Boundary Correlation Functions Research Articles

Holographic Analysis of Boundary Correlation Functions for the Hyperbolic-Lattice Ising Model

Abstract We analyze boundary spin correlation functions of the hyperbolic-lattice Ising model from the holographic point of view. Using the corner-transfer-matrix renormalization-group (CTMRG) method, we demonstrate that the boundary correlation function exhibits power-law decay with quasiperiodic oscillation, while the bulk correlation function always decays exponentially. On the basis of the geometric relation between the bulk correlation path and distance along the outer edge boundary, we find that scaling dimensions for the boundary correlation function can be well explained by a combination of the bulk correlation length and background curvatures inherent to the hyperbolic lattice. We also investigate the cutoff effect of the bond dimension in CTMRG, revealing that the long-distance behavior of the boundary spin correlation is accurately described even with a small bond dimension. In contrast, the short-distance behavior rapidly loses its accuracy.

Evaluation of integrals for the emptiness formation probability in the square-ice model

We study the emptiness formation probability (EFP) in the six-vertex model with domain wall boundary conditions. We present a conjecture according to which at the ice point, i.e., when all the Boltzmann weights are equal, the known multiple integral representation (MIR) for the EFP can be given as a finite-size matrix determinant of Fredholm type. Our conjecture is based on the explicit evaluation of the MIR for particular values of geometric parameters and on two kinds of identities for the boundary correlation function. The obtained representation can be further written as the Fredholm determinant of some linear integral operator. We show that as the geometric parameters of the EFP are tuned to the vicinity of the arctic curve arising in the scaling limit, the conjectured determinant turns into the GUE Tracy–Widom distribution.

A bootstrap study of minimal model deformations

For QFTs in AdS the boundary correlation functions remain conformal even if the bulk theory has a scale. This allows one to constrain RG flows with numerical conformal bootstrap methods. We apply this idea to flows between two-dimensional CFTs, focusing on deformations of the tricritical and ordinary Ising model. We provide non-perturbative constraints for the boundary correlation functions of these flows and compare them with conformal perturbation theory in the vicinity of the fixed points. We also reproduce a completely general constraint on the sign of the \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T\\overline{T }$$\\end{document} deformation in two dimensions.

Inflation correlators with multiple massive exchanges

The most general tree-level boundary correlation functions of quantum fields in inflationary spacetime involve multiple exchanges of massive states in the bulk, which are technically difficult to compute due to the multi-layer nested time integrals in the Schwinger-Keldysh formalism. On the other hand, correlators with multiple massive exchanges are well motivated in cosmological collider physics, with the original quasi-single-field inflation model as a notable example. In this work, with the partial Mellin-Barnes representation, we derive a simple rule, called family-tree decomposition, for directly writing down analytical answers for arbitrary nested time integrals in terms of multi-variable hypergeometric series. We present the derivation of this rule together with many explicit examples. This result allows us to obtain analytical expressions for general tree-level inflation correlators with multiple massive exchanges. As an example, we present the full analytical results for a range of tree correlators with two massive exchanges.

On time-dependent backgrounds in 1 + 1 dimensional string theory

In perturbative string theory, one is generally interested in asymptotic observables, such as the S-matrix in flat spacetime, and boundary correlation functions in anti-de Sitter spacetime. However, there are backgrounds in which such observables do not exist. We study examples of such backgrounds in 1 + 1 dimensional string theory. In these examples, the Liouville wall accelerates and can become spacelike in the past and/or future. When that happens, the corresponding null infinity, at which the standard scattering states are defined, is shielded by the Liouville wall. We compute scattering and particle production amplitudes in these backgrounds in the region in parameter space where the wall remains timelike, and discuss the continuation of this picture to the spacelike regime. We also discuss the physics from the point of view of the dynamics of free fermions in backgrounds with a time-dependent Fermi surface.

