Boundary Correlation Functions Research Articles

The role of orthogonal polynomials in the six-vertex model and its combinatorial applications

The Hankel determinant representations for the partition function and boundary correlation functions of the six-vertex model with domain wall boundary conditions are investigated by the methods of orthogonal polynomial theory. For specific values of the parameters of the model, corresponding to 1-, 2- and 3-enumerations of Alternating Sign Matrices (ASMs), these polynomials specialize to classical ones (Continuous Hahn, Meixner-Pollaczek, and Continuous Dual Hahn, respectively). As a consequence, a unified and simplified treatment of ASMs enumerations turns out to be possible, leading also to some new results such as the refined 3-enumerations of ASMs. Furthermore, the use of orthogonal polynomials allows us to express, for generic values of the parameters of the model, the partition function of the (partially) inhomogeneous model in terms of the one-point boundary correlation functions of the homogeneous one.

Formula omitted] from [formula omitted]: Bulk and boundary correlators

We study the c L = 25 limit, which corresponds to c = 1 string theory, of bulk and boundary correlation functions of Liouville theory with FZZT boundary conditions. This limit is singular and requires a renormalization of vertex operators. We formulate a regularization procedure which allows to extract finite physical results. A particular attention is paid to c = 1 string theory compactified at the self-dual radius R = 1 . In this case, the boundary correlation functions diverge even after the multiplicative renormalization. We show that all infinite contributions can be interpreted as contact terms arising from degenerate world sheet configurations. After their subtraction, one gets a well defined set of correlation functions. We also obtain several new results for correlation functions in Liouville theory at generic central charge.

Topological and Ghost Interaction Methods in Equilibrium and Non-Equilibrium Systems

A topological interaction method is proposed to study boundary correlation functions and spontaneous magnetization by introducing a topological interaction J to connect spins at two opposite boundaries and by taking the limit J → 0. Conceptual relations between the present scheme and the ghost spin method are discussed. The present paper gives a general starting point for subsequent papers on the applications of the present idea. This method is suggested to be applicable to studying boundary-boundary spin correlations, boundary critical dynamics and non-equilibrium stationary flows. Nonlinear mesoscopic phenomena in infinite systems are quite interesting to study theoretically as well as experimentally. Boundary or surface phase transitions and critical phenomena are still difficult to study rigorously even in low dimensions. A topological treatment is proposed 1) to study spontaneous symmetry breaking (SSB), particularly to calculate the spontaneous magnetization mb at the boundary of the relevant system and boundary-boundary correlation functions. In

Exact oblique boundary–boundary correlation functions of the two-dimensional ising model

The boundary–boundary correlation in an arbitrary oblique direction for the two-dimensional Ising model is calculated exactly using the topological interaction method. It is shown that these correlation functions satisfy Morita's sum rule not only for T > T c but also for T ⩽ T c . The asymptotic form of the oblique boundary–boundary correlation functions is derived. Non-monotonic behaviour of the oblique boundary–boundary correlation functions with respect to the system-size is found by using numerical calculations of an exact integral representation of the oblique correlation.

Study on boundary–boundary correlation functions in the transverse Ising chain and [formula omitted] chain

The topological interaction method proposed by Suzuki is applied to the transverse Ising chain and XY chain in order to study the behaviour of the boundary–boundary correlations. The correlation function 〈 σ 1 x σ N x 〉 for the thermodynamic limit is derived by this method as a function of the topological interaction parameter J 2 for finite temperatures. This analysis of the correlation function clarifies the emergence of the long-range order as temperature goes to zero. At zero temperature, a more general spin chain can be analysed analytically for N → ∞ and numerically for finite N to obtain the following results. The correlation function 〈 σ 1 x σ N x 〉 shows various kinds of behaviour, namely, (a) monotonic, (b) non-monotonic or (c) globally non-monotonic with small oscillation, with respect to the system-size N, in different regions of the parameters. The function 〈 σ 1 y σ N y 〉 shows also the above behaviour (a), or shows (d) globally monotonic decay with small oscillation.

Study on boundary–boundary correlation functions of the two-dimensional Ising model using topological interaction method

The topological interaction method is applied to the evaluation of boundary–boundary correlation functions, which yield the square of the spontaneous magnetization mb at the boundary in the thermodynamic limit. One of the remarkable results is that the boundary–boundary correlation function CM is found to be nonmonotonic with respect to the system size M for T<Tc, because of boundary effects. On the other hand, it is monotonic above Tc, CM≃(A+(T)∕M)exp(−M∕ξ) with the correlation length ξ and with A+(T)∼T−Tc near the critical point Tc. At the critical point, CM∼1∕M, and below Tc, CM≃mb2+(−A−(T)M+B(T)+C(T)∕M)exp(−M∕ξ), where A−(T)∼mb3, B(T)∼mb2, C(T)∼mb∼T−Tc. Fisher’s finite-size scaling is confirmed in this respect.

