Replica Field Theory Research Articles

Replica field theory for anharmonic sound attenuation in glasses

A saddle-point treatment of interacting phonons in a disordered environment is developed. In contrast to crystalline solids, anharmonic attenuation of density fluctuations becomes important in the hydrodynamic regime, due to a broken momentum conservation. The variance of the shear modulus Δ 2 turns out to be the strength of the disorder enhanced phonon–phonon interaction. In the low-frequency regime (below the boson peak frequency) we obtain an Akhiezer-like sound attenuation law Γ ∝ Τω 2. Together with the usual Rayleigh scattering mechanism this yields a crossover of the Brillouin linewidth from a ω 2 to a ω 4 regime. The crossover frequency ω c is fully determined by the boson peak frequency and the temperature. For network glasses like SiO 2 at room temperature this crossover is predicted to be situated one order of magnitude below the boson peak frequency.

TOPOLOGICAL PRINCIPLES IN THE THEORY OF ANDERSON LOCALIZATION

Scaling ideas in the theory of the quantum Hall effect are fundamentally based on topological principles in Anderson localization theory. These concepts have a very general significance and are not limited to replica field theory or disordered systems alone. In this chapter, we will discuss these ideas in several distinctly different physical contexts. We start with a brief overview that spans two and a half decades of experimental research on quantum criticality in strong magnetic fields. Secondly, we address the new understanding of universality that has emerged from the theory of Anderson localization and interaction phenomena. In the last part we show how the experimentally observed quantum phenomena fundamentally alter the way in which strong coupling problems in theoretical physics are perceived.

Random networks of cross-linked directed polymers

We explore the effect of random permanent cross-links on a system of directed polymers confined between two planes with their end points free to slide on them. We treat the cross-links as quenched disorder and we use a semimicroscopic replica field theory to study the structure and elasticity of this system. Upon increasing the cross-link density, we get a continuous gelation transition signaled by the emergence of a finite in-plane localization length. The distribution of localization length turns out to depend on the height along the preferred direction of the directed polymers. The gelation transition also gives rise to a finite in-plane shear modulus which we calculate and turns out to be universal, i.e., independent of the energy and length scales of the polymers and the cross-links. Using a symmetry argument, we show that cross-links of negligible extent along the preferred axis of the directed polymers do not cause any renormalization to the tilt modulus of the uncross-linked system.

The Ising spin glass in finite dimensions: A perturbative study of the free energy

Replica field theory is used to study the n-dependent free energy of the Ising spin glass in a first order perturbative treatment. Large sample-to-sample deviations of the free energy from its quenched average prove to be Gaussian, independently of the special structure of the order parameter. The free energy difference between the replica symmetric and (infinite level) replica symmetry broken phases is studied in details: the line n ( T ) where it is zero coincides with the Almeida–Thouless line for d > 8 . The dimensional domain 6 < d < 8 is more complicated, and several scenarios are possible.

Soft random solids and their heterogeneous elasticity

Spatial heterogeneity in the elastic properties of soft random solids is examined via vulcanization theory. The spatial heterogeneity in the structure of soft random solids is a result of the fluctuations locked-in at their synthesis, which also brings heterogeneity in their elastic properties. Vulcanization theory studies semimicroscopic models of random-solid-forming systems and applies replica field theory to deal with their quenched disorder and thermal fluctuations. The elastic deformations of soft random solids are argued to be described by the Goldstone sector of fluctuations contained in vulcanization theory, associated with a subtle form of spontaneous symmetry breaking that is associated with the liquid-to-random-solid transition. The resulting free energy of this Goldstone sector can be reinterpreted as arising from a phenomenological description of an elastic medium with quenched disorder. Through this comparison, we arrive at the statistics of the quenched disorder of the elasticity of soft random solids in terms of residual stress and Lamé-coefficient fields. In particular, there are large residual stresses in the equilibrium reference state, and the disorder correlators involving the residual stress are found to be long ranged and governed by a universal parameter that also gives the mean shear modulus.

Stochastic inflation and replica field theory

We adopt methods from statistical field theory to stochastic inflation. For the example of a free test field in de Sitter and power-law inflation, the power spectrum of long-wavelength fluctuations is computed. We study its dependence on the shape of the filter that separates long from short wavelength modes. While for filters with infinite support the phenomenon of dimensional reductions is found on large super-horizon scales, filters with compact support return a scale-invariant power spectrum in the infra-red. Features of the power spectrum, induced by the filter, decay within a few e-foldings. Thus the late-time power spectrum is independent of the filter details.

