arXiv subjects

Ling-Yan Hung

Publications and source records attributed to Ling-Yan Hung.

At least 19 recordsLinked to original sources

arXiv2025

Note on searching for critical lattice models as entropy critical points from strange correlator

An entropy function is proposed in [Phys. Rev. Lett. 131, 251602] as a way to detect criticality even when the system size is small. In this note we apply this strategy in the search for criticality of lattice transfer matrices constructed based on the topological holographic principle. We find that the combination of strategy is indeed a cost-effective and efficient way of identifying critical boundary conditions, estimating central charges and moreover, plotting entire phase diagrams in a multi-dimensional phase space.

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arXiv2025

A 2D-CFT Factory: Critical Lattice Models from Competing Anyon Condensation Processes in SymTO/SymTFT

In this paper, we introduce a ``CFT factory'' : a novel algorithm of methodically generating 2D lattice models that would flow to 2D conformal fixed points in the infrared. These 2D models are realised by giving critical boundary conditions to 3D topological orders (symTOs/symTFTs) described by string-net models, often called the strange correlators. We engineer these critical boundary conditions by introducing a commensurate amount of non-commuting anyon condensates. The non-invertible symmetries preserved at the critical point can be controlled by studying a novel ``refined condensation tree''. Our structured method generates an infinite family of critical lattice models, including the A-series minimal models, and uncovers previously unknown critical points. Notably, we find at least three novel critical points (c$\approx 1.3$, $1.8$, and $2.5$ respectively) preserving the Haagerup symmetries, in addition to recovering previously reported ones. The condensation tree, together with a generalised Kramers-Wannier duality, predicts precisely large swathes of phase boundaries, fixes almost completely the global phase diagram, and sieves out second order phase transitions. This is not only illustrated in well-known examples (such as the 8-vertex model related to the $A_5$ category) but also further verified with precision numerics, using our improved (non-invertible) symmetry-preserving tensor-network RG, in novel examples involving the Haagerup symmetries. We show that critical couplings can be precisely encoded in the categorical data (Frobenius algebras and quantum dimensions in unitary fusion categories), thus establishing a powerful, systematic route to discovering and potentially classifying new conformal field theories.

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arXiv2025

It from ETH: Multi-interval Entanglement and Replica Wormholes from Large-$c$ BCFT Ensemble

We provide a derivation of the Ryu-Takayanagi (RT) formula in 3D gravity for generic boundary subregion--including RT surface phase transitions--directly from the dual two-dimensional conformal field theory (CFT). Our approach relies on the universal statistics of the algebraic conformal data and the large-$c$ behavior of conformal blocks with Cardy boundaries involved. We observe the emergence of 3D multi-boundary black holes with Karch-Randall branes from entangled states of any number of CFT's with and without Cardy boundaries. The RT formula is obtained directly from the CFT in the high-temperature regime. Two direct applications are: $\textbf{1)}$ A simple derivation of the multi-interval entanglement entropy for the vacuum state of a single CFT; $\textbf{2)}$ A CFT-based detection of the emergence of replica wormholes in the context of entanglement islands and black hole microstate counting. Our framework yields the first holographic random tensor network that faithfully captures the entanglement structure of holographic CFTs. These results imply that bulk spacetime geometries indeed emerge from the eigenstate thermalization hypothesis (ETH) in the dual field theory in the large-$c$ limi--a paradigm we refer to as $\textit{It from ETH}$.

hep-th↗

arXiv2025

Universal Structures and Emergent Geometry from Large-$c$ BCFT Ensemble

In this paper, we study the ensemble average of boundary CFT (BCFT) data consistent with the bootstrap equations. We apply the results to computing ensemble average of copies of multi-point correlation functions of boundary changing operators (BCO), and find the results in agreement with one copy of the Virasoro TQFT. Further, we consider ensemble average of CFT path-integrals expressed as tensor networks of BCO correlation functions using the formalism developed in arXiv:2210.12127, arXiv:2311.18005 and arXiv:2403.03179. We find a natural emergence of locality and a loop-sum structure reminiscent of lattice integrable models. We illustrate this universal structure through explicit examples at genus zero and genus one. Moreover, we provide strong evidence that, at leading order in large-$c$, the results match those of three-dimensional Einstein gravity. In the presence of closed CFT operator insertions, generalized free fields emerge, with their correlation functions governed by the shortest paths connecting the insertions.

