Exponents Of Correlation Functions Research Articles

The geometrical approach of the manganitise compound deposition on the surface of manganisite ore

In this study, the mechanism for the growth of manganese deposition on surfaces of magnesite ore is studied. For this purpose pictures of naturally occurring manganese dendrites of different forms encountered on samples of magnesite ore are obtained by a scanner; these pictures are used for computing the exponents of density correlation functions and fractal dimensions. The results imply that the limiting step in the formation of manganese deposition (dendrites) is a diffusion process. In order to test this hypothesis we have performed a fractal analysis of manganese dendrites using two different methods. The result yields that manganese dendrites have a self-similar structure and fractal dimensions vary within the interval 1.61–1.88.

Nematic phase transitions in two-dimensional systems

Simulations of nematic-isotropic transition of liquid crystals in two dimensions are performed using an $O(2)$ vector model characterised by non linear nearest neighbour spin interaction governed by the fourth Legendre polynomial $P_4$. The system is studied through standard Finite-Size Scaling and conformal rescaling of density profiles or correlation functions. The low temperature limit is discussed in the spin wave approximation and confirms the numerical results, while the value of the correlation function exponent at the deconfining transition seems controversial.

Gauge-invariant critical exponents for the Ginzburg-Landau model.

The critical behavior of the Ginzburg-Landau model is described in a manifestly gauge-invariant manner. The gauge-invariant correlation-function exponent is computed to first order in the 4-d and 1/n expansion, and found to agree with the ordinary exponent obtained in the covariant gauge, with the parameter alpha=1-d in the gauge-fixing term ( partial differential (mu)A(mu))(2)/2alpha.

The reflection-antisymmetric counterpart of the Kármán–Howarth dynamical equation

We study the isotropic, helical component in homogeneous turbulence using statistical objects which have the correct symmetry and parity properties. Using these objects we derive an analogue of the Kármán–Howarth equation, that arises due to lack of mirror-reflection-symmetry in isotropic flows. The main equation we obtain is consistent with the results of Chkhetiani [JETP 63 (1996) 768] and L’vov et al. [Exact result for the 3rd order correlations of velocity in turbulence with helicity, 1997. http://xxx.lanl.gov/abs/chao-dyn/9705016] but is derived using only velocity correlations, with no direct consideration of the vorticity or helicity. This alternative formulation offers an advantage to both experimental and numerical measurements. We also postulate, under the assumption of self-similarity, the existence of a hierarchy of scaling exponents for helical velocity correlation functions of arbitrary order, analogous to the Kolmogorov prediction for the scaling exponents of velocity structure function.

Integrable one-dimensional N-component fermion model with correlated hopping and hard-core repulsion

The N-component Bariev model for correlated hopping and a hard-core repulsion is shown to be integrable in one dimension. The solution of the model is obtained within the framework of nested Bethe ansatze. The ground state integral equations for the densities of the rapidities are derived for repulsive and attractive correlations. In zero-field and for a repulsive interaction the spin excitations are gapped and only the charge sector has a Fermi surface. The properties are then those of a one-component Luttinger liquid. The spin-gaps are gradually closed with increasing magnetic field. For an attractive interaction potential charge bound states (generalized non-local Cooper pairs) are formed and the spin excitations are gapped in zero magnetic field. The ground state properties and the critical exponents of correlation functions are discussed for both, repulsive and attractive, potentials. The string hypothesis is invoked to derive the thermodynamic Bethe ansatz equations. Some special limits of the thermodynamic equations are analyzed, e.g., the weak and strong interaction cases, and the low and high temperature limits.