Boundary ground ring in 2D string theory

The 2D quantum gravity on a disc, or the non-critical theory of open strings, is known to exhibit an integrable structure, the boundary ground ring, which determines completely the boundary correlation functions. Inspired by the recent progress in boundary Liouville theory, we extend the ground ring relations to the case of non-vanishing boundary Liouville interaction known also as FZZT brane in the context of the 2D string theory. The ring relations yield an over-determined set of functional recurrence equations for the boundary correlation functions. The ring action closes on an infinite array of equally spaced FZZT branes for which we propose a matrix model realization. In this matrix model the boundary ground ring is generated by a pair of complex matrix fields.

Boundary Liouville theory and 2D quantum gravity

We study the boundary correlation functions in Liouville theory and in solvable statistical models of 2D quantum gravity. In Liouville theory we derive functional identities for all fundamental boundary structure constants, similar to the one obtained for the boundary two-point function by Fateev, Zamolodchikov and Zamolodchikov. All these functional identities can be written as difference equations with respect to one of the boundary parameters. Then we switch to the microscopic realization of 2D quantum gravity as a height model on a dynamically triangulated disc and consider the boundary correlation functions of electric, magnetic and twist operators. By cutting open the sum over surfaces along a domain wall, we derive difference equations identical to those obtained in Liouville theory. We conclude that there is a complete agreement between the predictions of Liouville theory and the discrete approach.

D3-branes on the Coulomb branch and instantons

The relative coefficients of higher derivative interactions of the IIB effective action of the form C^4, (D F_5)^4, F_5^8, ... (where C is the Weyl tensor and F_5 is the five-form field strength) are motivated by supersymmetry arguments. It is shown that the classical supergravity solution for N parallel D3-branes is unaltered by this combination of terms. The non-vanishing of \zeroC^2 in this background (where \zero C is the background value of the Weyl tensor) leads to effective O(1/alpha') interactions, such as C^2 and Lambda^8 (where Lambda is the dilatino). These contain D-instanton contributions in addition to tree and one-loop terms. The near horizon limit of the N D3-brane system describes a multi-AdS_5xS^5 geometry that is dual to \calN=4 SU(N) Yang-Mills theory spontaneously broken to S(U(M_1)x...xU(M_r)). Here, the N D3-branes are grouped into r coincident bunches with M_r in each group, with M_r/N = m_r fixed as N goes to infinity. The boundary correlation function of eight Lambda's is constructed explicitly. The second part of the paper considers effects of a constrained instanton in this large-N Yang-Mills theory by an extension of the analysis of Dorey, Hollowood and Khoze of the one-instanton measure at finite N. This makes precise the correspondence with the supergravity D-instanton measure at leading order in the 1/N expansion. However, the duality between instanton-induced correlation functions in Yang-Mills theory and the dual supergravity is somewhat obscured by complications relating to the structure of constrained instantons.

Free boson formulation of boundary states inW3minimal models and the critical Potts model

We develop a Coulomb gas formalism for boundary conformal field theory having a $W$ symmetry and illustrate its operation using the three state Potts model. We find that there are free-field representations for six $W$ conserving boundary states, which yield the fixed and mixed physical boundary conditions, and two $W$ violating boundary states which yield the free and new boundary conditions. Other $W$ violating boundary states can be constructed but they decouple from the rest of the theory. Thus we have a complete free-field realization of the known boundary states of the three state Potts model. We then use the formalism to calculate boundary correlation functions in various cases. We find that the conformal blocks arising when the two point function of $\phi_{2,3}$ is calculated in the presence of free and new boundary conditions are indeed the last two solutions of the sixth order differential equation generated by the singular vector.

Free-field realization of boundary states and boundary correlation functions of minimal models

We propose a general formalism to compute exact correlation functions for Cardy's boundary states. Using the free-field construction of boundary states and applying the Coulomb-gas technique, it is shown that charge neutrality conditions pick up particular linear combinations of conformal blocks. As an example we study the critical Ising model with free and fixed boundary conditions, and demonstrate that conventional results are reproduced. This formalism thus directly associates algebraically constructed boundary states with correlation functions which are in principle observable or numerically calculable.

