States In Conformal Field Theories Research Articles

Studying the 3d Ising surface CFTs on the fuzzy sphere

Boundaries not only are fundamental elements in nearly all realistic physical systems, but also greatly enrich the structure of quantum field theories. In this paper, we demonstrate that conformal field theory (CFT) with a boundary, known as surface CFT in three dimensions, can be studied with the setup of fuzzy sphere. We consider the example of surface criticality of the 3d Ising CFT. We propose two schemes by cutting a boundary in the orbital space and the real space respectively to realise the ordinary and the normal surface CFTs on the fuzzy sphere. We obtain the operator spectra through state-operator correspondence. We observe integer spacing of the conformal multiplets, and thus provide direct evidence of conformal symmetry. We identify the ordinary surface primary oo, the displacement operator \mathrm{D}D and their conformal descendants and extract their scaling dimensions. We also study the one-point and two-point correlation functions and extract the bulk-to-surface OPE coefficients, some of which are reported for the first time. In addition, using the overlap of the bulk CFT state and the polarised state, we calculate the boundary central charges of the 3d Ising surface CFTs non-perturbatively. Other conformal data obtained in this way also agrees with prior methods. We also discuss future directions, such as special surface CFT, cone defects, and surfaces of other bulk CFTs.

Imprints of phase transitions on Kasner singularities

Under the AdS/CFT correspondence, asymptotically anti–de Sitter geometries with backreaction can be viewed as conformal field theory states subject to a renormalization group (RG) flow from an ultraviolet (UV) description toward an infrared (IR) sector. For black holes, however, the IR point is the horizon, so one way to interpret the interior is as an analytic continuation to a “trans-IR” imaginary-energy regime. In this paper, we demonstrate that this analytic continuation preserves some imprints of the UV physics, particularly near its “end point” at the classical singularity. We focus on holographic phase transitions of geometric objects in round black holes. We first assert the consistency of interpreting such black holes, including their interiors, as RG flows by constructing a monotonic a function. We then explore how UV phase transitions of entanglement entropy and scalar two-point functions, each of which are encoded by bulk geometry under the holographic mapping, are related to the structure of the near-singularity geometry, which is quantified by Kasner exponents. Using 2D holographic flows triggered by relevant scalar deformations as test beds, we find that the 3D bulk’s near-singularity Kasner exponents can be viewed as functions of the UV physics precisely when the deformation is nonzero. Published by the American Physical Society 2024

Holographic thermodynamics of rotating black holes

We provide mass/energy formulas for the extended thermodynamics, mixed thermodynamics, and holographic conformal field theory (CFT) thermodynamics for the charged and rotating Kerr-Newman Anti-de Sitter black holes. Then for the CFT thermal states dual to the black hole, we find the first-order phase transitions and criticality phenomena in the canonical ensemble with fixed angular momentum, volume, and central charge. We observe that the CFT states cannot be analogous to the Van der Waals fluids, despite the critical exponents falling into the universality class predicted by the mean field theory. Additionally, we examine the (de)confinement phase transitions within the grand canonical ensemble with fixed angular velocity, volume, and central charge of the CFT. Our findings suggest that the near zero temperature (de)confinement phase transitions can occur with the angular velocity of the CFT that solely depends on the CFT volume.

Area operator and fixed area states in conformal field theories

The fixed area states are constructed by gravitational path integrals in previous studies.In this paper we show the dual of the fixed area states in conformal field theories (CFTs).These CFT states are constructed by using spectrum decomposition of reduced density matrix $\rho_A$ for a subsystem $A$. For 2 dimensional CFTs we directly construct the bulk metric, which is consistent with the expected geometry of the fixed area states. For arbitrary pure geometric state $|\psi\rangle$ in any dimension we also find the consistency by using the gravity dual of R\'enyi entropy. We also give the relation of parameters for the bulk and boundary state. The pure geometric state $|\psi\rangle$ can be expanded as superposition of the fixed area states. Motivated by this, we propose an area operator $\hat A^\psi$. The fixed area state is the eigenstate of $\hat A^\psi$, the associated eigenvalue is related to R\'enyi entropy of subsystem $A$ in this state. The Ryu-Takayanagi formula can be expressed as the expectation value $\langle \psi| {\hat A}^\psi|\psi\rangle$ divided by $4G$, where $G$ is the Newton constant. We also show the fluctuation of the area operator in the geometric state $|\psi\rangle$ is suppressed in the semiclassical limit $G\to0$.

