Recent developments on emergence of logarithmic terms in correlators or response functions of models which exhibit dynamical symmetries analogous to conformal invariance in not necessarily relativistic systems are reviewed. The main examples of these are logarithmic Schrödinger-invariance and logarithmic conformal Galilean invariance. Some applications of these ideas to statistical physics are described.
We present the detailed calculations of the asymptotics of two-site correlation functions for height variables in the two-dimensional Abelian sandpile model. By using combinatorial methods for the enumeration of spanning trees, we extend the well-known result for the correlation $σ_{1,1} \simeq 1/r^4$ of minimal heights $h_1=h_2=1$ to $σ_{1,h} = P_{1,h}-P_1P_h$ for height values $h=2,3,4$. These results confirm the dominant logarithmic behaviour $σ_{1,h} \simeq (c_h\log r + d_h)/r^4 + {\cal O}(r^{-5})$ for large $r$, predicted by logarithmic conformal field theory based on field identification…
We show how logarithmic terms may arise in the correlators of fields which belong to the representation of the Schrodinger-Virasoro algebra (SV) or the affine Galilean Conformal Algebra (GCA). We show that in GCA, only scaling operator can have a Jordanian form and rapidity can not. We observe that in both algebras logarithmic dependence appears along the time direction alone.
We will describe how logarithmic singularities arise as limits of Coulomb Gas integrals. Our approach will combine analytic properties of the time-like Liouville structure constants, together with the recursive formula of the Virasoro conformal blocks. Although the Coulomb Gas formalism forces a diagonal coupling between the chiral and antichiral sectors of the Conformal Field Theory (CFT), we present new results for the multi-screening integrals which are potentially interesting for applications to critical statistical systems described by Logarithmic CFTs. In particular our findings extend a…
We study a particular type of logarithmic extension of SL(2,R) Wess-Zumino-Witten models. It is based on the introduction of affine Jordan cells constructed as multiplets of quasi-primary fields organized in indecomposable representations of the Lie algebra sl(2). We solve the simultaneously imposed set of conformal and SL(2,R) Ward identities for two- and three-point chiral blocks. These correlators will in general involve logarithmic terms and may be represented compactly by considering spins with nilpotent parts. The chiral blocks are found to exhibit hierarchical structures revealed by com…
We study the logarithmic conformal field theories in which conformal weights are continuous subset of real numbers. A general relation between the correlators consisting of logarithmic fields and those consisting of ordinary conformal fields is investigated. As an example the correlators of the Coulomb-gas model are explicitly studied.
We study two-dimensional non-abelian BF theory in Lorenz gauge and prove that it is a topological conformal field theory. This opens the possibility to compute topological string amplitudes (Gromov-Witten invariants). We found that the theory is exactly solvable in the sense that all correlators are given by finite-dimensional convergent integrals. Surprisingly, this theory turns out to be logarithmic in the sense that there are correlators given by polylogarithms and powers of logarithms. Furthermore, we found fields with "logarithmic conformal dimension" (elements of a Jordan cell for $L_0$)…
The analytic structure of non-Gaussian correlators in inflationary cosmologies has recently been proposed as a test of the quantum origin of structure in the universe. To further understand this proposal, we explore the analogous equal-time in-in correlators in flat space and show they exhibit the same features as their cosmological counterparts. The quantum vacuum is uniquely identified by in-in correlators with a total energy pole and no additional poles at physical momenta. We tie this behavior directly to the S-matrix and show that poles at physical momenta always arise from scattering of …
We propose a contour integral representation for the one-point correlators at genus one of the primaries of a family of rational logarithmic conformal field theories.
The Einstein-Proca action is known to have asymptotically locally Lifshitz spacetimes as classical solutions. For dynamical exponent z=2, two-point correlation functions for fluctuations around such a geometry are derived analytically. It is found that the retarded correlators are stable in the sense that all quasinormal modes are situated in the lower half-plane of complex frequencies. Correlators in the longitudinal channel exhibit features that are reminiscent of a structure usually obtained in field theories that are logarithmic, i.e. contain an indecomposable highest weight representation…
We derive the analogues of the Harer-Zagier formulas for single- and double-trace correlators in the q-deformed Hermitian Gaussian matrix model. This fully describes single-trace correlators and opens a road to $q$-deformations of important matrix models properties, such as genus expansion and Wick theorem.