Magnetic impurities in a correlated electron system with spin-triplet pairing

We consider an exactly solvable two-band model of electrons moving in one dimension and interacting with a δ -function spin-exchange potential. The interaction leads to the formation of spin-triplet bound states of the Cooper type (hard-core bosons that do not condensate and exist at all temperatures). Without destroying the integrability we introduce a finite concentration of mixed-valent impurities with two magnetic configurations of spin S and S− 1 2 , respectively, which hybridize with the conduction states of the host. We derive the Bethe ansatz equations diagonalizing the correlated host with impurities and discuss the ground state properties as a function of magnetic field, the crystalline field splitting of the bands, the concentration of the impurities and the Kondo exchange coupling. In the zero-field ground state the spin-triplet states are effectively free bosons and a threshold band splitting Δ is required to break up a triplet pair. The gap of the unpaired electron excitations is gradually reduced by the band splitting and closes at Δ . For a splitting larger than Δ a fraction of the electrons is spin-polarized. The impurities are antiferromagnetically correlated with each other and the effect of a magnetic field is to suppress these correlations as well as to gradually align the spins of the spin-triplets with the field. Spin- 1 2 impurities always enhance the gap Δ , while with impurities of higher spin the gap decreases if the coupling between the triplets and impurities is weak (small Kondo temperature), but increases otherwise. For small T K the gap is closed at a critical concentration x cr and the two electron bands have different populations. For x > x cr we obtain a zero-magnetization phase with three coexisting components, namely, itinerant ferromagnetism of unpaired electrons, antiferromagnetism of the impurities and superconducting fluctuations of ferromagnetically correlated spin triplets. The critical exponents of correlation functions are also briefly discussed in the conformal field theory limit.

Breakdown of Luttinger liquid state in a one-dimensional frustrated spinless fermion model

The Haldane hypothesis about the universality of the Luttinger liquid (LL) behavior in conducting one-dimensional fermion systems is checked numerically for the spinless fermion model with next-nearest-neighbor interactions. It is shown that for the large enough interactions the ground state can be gapless due to frustrations but at the same time might not belong to the same universality class as a simple LL. The exponents of the correlation functions for this unusual conducting state are found numerically by a finite-size method.

Spin-gapped two-band correlated electron system with magnetic impurities

We consider an integrable model consisting of two parabolic bands of electrons of equal mass interacting via a δ-function exchange potential. The exchange is attractive or repulsive depending on whether the interacting particles are in a spin-singlet or spin-triplet. Without destroying the integrability we introduce a finite concentration of mixed-valent impurities with two magnetic configurations of spin S and (S+ 1 2 ) , respectively, which hybridize with the conduction states of the host. We derive the Bethe ansatz equations diagonalizing the correlated host with impurities and discuss the ground state properties as a function of magnetic field, the crystalline field splitting of the bands, the concentration of impurities and the Kondo exchange coupling. The attractive interaction in the host leads to Cooper-pair-like bound states, and a magnetic field larger than a critical one H c is required to break up the singlet pairs. The spin-gap is gradually reduced by a magnetic field and closes at H c. For H> H c a fraction of the electrons is spin-polarized. The impurities are antiferromagnetically correlated with each other and the effect of a magnetic field is to suppress these correlations, which are totally quenched at the saturation field H s. Depending on the off-resonance shift θ of the impurities the spin-gap may decrease or increase linearly with the concentration of impurities. For small θ and a weak interaction among the electrons the spin-gap is closed at a critical concentration x cr. For x> x F> x cr ferromagnetism is induced, so that three phases coexist, namely itinerant ferromagnetism, ferrimagnetism of the impurity spins and superconducting fluctuations. The constructive interference between antiferromagnetic and superconducting fluctuations for large θ could be related to the spin-compensation of the impurity spins among each other and the dominating forward scattering of the impurities. The critical exponents of correlation functions are also briefly discussed in the conformal field theory limit.

Boundary spatiotemporal correlations in a self-organized critical model of punctuated equilibrium

In a semi-infinite geometry, a one-dimensional, M-component model of biological evolution realizes microscopically an inhomogeneous branching process for M-->infinity. This implies a size distribution exponent tau(')=7/4 for avalanches starting at a free, "dissipative" end of the evolutionary chain. A bulklike behavior with tau(')=3/2 is restored by "conservative" boundary conditions. These are such as to strictly fix to its critical, bulk value the average number of species directly involved in an evolutionary avalanche by the mutating species located at the chain end. A two-site correlation function exponent tau(')(R)=4 is also calculated exactly in the "dissipative" case, when one of the points is at the border. Together with accurate numerical determinations of the time recurrence exponent tau(')(first), these results show also that, no matter whether dissipation is present or not, boundary avalanches have the same space and time fractal dimensions as those in the bulk, and their distribution exponents obey the basic scaling laws holding there.

