thermal fluctuations and quenched disorder. The manuscript is based on an extensive draft dating back to 1993, that was circulated privately. It presents the general theoretical framework and calculational details of numerous results, partial forms of which have been published in brief Letters (Le Doussal and Radzihovsky, 1992; 1993). The experimental realization atom-thin graphene sheets (Novoselov et al., 2004) have driven a resurgence in this fascinating subject, making our dated predictions and their detailed derivations timely. To this end we analyze the statistical mechanics of a generalized D-dimensional elastic “membrane” embedded in d dimensions using a self-consistent screening approximation (SCSA), that has proved to be unprecedentedly accurate in this system, exact in three complementary limits: (i) d→∞, (ii) D→4, and (iii) D=d. Focusing on the critical “flat” phase, for a homogeneous two-dimensional (D=2) membrane embedded in three dimensions (d=3), we predict its universal roughness exponent ζ=0.590, length-scale dependent elastic moduli exponents η=0.821 and ηu=0.358, and an anomalous Poisson ratio, σ=−1∕3. In the presence of random uncorrelated heterogeneity the membrane exhibits a glassy wrinkled ground state, characterized by ζ′=0.775,η′=0.449,ηu′=1.101 and a Poisson ratio σ′=−1∕3. Motivated by a number of physical realizations (charged impurities, disclinations and dislocations) we also study power-law correlated quenched disorder that leads to a variety of distinct glassy wrinkled phases. Finally, neglecting self-avoiding interaction we demonstrate that at high temperature a “phantom” sheet undergoes a continuous crumpling transition, characterized by a radius of gyration exponent, ν=0.732 and η=0.535. Many of these universal predictions have received considerable support from simulations. We hope that this detailed presentation of the SCSA theory will be useful to further theoretical developments and corresponding experimental investigations on freely suspended graphene.

Scaling Behavior and Strain Dependence of In-Plane Elastic Properties of Graphene.

We show by atomistic simulations that, in the thermodynamic limit, the in-plane elastic moduli of graphene at finite temperature vanish with system size L as a power law L(-η(u)) with η(u)≃0.325, in agreement with the membrane theory. We provide explicit expressions for the size and strain dependence of graphene's elastic moduli, allowing comparison to experimental data. Our results explain the recently experimentally observed increase of the Young modulus by more than a factor of 2 for a tensile strain of only a few per mill. The difference of a factor of 2 between the measured asymptotic value of the Young modulus for tensilely strained systems and the value from ab initio calculations remains, however, unsolved. We also discuss the asymptotic behavior of the Poisson ratio, for which our simulations disagree with the predictions of the self-consistent screening approximation.

Rippling transition from electron-induced condensation of curvature field in graphene

A quantum field theory approach is applied to investigate the dynamics of flexural phonons in a metallic membrane like graphene, looking for the effects deriving from the strong interaction between the electronic excitations and elastic deformations. Relying on a self-consistent screening approximation to the phonon self-energy, we show that the theory has a critical point characterized by the vanishing of the effective bending rigidity of the membrane at low momentum. We also check that the instability in the sector of flexural phonons takes place without the development of an in-plane static distortion, which is avoided due to the significant reduction of the electron-phonon couplings for in-plane phonons at large momenta. Furthermore, we analyze the scaling properties of the many-body theory to identify the order parameter that opens up at the point of the transition. We find that the vanishing of the effective bending rigidity and the onset of a nonzero expectation value of the mean curvature are concurrent manifestations of the critical behavior. The results presented here imply that, even in the absence of tension, the theory has a critical point at which the flat geometry becomes an unstable configuration of the metallic membrane, with a condensation of the mean curvature field that may well reproduce the smooth distribution of ripples observed in free-standing graphene.

Nematic phase in stripe-forming systems within the self-consistent screening approximation

We show that in order to describe the isotropic-nematic transition in stripe-forming systems with isotropic competing interactions of the Brazovskii class it is necessary to consider the next to leading order in a 1/N approximation for the effective Hamiltonian. This can be conveniently accomplished within the self-consistent screening approximation. We solve the relevant equations and show that the self-energy in this approximation is able to generate the essential wave vector dependence to account for the anisotropic character of a two-point correlation function characteristic of a nematic phase.

Suppression of anharmonicities in crystalline membranes by external strain

In practice, physical membranes are exposed to a certain amount of external strain (tension or compression), due to the environment where they are placed. As a result, the behavior of the phonon modes of the membrane is modified. We show that anharmonic effects in stiff two-dimensional membranes are highly suppressed under the application of tension. For this, we consider the anharmonic coupling between bending and stretching modes in the self-consistent screening approximation (SCSA), and compare the obtained height-height correlation function in the SCSA to the corresponding harmonic propagator. The elasticity theory results are compared to atomistic Monte Carlo simulations for a graphene membrane under tension. We find that, while rather high values of strain are needed to avoid anharmonicity in soft membranes, strain fields less than 1% are enough to suppress all the anharmonic effects in stiff membranes, as graphene.