Universality in unitary invariant random matrix ensembles with complex matrix elements is considered. We treat two general ensembles which have a determinant factor in the weight. These ensembles are relevant, e.g., for spectra of the Dirac operator in QCD. In addition to the well established universality with respect to the choice of potential, we prove that microscopic spectral correlators are unaffected when the matrix in the determinant is replaced by an expansion in powers of the matrix. We refer to this invariance as logarithmic universality. The result is used in proving that a simple r…
We compute lattice correlation functions for the model of critical dense polymers on a semi-infinite cylinder of perimeter $n$. In the lattice loop model, contractible loops have a vanishing fugacity whereas non-contractible loops have a fugacity $α\in(0,\infty)$. These correlators are defined as ratios $Z(x)/Z_0$ of partition functions, where $Z_0$ is a reference partition function wherein only simple arcs are attached to the boundary of the cylinder. For $Z(x)$, the boundary is also decorated with simple arcs, but it also has two positions $1$ and $x$ where the boundary condition is differen…
JIMWLK equation tells how gauge invariant higher order Wilson line correlators would evolve at high energy. In this article we present a convenient integro-differential form of this equation, for 2n-tuple correlator, where all real and virtual terms are explicit. The `real' terms correspond to splitting (say at position z) of this 2n-tuple correlator to various pairs of 2m-tuple and (2n+2-2m)-tuple correlators whereas `virtual' terms correspond to splitting into pairs of 2m-tuple and (2n-2m)-tuple correlators. Kernels of virtual terms with m=0 (no splitting) and of real terms with m=1 (splitti…
We generalize the one-to one correspondence between quasi normal modes in 3- dimensional anti deSitter black holes and the poles of the retarded correlators in the boundary conformal field theory to include logarithmic operators in the latter. This analysis is carried out explicitly for the logarithmic mode in chiral topologically massive gravity.
We study correlators of four protected (half-BPS) operators in strongly coupled supersymmetric Yang-Mills theory. These are dual to tree-level supergravity amplitudes on AdS${}_5\times$S${}_5$ for various spherical harmonics on the five-sphere. We use conformal field theory methods, in particular a recently obtained Lorentzian inversion formula, to analytically bootstrap these correlators. The extracted $1/N^2$ double-trace anomalous dimensions confirm a simple pattern recently conjectured by Aprile, Drummond, Heslop and Paul. We explain this pattern by an unexpected ten-dimensional conformal …
Although logarithmic conformal field theories (LCFTs) are known not to factorise many previous findings have only been formulated on their chiral halves. Making only mild and rather general assumptions on the structure of an chiral LCFT we deduce statements about its local non-chiral equivalent. Two methods are presented how to construct local representations as subrepresentations of the tensor product of chiral and anti-chiral Jordan cells. Furthermore we explore the assembly of generic non-chiral correlation functions from generic chiral and anti-chiral correlators. The constraint of localit…
It is now well known that non-local observables in critical statistical lattice models, polymers and percolation for example, may be modelled in the continuum scaling limit by logarithmic conformal field theories. Fusion rules for such theories, sometimes referred to as logarithmic minimal models, have been intensively studied over the last ten years in order to explore the representation-theoretic structures relevant to non-local observables. Motivated by recent lattice conjectures, this work studies the fusion rules of the $N=1$ supersymmetric analogues of these logarithmic minimal models in…
We use the process of quantum hamiltonian reduction of SU(2)_k, at rational level k, to study explicitly the correlators of the h_{1,s} fields in the c_{p,q} models. We find from direct calculation of the correlators that we have the possibility of extra, chiral and non-chiral, multiplet structure in the h_{1,s} operators beyond the `minimal' sector. At the level of the vacuum null vector h_{1,2p-1}=(p-1)(q-1) we find that there can be two extra non-chiral fermionic fields. The extra indicial structure present here permeates throughout the entire theory. In particular we find we have a chiral …
The generic structure of 4-point functions of fields residing in indecomposable representations of arbitrary rank is given. The used algorithm is described and we present all results for Jordan-rank $r=2$ and $r=3$ where we make use of permutation symmetry and use a graphical representation for the results. A number of remaining degrees of freedom which can show up in the correlator are discussed in detail. Finally we present the results for two-logarithmic fields for arbitrary Jordan-rank.
We investigate quantum entanglement in a non-relativistic critical system by calculating the logarithmic negativity of a class of mixed states in the quantum Lifshitz model in one and two spatial dimensions. In 1+1 dimensions we employ a correlator approach to obtain analytic results for both open and periodic biharmonic chains. In 2+1 dimensions we use a replica method and consider spherical and toroidal spatial manifolds. In all cases, the universal finite part of the logarithmic negativity vanishes for mixed states defined on two disjoint components. For mixed states defined on adjacent com…
A general discussion of the conformal Ward identities is presented in the context of logarithmic conformal field theory with conformal Jordan cells of rank two. The logarithmic fields are taken to be quasi-primary. No simplifying assumptions are made about the operator-product expansions of the primary or logarithmic fields. Based on a very natural and general ansatz about the form of the two- and three-point functions, their complete solutions are worked out. The results are in accordance with and extend the known results. It is demonstrated, for example, that the correlators exhibit hierarch…
Based on the modular functor associated with a -- not necessarily semisimple -- finite non-degenerate ribbon category $\mathcal D$, we present a definition of a consistent system of bulk field correlators for a conformal field theory which comprises invariance under mapping class group actions and compatibility with the sewing of surfaces. We show that when restricting to surfaces of genus zero such systems are in bijection with commutative symmetric Frobenius algebras in $\mathcal D$, while for surfaces of any genus they are in bijection with modular Frobenius algebras in $\mathcal D$. This p…
The analogue of the charge-conjugation modular invariant for rational logarithmic conformal field theories is constructed. This is done by reconstructing the bulk spectrum from a simple boundary condition (the analogue of the Cardy `identity brane'). We apply the general method to the c_1,p triplet models and reproduce the previously known bulk theory for p=2 at c=-2. For general p we verify that the resulting partition functions are modular invariant. We also construct the complete set of 2p boundary states, and confirm that the identity brane from which we started indeed exists. As a by-prod…
We study O(n)-symmetric two-dimensional conformal field theories (CFTs) for a continuous range of n below two. These CFTs describe the fixed point behavior of self-avoiding loops. There is a pair of known fixed points connected by an RG flow. When n is equal to two, which corresponds to the Kosterlitz-Thouless critical theory, the fixed points collide. We find that for n generic these CFTs are logarithmic and contain negative norm states; in particular, the O(n) currents belong to a staggered logarithmic multiplet. Using a conformal bootstrap approach we trace how the negative norm states deco…
Inspired by a recent experiment by Steinhauer and co-workers, we present a simple model which describes the formation of an acoustic black hole in a Bose-Einstein condensate, allowing an analytical computation of the evolution in time of the corresponding density-density correlator. We show the emergence of analog Hawking radiation out of a "quantum atmosphere" region significantly displaced from the horizon. This is quantitatively studied both at $T=0$ and even in the presence of an initial temperature T, as is always the case experimentally.