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arXiv2024

QG from SymQRG: AdS$_3$/CFT$_2$ Correspondence as Topological Symmetry-Preserving Quantum RG Flow

By analyzing non-perturbative RG flows that explicitly preserve given topological symmetries, we show that each of their coarse-graining steps can be expressed as a quantum path integral of the SymTFT in one higher dimension. When the symmetries involved include the Virasoro defect lines, such as in the case of $T\bar{T}$ deformations, this RG kernel is the 3D quantum gravitational path integral. For 2D CFTs whose structure constants are under control, we identify the corresponding ground state of the SymTFT, from which the Wheeler-DeWitt equation emerges as the non-perturbative no-flux constraint: the gravitational path integral acts on this state as the projector that renders symmetry-preserving coarse-graining exact. These observations are summarized in the slogan: $\textbf{SymQRG = QG}$. The exact discrete formulation of Liouville theory in \cite{Chen:2024unp} allows us to identify a universal SymQRG kernel, constructed from quantum $6j$ symbols of $U_q(SL(2,\mathbb{R}))$: it provides a discrete realization of two copies of the Virasoro TQFT on $\Sigma\times I$, and manifests as an exact and analytic 3D background-independent MERA-type holographic tensor network. Many aspects of the AdS/CFT correspondence, including the factorization puzzle, admit a natural interpretation within this framework. We propose that the non-perturbative AdS$_3$/CFT$_2$ correspondence is a \textit{maximal} form of topological holography, in which the physical boundary is fixed by kinematics alone.

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arXiv2024

Simplex tensor network renormalization group for boundary theory of 3+1D symTFT

Following the construction in arXiv:2210.12127, we develop a symmetry-preserving renormalization group (RG) flow for 3D symmetric theories. These theories are expressed as boundary conditions of a symTFT, which in our case is a 3+1D Dijkgraaf-Witten topological theory in the bulk. The boundary is geometrically organized into tetrahedra and represented as a tensor network, which we refer to as the "simplex tensor network" state. Each simplex tensor is assigned indices corresponding to its vertices, edges, and faces. We propose a numerical algorithm to implement RG flows for these boundary conditions, and explicitly demonstrate its application to a $\mathbb{Z}_2$ symmetric theory. By linearly interpolating between three topological fixed-point boundaries, we map the phase transitions characterized by local and non-local order parameters, which respectively detects the breaking of a 0-form and a 2-form symmetry. This formalism is readily extendable to other discrete symmetry groups and, in principle, can be generalized to describe 3D symmetric topological orders.

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arXiv2024

Building up quantum spacetimes with BCFT Legos

Is it possible to read off the quantum gravity dual of a CFT directly from its operator algebra? In this essay, we present a step-by-step recipe synthesizing results and techniques from conformal bootstrap, topological symmetries, tensor networks, a novel symmetry-preserving real-space renormalization algorithm devised originally in lattice models, and the asymptotics of quantum $6j$ symbols, thereby providing an answer in the affirmative. Quantum 2D Liouville theory serves as a simple and explicit example, illustrating how the quantum gravitational path integral can be built up from local pieces of BCFT correlation functions, which we call the ``BCFT Legos''. The constructive map between gravity and CFT naturally and explicitly bridges local geometrical data, algebraic structures, and quantum entanglement, as envisaged by the $\it{It \, from \, Qubit}$ motto.

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arXiv2024

CFT$_D$ from TQFT$_{D+1}$ via Holographic Tensor Network, and Precision Discretisation of CFT$_2$

We show that the path-integral of conformal field theories in $D$ dimensions (CFT$_D$) can be constructed by solving for eigenstates of an RG operator following from the Turaev-Viro formulation of a topological field theory in $D+1$ dimensions (TQFT$_{D+1}$), explicitly realising the holographic sandwich relation between a symmetric theory and a TQFT. Generically, exact eigenstates corresponding to symmetric-TQFT$_D$ follow from Frobenius algebra in the TQFT$_{D+1}$. For $D=2$, we constructed eigenstates that produce 2D rational CFT path-integral exactly, which, curiously connects a continuous field theoretic path-integral with the Turaev-Viro state sum. We also devise and illustrate numerical methods for $D=2,3$ to search for CFT$_D$ as phase transition points between symmetric TQFT$_D$. Finally since the RG operator is in fact an exact analytic holographic tensor network, we compute ``bulk-boundary'' correlator and compare with the AdS/CFT dictionary at $D=2$. Promisingly, they are numerically compatible given our accuracy, although further works will be needed to explore the precise connection to the AdS/CFT correspondence.