Self-consistent screening approximation for flexible membranes: Application to graphene

Crystalline membranes at finite temperatures have an anomalous behavior of the bending rigidity that makes them more rigid in the long-wavelength limit. This issue is particularly relevant for applications of graphene in nanoelectromechanical and microelectromechanical systems. We calculate numerically the height-height correlation function $G(q)$ of crystalline two-dimensional membranes, determining the renormalized bending rigidity, in the range of wave vectors $q$ from ${10}^{\ensuremath{-}7}\text{ }{\text{\AA{}}}^{\ensuremath{-}1}$ till $10\text{ }{\text{\AA{}}}^{\ensuremath{-}1}$ in the self-consistent screening approximation (SCSA). For parameters appropriate to graphene, the calculated correlation function agrees reasonably with the results of atomistic Monte Carlo simulations for this material within the range of $q$ from ${10}^{\ensuremath{-}2}\text{ }{\text{\AA{}}}^{\ensuremath{-}1}$ till $1\text{ }{\text{\AA{}}}^{\ensuremath{-}1}$. In the limit $q\ensuremath{\rightarrow}0$ our data for the exponent $\ensuremath{\eta}$ of the renormalized bending rigidity ${\ensuremath{\kappa}}_{R}(q)\ensuremath{\propto}{q}^{\ensuremath{-}\ensuremath{\eta}}$ is compatible with the previously known analytical results for the SCSA $\ensuremath{\eta}\ensuremath{\simeq}0.82$. However, this limit appears to be reached only for $q<{10}^{\ensuremath{-}5}\text{ }{\text{\AA{}}}^{\ensuremath{-}1}$ whereas at intermediate $q$ the behavior of $G(q)$ cannot be described by a single exponent.

Structure of physical crystalline membranes within the self-consistent screening approximation

The anomalous exponents governing the long-wavelength behavior of the flat phase of physical crystalline membranes are calculated within a self-consistent screening approximation (SCSA) applied to second order expansion in 1/dC ( dC is the codimension), extending the seminal work of Le Doussal and Radzihovsky [Phys. Rev. Lett. 69, 1209 (1992)]. In particular, the bending rigidity is found to harden algebraically in the long-wavelength limit with an exponent eta=0.789... , which is used to extract the elasticity softening exponent eta(u)=0.422... , and the roughness exponent zeta=0.605... . The scaling relation eta(u)=2-2eta is proven to hold to all orders in SCSA. Further, applying the SCSA to an expansion in 1/dC , is found to be essential, as no solution to the self-consistent equations is found in a two-bubble level, which is the naive second-order expansion. Surprisingly, even though the expansion parameter for physical membrane is 1/dC=1 , the SCSA applied to second-order expansion deviates only slightly from the first order, increasing zeta by mere 0.016. This supports the high quality of the SCSA for physical crystalline membranes, as well as improves the comparison to experiments and numerical simulations of these systems. The prediction of SCSA applied to first order expansion for the Poisson ratio is shown to be exact to all orders.

Langevin dynamics of the Coulomb frustrated ferromagnet: A mode-coupling analysis.

We study the Langevin dynamics of the soft-spin, continuum version of the Coulomb-frustrated Ising ferromagnet. By using the dynamical mode-coupling approximation, supplemented by reasonable approximations for describing the equilibrium static correlation function, and the somewhat improved dynamical self-consistent screening approximation, we find that the system displays a transition from an ergodic to a nonergodic behavior. This transition is similar to that obtained in the idealized mode-coupling theory of glass-forming liquids and in the mean-field generalized spin glasses with one-step replica symmetry breaking. The significance of this result and the relation to the appearance of a complex free-energy landscape are also discussed.

Replica symmetry breaking in a quantum spin system with random fields

The zero temperature quantum transition for the random-field Ising model in a transverse field is studied. Using the self-consistent screening approximation for the correlation functions, we show that the replica symmetry is broken at the transition point.

Self-consistent screening approximation for critical dynamics

We generalize Bray's self-consistent screening approximation to describe the critical dynamics of the theory. In order to obtain the dynamical exponent z, we have to make an ansatz for the form of the scaling functions, which fortunately can be restricted by general arguments. Numerical values of z for d = 3, and n = 1, ..., 10 are obtained using two different ansatz, and differ by a very small amount. In particular, the value of obtained for the three-dimensional Ising model agrees well with recent Monte Carlo simulations.

Self-consistent theory of polymerized membranes.

We study $D$-dimensional polymerized membranes embedded in $d$ dimensions using a self-consistent screening approximation. It is exact for large $d$ to order $1/d$, for any $d$ to order $\epsilon=4-D$ and for $d=D$. For flat physical membranes ($D=2,d=3$) it predicts a roughness exponent $\zeta=0.590$. For phantom membranes at the crumpling transition the size exponent is $\nu=0.732$. It yields identical lower critical dimension for the flat phase and crumpling transition $D_{lc}(d)={2 d \over {d+1}}$ ($D_{lc}={\sqrt{2}}$ for codimension 1). For physical membranes with ${\it random}$ quenched curvature $\zeta=0.775$ in the new $T=0$ flat phase in good agreement with simulations.

Self-consistent screening approximation, long range interactions and the critical exponent η

The self-consistent screening approximation of Bray is used in the presence of long range interactions to suggest that the critical exponent η might differ from its classical value of 2 - σ, under certain circumstances.

Resummation of the1nexpansion through a self-consistent approach

Bray's self-consistent screening approximation for calculating critical exponents is extended to higher orders, resulting in a scheme equivalent to a resummation of the $\frac{1}{n}$ expansion. While the integration techniques borrowed from conformal invariant field theory reduce the task of evaluating Feynman integrals appearing in this scheme to about the level of a straight $\frac{1}{n}$ calculation, the second-order results obtained here for critical exponents as functions of the order-parameter dimensionality $n$ and space dimensionality $d$ are definitely superior to those of a direct $\frac{1}{n}$ calculation and produce acceptable estimates down to about $n\ensuremath{\simeq}1\ensuremath{-}2$ for $d=3$.