We introduce an infinite set of jet substructure observables, derived as projections of $N$-point energy correlators, that are both convenient for experimental studies and maintain remarkable analytic properties derived from their representations in terms of a finite number of light ray operators. We show that these observables can be computed using tracking or charge information with a simple reweighting by integer moments of non-perturbative track or fragmentation functions. Our results for the projected $N$-point correlators are analytic functions of $N$, allowing us to derive resummed resu…
We calculate the four point correlation function for scalar perturbations in the canonical model of slow-roll inflation. We work in the leading slow-roll approximation where the calculation can be done in de Sitter space. Our calculation uses techniques drawn from the AdS/CFT correspondence to find the wave function at late times and then calculate the four point function from it. The answer we get agrees with an earlier result in the literature, obtained using different methods. Our analysis reveals a subtlety with regard to the Ward identities for conformal invariance, which arises in de Sit…
Generalizing the concept of primary fields, we find a new representation of the Virasoro algebra, which we call it a pseudo-conformal representation. In special cases, this representation reduces to ordinary- or logarithmic-conformal field theory. There are, however, other cases in which the Green functions differ from those of ordinary- or logarithmic-conformal field theories. This representation is parametrized by two matrices. We classify these two matrices, and calculate some of the correlators for a simple example.
We study the extent to which diffeomorphism invariance restricts the properties of the primordial perturbations in single scalar field models. We derive a set of identities that constrain the connected correlators of the cosmological perturbations, as well as the one-particle-irreducible vertices of the theory in any gauge. These identities are the analogues of Slavnov-Taylor identities in gauge theories, and follow essentially from diffeomorphism invariance alone. Yet because quantization requires diffeomorphism invariance to be broken, they not only reflect invariance under diffeomorphisms, …
The onset of quantum chaos in quantum field theory may be studied using out-of-time-order correlators at finite temperature. Recent work argued that a timescale logarithmic in the central charge emerged in the context of two-dimensional conformal field theories, provided the intermediate channel was dominated by the Virasoro identity block. This suggests a wide class of conformal field theories exhibit a version of fast scrambling. In the present work we study this idea in more detail. We begin by clarifying to what extent correlators of wavepackets built out of superpositions of primary opera…
We analyse the SU(2)_k WZNW models beyond the integrable representations and in particular the case of SU(2)_0. We find that these are good examples of logarithmic conformal field theories as indecomposable representations are naturally produced in the fusion of discrete irreducible representations. We also find extra, chiral and non-chiral, multiplet structure in the theory. The chiral fields, which we construct explicitly in SU(2)_0, generate extended algebras within the model. We also study the process of quantum hamiltonian reduction of SU(2)_0, giving the c=-2 triplet model, in both the f…
Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Napoli（意大利国家核物理研究所（INFN）那不勒斯分部）；University of Southampton（南安普顿大学）；Indian Institute of Technology Indore（印度理工学院印多尔分校）
In recent years, various aspects of theoretical models with long range interactions have attracted attention, ranging from out-of-time-ordered correlators to entanglement. In the present paper, entanglement properties of a simple non-local model with long-range interactions in the form of a fractional Laplacian is investigated in both static and a quantum quench scenario. Logarithmic negativity, which is a measure for entanglement in mixed states is calculated numerically. In the static case, it is shown that the presence of long-range interaction ensures that logarithmic negativity decays muc…
Correlators of local operators inserted on a straight Wilson loop in a conformal gauge theory have the structure of a one-dimensional "defect" CFT. As was shown in arXiv:1706.00756, in the case of supersymmetric Wilson-Maldacena loop in $\mathcal{N}=4$ SYM one can compute the strong-coupling contributions to 4-point correlators of operator insertions by starting with the AdS$_5 \times S^5$ string action expanded near the AdS$_2$ minimal surface and evaluating the corresponding AdS$_2$ Witten diagrams. We perform the analogous computations in the non-supersymmetric case of the standard Wilson l…