Perturbative RG flows in AdS. An étude

We discuss general properties of perturbative RG flows in AdS with a focus on the treatment of boundary conditions and infrared divergences. In contrast with flat-space boundary QFT, general covariance in AdS implies the absence of independent boundary flows. We illustrate how boundary correlation functions remain conformally covariant even if the bulk QFT has a scale. We apply our general discussion to the RG flow between consecutive unitary diagonal minimal models which is triggered by the ϕ(1,3) operator. For these theories we conjecture a flow diagram whose form is significantly simpler than that in flat-space boundary QFT. In several stand-alone appendices we discuss two-dimensional BCFTs in general and the minimal model BCFTs in particular. These include both an extensive review as well as the computation of several new BCFT correlation functions.

Holographic codes from hyperinvariant tensor networks

Holographic quantum-error correcting codes are models of bulk/boundary dualities such as the anti-de Sitter/conformal field theory (AdS/CFT) correspondence, where a higher-dimensional bulk geometry is associated with the code’s logical degrees of freedom. Previous discrete holographic codes based on tensor networks have reproduced the general code properties expected from continuum AdS/CFT, such as complementary recovery. However, the boundary states of such tensor networks typically do not exhibit the expected correlation functions of CFT boundary states. In this work, we show that a new class of exact holographic codes, extending the previously proposed hyperinvariant tensor networks into quantum codes, produce the correct boundary correlation functions. This approach yields a dictionary between logical states in the bulk and the critical renormalization group flow of boundary states. Furthermore, these codes exhibit a state-dependent breakdown of complementary recovery as expected from AdS/CFT under small quantum gravity corrections.

From celestial correlators to AdS, and back

We present a general relation between celestial correlation functions in d-dimensions and Witten diagrams in (d + 1)-dimensional Euclidean anti-de Sitter (EAdS) space, to all orders in perturbation theory. Contact diagram processes are proportional to contact Witten diagrams and particle exchanges can be recast as a continuum of particle exchanges in EAdS where the exchanged particles carrying unitary Principal Series representations of SO(d + 1, 1). One can then try to import familiar EAdS techniques to study the properties of celestial correlators. In this work we use this relation to infer the analytic structure of the spectral density in the conformal partial wave expansion of celestial correlators which, at least perturbatively, should be a meromorphic function of the spectral parameter. We also discuss non-perturbative constraints from unitarity in Euclidean Conformal Field Theory, which requires positivity of the spectral density. This extends similar relations recently uncovered between boundary correlation functions in de Sitter space and Witten diagrams in EAdS, suggesting that EAdS could play a central role in efforts towards holography for all lambdas.

Quantum Field Theory in AdS Space instead of Lehmann-Symanzik-Zimmerman Axioms.

The boundary correlation functions for a quantum field theory (QFT) in a fixed anti-de Sitter (AdS) background should reduce to S-matrix elements in the flat-space limit. We consider this procedure in detail for four-point functions. With minimal assumptions we rigorously show that the resulting S-matrix element obeys a dispersion relation, the nonlinear unitarity conditions, and the Froissart-Martin bound. QFT in AdS thus provides an alternative route to fundamental QFT results that normally rely on the LSZ axioms.

Markov chain Monte Carlo methods for graph refinement in spinfoam cosmology

We study the behavior of the Lorentzian Engle-Pereira-Rovelli-Livine spinfoam amplitude with homogeneous boundary data, under a graph refinement going from five to twenty boundary tetrahedra. This can be interpreted as a wave function of the Universe, for which we compute boundary geometrical operators, correlation functions, and entanglement entropy. The numerical calculation is made possible by adapting the Metropolis-Hastings algorithm, along with recently developed computational methods appropriate for the deep quantum regime. We confirm that the transition amplitudes are stable against such refinement. We find that the average boundary geometry does not change, but the new degrees of freedom correct the quantum fluctuations of the boundary and the correlations between spatial patches. The expectation values are compatible with their geometrical interpretation and the correlations between neighboring patches decay when computed across different spinfoam vertices.

Holography for Ising spins on the hyperbolic plane

Motivated by the AdS/CFT correspondence, we use Monte Carlo simulation to investigate the Ising model formulated on tessellations of the two-dimensional hyperbolic disk. We focus in particular on the behavior of boundary-boundary correlators, which exhibit power-law scaling both below and above the bulk critical temperature indicating scale invariance of the boundary theory at any temperature. This conclusion is strengthened by a finite-size scaling analysis of the boundary susceptibility which yields a scaling exponent consistent with the scaling dimension extracted from the boundary correlation function. This observation provides evidence that the connection between continuum boundary conformal symmetry and isometries of the bulk hyperbolic space survives for simple interacting field theories even when the bulk is approximated by a discrete tessellation.