Anisotropic Random Networks of Semiflexible Polymers

Motivated by the organization of cross-linked cytoskeletal biopolymers, we present a semimicroscopic replica field theory for the formation of anisotropic random networks of semiflexible polymers. The networks are formed by introducing random permanent cross-links which fix the orientations of the corresponding polymer segments to align with one another. Upon increasing the cross-link density, we obtain a continuous gelation transition from a fluid phase to a gel where a finite fraction of the system gets localized at random positions. For sufficiently stiff polymers, this positional localization is accompanied by a continuous isotropic-to-nematic (IN) transition occurring at the same cross-link density. As the polymer stiffness decreases, the IN transition becomes first order, shifts to a higher cross-link density, and is preceded by an amorphous solid where the average polymer orientations freeze in random directions.

Is the droplet theory for the Ising spin glass inconsistent with the replica field theory?

Symmetry arguments are used to derive a set of exact identities between irreducible vertex functions for the replica symmetric field theory of the Ising spin glass in zero magnetic field. Their range of applicability spans from mean field to short ranged systems in physical dimensions. The replica symmetric theory is unstable for d>8, just like in mean field theory. For 6<d<8 and d<6 the resummation of an infinite number of terms is necessary to settle the problem. When d<8, these Ward-like identities must be used to distinguish an Almeida-Thouless line from the replica symmetric droplet phase.

Toda lattice representation for random matrix model with logarithmic confinement

We construct a replica field theory for a random matrix model with logarithmic confinement [K.A. Muttalib et al., Phys. Rev. Lett. 71 (1993) 471]. The corresponding replica partition function is calculated exactly for any size of matrix N. We make a color–flavor transformation of the original model and find corresponding Toda lattice equations for the replica partition function in both formulations. The replica partition function in the flavor space is defined by generalized Itzikson–Zuber (IZ) integral over homogeneous factor space of pseudounitary supergroups SU ( n | M , M ) / SU ( n | M − N , M ) (Stiefel manifold) with M → ∞ , which is evaluated and represented in a compact form.

Spatial correlation functions in three-dimensional Ising spin glasses

We investigate spin-spin correlation functions in the low temperature phase of spin-glasses. Using the replica field theory formalism, we examine in detail their infrared (long distance) behavior. In particular we identify a longitudinal mode that behaves as massive at intermediate length scales (near-infrared region). These issues are then addressed by numerical simulation; the analysis of our data is compatible with the prediction that the longitudinal mode appears as massive, i.e., that it undergoes an exponential decay, for distances smaller than an appropriate correlation length.

Nature of perturbation theory in spin glasses

The high-order behavior of the perturbation expansion in the cubic replica field theory of spin glasses in the paramagnetic phase has been investigated. The study starts with the zero-dimensional version of the replica field theory and this is shown to be equivalent to the problem of finding finite size corrections in a modified spherical spin glass near the critical temperature. We find that the high-order behavior of the perturbation series is described, to leading order, by coefficients of alternating signs (suggesting that the cubic field theory is well-defined) but that there are also subdominant terms with a complicated dependence of their sign on the order. Our results are then extended to the d-dimensional field theory and in particular used to determine the high-order behavior of the terms in the expansion of the critical exponents in a power series in epsilon=6-d. We have also corrected errors in the existing epsilon expansions at third order.

Integral Equation Theory of Random Copolymer Melts

An integral equation theory is developed for the structure and thermodynamics of melts of random copolymers. The molecules are modeled as flexible chains with a random sequence of two types of sites, with square well potentials between sites. The polymer reference interaction site model (PRISM) integral equation theory is extended to random copolymers and used to calculate the static correlations and spinodal lines. For threadlike chains and with the incompressible random phase approximation (IRPA), the PRISM theory reduces to a replica field theory or Landau-type theory presented earlier by others. In this limit the theory predicts a macrophase separation for values of the monomer correlation strength, λ, greater than a critical value. For smaller values of λ a microphase separation is predicted with strong concentration fluctuations at finite wave vectors. With the hard-core interactions included, however, the predictions of the theory with different closure approximations are qualitatively different from each other. With atomic closure approximations the nature of the spinodal diagram is similar to the IRPA, although fluctuations are predicted to promote microphase separation. With molecular closure approximations the theory predicts that no microphase separation occurs in these systems and that there is a macrophase separation for all values of λ.