Generalized entanglement entropies in two-dimensional conformal field theory

We introduce and study generalized Rényi entropies defined through the traces of products of TrB(| Ψi⟩⟨Ψj| ) where ∣Ψi⟩ are eigenstates of a two-dimensional conformal field theory (CFT). When ∣Ψi⟩ = ∣Ψj⟩ these objects reduce to the standard Rényi entropies of the eigenstates of the CFT. Exploiting the path integral formalism, we show that the second generalized Rényi entropies are equivalent to four point correlators. We then focus on a free bosonic theory for which the mode expansion of the fields allows us to develop an efficient strategy to compute the second generalized Rényi entropy for all eigenstates. As a byproduct, our approach also leads to new results for the standard Rényi and relative entropies involving arbitrary descendent states of the bosonic CFT.

НИЗКОЛЕЖАЩИЕ ЧАСТОТЫ КВАЗИНОРМАЛЬНЫХ МОД ЧЕРНОЙ ДЫРЫ ТАУБ-НУТ

Quasi-normal modes of black holes determine the damping of disturbances at intermediate times and are important in studying the dynamics of black holes and external fields around them. Currently, interest in quasi-normal modes is due to three of their features. The first one is connected with the possibility of observing quasi-normal modes and obtaining a "trace" of a black hole with the help of new-generation gravitational antennas under construction. Recently, collaborations between the Laser Interferometric Gravitational Wave Observatory (LIGO) and the French-Italian gravitational wave detector located at the European Gravitational Observatory (VIRGO) reported the observation of a gravitational wave signal corresponding to the spiral and merging of two black holes, resulting in the formation of a single black hole. It was shown that the observations agree with Einstein's theory of gravity with a high accuracy, limited mainly by a statistical error. The second concerns the anti-de Sitter / Conformal field theory correspondence, which implies that a large black hole in anti-de Sitter space corresponds roughly to thermal states in Conformal field theory. Thus, the damping of black hole perturbations can be associated with the return to thermal equilibrium of the perturbed state in the Conformal field theory. Note that in anti-de Sitter space, the quasi-normal modes of black holes control the decay of the field at later times, since there are no power-law tails and the decay is always exponential. The third feature is associated with the possible connection of quasinormal modes of black holes in some space-time geometries with the Choptyuk scaling. This paper investigates the low-lying frequencies of the quasi-normal modes of the Taub-NUT (Newman-Unti-Tamburino) black hole for scalar, electromagnetic, and Dirac perturbations using the Wenzel, Kramers, and Brillouin (WKB) approximation of the third order. The influence of the NUT metric parameter on the considered types of frequencies of quasi-normal modes is shown.

Tensor network models of AdS/qCFT

The study of critical quantum many-body systems through conformal field theory (CFT) is one of the pillars of modern quantum physics. Certain CFTs are also understood to be dual to higher-dimensional theories of gravity via the anti-de Sitter/conformal field theory (AdS/CFT) correspondence. To reproduce various features of AdS/CFT, a large number of discrete models based on tensor networks have been proposed. Some recent models, most notably including toy models of holographic quantum error correction, are constructed on regular time-slice discretizations of AdS. In this work, we show that the symmetries of these models are well suited for approximating CFT states, as their geometry enforces a discrete subgroup of conformal symmetries. Based on these symmetries, we introduce the notion of a quasiperiodic conformal field theory (qCFT), a critical theory less restrictive than a full CFT and with characteristic multi-scale quasiperiodicity. We discuss holographic code states and their renormalization group flow as specific implementations of a qCFT with fractional central charges and argue that their behavior generalizes to a large class of existing and future models. Beyond approximating CFT properties, we show that these can be best understood as belonging to a paradigm of discrete holography.

Complexity for Conformal Field Theories in General Dimensions.

We study circuit complexity for conformal field theory states in an arbitrary number of dimensions. Our circuits start from a primary state and move along a unitary representation of the Lorentzian conformal group. Different choices of distance functions can be understood in terms of the geometry of coadjoint orbits of the conformal group. We explicitly relate our circuits to timelike geodesics in anti-de Sitter space and the complexity metric to distances between these geodesics. We extend our method to circuits in other symmetry groups using a group theoretic generalization of the notion of coherent states.

Lorentzian Threads as Gatelines and Holographic Complexity.

The continuous min flow-max cut principle is used to reformulate the "complexity=volume" conjecture using Lorentzian flows-divergenceless norm-bounded timelike vector fields whose minimum flux through a boundary subregion is equal to the volume of the homologous maximal bulk Cauchy slice. The nesting property is used to show the rate of complexity is bounded below by "conditional complexity," describing a multistep optimization with intermediate and final target states. Conceptually, discretized Lorentzian flows are interpreted in terms of threads or gatelines such that complexity is equal to the minimum number of gatelines used to prepare a conformal field theory (CFT) state by an optimal tensor network (TN) discretizing the state. We propose a refined measure of complexity, capturing the role of suboptimal TNs, as an ensemble average. The bulk symplectic potential provides a "canonical" thread configuration characterizing perturbations around arbitrary CFT states. Its consistency requires the bulk to obey linearized Einstein's equations, which are shown to be equivalent to the holographic first law of complexity, thereby advocating a notion of "spacetime complexity."