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arXiv2024

Deriving the non-perturbative gravitational dual of quantum Liouville theory from BCFT operator algebra

We demonstrate that, by utilizing the boundary conformal field theory (BCFT) operator algebra of the Liouville CFT, one can express its path-integral on any Riemann surface as a three dimensional path-integral with appropriate boundary conditions, generalising the recipe for rational CFTs \cite{Hung:2019bnq, Brehm:2021wev, Chen:2022wvy, Cheng:2023kxh}. This serves as a constructive method for deriving the \textit{quantum} holographic dual of the CFT, which reduces to Einstein gravity in the large central charge limit. As a byproduct, the framework provides an explicit discrete state-sum of a 3D non-chiral topological theory constructed from quantum $6j$ symbols of $\mathcal{U}_q(sl(2,\mathbb{R}))$ with non-trivial boundary conditions, representing a long-sought non-perturbative discrete formulation of 3D pure gravity with negative cosmological constant, at least within a class of three manifolds. This constitutes the first example of an exact holographic tensor network that reproduces a known irrational CFT with a precise quantum gravitational interpretation.

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arXiv2023

Precision reconstruction of rational CFT from exact fixed point tensor network

The novel concept of entanglement renormalization and its corresponding tensor network renormalization technique have been highly successful in developing a controlled real space renormalization group (RG) scheme. Numerically approximate fixed-point (FP) tensors are widely used to extract the conformal data of the underlying conformal field theory (CFT) describing critical phenomena. In this paper, we present an explicit analytical construction of the FP tensor for 2D rational CFT. We define it as a correlation function between the "boundary-changing operators" (BCO) on triangles. Our construction fully captures all the real-space RG conditions. We also provide concrete examples, such as Ising, Yang-Lee and Tri-critical Ising models to compute the scaling dimensions explicitly based on the corresponding FP tensor. The BCO descendants turn out to be an optimal basis such that truncation in bond dimensions naturally produces comparable accuracies with the leading existing FP algorithms. Interestingly, our construction of FP tensors is closely related to a strange correlator, where the holographic picture naturally emerges. Our results also open a new door towards understanding CFT in higher dimensions.

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arXiv2023

Bulk Operator Reconstruction in Topological Tensor Network and Generalized Free Fields

In this paper, we would like to study operator reconstruction in a class of holographic tensor networks describing renormalization group flows studied in arXiv:2210.12127. We study examples of 2d bulk holographic tensor networks constructed from Dijkgraaf-Witten theories and found that for both $\mathbb{Z}_n$ group and $S_3$ group the number of bulk operators behaving like a generalized free field in the bulk scales as the order of the group. We also generalize our study to 3d bulks and found the same scaling for $\mathbb{Z}_n$ theories. However, there is no generalized free field when the bulk comes from more generic fusion categories such as the Fibonacci model.

hep-th↗

arXiv2023

Scrambling and Entangling Spinning Particles

In this paper we revisit the gravitational eikonal amplitudes of two scattering spinning particles and inspect their scrambling power in the spin spaces that is quantified through the tripartite information. We found that in the non-relativistic limit and a special high-energy limit the leading contribution is a quantity that is universal and theory independent. The minimal coupling is singled out with minimal scrambling in a different high momenta limit. We also inspected the initial state dependence of entanglement generation and found that the spin coherent state with vanishing spin may not necessarily be the hardest to entangle. Interestingly, among a family of mixed states, the only P-rep state there known to be the best approximation of classical mixed states was singled out as one with minimal entanglement generated.

