Correlation Functions Of Twist Fields Research Articles

Thermal entanglement in conformal junctions

We consider a quantum junction described by a 1+1-dimensional boundary conformal field theory (BCFT). Our analysis focuses on correlations emerging at finite temperature, achieved through the computation of entanglement measures. Our approach relies on characterizing correlation functions of twist fields using BCFT techniques. We provide non-perturbative predictions for the crossover between low and high temperatures. An intriguing interplay between bulk and boundary effects, associated with the bulk/boundary scaling dimensions of the fields above, is found. In particular, the entanglement entropy is primarily influenced by bulk thermal fluctuations, exhibiting extensiveness for large system sizes with a prefactor independent of the scattering properties of the defect. In contrast, negativity is governed by fluctuations across the entangling points only, adhering to an area law; its value depends non-trivially on the defect, and it diverges logarithmically as the temperature is decreased. To validate our predictions, we numerically check them for free fermions on the lattice, finding good agreement.

Correlators of the symmetric product orbifold

We exploit null vectors of the fractional Virasoro algebra of the symmetric product orbifold to compute correlation functions of twist fields in the large N limit. This yields a new method to derive correlation functions in these orbifold CFTs that is purely based on the symmetry algebra. We explore various generalisations, such as subleading (torus) contributions or correlation functions of other fields than the bare twist fields. We comment on the consequences of our computation for the AdS3/CFT2 correspondence.

Entanglement entropy in warped conformal field theories

We present a detailed discussion of entanglement entropy in (1+1)-dimensional Warped Conformal Field Theories (WCFTs). We implement the Rindler method to evaluate entanglement and Renyi entropies for a single interval and along the way we interpret our results in terms of twist field correlation functions. Holographically a WCFT can be described in terms of Lower Spin Gravity, a SL(2,R)xU(1) Chern-Simons theory in three dimensions. We show how to obtain the universal field theory results for entanglement in a WCFT via holography. For the geometrical description of the theory we introduce the concept of geodesic and massive point particles in the warped geometry associated to Lower Spin Gravity. In the Chern-Simons description we evaluate the appropriate Wilson line that captures the dynamics of a massive particle.

Entanglement entropy of non-unitary conformal field theory

Here we show that the Rényi entanglement entropy of a region of large size ℓ in a one-dimensional critical model whose ground state breaks conformal invariance (such as in those described by non-unitary conformal field theories), behaves as , where is the effective central charge, c (which may be negative) is the central charge of the conformal field theory and is the lowest holomorphic conformal dimension in the theory. We also obtain results for models with boundaries, and with a large but finite correlation length, and we show that if the lowest conformal eigenspace is logarithmic ( with N nilpotent), then there is an additional term proportional to . These results generalize the well known expressions for unitary models. We provide a general proof, and report on numerical evidence for a non-unitary spin chain and an analytical computation using the corner transfer matrix method for a non-unitary lattice model. We use a new algebraic technique for studying the branching that arises within the replica approach, and find a new expression for the entanglement entropy in terms of correlation functions of twist fields for non-unitary models.

More general correlation functions of twist fields from Ward identities in the massive Dirac theory

Following on from previous work we derive the nonlinear differential equations of more general correlators of U(1) twist fields in two-dimensional massive Dirac theory. Using the conserved charges of the double-copy model equations parametrizing the correlators of twist fields with arbitrary twist parameter are found. This method also gives a parametrization of the correlation functions of general, fermionic, descendent twist fields. The equations parametrizing correlators of primary twist fields are compared to those of the literature and evidence is presented to confirm that these equations represent the correct parametrization.

Correlation functions of twist fields from Ward identities in the massive Dirac theory

We derive non-linear differential equations for correlation functions of U(1) twist fields in the two-dimensional massive Dirac theory. Primary U(1) twist fields correspond to exponential fields in the sine-Gordon model at the free-fermion point, and it is well-known that their vacuum two-point functions are determined by integrable differential equations. We extend part of this result to more general quantum states (pure or mixed) and to certain descendents, showing that some two-point functions are determined by the sinh-Gordon differential equations whenever there is translation and parity invariance, and the density matrix is the exponential of a bilinear expression in fermions. We use methods involving Ward identities associated to the copy-rotation symmetry in a model with two independent, anti-commuting copies. Such methods were used in the context of the thermally perturbed Ising quantum field theory model. We show that they are applicable to the Dirac theory as well, and we suggest that they are likely to have a much wider applicability to free fermion models in general. Finally, we note that our form-factor study of descendents twist fields combined with a CFT analysis provides a new way of evaluating vacuum expectation values of primary U(1) twist fields: by deriving and solving a recursion relation.