Towards bootstrapping RG flows: sine-Gordon in AdS

The boundary correlation functions for a Quantum Field Theory (QFT) in an Anti-de Sitter (AdS) background can stay conformally covariant even if the bulk theory undergoes a renormalization group (RG) flow. Studying such correlation functions with the numerical conformal bootstrap leads to non-perturbative constraints that must hold along the entire flow. In this paper we carry out this analysis for the sine-Gordon RG flows in AdS2, which start with a free (compact) scalar in the UV and end with well-known massive integrable theories that saturate many S-matrix bootstrap bounds. We numerically analyze the correlation functions of both breathers and kinks and provide a detailed comparison with perturbation theory near the UV fixed point. Our bounds are often saturated to one or two orders in perturbation theory, as well as in the flat-space limit, but not necessarily in between.

From AdS to dS exchanges: Spectral representation, Mellin amplitudes, and crossing

We present a simple general relation between tree-level exchanges in AdS and dS. This relation allows to directly import techniques and results for AdS Witten diagrams, both in position and momentum space, to boundary correlation functions in dS. In this work we apply this relation to define Mellin amplitudes and a spectral representation for exchanges in dS. We also derive the conformal block decomposition of a dS exchange, both in the direct and crossed channels, from their AdS counterparts. The relation between AdS and dS exchanges itself is derived using a recently introduced Mellin-Barnes representation for boundary correlators in momentum space, where (A)dS exchanges are straightforwardly fixed by a combination of factorisation, conformal symmetry and boundary conditions.

Integrable lattice models and holography

We study four-dimensional Chern-Simons theory on D × ℂ (where D is a disk), which is understood to describe rational solutions of the Yang-Baxter equation from the work of Costello, Witten and Yamazaki. We find that the theory is dual to a boundary theory, that is a three-dimensional analogue of the two-dimensional chiral WZW model. This boundary theory gives rise to a current algebra that turns out to be an “analytically-continued” toroidal Lie algebra. In addition, we show how certain bulk correlation functions of two and three Wilson lines can be captured by boundary correlation functions of local operators in the three-dimensional WZW model. In particular, we reproduce the leading and subleading nontrivial contributions to the rational R-matrix purely from the boundary theory.

Holography on tessellations of hyperbolic space

We compute boundary correlation functions for scalar fields on tessellations of two- and three-dimensional hyperbolic geometries. We present evidence that the continuum relation between the scalar bulk mass and the scaling dimension associated with boundary-to-boundary correlation functions survives the truncation of approximating the continuum hyperbolic space with a lattice.

Holographic scattering requires a connected entanglement wedge

In AdS/CFT, there can exist local 2-to-2 bulk scattering processes even when local scattering is not possible on the boundary; these have previously been studied in con- nection with boundary correlation functions. We show that boundary regions associated with these scattering configurations must have O(1/GN) mutual information, and hence a connected entanglement wedge. One of us previously argued for this statement from the boundary theory using operational tools in quantum information theory. We improve that argument to make it robust to small errors and provide a proof in the bulk using focusing arguments in general relativity. We also provide a direct link to entanglement wedge reconstruction by showing that the bulk scattering region must lie inside the con- nected entanglement wedge. Our construction implies the existence of nonlocal quantum computation protocols that are exponentially more efficient than the optimal protocols currently known.