Correlation functions for a two-dimensional electron system with bosonic interactions and a square Fermi surface

We calculate zero-temperature correlation functions for a model of 2D interacting electrons with short-range interactions and a square Fermi surface. The model was arrived at by mapping electronic states near a square Fermi surface with Hubbard-like interactions onto one-dimensional quantum chains, retaining terms which can be written in terms of bosonic density operators. Interactions between orthogonal chains, corresponding to orthogonal faces of the square Fermi surface, are neglected. The correlation functions become sums of Luttinger-type correlation functions due to the bosonic model. However, the correlation function exponents differ in form from those of the Luttinger model. As a consequence, the simple scaling relations found to exist between the Luttinger model exponents, do not carry over to the leading exponents of our model. We find that for repulsive effective interactions, charge-density wave/spin-density wave instabilities are dominant. We do not consider d-wave instabilities here.

The Hausdorff dimension in polymerized quantum gravity

We calculate the Hausdorff dimension, d H , and the correlation function exponent, η, for polymerized two dimensional quantum gravity models. If the non-polymerized model has correlation function exponent η 0>3 then d H = γ −1 where γ is the susceptibility exponent. This suggests that these models may be in the same universality class as certain non-generic branched polymer models.

Influence of long-range interactions on the critical behavior of systems with a negative Fisher exponent

The influence of long-range interactions decaying in d dimensions as ${1/R}^{d+\ensuremath{\sigma}}$ on the critical behavior of systems with negativeFisher's correlation-function exponent for short-range interactions, ${\ensuremath{\eta}}_{\mathrm{SR}}<0,$ is reexamined. Such systems, typically described by ${\ensuremath{\varphi}}^{3}$-field theories, are, e.g., the Potts model in the percolation limit, the Edwards-Anderson spin-glass model, and the Yang-Lee edge singularity. In contrast to preceding studies, it is shown by means of Wilson's momentum-shell renormalization-group recursion relations that the long-range interactions dominate as long as $\ensuremath{\sigma}<2\ensuremath{-}{\ensuremath{\eta}}_{\mathrm{SR}}.$ Exponents change continuously to their short-range values at the boundary of this region.

Critical behavior of Tomonaga-Luttinger liquids with a mobile impurity

The critical properties of Tomonaga-Luttinger liquids including a mobile impurity are studied by means of the Bethe ansatz method, conformal field theory (CFT), and bosonization. We calculate conformal dimensions by performing the CFT analysis of the energy spectrum for an integrable impurity model, and show that this system is classified as $c=1$ shifted Gaussian CFT. The exact critical exponents of various correlation functions are obtained as functions of the impurity mass and its momentum. Passing to the continuum limit, we then construct an effective field theory in terms of bosonization, and directly calculate some correlation functions. A possible application to the Fermi-edge singularity in quantum wires is also discussed.

Critical properties of frustrated spin chain with bond alternation in magnetic fields

Abstract The critical exponents of spin correlation functions of the frustrated S = 1 2 antiferromagnetic Heisenberg chain with bond alternation are investigated numerically by the exact diagonalization method combined with finite-size-scaling analysis. The magnetization curve in the thermodynamic limit is also presented.

Invariants for Correlations of Velocity Differences in Turbulent Fields

The phenomenology of the scaling behavior of higher order structure functions of velocity differences across a scale $R$ in turbulence should be built around the irreducible representations of the rotation symmetry group. Every irreducible representation is associated with a scalar function of $R$ which may exhibit different scaling exponents. The common practice of using moments of longitudinal and transverse fluctuations mixes different scalar functions and therefore may mix different scaling exponents. It is shown explicitly how to extract pure scaling exponents for correlation functions of arbitrary orders.

