Time and temperature dependent correlation functions of 1D models of quantum statistical mechanics

We consider exactly solvable, gapless models of statistical mechanics. We found the universal formula which describes the asymptotics of correlation functions at any temperature and an arbitrary coupling constant.     

Conformal correlation functions,Frobenius algebras and triangulations

We formulate two-dimensional rational conformal field theory as a natural generalization of two-dimensional lattice topological field theory. To this end we lift various structures from complex vector spaces to modular tensor categories. The central ingredient is a special Frobenius algebra object A in the modular category that encodes the Moore–Seiberg data of the underlying chiral CFT. Just like for lattice TFTs, this algebra is itself not an observable quantity. Rather, Morita equivalent algebras give rise to equivalent theories. Morita equivalence also allows for a simple understanding of T-duality.We present a construction of correlators, based on a triangulation of the world sheet, that generalizes the one in lattice TFTs. These correlators are modular invariant and satisfy factorization rules. The construction works for arbitrary orientable world sheets, in particular, for surfaces with boundary. Boundary conditions correspond to representations of the algebra A. The partition functions on the torus and on the annulus provide modular invariants and NIM-reps of the fusion rules, respectively.     

Renormalisation-group calculation of correlation functions for the 2D random bond Ising and Potts models

We find the crossover behaviour of the disorder averaged spin-spin correlation function for the 2D Ising and 3-state Potts model with random bonds at the critical point. The procedure employed is the renormalisation approach of the perturbation series around the conformal field theories representing the pure models. We obtain a crossover in the amplitude for the correlation function for the Ising model, which does not change the critical exponent, and a shift in the critical exponent produced by randomness in the case of the Potts model. A comparison with numerical data is discussed briefly.     

Random-walk theory and correlation functions in classical statistical mechanics

As a step towards a random-process formulation for classical fluids which ivolve many-body correlations, a random-walk formulation is presented wherein, for both lattice-gas and continuum models, the Green function and weight function describing the random walk are related to the total pair-correlation function of the model and to either the direct pair-correlation function or the first element of the direct correlation matrix.     

Multiscale representation of generating and correlation functions for some models of statistical mechanics and quantum field theory

We consider models of statistical mechanics and quantum field theory (in the Euclidean formulation) which are treated using renormalization group methods and where the action is a small perturbation of a quadratic action. We obtain multiscale formulas for the generating and correlation functions aftern renormalization group transformations which bring out the relation with thenth effective action. We derive and compare the formulas for different RGs. The formulas for correlation functions involve (1) two propagators which are determined by a sequence of approximate wave function renormalization constants and renormalization group operators associated with the decomposition into scales of the quadratic form and (2) field derivatives of the nth effective action. For the case of the block field -function RG the formulas are especially simple and for asymptotic free theories only the derivatives at zero field are needed; the formulas have been previously used directly to obtain bounds on correlation functions using information obtained from the analysis of effective actions. The simplicity can be traced to an orthogonality-of-scales property which follows from an implicit wavelet structure. Other commonly used RGs do not have the orthogonality of scales property.     

The exact solution of 2D ZN invariant statistical models

The exact solution of the hamiltonian version of Z_N invariant statistical models at τ ≠ 0 is found by means of the Bethe-ansatz technique.     

Conformal invariance,current algebra and modular invariance

A large class of conformally invariant models in two dimensions is realised by constraining free fermion theories. The Fock spaces of the constrained theories are described, using the representation theory of affine Kac-Moody algebras. The results are extended to superconformally invariant theories. Projections of the models, producing consistent two-dimensional field theories, are discussed.     

The twisted quantum affine algebra Uq(A2(2)) and correlation functions of the Izergin-Korepin model

We derive the exchange relations of the vertex operators of U_q(A₂⁽²⁾) and show that these vertex operators give the bosonization of the Izergin-Korepin model. We give an integral expression of the correlation functions of the Izergin-Korepin model and derive the difference equations which they satisfy.     