Goldstone-type fluctuations and their implications for the amorphous solid state

In sufficiently high spatial dimensions, the formation of the amorphous (i.e., random) solid state of matter, e.g., upon sufficent crosslinking of a macromolecular fluid, involves particle localization and, concomitantly, the spontaneous breakdown of the (global, continuous) symmetry of translations. Correspondingly, the state supports Goldstone-type low energy, long wavelength fluctuations, the structure and implications of which are identified and explored from the perspective of an appropriate replica field theory. In terms of this replica perspective, the lost symmetry is that of relative translations of the replicas; common translations remain as intact symmetries, reflecting the statistical homogeneity of the amorphous solid state. What emerges is a picture of the Goldstone-type fluctuations of the amorphous solid state as shear deformations of an elastic medium, along with a derivation of the shear modulus and the elastic free energy of the state. The consequences of these fluctuations---which dominate deep inside the amorphous solid state---for the order parameter of the amorphous solid state are ascertained and interpreted in terms of their impact on the statistical distribution of localization lengths, a central diagnostic of the state. The correlations of these order parameter fluctuations are also determined, and are shown to contain information concerning further diagnostics of the amorphous solid state, such as spatial correlations in the statistics of the localization characteristics. Special attention is paid to the properties of the amorphous solid state in two spatial dimensions, for which it is shown that Goldstone-type fluctuations destroy particle localization, the order parameter is driven to zero, and power-law order-parameter correlations hold.

Functional renormalization group at largeNfor disordered elastic systems, and relation to replica symmetry breaking

We study the replica field theory which describes the pinning of elastic manifolds of arbitrary internal dimension d in a random potential, with the aim of bridging the gap between mean field and renormalization theory. The full effective action is computed exactly in the limit of large embedding space dimension N. The second cumulant of the renormalized disorder obeys a closed self-consistent equation. It is used to derive a Functional Renormalization Group (FRG) equation valid in any dimension d, which correctly matches the Balents-Fisher result to first order in epsilon=4-d. We analyze in detail the solutions of the large-N FRG for both long-range and short-range disorder, at zero and finite temperature. We find consistent agreement with the results of Mezard Parisi (MP) from the Gaussian variational method (GVM) in the case where full replica symmetry breaking (RSB) holds there. We prove that the cusplike non-analyticity in the large N FRG appears at a finite scale, corresponding to the instability of the replica symmetric solution of MP. We show that the FRG exactly reproduces, for any disorder correlator and with no need to invoke Parisi's spontaneous RSB, the non-trivial result of the GVM for small overlap. A formula is found yielding the complete RSB solution for all overlaps. Since our saddle-point equations for the effective action contain both the MP equations and the FRG, it can be used to describe the crossover from FRG to RSB. A qualitative analysis of this crossover is given, as well as a comparison with previous attempts to relate FRG to GVM. Finally, we discuss applications to other problems and new perspectives.

Replica field theory and renormalization group for the Ising spin glass in an external magnetic field.

We use the generic replica symmetric cubic field theory to study the transition of short-range Ising spin glasses in a magnetic field around the upper critical dimension. A novel fixed point is found from the application of the renormalization group. In the spin-glass limit, this fixed point governs the critical behavior of a class of systems characterized by a single cubic parameter. For this universality class, the spin-glass susceptibility diverges at criticality, whereas the longitudinal mode remains massive. The third mode, however, behaves unusually. The physical consequences of this unusual behavior are discussed, and a comparison with the conventional de Almeida-Thouless scenario is presented.

Thermodynamics and mechanical properties of random copolymer networks

We analyze of the thermodynamic properties of random crosslinked polymer networks made of phantom chains with compositional disorder. We start our investigations from an extended Edwards Hamiltonian and end with the discussion of the free energy and mechanical properties. Using replica field theory we determine and solve the saddle point equations. The physically relevant ground state is invariant against translations in the real space, but it depends strongly on the coupling parameters for interactions, topology and compositional disorder of the network. We predict three different thermodynamic states for random copolymer networks. For two regimes the saddle point shows a rotational symmetry after elimination of translation effects. Here the network behaves similar to a homogeneous network. The third regime corresponds to a ground state with broken replica symmetry. Here, the behavior of the random copolymer network shows similarities to the thermodynamical properties of spin glasses. The network state is comparable diluted proteins.

