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Physical review. B./Physical review. B

Entropic F function of three-dimensional Ising conformal field theory via fuzzy sphere regularization

Liangdong Hu, Wei Kang Zhu, Yin-Chen He

Published Apr 28, 2025• 19 citations
Publisher

Abstract

The $F$ function, the three-dimensional counterpart of the central charge in the two-dimensional (2D) conformal field theory, measures the effective number of degrees of freedom in three-dimensional (3D) quantum field theory, and it is monotonically decreasing under the renormalization group flow. However, unlike the 2D central charge, the $F$ function is a nonlocal quantity and cannot be computed using correlators of local operators. Utilizing the recently proposed fuzzy sphere regularization, we have performed the first nonperturbative computation of the $F$ function for the paradigmatic 3D Ising conformal field theory through entanglement entropy. Our computation yields ${F}_{\text{Ising}}=0.0612(5)$, slightly smaller than the $F$ function of a real free scalar, ${F}_{\text{free}}=\frac{log2}{8}\ensuremath{-}\frac{3\ensuremath{\zeta}(3)}{16{\ensuremath{\pi}}^{2}}\ensuremath{\approx}0.0638$, consistent with the $F$ theorem, and close to the $4\ensuremath{-}\ensuremath{\epsilon}$ expansion estimates of ${F}_{\text{Ising}}\ensuremath{\approx}0.061\ensuremath{\sim}0.062$.