Entanglement entropy and twist fields

The entanglement entropy of a subsystem of a quantum system is expressed, in the replica approach, through analytic continuation with respect to n of the trace of the n-th power of the reduced density matrix. This trace can be thought of as the vacuum expectation value of a suitable observable in a system made with n independent copies of the original system. We use this property to numerically evaluate it in some two-dimensional critical systems, where it can be compared with the results of Calabrese and Cardy, who wrote the same quantity in terms of correlation functions of twist fields of a conformal field theory. Although the two calculations match perfectly even in finite systems when the analyzed subsystem consists of a single interval, they disagree whenever the subsystem is composed of more than one connected part. The reasons of this disagreement are explained.

Integral equations and long-time asymptotics for finite-temperature Ising chain correlationfunctions

This work concerns the dynamical two-point spin correlation functions of the transverseIsing quantum chain at finite (non-zero) temperature, in the universal region near thequantum critical point. They are correlation functions of twist fields in the massiveMajorana fermion quantum field theory. At finite temperature, these are known to satisfy aset of integrable partial differential equations, including the sinh–Gordon equation. Weapply the classical inverse scattering method to study them, finding that the ‘initialscattering data’ corresponding to the correlation functions are simply related to theone-particle finite-temperature form factors calculated recently by one of the persentauthors. The set of linear integral equations (Gelfand–Levitan–Marchenko equations)associated with the inverse scattering problem then gives, in principle, the two-pointfunctions at all space and time separations, and all temperatures. From these, we evaluatethe long-time asymptotic expansion ‘near the light cone’, in the region where thedifference between the space and time separations is of the order of the correlationlength.

DLCQ strings, twist fields and one-loop correlators on a permutation orbifold

We investigate some aspects of the relationship between matrix string theory and light-cone string field theory by analysing the correspondence between the two-loop thermal partition function of DLCQ strings in flat space and the integrated two-point correlator of twist fields in a symmetric product orbifold conformal field theory at one-loop order. This is carried out by deriving combinatorial expressions for generic twist field correlation functions in permutation orbifolds using the covering surface method, by deriving the one-loop modification of the twist field interaction vertex, and by relating the two-loop finite temperature DLCQ string theory to the theory of Prym varieties for genus two covers of an elliptic curve. The case of bosonic Z 2 orbifolds is worked out explicitly and precise agreement between both amplitudes is found. We use these techniques to derive explicit expressions for Z 2 orbifold spin twist field correlation functions in the type II and heterotic string theories.

Topological orbifold models and quantum cohomology rings

We discuss the toplogical sigma model on an orbifold target space. We describe the moduli space of classical minima for computing correlation functions involving twisted operators, and show, through a detailed computation of an orbifold of ${\bf CP}^1$ by the dihedral group $D_{4},$ how to compute the complete ring of observables. Through this procedure, we compute all the rings from dihedral ${\bf CP}^1$ orbifolds; we note a similarity with rings derived from perturbed $D-$series superpotentials of the $A-D-E$ classification of $N = 2$ minimal models. We then consider ${\bf CP}^2/D_4,$ and show how the techniques of topological-anti-topological fusion might be used to compute twist field correlation functions for nonabelian orbifolds.

Higher twisted sector couplings of [formula omitted] orbifolds

We derive the basic correlation functions of twist fields coming from arbitrary twisted sectors in symmetric Z N orbifold conformal field theories, keeping all the admissible marginal perturbations, in particular those corresponding to the antisymmetric-tensor background field. This allows a thorough investigation of modular symmetries in this type of string compactification. Such a study is explicitly carried out for the group generated by duality transformations. Thus, apart from being of phenomenological use, our couplings are also interesting from the mathematical point of view as they represent automorphic functions for a large class of discrete groups.

Multi-loop twist field correlation functions for ZN orbifolds

We calculate the correlation functions of bosonic twist fields for Z N orbifolds on Riemann surfaces of arbitrary genus.

Analytic fields on Riemann surfaces. II

The properties of analytic fields on a Riemann surface represented by a branch covering of ℂℙ1 are investigated in detail. Branch points are shown to correspond to the vertex operators with simple conformal properties. As applications we compute determinants of\(\bar \partial _j\) operators forZn-symmetric surfaces and obtain various representations for the two-loop measure in the bosonic string theory together with various identities for theta-functions of hyperelliptic surfaces. We also present an integral representation for the quantum part of the twist field correlation functions, which describe propagation of the string on the orbifold background. We also calculate the quantum part of the structure constants of the twist-field operator algebra, generalizing the results of Dixon, Friedan, Martinec, and Shenker.

Vacuum amplitudes without twist fields for the Z N orbifold and correlation functions of twist fields for the Z 2 orbifold

We calculate the vacuum amplitudes without twist fields for the Z N ( N=2, 3, 4, 6) orbifold. As a by-product, we show the Z 2 boson-fermion equivalence in arbitrary genus. By factorization, we obtain correlation functions of twist fields for the Z 2 orbifold.