Bulk locality in the AdS/CFT correspondence

In this paper, we study the bulk local states in the $\mathrm{AdS}/\mathrm{CFT}$ correspondence in the large $N$ limit using the formula explicitly relating the bulk local operators and the conformal field theory (CFT) local operators. We identify the bulk local state in terms of CFT local states and find that the bulk local state corresponds to a CFT state supported in the whole space, which means a version of the subregion duality is not valid. On the other hand, CFT states supported in a space region are expressed in terms of the bulk states supported in a certain region. We find that the quantum error correction proposal is not realized although the puzzles of the radial locality which motivated the proposal are resolved in our analysis. For the understating of the bulk local states, an explicit realization of an analog of the Reeh-Schlieder theorem plays an important role.

Spectrum of end of the world branes in holographic BCFTs

We study overlaps between two regularized boundary states in conformal field theories. Regularized boundary states are dual to end of the world branes in an AdS black hole via the AdS/BCFT. Thus they can be regarded as microstates of a single sided black hole. Owing to the open-closed duality, such an overlap between two different regularized boundary states is exponentially suppressed as leftlangle left.{psi}_aright|{psi}_brightrangle sim {e}^{-Oleft({h}_{ab}^{left(min right)}right)} , where {h}_{ab}^{left(min right)} is the lowest energy of open strings which connect two different boundaries a and b. Our gravity dual analysis leads to {h}_{ab}^{left(min right)} = c/24 for a pure AdS3 gravity. This shows that a holographic boundary state is a random vector among all left-right symmetric states, whose number is given by a square root of the number of all black hole microstates. We also perform a similar computation in higher dimensions, and find that {h}_{ab}^{left(min right)} depends on the tensions of the branes. In our analysis of holographic boundary states, the off diagonal elements of the inner products can be computed directly from on-shell gravity actions, as opposed to earlier calculations of inner products of microstates in two dimensional gravity.

Bulk reconstruction beyond the entanglement wedge

We study the portion of an asymptotically Anti de Sitter geometry's bulk where the metric can be reconstructed, given the areas of minimal 2-surfaces anchored to a fixed boundary subregion. We exhibit situations in which this region can reach parametrically far outside of the entanglement wedge. If the setting is furthermore holographic, so that the bulk geometry is dual to a state in a conformal field theory (CFT), these minimal 2-surface areas can be deduced from the expectation values of operators localized within the boundary subregion. This presents us with an alternative: Either the reduced CFT state encodes significant information about the bulk beyond the entanglement wedge, challenging conventional intuition about holographic subregion duality; or the reduced CFT state fails to contain information about operators whose expectation values give the areas of minimal 2-surfaces anchored within that subregion, challenging conventional intuition about the holographic dictionary.

Holographic complexity for disentangled states

Abstract We consider the maximal volume and the action, which are conjectured to be gravity duals of the complexity, in the black hole geometries with end-of-the-world branes. These geometries are duals of boundary states in conformal field theories which have small real space entanglement. When we raise the black hole temperature while keeping the cutoff radius, black hole horizons or end-of-the-world branes come in contact with the cutoff surface. In this limit, holographic entanglement entropy reduces to zero. We study the behavior of the volume and the action, and find that the volume reduces to zero in this limit. The behavior of the action depends on their regularization. We study the implication of these results to the reference state of the holographic complexity both in the complexity = volume or the complexity = action conjectures.

Holographic interpretation of Shannon entropy of coherence of quantum pure states

For a quantum pure state in conformal field theory, we generate the Shannon entropy of its coherence, that is, the von Neumann entropy obtained by introducing quantum measurement errors. We give a holographic interpretation of this Shannon entropy, based on Swingle's interpretation of anti-de Sitter space/conformal field theory (AdS/CFT) correspondence in the context of AdS3/CFT2. As a result of this interpretation, we conjecture a differential geometrical formula for the Shannon entropy of the coherence of a quantum pure or purified state in CFT2 at thermal and momentum equilibrium as the sum of the holographic complexity and the abbreviated action, divided by πℏ, in the bulk domain enclosed by the Ryu-Takayanagi curve. This result offers a definition of the action of a bulk model of qubits dual to the boundary CFT2 at this equilibrium.

Does boundary distinguish complexities?

Recently, Chapman et al. argued that holographic complexities for defects distinguish action from volume. Motivated by their work, we study complexity of quantum states in conformal field theory with boundary. In generic two-dimensional BCFT, we work on the path-integral optimization which gives one of field-theoretic definitions for the complexity. We also perform holographic computations of the complexity in Takayanagi’s AdS/BCFT model following by the “complexity = volume” conjecture and “complexity = action” conjecture. We find that increments of the complexity due to the boundary show the same divergent structures in these models except for the CA complexity in the AdS3/BCFT2 model as the argument by Chapman et al. . Thus, we conclude that boundary does not distinguish the complexities in general.