Direct correlation functions for three-site and four-site water models

ABSTRACT

Direct correlation functions (DCFs) play a pivotal role in modern liquid-state theories and are virtually indispensable for non-mean-field implementation of the classical density functional theory (cDFT). Whereas analytical expressions have been derived for the DCFs of simple fluids and electrolytes, DCFs for molecular systems are attainable only through numerical solution of the integral-equation theories in combination with molecular simulation or an approximate closure. Unlike radial distribution functions, DCFs reflect the variational of the local chemical potential of individual atoms/interaction sites with the density-profile fluctuations, which are difficult to be sampled from the simulation trajectory. This article presents an improved numerical procedure to calculate the DCFs of three-site (SPC/E and TIP3P) and four-site (TIP4P-Ew) water models based on the reference interaction site model and molecular dynamics simulations. In combination with the modified fundamental measure theory for the bridge functionals, the DCFs have been utilised to predict the hydration free energies of 504 small organic molecules. The theoretical predictions yield an average unsigned error of 0.66 kcal/mol in comparison with the simulation data, better than that (～1 kcal/mol) reported in previous cDFT calculations.     

A new quantum statistical evaluation method for time correlation functions

Considering a system ofN identical interacting particles, which obey Fermi-Dirac or Bose-Einstein statistics, we derive new formulas for correlation functions of the type (whereB _j is diagonal in the free-particle states) in the thermodynamic limit. Thereby we apply and extend a superoperator formalism, recently developed for the derivation of long-time tails in semiclassical systems. As an illustrative application, the Boltzmann equation value of the time-integrated correlation functionC(t) is derived in a straightforward manner. Due to exchange effects, the obtained t-matrix and the resulting scattering cross section, which occurs in the Boltzmann collision operator, are now functionals of the Fermi-Dirac or Bose-Einstein distribution.     

Conformal blocks and generalized theta functions

LetSU _X r be the moduli space of rankr vector bundles with trivial determinant on a Riemann surfaceX. This space carries a natural line bundle, the determinant line bundleL. We describe a canonical isomorphism of the space of global sections ofL ^k with the space of conformal blocks defined in terms of representations of the Lie algebrasl _r (C((z))). It follows in particular that the dimension ofH ⁰(SU _X r,L ^k ) is given by the Verlinde formula.Both authors were partially supported by the European Science Project Geometry of Algebraic Varieties, Contract no. SCI-0398-C(A)     

Conformal anomaly,virasoro algebra and Chern—Simons cohomology

We make use of Chern-Simons cohomology and the family index theorem to deal with two-dimensional anomalies, infinite-dimensional algebras and their relations. We also take advantage of Zweibein formulation to treat the world sheet trace anomaly and Virasoro algebra in Polyakov string and to show how the anaomaly-free condition gives rise to the critical dimensions.     

Completely X-symmetric S-matrices corresponding to theta functions and models of statistical mechanics

We consider general expressions of factorized S-matrices with Abelian symmetry expressed in terms of -functions. These expressions arise from representations of the Heisenberg group. New examples of factorized S-matrices lead to a large class of completely integrable models of statistical mechanics which generalize the XYZ-model of the eight-vertex model.     

Correlation functions and correlation times for models with multiplicative white noise

Correlation functions and correlation times for the Stratonovich and Verhulst model are investigated. By transforming the Fourier transform of the corresponding Fokker-Planck equation into a tridiagonal vector recurrence relation, the Fourier transform of the correlation function and the correlation time are expressed in terms of matrix continued fractions or by similar iterations and are thus obtained numerically. By using the inverse Fourier transform, the correlation function itself is calculated. Furthermore an analytic expression in terms of an integral is obtained for the correlation time, which is evaluated exactly in the Verhulst model and asymptotically for large and weak noise strength in the Stratonovich model. A Padé expansion approximating the correlation time for all noise strength is also given.