DOI
10.1103/physrevb.111.155151

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The FFF function, the three-dimensional counterpart of the central charge in the two-dimensional (2D) conformal field theory, measures the effective number of degrees of freedom in three-dimensional (3D) quantum field theory, and it is monotonically decreasing under the renormalization group flow. However, unlike the 2D central charge, the FFF function is a nonlocal quantity and cannot be computed using correlators of local operators. Utilizing the recently proposed fuzzy sphere regularization, we have performed the first nonperturbative computation of the FFF function for the paradigmatic 3D Ising conformal field theory through entanglement entropy. Our computation yields FIsing=0.0612(5){F}_{\text{Ising}}=0.0612(5)FIsing​=0.0612(5), slightly smaller than the FFF function of a real free scalar, Ffree=log28\ensuremath−3\ensuremathζ(3)16\ensuremathπ2\ensuremath≈0.0638{F}_{\text{free}}=\frac{log2}{8}\ensuremath{-}\frac{3\ensuremath{\zeta}(3)}{16{\ensuremath{\pi}}^{2}}\ensuremath{\approx}0.0638Ffree​=8log2​\ensuremath−16\ensuremathπ23\ensuremathζ(3)​\ensuremath≈0.0638, consistent with the FFF theorem, and close to the 4\ensuremath−\ensuremathϵ4\ensuremath{-}\ensuremath{\epsilon}4\ensuremath−\ensuremathϵ expansion estimates of FIsing\ensuremath≈0.061\ensuremath∼0.062{F}_{\text{Ising}}\ensuremath{\approx}0.061\ensuremath{\sim}0.062FIsing​\ensuremath≈0.061\ensuremath∼0.062.
Entropic FFF-function of 3D Ising conformal field theory via the fuzzy sphere regularization
The FFF-function, the three-dimensional counterpart of the central charge in the 2D conformal field theory, measures the effective number of degrees of freedom in 3D quantum field theory, and it is monotonically decreasing under the renormalization group flow. However, unlike the 2D central charge, the FFF-function is a non-local quantity and cannot be computed using correlators of local operators. Utilizing the recently proposed fuzzy sphere regularization, we have performed the first non-perturbative computation of the FFF-function for the paradigmatic 3D Ising conformal field theory through entanglement entropy. Our estimate yields FIsing≈0.0612(5)F_{\text{Ising}} \approx 0.0612(5)FIsing​≈0.0612(5), slightly smaller than the FFF-function of a real free scalar, Ffree=log⁡28−3ζ(3)16π2≈0.0638F_{\text{free}} = \frac{\log 2}{8} - \frac{3ζ(3)}{16π^2} \approx 0.0638Ffree​=8log2​−16π23ζ(3)​≈0.0638, consistent with the FFF-theorem, and close to the 4−ε4-ε4−ε expansion estimates of FIsing≈0.0610∼0.0623F_{\text{Ising}} \approx 0.0610 \sim 0.0623FIsing​≈0.0610∼0.0623.
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.
The three-dimensional cubic conformal field theory governs the critical behavior of Heisenberg magnets with cubic anisotropy. Studying this theory nonperturbatively is challenging because its most easily accessible observables are numerically very close to those of the more symmetric O ( 3 ) model. In this work, we overcome this difficulty using the fuzzy sphere regularization method. By adding a cubic-invariant two-body interaction to the quantum rotor Hamiltonian used for the O ( 3 ) model, we break the continuous rotational symmetry by construction and unambiguously isolate the cubic critical point. Using exact diagonalization and the density matrix renormalization group, we calculate the scaling dimensions of several key operators, including the leading scalar singlets, and resolve the splitting of the O ( 3 ) rank-two traceless symmetric tensor into the E g and T 2 g representations of the cubic group. Our results are consistent with existing Monte Carlo, conformal perturbation theory, and ɛ expansion benchmarks, demonstrating the power of the fuzzy sphere in resolving closely spaced universality classes.
We revisit the problem of \textit{real-time} quantum dynamics of the paradigmatic two dimensional transverse-field Ising model using the recently developed fuzzy sphere regularization scheme. By linearly ramping the transverse field from the paramagnetic phase to criticality, we study the finite-time scaling behavior of the squared order parameter ⟨mz2⟩\langle m_z^2 \rangle⟨mz2​⟩, the excitation energy density QQQ, and the two-point correlation function of mzm_zmz​. We establish numerically that, at intermediate quench rate, ⟨mz2⟩\langle m_z^2 \rangle⟨mz2​⟩ follows the conventional Kibble-Zurek prediction set by the critical exponents of the 333D Ising universality class, and the correlation function exhibits the expected exponential decay whose correlation length can be used to estimate the non-universal scaling coefficient in the freeze-out time/length. In contrast, the excitation energy density QQQ does not reach the same scaling regime at available system sizes due to large effective finite-size gap from symmetry-enforced level sparsity in the energy spectrum. At slow quench rates the universal quasi-adiabatic scaling for both ⟨mz2⟩\langle m_z^2 \rangle⟨mz2​⟩ and QQQ is recovered. Since the fuzzy sphere construction can realize not only the Ising conformal field theory (CFT), but a broad family of (2+1)d(2+1)d(2+1)d CFTs, our results establish a route to the real-time critical dynamics of strongly coupled CFTs that are otherwise computationally challenging to study.
Free theories are landmarks in the landscape of quantum field theories: their exact solvability serves as a pillar for perturbative constructions of interacting theories. Fuzzy sphere regularization, which combines quantum Hall physics with state-operator correspondence, has recently been proposed as a promising framework for simulating three-dimensional conformal field theories (CFTs), but so far it has not provided access to free theories. We overcome this limitation by designing a bilayer quantum Hall system that hosts an Ising tricritical point-a nontrivial fixed point where first-order and second-order transitions meet-which flows to the conformally coupled scalar theory in the infrared. The critical energy spectrum and operator structure match those at the Gaussian fixed point, providing nonperturbative evidence for the emergence of a free scalar CFT. Our results expand the landscape of CFTs realizable on the fuzzy sphere and demonstrate that even free bosonic theories-previously inaccessible-can emerge from interacting electrons in this framework.
We study the natural scheme-independent quantity obtained from the three-sphere partition function of a ( 2 + 1 )-dimensional quantum field theory by removing all local counterterm ambiguities. At conformal fixed points this quantity equals the standard F -theorem invariant. Conformal perturbation theory shows that it locally decreases at O ( g 2 ) under any relevant scalar deformation of a three-dimensional Conformal Field Theory. However, an exact analysis of the free massive scalar on S 3 shows that this sphere-free-energy interpolant is not monotone along the full renormalization-group flow: it dips below its infrared value and then returns to it. Thus the natural counterterm-subtracted quantity built from sphere thermodynamics is not, by itself, a monotone F -function. We trace the obstruction to the second-order differential operator required to eliminate the local ambiguities.