Bootstrapping inflationary correlators in Mellin space

We develop a Mellin space approach to boundary correlation functions in anti-de Sitter (AdS) and de Sitter (dS) spaces. Using the Mellin-Barnes representation of correlators in Fourier space, we show that the analytic continuation between AdSd+1 and dSd+1 is encoded in a collection of simple relative phases. This allows us to determine the late-time tree-level three-point correlators of spinning fields in dSd+1 from known results for Witten diagrams in AdSd+1 by multiplication with a simple trigonometric factor. At four point level, we show that Conformal symmetry fixes exchange four-point functions both in AdSd+1 and dSd+1 in terms of the dual Conformal Partial Wave (which in Fourier space is a product of boundary three-point correlators) up to a factor which is determined by the boundary conditions. In this work we focus on late-time four-point correlators with external scalars and an exchanged field of integer spin-ℓ. The Mellin-Barnes representation makes manifest the analytic structure of boundary correlation functions, providing an analytic expression for the exchange four-point function which is valid for general d and generic scaling dimensions, in particular massive, light and (partially-)massless fields. It moreover naturally identifies boundary correlation functions for generic fields with multi-variable Meijer-G functions. When d = 3 we reproduce existing explicit results available in the literature for external conformally coupled and massless scalars. From these results, assuming the weak breaking of the de Sitter isometries, we extract the corresponding correction to the inflationary three-point function of general external scalars induced by a general spin- ℓ field at leading order in slow roll. These results provide a step towards a more systematic understanding of de Sitter observables at tree level and beyond using Mellin space methods.

Conformally covariant boundary correlation functions with a quantum group

Particular boundary correlation functions of conformal field theory are needed to answer some questions related to random conformally invariant curves known as Schramm–Loewner evolutions (SLE). In this article, we introduce a correspondence and establish its fundamental properties, which are used in the companion articles [JJK16, KP16] to explicitly solve two such problems. The correspondence associates Coulomb gas type integrals to vectors in a tensor product representation of a quantum group, a q -deformation of the Lie algebra \mathfrak {sl}_2 . We show that the desired properties of the functions are guaranteed by natural representation-theoretical properties of the vectors.

Boundary correlations for the six-vertex model with reflecting end boundary condition

We consider the six-vertex model with reflecting end boundary condition. We compute analytically boundary correlation functions, such as the boundary polarization and the emptiness formation probability. In order to do that, we use the Sklyanin’s reflection algebra to derive recursion relations for the partition function of the model as well as for the boundary correlations in terms of the partition function. Thanks to the Tsuchiya determinant formula, these recursion relations allow the boundary correlations to be also efficiently written in determinant form.

Two-point boundary correlation functions of dense loop models

We investigate six types of two-point boundary correlation functions in the dense loop model. These are defined as ratios Z/Z^0Z/Z0 of partition functions on the m\times nm×n square lattice, with the boundary condition for ZZ depending on two points xx and yy. We consider: the insertion of an isolated defect (a) and a pair of defects (b) in a Dirichlet boundary condition, the transition (c) between Dirichlet and Neumann boundary conditions, and the connectivity of clusters (d), loops (e) and boundary segments (f) in a Neumann boundary condition.For the model of critical dense polymers, corresponding to a vanishing loop weight (\beta = 0β=0), we find determinant and pfaffian expressions for these correlators. We extract the conformal weights of the underlying conformal fields and find \Delta = -\frac18Δ=−18, 00, -\frac3{32}−332, \frac3838, 11, \tfrac \theta \pi (1+ \tfrac{2\theta}\pi)θπ(1+2θπ), where \thetaθ encodes the weight of one class of loops for the correlator of type f. These results are obtained by analysing the asymptotics of the exact expressions, and by using the Cardy-Peschel formula in the case where xx and yy are set to the corners. For type b, we find a \ln|x-y|ln|x−y| dependence from the asymptotics, and a \ln (\ln n)ln(lnn) term in the corner free energy. This is consistent with the interpretation of the boundary condition of type b as the insertion of a logarithmic field belonging to a rank two Jordan cell.For the other values of \beta = 2 \cos \lambdaβ=2cosλ, we use the hypothesis of conformal invariance to predict the conformal weights and find \Delta = \Delta_{1,2}Δ=Δ1,2, \Delta_{1,3}Δ1,3, \Delta_{0,\frac12}Δ0,12, \Delta_{1,0}Δ1,0, \Delta_{1,-1}Δ1,−1 and \Delta_{\frac{2\theta}\lambda+1,\frac{2\theta}\lambda+1}Δ2θλ+1,2θλ+1, extending the results of critical dense polymers. With the results for type f, we reproduce a Coulomb gas prediction for the valence bond entanglement entropy of Jacobsen and Saleur.