Exact Results for Highly Correlated Electron Systems in One Dimension

One-dimensional conductors are a long-standing topic of research with direct applications to organic conductors and mesoscopic rings. The discovery of the ceramic high-temperature superconductors has revitalized the interest in low-dimensional charge and spin fluctuations of highly correlated electron systems. Several mechanisms proposed to explain the high-T c superconductors invoke properties of the two-dimensional Hubbard model, but probably also some one-dimensional aspects are relevant. Numerous one-dimensional models for correlated electrons have been studied with various approximate, asymptotically exact and exact methods. These results lead to the concept of Luttinger liquid for interacting electron gases without excitation gaps (metallic systems). Characteristic of Luttinger liquids are the charge and spin separation, marginal Fermi liquid properties, e.g. the absence of quasiparticles in the vicinity of the Fermi surface, nonuniversal power-law singularities in the one-particle spectral function and the related absence of a discontinuity in the momentum distribution at the Fermi level, the power-law decay of correlation functions for long times and large distances, persistent currents in finite rings, etc. Due to the peculiarities of the phase space in one dimension some of the models have sufficient conserved currents to be completely integrable. We review exact results derived within the framework of Bethe's ansatz for integrable one-dimensional models of correlated electrons. The Bethe-ansatz method is presented by explicitly showing the steps leading to the solution of the N-component electron gas interacting via a δ-function potential (repulsive and attractive interaction), which is probably the simplest model of correlated electrons. Emphasis is given to the procedure to extract the groundstate properties, the classification of states, the excitation spectrum, the thermodynamics and finite size effects, such as critical exponents of correlation functions and persistent currents. The method is then applied to numerous other models, e.g. (i) a two-band model involving attractive and repulsive potentials and crystalline fields splitting the bands, (ii) the traditional Hubbard chain with attractive and repulsive U, (iii) the degenerate Hubbard model with repulsive U, which displays a metal–insulator transition at a finite U, (iv) a two-band Hubbard model with repulsive U, (v) the traditional supersymmetric t–J model (vi) a two-band supersymmetric t–J model with band-splitting and (vii) the N-component supersymmetric t–J model. Finally, results for models with long-range interactions, in particular r-2 and sinh -2(r) potentials, are briefly reviewed.

Geometrical interpretation of the Knizhnik-Polyakov-Zamolodchikov exponents

We provide evidence that the KPZ exponents in two-dimensional quantum gravity can be interpreted as scaling exponents of correlation functions which are functions of the invariant geodesic distance between the fields.

Exact critical properties of the multicomponent interacting fermion model with boundaries.

Exact critical properties of the one-dimensional SU(N) interacting fermion model with open boundaries are studied by using the Bethe ansatz method. We derive the surface critical exponents of various correlation functions using boundary conformal field theory. They are classified into two types, i.e., the exponents for the chiral SU(N) Tomonaga-Luttinger liquid and those related to the orthogonality catastrophe. We discuss a possible application of the results to the photoemission (absorption) in the edge state of the fractional quantum Hall effect. \textcopyright{} 1996 The American Physical Society.

Critical exponents for the sinh-cosh interaction model in the zero sector.

In this paper, we continue our investigation of a one-dimensional, two-component, quantum many-body system in which like particles interact with a pair potential $s(s+1)/{\rm sinh}^{2}(r)$, while unlike particles interact with a pair potential $-s(s+1)/{\rm cosh}^{2}(r)$. For an equal number of particles of the two components, the ground state for $s>0$ corresponds to an antiferromagnet/insulator. Excitations consist of a gapless pair-hole--pair continuum, a two-particle continuum with gap and excitons with gap. For $-1<s<0$, the system has two gapless excitations --- a particle-hole continuum and a two spin-wave continuum. Using finite-size scaling methods of conformal field theory, we calculate the asymptotic expressions and critical exponents for correlation functions of these gapless excitations at zero temperature. The conformal structure is closely related to the Hubbard model with repulsive on-site interaction.

Correlation Exponents for Multicomponent Jastrow Wave Functions and Effects of Chiral Constraint in One Dimension

The long-distance critical behavior of correlation functions is studied for the multicomponent Jastrow wave function in one dimension (1D), which is a 1D analogue of generalized Laughlin's wave function for the fractional quantum Hall effect. The critical exponents for various correlation functions are obtained analytically. We then impose the chiral constraint and discuss the relationship to the chiral Luttinger liquid for the edge states of the quantum Hall effect.