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arXiv2023

String Condensations in 3+1D and Lagrangian Algebras

We present three Lagrangian algebras in the modular 2-category associated to the 3+1D $\mathbb{Z}_2$ topological order and discuss their physical interpretations, connecting algebras with gapped boundary conditions, and interestingly, maps (braided autoequivalences) exchanging algebras with bulk domain walls. A Lagrangian algebra, together with its modules and local modules, encapsulates detailed physical data of strings condensing at a gapped boundary. In particular, the condensed strings can terminate at boundaries in non-trivial ways. This phenomenon has no lower dimensional analogue and corresponds to novel mathematical structures associated to higher algebras. We provide a layered construction and also explicit lattice realizations of these boundaries and illustrate the correspondence between physics and mathematics of these boundary conditions. This is a first detailed study of the mathematics of Lagrangian algebras in modular 2-categories and their corresponding physics, that brings together rich phenomena of string condensations, gapped boundaries and domain walls in 3+1D topological orders.

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arXiv2023

Virasoro Generators in the Fibonacci Model Tensor Network -- Tackling Finite Size Effects

In this paper, we extend the method implementing Virasoro operators in a tensor network we proposed in arXiv:2205.04500 and test it on the Fibonacci model, which is known to suffer from far more finite size effects. To pick up the "seed" state that would flow to the stress tensor in the thermodynamic limit, we make use of the topological idempotent that projects the transfer matrix to the trivial sector. Combined with an optimization method, the seed state can be identified. We demonstrate that the descendant states in the Fibonacci model can be correctly generated with this approximate stress tensor, giving further evidence that the method applies more generally.

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arXiv2022

Modular parallel transport of multiple intervals in 1+1-dimensional free fermion theory

Modular parallel transport is a generalization of Berry phases, applied to modular (entanglement) Hamiltonians. Here we initiate the study of modular parallel transport for disjoint field theory regions. We study modular parallel transport in the kinematic space of multi-interval regions in the vacuum of 1+1-dimensional free fermion theory--one of the few theories for which modular Hamiltonians on disjoint regions are known. We compute explicitly the generators of modular parallel transport, and explain why their relatively simple form follows from a half-sided modular inclusion. We also compute explicitly the curvature two-form of modular parallel transport. We contrast all calculations with the expected behavior of modular parallel transport in holographic theories, emphasizing the role of non-local terms that couple distinct intervals.

hep-th↗

arXiv2022

Virasoro and Kac-Moody algebra in generic tensor network representations of 2d critical lattice partition functions

In this paper, we propose a general implementation of the Virasoro generators and Kac-Moody currents in generic tensor network representations of 2-dimensional critical lattice models. Our proposal works even when a quantum Hamiltonian of the lattice model is not available, which is the case in many numerical computations involving numerical blockings. We tested our proposal on the 2d Ising model, and also the dimer model, which works to high accuracy even with a fairly small system size. Our method makes use of eigenstates of a small cylinder to generate descendant states in a larger cylinder, suggesting some intricate algebraic relations between lattice of different sizes.

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arXiv2022

Information propagation in a non-local model with emergent locality

In this paper, we revisit a "relatively local" model proposed in arXiv:1811.07241, where locality and dimensionality of space only emerges from the entanglement structure of the state the system is in. Various quantities such as butterfly velocity/ entanglement speed can be defined similarly, at least in the regime where locality is well defined and a light cone structure emerges in the correlation between sites. We find that the relations observed between them in local models arXiv:1908.06993 are not respected. In particular, we conjecture that the hierarchy of the interaction over different distances provides different "layers" of light-cones. When long range interactions are sufficiently suppressed, the effective light cones are dominated by linear behaviors with little remnant of non-locality. This could potentially be used as a physical smoking gun for emergent locality in non-local models.

hep-th↗

arXiv2022

Galois conjugates of String-net Model

We revisit a class of non-Hermitian topological models that are Galois conjugates of their Hermitian counter parts. Particularly, these are Galois conjugates of unitary string-net models. We demonstrate these models necessarily have real spectra, and that topological numbers are recovered as matrix elements of operators evaluated in appropriate bi-orthogonal basis, that we conveniently reformulate as a concomitant Hilbert space here. We also compute in the bi-orthogonal basis thetopological entanglement entropy, demonstrating that its real part is related to the quantum dimension of the topological order. While we focus mostly on the Yang-Lee model, the results in the paper apply generally to Galois conjugates.

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