Conformal Dimension Research Articles

Entanglement distance between quantum states and its implications for a density-matrix renormalization group study of degenerate ground states

We study the concept of entanglement distance between two quantum states which quantifies the amount of information shared between their reduced density matrices (RDMs). Using analytical arguments combined with density-matrix-renormalization-group (DMRG) and exact diagonalization (ED) calculations, we show that for gapless systems the entanglement distance has power law dependence on the energy separation and subsystem size with $\alpha_E$ and $\alpha_{\ell}$ exponents, respectively. Using conformal field theory (CFT) we find $\alpha_E = 2$ and $\alpha_{\ell} = 4$ for Abelian theories with $c=1$ such as free fermions. For non-Abelian CFTs $\alpha_E = 0$ , and $\alpha_{\ell}$ is twice the conformal dimension of the thermal primary fields. For instance for $Z_3$ parafermion CFT $\alpha_E = 1$ and $\alpha_{\ell} = 4/5$. For gapped 1+1D fermion systems, we show that the entanglement distance divides the low energy excitations into two branches with different values of $\alpha_E$ and $\alpha_{\ell}$. These two branches are related to momentum transfers near zero and $\pi$. We also demonstrate that the entanglement distance reaches its maximum for degenerate states related through nonlocal operators such as Wilson loops. For example, degenerate ground-states (GSs) of 2+1 D topological states have maximum entanglement distance. On the contrary, degenerate GSs related through confined anyon excitations such as genons have minimum entanglement distance. Various implications of this concept for quantum simulations are discussed. Finally, based on the ideas developed we discuss the computational complexity of DMRG algorithms that are capable of finding all degenerate GSs.

Holographic 4-point correlators with heavy states

The AdS/CFT duality maps supersymmetric heavy operators with conformal dimension of the order of the central charge to asymptotically AdS supergravity solutions. We show that by studying the quadratic fluctuations around such backgrounds it is possible to derive the 4-point correlators of two light and two heavy states in the supergravity approximation. We provide an explicit example in the AdS3 setup relevant for the duality with the D1-D5 CFT. Contrary to previously studied examples, the supergravity correlator derived in this work differs from the result obtained at the CFT orbifold point. Our method bypasses the difficulties of applying the standard Witten’s diagrams approach to correlators with operators of large conformal dimension and also avoids some technical steps that have made the computation of dynamical 4-point correlators in the AdS3/CFT2 context unfeasible until now.

Holographic duals of large-c torus conformal blocks

We study CFT2 conformal blocks on a torus and their holographic realization. The classical conformal blocks arising in the regime where conformal dimensions grow linearly with the large central charge are shown to be holographically dual to the geodesic networks stretched in the thermal AdS bulk space. We discuss the n-point conformal blocks and their duals, the 2-point case is elaborated in full detail. We develop various techniques to calculate both quantum and classical conformal block functions. In particular, we show that exponentiated global torus blocks reproduce classical torus blocks in the specific perturbative regimes of the conformal parameter space.

Conformal basis for flat space amplitudes

We study solutions of the Klein-Gordon, Maxwell, and linearized Einstein equations in $\mathbb{R}^{1,d+1}$ that transform as $d$-dimensional conformal primaries under the Lorentz group $SO(1,d+1)$. Such solutions, called conformal primary wavefunctions, are labeled by a conformal dimension $\Delta$ and a point in $\mathbb{R}^d$, rather than an on-shell $(d+2)$-dimensional momentum. We show that the continuum of scalar conformal primary wavefunctions on the principal continuous series $\Delta\in \frac d2+ i\mathbb{R}$ of $SO(1,d+1)$ spans a complete set of normalizable solutions to the wave equation. In the massless case, with or without spin, the transition from momentum space to conformal primary wavefunctions is implemented by a Mellin transform. As a consequence of this construction, scattering amplitudes in this basis transform covariantly under $SO(1,d+1)$ as $d$-dimensional conformal correlators.

Conformational Heterogeneity and FRET Data Interpretation for Dimensions of Unfolded Proteins

A mathematico-physically valid formulation is required to infer properties of disordered protein conformations from single-molecule Förster resonance energy transfer (smFRET). Conformational dimensions inferred by conventional approaches that presume a homogeneous conformational ensemble can be unphysical. When all possible—heterogeneous as well as homogeneous—conformational distributions are taken into account without prejudgment, a single value of average transfer efficiency 〈E〉 between dyes at two chain ends is generally consistent with highly diverse, multiple values of the average radius of gyration 〈Rg〉. Here we utilize unbiased conformational statistics from a coarse-grained explicit-chain model to establish a general logical framework to quantify this fundamental ambiguity in smFRET inference. As an application, we address the long-standing controversy regarding the denaturant dependence of 〈Rg〉 of unfolded proteins, focusing on Protein L as an example. Conventional smFRET inference concluded that 〈Rg〉 of unfolded Protein L is highly sensitive to [GuHCl], but data from SAXS suggested a near-constant 〈Rg〉 irrespective of [GuHCl]. Strikingly, our analysis indicates that although the reported 〈E〉 values for Protein L at [GuHCl] = 1 and 7 M are very different at 0.75 and 0.45, respectively, the Bayesian Rg2 distributions consistent with these two 〈E〉 values overlap by as much as 75%. Our findings suggest, in general, that the smFRET-SAXS discrepancy regarding unfolded protein dimensions likely arise from highly heterogeneous conformational ensembles at low or zero denaturant, and that additional experimental probes are needed to ascertain the nature of this heterogeneity.