Replica field theory for a polymer in random media

In this paper we revisit the problem of a (non-self-avoiding) polymer chain in a random medium which was previously investigated by Edwards and Muthukumar (EM) [J. Chem. Phys. 89, 2435 (1988)]. As noticed by Cates and Ball (CB) [J. Phys. (France) 49, 2009 (1988)] there is a discrepancy between the predictions of the replica calculation of EM and the expectation that in an infinite medium the quenched and annealed results should coincide (for a chain that is free to move) and a long polymer should always collapse. CB argued that only in a finite volume one might see a "localization transition" (or crossover) from a stretched to a collapsed chain in three spatial dimensions. Here we carry out the replica calculation in the presence of an additional confining harmonic potential that mimics the effect of a finite volume. Using a variational scheme with five variational parameters we derive analytically for d<4 the result R approximately (g|ln &mgr;|)(-1/(4-d)) approximately (g ln V)(-1/(4-d)), where R is the radius of gyration, g is the strength of the disorder, &mgr; is the spring constant associated with the confining potential, and V is the associated effective volume of the system. Thus the EM result is recovered with their constant replaced by ln V as argued by CB. We see that in the strict infinite volume limit the polymer always collapses, but for finite volume a transition from a stretched to a collapsed form might be observed as a function of the strength of the disorder. For d<2 and for large V>V' approximately exp(g(2/(2-d))L((4-d)/(2-d))) the annealed results are recovered and R approximately (Lg)(1/(d-2)), where L is the length of the polymer. Hence the polymer also collapses in the large L limit. The one-step replica symmetry breaking solution is crucial for obtaining the above results.

Scaling and infrared divergences in the replica field theory of the Ising spin glass

Replica field theory for the Ising spin glass in zero magnetic field is studied around the upper critical dimension d=6. A scaling theory of the spin glass phase, based on Parisi's ultrametrically organised order parameter, is proposed. We argue that this infinite step replica symmetry broken (RSB) phase is nonperturbative in the sense that amplitudes of scaling forms cannot be expanded in term of the coupling constant w^2. Infrared divergent integrals inevitably appear when we try to compute amplitudes perturbatively, nevertheless the \epsilon-expansion of critical exponents seems to be well-behaved. The origin of these problems can be traced back to the unusual behaviour of the free propagator having two mass scales, the smaller one being proportional to the perturbation parameter w^2 and providing a natural infrared cutoff. Keeping the free propagator unexpanded makes it possible to avoid producing infrared divergent integrals. The role of Ward-identities and the problem of the lower critical dimension are also discussed.

Disordered heteropolymers with cross-links – phase diagram and conformational transitions

The phase structure of globular random heteropolymers with quenched cross-links is studied by replica field theory. Our analysis predicts the occurrence of complex freezing patterns with few conformations sampled. The low cross-linking temperature phase is attributed to microphase separation of the pre-gel state with cross-links forming at the microphase interfaces. A first-order transition occurs at higher cross-linking temperatures by an increase of the microdomain size by statistical matching of microphases bounded by cross-links. At higher cross-linking temperatures, a second-order transition is predicted to a sequence-stabilized globular state.

Phase diagram analysis of random heteropolymers with composition specific and quenched cross-links

We study the conformational organization of a novel class of soft matter, random heteropolymers (RHPs) with composition specific and quenched cross-links by replica field theory. These technologically relevant materials carry an essentially quenched sequence distribution of disparate segments, and also a fixed distribution of composition specific cross-links. The chain connectivity of the linear RHPs is captured most generally by a continuous microscopic RHP Hamiltonian, while the quenched character of the cross-links is enforced by spatial constraints on segments of a prescribed composition, and by allowing fluctuations in the total number of composition specific cross-links around an experimentally controlled average. The replica theory, a formalism widely used in the study of linear RHPs, is extended in the present work to analysis of systems with multiform disorder that have interdependent disorder components. By numerically analyzing the free energy and its stability we predict the occurrence of frozen phase formation wherein few conformations are sampled. By systematically varying the sequence/cross-link fluctuations, temperature, and the RHP interaction parameters we show that two conformational transitions can occur in the frozen phase. The low temperature conformational transition resembles to a large extent the reduction in chain conformations which occurs during the folding of proteins with fixed disulfide bonds. The frozen phase domain microstructure formation is discussed in context of the recently elucidated phase diagram of a close related system, RHPs with annealed cross-links. Explicit connection is made with possible experiments and computer simulations methods to test our predictions.