Boundary correlators in 2D quantum gravity: Liouville versus discrete approach

We calculate a class of two-point boundary correlators in 2D quantum gravity using its microscopic realization as loop gas on a random surface. We find a perfect agreement with the two-point boundary correlation function in Liouville theory, obtained by V. Fateev, A. Zamolodchikov and Al. Zamolodchikov. We also give a geometrical meaning of the functional equation satisfied by this two-point function.

Boundary correlation functions of the six-vertex model with open boundary

The six-vertex model with open boundary conditions is discussed.The correlation functions for the model are calculated and expressed as some determinants.

Strings in the Extended BTZ Spacetime

We study string theory on the extended spacetime of the BTZ black hole, as described by an orbifold of the SL(2,R) WZW model. The full spacetime has an infinite number of disconnected boundary components, each corresponding to a dual CFT. We discuss the computation of bulk and boundary correlation functions for operators inserted on different components. String theory correlation functions are obtained by analytic continuation from an orbifold of the SL(2,C)/SU(2) coset model. This yields two-point functions for general operators, including those describing strings that wind around the horizon of the black hole.

Boundary correlation functions of the six-vertex model

We consider the six-vertex model on an $N \times N$ square lattice with the domain wall boundary conditions. Boundary one-point correlation functions of the model are expressed as determinants of $N\times N$ matrices, generalizing the known result for the partition function. In the free fermion case the explicit answers are obtained. The introduced correlation functions are closely related to the problem of enumeration of alternating sign matrices and domino tilings.

Sum Rule Identities and the Duality Relation for the Pottsn-Point Boundary Correlation Function

It is shown that certain sum rule identities exist which relate correlation functions for $n$ Potts spins on the boundary of a planar lattice for $n\geq 4$. Explicit expressions of the identities are obtained for $n=4,5$. It is also shown that the identities provide the missing link needed for a complete determination of the duality relation for the $n$-point correlation function. The $n=4$ duality relation is obtained explicitly. More generally we deduce the number of correlation identities for any $n$ as well as an inversion relation and a conjecture on the general form of the duality relation.

Hydrodynamic boundary conditions, correlation functions, and Kubo relations for confined fluids.

Dynamical correlation functions of a fluid slab confined between two solid walls are computed using a phenomenologial, hydrodynamic appoach that generalizes Onsager's principle of linear regression of fluctuations to inhomogeneous systems. The phenomenological results are compared to exact molecular-dynamics simulations on simple model systems. This comparison permits a determination of the phenomenological parameters that describe the hydrodynamics of the fluid slab and especially of the boundary conditions (BC's) that account for the presence of solid walls. In most cases, the hydrodynamic BC for the tangential velocity field is found to be a no-slip BC and must be applied in a plane that is separated from the solid by about one layer of fluid atoms. A set of formal relations between the parameters that characterize the hydrodynamic BC and the equilibrium correlation functions of the inhomogeneous fluid is also derived. These relations are analogous to the usual Green-Kubo equalities for the transport coefficients of bulk fluids.

Hydrodynamic boundary conditions and correlation functions of confined fluids.

A phenomenological description of the dynamical correlations in a fluid confined between two parallel solid walls, using a generalization of Onsager's theory of fluctuations, is presented. Besides the bulk viscosity, this description involves parameters characterizing the boundary conditions. Correlations functions obtained in equilibrium molecular dynamics simulations are very well described within this framework. Featureless walls yield perfect slip boundary conditions, while corrugated walls in most cases correspond to stick boundary conditions. Only for very small corrugation amplitudes is it necessary to introduce a finite slipping length.

Boundary correlation functions and interface formation in two-dimensional solids

Two-point correlation functions at the rectilinear boundary of a two-dimensional crystal half-space on a reference lattice are studied for the case of free boundary conditions. We show that the power of the decay law of the positional correlation function at the surface satisfies η surf = 2η bulk, within the approximation that dislocation pair correlations are taken into account. Dislocations at the boundary form a quasi-one-dimensional plasma for T ≠ 0 and this may lead to a dynamic roughening of the surface. Formation of dynamic interfaces in grain boundaries and near free boundaries is discussed in terms of 1-dimensional charged gas models with long-range interaction. Melting of boundary layers at T surf c< T bulk c is suggested to be possible only if topological effects of moving dislocations on the reference lattice are taken into account.

Qualitative study of the boundary correlation function in the two-dimensional XY-model

Abstract The two-point correlation function at the rectilinear boundary of the two-dimensional half-space is studied for the XY-model for the case of free boundary conditions. It is shown that defects of vortex-type at the boundary form a one-dimensional plasma for T ≠ 0, but that this does not lead to exponential decay of the boundary correlation function for T