Thermality of eigenstates in conformal field theories.

The eigenstate thermalization hypothesis (ETH) provides a way to understand how an isolated quantum mechanical system can be approximated by a thermal density matrix. We find a class of operators in (1+1)-dimensional conformal field theories, consisting of quasiprimaries of the identity module, which satisfy the hypothesis only at the leading order in large central charge. In the context of subsystem ETH, this plays a role in the deviation of the reduced density matrices, corresponding to a finite energy density eigenstate from its hypothesized thermal approximation. The universal deviation in terms of the square of the trace-square distance goes as the eighth power of the subsystem fraction and is suppressed by powers of inverse central charge (c). Furthermore, the nonuniversal deviations from subsystem ETH are found to be proportional to the heavy-light-heavy structure constants which are typically exponentially suppressed in sqrt[h/c], where h is the conformal scaling dimension of the finite energy density state. We also examine the effects of the leading finite-size corrections.

Far-from-equilibrium initial conditions probed by a nonlocal observable

Using the gauge/gravity duality, we investigate the evolution of an out-of-equilibrium strongly-coupled plasma from the viewpoint of the two-point function of scalar gauge-invariant operators with large conformal dimension. This system is out of equilibrium due to the presence of anisotropy and/or a massive scalar field. Considering various functions for the initial anisotropy and scalar field, we conclude that the effect of the anisotropy on the evolution of the two-point function is considerably more than the effect of the scalar field. We also show that the ordering of the equilibration time of the one-point function for the non-probe scalar field and the correlation function between two points with a fixed separation can be reversed by changing the initial configuration of the plasma, when the system is out of the equilibrium due to the presence of at least two different sources like our problem. In addition, we find the equilibration time of the two-point function to be linearly increasing with respect to the separation of the two points with a fixed slope, regardless of the initial configuration that we start with. Finally we observe that, for larger separations the geodesic connecting two points on the boundary crosses the event horizon after it has reached its final equilibrium value, meaning that the two-point function can probe behind the event horizon.

Large N correlation functions mathcal{N} = 2 superconformal quivers

Using supersymmetric localization, we consider four-dimensional mathcal{N} = 2 superconformal quiver gauge theories obtained from {mathbb{Z}}_n orbifolds of mathcal{N} = 4 Super Yang-Mills theory in the large N limit at weak coupling. In particular, we show that: 1) The partition function for arbitrary couplings can be constructed in terms of universal building blocks. 2) It can be computed in perturbation series, which converges uniformly for |λI | < π2, where λI are the ’t Hooft coupling of the gauge groups. 3) The perturbation series for two-point functions can be explicitly computed to arbitrary orders. There is no universal effective coupling by which one can express them in terms of correlators of the mathcal{N} = 4 theory. 4) One can define twisted and untwisted sector operators. At the perturbative orbifold point, when all the couplings are the same, the correlators of untwisted sector operators coincide with those of mathcal{N} = 4 Super Yang-Mills theory. In the twisted sector, we find remarkable cancellations of a certain number of planar loops, determined by the conformal dimension of the operator.

Geodesic description of heavy-light Virasoro blocks

We continue to investigate the dual description of the Virasoro conformal blocks arising in the framework of the classical limit of the AdS3/CFT2 correspondence. To give such an interpretation in previous studies, certain restrictions were necessary. Our goal here is to consider a more general situation available through the worldline approximation to the dual AdS gravity. Namely, we are interested in computing the spherical conformal blocks without the previously imposed restrictions on the conformal dimensions of the internal channels. The duality is realised as an equality of the so-called heavy-light limit of the n-point conformal block and the action of n−2 particles propagating in some AdS-like background with either a conical singularity or a BTZ black hole. We describe a procedure that allows relaxing the constraint on the intermediate channels. To obtain an explicit expression for the conformal block on the CFT side, we use a recently proposed recursion procedure and find full agreement between the results of the boundary and bulk computations.