Beyond toy models: distilling tensor networks in full AdS/CFT

We present a general procedure for constructing tensor networks that accurately reproduce holographic states in conformal field theories (CFTs). Given a state in a large-N CFT with a static, semiclassical gravitational dual, we build a tensor network by an iterative series of approximations that eliminate redundant degrees of freedom and minimize the bond dimensions of the resulting network. We argue that the bond dimensions of the tensor network will match the areas of the corresponding bulk surfaces. For “tree” tensor networks (i.e., those that are constructed by discretizing spacetime with non­ intersecting Ryu-Takayanagi surfaces), our arguments can be made rigorous using a version of one-shot entanglement distillation in the CFT. Using the known quantum error correcting properties of AdS/CFT, we show that bulk legs can be added to the tensor networks to create holographic quantum error correcting codes. These codes behave similarly to previous holographic tensor network toy models, but describe actual bulk excitations in continuum AdS/CFT. By assuming some natural generalizations of the “holographic entanglement of purification” conjecture, we are able to construct tensor networks for more general bulk discretizations, leading to finer-grained networks that partition the information content of a Ryu-Takayanagi surface into tensor-factorized subregions. While the granularity of such a tensor network must be set larger than the string/Planck scales, we expect that it can be chosen to lie well below the AdS scale. However, we also prove a no-go theorem which shows that the bulk-to-boundary maps cannot all be isometries in a tensor network with intersecting Ryu-Takayanagi surfaces.

Nongeometric states in a holographic conformal field theory

In the AdS$_3$/CFT$_2$ correspondence, we find some conformal field theory (CFT) states that have no bulk description by the Ba\~nados geometry. We elaborate the constraints for a CFT state to be geometric, i.e., having a dual Ba\~nados metric, by comparing the order of central charge of the entanglement/R\'enyi entropy obtained respectively from the holographic method and the replica trick in CFT. We find that the geometric CFT states fulfill Bohr's correspondence principle by reducing the quantum KdV hierarchy to its classical counterpart. We call the CFT states that satisfy the geometric constraints geometric states, and otherwise non-geometric states. We give examples of both the geometric and non-geometric states, with the latter case including the superposition states and descendant states.

Towards an entanglement measure for mixed states in CFTs based on relative entropy

Relative entropy of entanglement (REE) is an entanglement measure of bipartite mixed states, defined by the minimum of the relative entropy S(ρAB ||σAB ) between a given mixed state ρAB and an arbitrary separable state σAB . The REE is always bounded by the mutual information IAB = S(ρAB ||ρA ⊗ ρB) because the latter measures not only quantum entanglement but also classical correlations. In this paper we address the question of to what extent REE can be small compared to the mutual information in conformal field theories (CFTs). For this purpose, we perturbatively compute the relative entropy between the vacuum reduced density matrix ρAB0 on disjoint subsystems A ∪ B and arbitrarily separable state σAB in the limit where two subsystems A and B are well separated, then minimize the relative entropy with respect to the separable states. We argue that the result highly depends on the spectrum of CFT on the subsystems. When we have a few low energy spectrum of operators as in the case where the subsystems consist of finite number of spins in spin chain models, the REE is considerably smaller than the mutual information. However in general our perturbative scheme breaks down, and the REE can be as large as the mutual information.

An uplifting discussion of T-duality

It is well known that string theory has a T-duality symmetry relating circle compactifications of large and small radius. This symmetry plays a foundational role in string theory. We note here that while T-duality is order two acting on the moduli space of compactifications, it is order four in its action on the conformal field theory state space. More generally, involutions in the Weyl group W (G) which act at points of enhanced G symmetry have canonical lifts to order four elements of G, a phenomenon first investigated by J. Tits in the mathematical literature on Lie groups and generalized here to conformal field theory. This simple fact has a number of interesting consequences. One consequence is a reevaluation of a mod two condition appearing in asymmetric orbifold constructions. We also briefly discuss the implications for the idea that T-duality and its generalizations should be thought of as discrete gauge symmetries in spacetime.

Zero-energy states in conformal field theory with sine-square deformation

We study the properties of two-dimensional conformal field theories (CFTs) with sine-square deformation (SSD). We show that there are no eigenstates of finite norm for the Hamiltonian of a unitary CFT with SSD, except for the zero-energy vacuum state $|0\rangle$. We then introduce a regularized version of the SSD Hamiltonian which is related to the undeformed Hamiltonian via a unitary transformation corresponding to the Mobius quantization. The unitary equivalence of the two Hamiltonians allows us to obtain zero-energy states of the deformed Hamiltonian in a systematic way. The regularization also provides a way to compute the expectation values of observables in zero-energy states that are not necessarily normalizable.