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arXiv:2606.12540

Toward Entanglement Bootstrap for Conformal Field Theory in Any Dimension

Rolando Ramirez Camasca, Xiang Li, Ting-Chun Lin, John McGreevy

hep-thcond-mat.str-elquant-phSubmitted Jun 10, 2026• Announced Jun 12, 2026• 0 citations
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Abstract

Given a quantum critical wavefunction in any dimension, we propose a reconstructed Hamiltonian, analogous to the ones previously found for 1+1d CFT and for 2+1d bosonic liquid topologically-ordered states. We test numerically that, for known regularized approximate CFT groundstates (on the icosahedron and the fuzzy sphere), (1) they are close to the groundstate of their reconstructed Hamiltonian, and (2) the spectrum of their reconstructed Hamiltonian on the unit sphere has CFT properties (integer spacing of descendants) and matches known low-lying energies. We show that this provides an automated method to improve the finite-size effects in a fixed Hilbert space.

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5.5 + 20 pages

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Given a quantum critical wavefunction in any dimension, we propose a reconstructed Hamiltonian, analogous to the ones previously found for 1+1d CFT and for 2+1d bosonic liquid topologically-ordered states. We test numerically that, for known regularized approximate CFT groundstates (on the icosahedron and the fuzzy sphere), (1) they are close to the groundstate of their reconstructed Hamiltonian, and (2) the spectrum of their reconstructed Hamiltonian on the unit sphere has CFT properties (integer spacing of descendants) and matches known low-lying energies. We show that this provides an automated method to improve the finite-size effects in a fixed Hilbert space.
A bstract The fuzzy sphere method has enjoyed great success in the study of (2+1)-dimensional unitary conformal field theories (CFTs) by regularizing them as quantum Hall transitions on the sphere. Here, we extend this approach to the Yang-Lee CFT — the simplest non-unitary CFT. We use an Ising quantum-Hall ferromagnet Hamiltonian with a transverse field and an imaginary longitudinal field, the latter breaks the Hermiticity of the Hamiltonian and thus the unitarity of the associated quantum field theory. Non-unitary conformal field theories — particularly the Yang-Lee CFT — pose significant challenges to conventional fuzzy sphere approaches. To overcome these obstacles, here we utilize a different method for determining critical points that requires no a priori knowledge of CFT scaling dimensions. Our method instead leverages the state-operator correspondence while utilizing two complementary criteria: the conformality of the energy spectrum and its consistency with conformal perturbation theory. We also discuss a new finite-size scaling on the fuzzy sphere that makes the sources of systematic error in extracting conformal data explicit, and compare it with the conventional analysis using the (1+1)-dimensional Yang-Lee problem as an example. Our results show broad agreement with previous Monte-Carlo and conformal bootstrap results. We also uncover one previously unknown primary operator and several operator product expansion coefficients.
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The recently introduced “fuzzy-sphere” method has enabled accurate numerical regularizations of certain three-dimensional (3D) conformal field theories (CFTs). The regularization is provided by the noncommutative geometry of the lowest Landau level filled by electrons, such that the charge is trivially gapped due to the Pauli exclusion principle at filling factor ν = 1 , while the electron spins encode the desired CFT. Successful applications of the fuzzy sphere to paradigmatic CFTs, such as the 3D Ising model, raise an important question: How finely tuned does the underlying electron system need to be? Here, we show that the 3D Ising CFT can also be realized at electron fillings. In such cases, the CFT spectrum is intertwined with the charge-neutral spectrum of the underlying fractional quantum Hall state—a feature that is trivially absent in the previously studied ν = 1 case. Remarkably, we show that the mixing between the CFT spectrum and the fractional quantum Hall spectrum is strongly suppressed within the numerically accessible system sizes. Moreover, we demonstrate that the CFT critical point is unaffected by the exchange statistics of the particles and by the nature of topological order in the charge sector. Our results set the stage for the fuzzy-sphere exploration of conformal critical points between topologically ordered states.
Abstract Entanglement or modular Hamiltonians play a crucial role in the investigation of correlations in quantum field theories. In particular, in 1 + 1 space–time dimensions, the spectra of entanglement Hamiltonians of conformal field theories (CFTs) for certain geometries are related to the spectra of the physical Hamiltonians of corresponding boundary CFTs. As a result, conformal invariance allows exact computation of the spectra of the entanglement Hamiltonians for these models. In this work, we perform this computation of the spectrum of the entanglement Hamiltonian for the free compactified boson CFT over a finite spatial interval. We compare the analytical results obtained for the continuum theory with numerical simulations of a lattice-regularized model for the CFT using density matrix renormalization group technique. To that end, we use a lattice regularization provided by superconducting quantum electronic circuits, built out of Josephson junctions and capacitors. Up to non-universal effects arising due to the lattice regularization, the numerical results are compatible with the predictions of the exact computations.
We study nonchiral wave functions for systems with continuous spins obtained from the conformal field theory (CFT) of a free, massless boson. In contrast to the case of discrete spins, these can be treated as bosonic Gaussian states, which allows us to efficiently compute correlations and entanglement properties both in one (1D) and in two spatial dimensions (2D). In 1D, the computed entanglement entropy and spectra are in agreement with the underlying CFT. Furthermore, we construct a 1D parent Hamiltonian with a low-energy spectrum corresponding to that of a free, massless boson. In 2D, we find edge excitations in the entanglement spectrum, although the states do not have intrinsic topological order, as revealed by a determination of the topological entanglement entropy.
Ground states of 1+1d conformal field theories (CFTs) satisfy a local entropic condition called the vector fixed point equation. This condition is surprisingly well satisfied by ground states of quantum critical lattice models even at small system sizes. We perform a search in the space of states of very small systems (4 qubits and 4 qutrits) and examine the states that satisfy this condition. By reconstructing a local Hamiltonian from each state, we are able to identify many of these solutions with known CFTs; others are gapped fixed points or involve large relevant perturbations, and others are CFTs we have not yet identified. These ideas are also useful for identifying continuous quantum phase transitions in a given family of Hamiltonians and for identifying the nature of the critical theory in small systems.