Two-point functions in AdS/dCFT and the boundary conformal bootstrap equations

We calculate the leading contributions to the connected two-point functions of protected scalar operators in the defect version of mathcal{N} = 4 SYM theory which is dual to the D5-D3 probe-brane system with k units of background gauge field flux. This involves several types of two-point functions which are vanishing in the theory without the defect, such as two-point functions of operators of unequal conformal dimension. We furthermore exploit the operator product expansion (OPE) and the boundary operator expansion (BOE), which form the basis of the boundary conformal bootstrap equations, to extract conformal data both about the defect CFT and about mathcal{N} = 4 SYM theory without the defect. From the knowledge of the one- and two-point functions of the defect theory, we extract certain structure constants of mathcal{N} = 4 SYM theory using the (bulk) OPE and constrain certain bulk-to-boundary couplings using the BOE. The extraction of the former relies on a non-trivial, polynomial k dependence of the one-point functions, which we explicitly demonstrate. In addition, it requires the knowledge of the one-point functions of SU(2) descendant operators, which we likewise explicitly determine.

Reconceptualizing the Role of Conformity Behaviors in Family Communication Patterns Theory

ABSTRACTFamily communication patterns theory is one of the most influential theories in the field of family communication. The current article examines the construct of conformity, arguing that current conceptualizations of conformity paint this behavior as largely a negative familial element. Instead, this research argues for the concept of warm conformity behavior, where parents communicate their ideas in ways that promote warmth and closeness rather than just adherence to hierarchy and rules. This move does not add to the two dimensions of the theory, but rather speaks to the different ways in which parents might communicate the dimension of conformity to their child. To further this idea, our article includes a series of three studies designed to create and validate new measures of warm and cold conformity. Overall, the studies found general support for the validity of the scales. Implications and directions for future research are discussed.

PENETAPAN POSISI DI BENAK KONSUMEN (POSITIONING) MUSEUM NEGERI NTB BERDASAR PERSEPSI PENGUNJUNG

This study aimed to determining the positioning of the State Museum of West Nusa Tenggara based on visitor’s perception. The research method used was qualitative approach with data collecting through interview, observation and documentation. Informants is selected by accidental sampling technique. Data from interviews with 20 informants were analyzed using content analysis method. The results showed that the State Museum of West Nusa Tenggara hada positioning as edutainment with dimensions of value, uniqueness, credibility, sustainability and conformity. The museum should consider the desire and needs of visitors with the main purpose of the museum, which is to present the mission of education as well as entertainment to the public. Thus, the Museum of West Nusa Tenggara is expected to be a place to gain knowledge and experience as well as recreation that reaches all levels of society. The most important contribution of this study is the positioning model of the State Museum of West Nusa Tenggara based on the perception of visitors i.e. edutainment concept to become a pioneer for further empirical research. It is expected that in the future the managerial of West Nusa Tenggara State Museum can increase visit to the museum. Keywords: Museum, positioning, visitor perceptions, edutainment

Conformal dimension on boundary of right-angled hyperbolic buildings

In this note, we use some combinatorial modulus on the boundary of a right-angled hyperbolic building to control its conformal dimension. The lower bound obtained is optimal in the case of Fuchsian buildings.

Operator mixing in deformed D1D5 CFT and the OPE on the cover

We consider the D1D5 CFT near the orbifold point and develop methods for computing the mixing of untwisted operators to first order by using the OPE on the covering surface. We argue that the OPE on the cover encodes both the structure constants for the orbifold CFT and the explicit form of the mixing operators. We show this explicitly for some example operators. We start by considering a family of operators dual to supergravity modes, and show that the OPE implies that there is no shift in the anomalous dimension to first order, as expected. We specialize to the operator dual to the dilaton, and show that the leading order singularity in the OPE reproduces the correct structure constant. Finally, we consider an unprotected operator of conformal dimension (2,2), and show that the leading order singularity and one of the subleading singularities both reproduce the correct structure constant. We check that the operator produced at subleading order using the OPE method is correct by calculating a number of three point functions using a Mathematica package we developed. Further development of this OPE technique should lead to more efficient calculations for the D1D5 CFT perturbed away from the orbifold point.

Conformal dimension and boundaries of planar domains

Building off of techniques that were recently developed by M. Carrasco, S. Keith, and B. Kleiner to study the conformal dimension of boundaries of hyperbolic groups, we prove that uniformly perfect boundaries of John domains in C ^ \hat {\mathbb {C}} have conformal dimension equal to 0 or 1. Our proof uses a discretized version of Carrasco’s “uniformly well-spread cut point” condition, which we call the discrete UWS property, that is well-suited to deal with metric spaces that are not linearly connected. More specifically, we prove that boundaries of John domains have the discrete UWS property and that any compact, doubling, uniformly perfect metric space with the discrete UWS property has conformal dimension equal to 0 or 1. In addition, we establish other geometric properties of metric spaces with the discrete UWS property, including connectivity properties of their weak tangents.

Exact holography of the mass-deformed M2-brane theory

We test the holographic relation between the vacuum expectation values of gauge invariant operators in {mathcal {N}} = 6 U_k(N)times mathrm{U}_{-k}(N) mass-deformed ABJM theory and the LLM geometries with {mathbb {Z}}_k orbifold in 11-dimensional supergravity. To do so, we apply the Kaluza–Klein reduction to construct a 4-dimensional gravity theory and implement the holographic renormalization procedure. We obtain an exact holographic relation for the vacuum expectation values of the chiral primary operator with conformal dimension Delta = 1, which is given by langle {mathcal {O}}^{(Delta =1)}rangle = N^{frac{3}{2}} , f_{(Delta =1)}, for large N and k=1. Here the factor f_{(Delta )} is independent of N. Our results involve an infinite number of exact dual relations for all possible supersymmetric Higgs vacua and so provide a non-trivial test of gauge/gravity duality away from the conformal fixed point. We extend our results to the case of kne 1 for LLM geometries represented by rectangular-shaped Young diagrams. We also discuss the exact mapping of the gauge/gravity at finite N for classical supersymmetric vacuum solutions in field theory side and corresponding classical solutions in gravity side.

Mass-deformed ABJM theory and LLM geometries: exact holography

We present a detailed account and extension of our claim in arXiv:1610.01490. We test the gauge/gravity duality between the mathcal{N} = 6 mass-deformed ABJM theory with Uk(N )×U−k(N ) gauge symmetry and the 11-dimensional supergravity on LLM geometries with SO(4)/ℤk × SO(4)/ℤk isometry, in the large N limit. Our analysis is based on the evaluation of vacuum expectation values of chiral primary operators from the supersymmetric vacua of mass-deformed ABJM theory and from the implementation of Kaluza-Klein holography to the LLM geometries. We focus on the chiral primary operator with conformal dimension Δ = 1. We show that leftlangle {mathcal{O}}^{left(varDelta =1right)}rightrangle ={N}^{frac{3}{2}}{f}_{left(varDelta =1right)} for all supersymmetric vacuum solutions and LLM geometries with k = 1, where the factor f(Δ) is independent of N. We also confirm that the vacuum expectation value of the energy momentum tensor is vanishing as expected by the supersymmetry. We extend our results to the case of k ≠ 1 for LLM geometries represented by rectangular-shaped Young-diagrams. In analogy with the Coulomb branch of the mathcal{N} = 4 super Yang-Mills theory, we argue that the discrete Higgs vacua of the mABJM theory as well as the corresponding LLM geometries are parametrized by the vevs of the chiral primary operators.

The analytic structure of conformal blocks and the generalized Wilson-Fisher fixed points

We describe in detail the method used in our previous work arXiv:1611.10344 to study the Wilson-Fisher critical points nearby generalized free CFTs, exploiting the analytic structure of conformal blocks as functions of the conformal dimension of the exchanged operator. Our method is equivalent to the mechanism of conformal multiplet recombination set up by null states. We compute, to the first non-trivial order in the ϵ-expansion, the anomalous dimensions and the OPE coefficients of infinite classes of scalar local operators using just CFT data. We study single-scalar and O(N)-invariant theories, as well as theories with multiple deformations. When available we agree with older results, but we also produce a wealth of new ones. Unitarity and crossing symmetry are not used in our approach and we are able to apply our method to non-unitary theories as well. Some implications of our results for the study of the non-unitary theories containing partially conserved higher-spin currents are briefly mentioned.

Various semiclassical limits of torus conformal blocks

We study four types of one-point torus blocks arising in the large central charge regime. There are the global block, the light block, the heavy-light block, and the linearized classical block, according to different regimes of conformal dimensions. It is shown that the blocks are not independent being connected to each other by various links. We find that the global, light, and heavy-light blocks correspond to three different contractions of the Virasoro algebra. Also, we formulate the c-recursive representation of the one-point torus blocks which is relevant in the semiclassical approximation.

Central charge from adiabatic transport of cusp singularities in the quantum Hall effect

We study quantum Hall (QH) states on a punctured Riemann sphere. We compute the Berry curvature under adiabatic motion in the moduli space in the large N limit. The Berry curvature is shown to be finite in the large N limit and controlled by the conformal dimension of the cusp singularity, a local property of the mean density. Utilizing exact sum rules obtained from a Ward identity, we show that for the Laughlin wave function, the dimension of a cusp singularity is given by the central charge, a robust geometric response coefficient in the QHE. Thus, adiabatic transport of curvature singularities can be used to determine the central charge of QH states. We also consider the effects of threaded fluxes and spin-deformed wave functions. Finally, we give a closed expression for all moments of the mean density in the integer QH state on a punctured disk.