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arXiv:2607.18028

Universal Dynamic Scaling of 2D Quantum Ising Transition on the Fuzzy Sphere

Meng Zeng, Shuai Yin, Roderich Moessner

cond-mat.str-elcond-mat.stat-mechquant-phSubmitted Jul 20, 2026• Announced Jul 21, 2026• 0 citations
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Abstract

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 $\langle m_z^2 \rangle$, the excitation energy density $Q$, and the two-point correlation function of $m_z$. We establish numerically that, at intermediate quench rate, $\langle m_z^2 \rangle$ follows the conventional Kibble-Zurek prediction set by the critical exponents of the $3$D 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 $Q$ 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 $\langle m_z^2 \rangle$ and $Q$ is recovered. Since the fuzzy sphere construction can realize not only the Ising conformal field theory (CFT), but a broad family of $(2+1)d$ CFTs, our results establish a route to the real-time critical dynamics of strongly coupled CFTs that are otherwise computationally challenging to study.

Comments
4.5 pages with 4 figures in main text + 1.5 pages of appendix

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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.
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.
We present a numerical quantum Monte Carlo (QMC) method for simulating the 3D phase transition on the recently proposed fuzzy sphere [Phys. Rev. X 13, 021009 (2023)]. By introducing an additional SU(2) layer degree of freedom, we reformulate the model into a form suitable for sign-problem-free QMC simulation. From the finite-size-scaling, we show that this QMC-friendly model undergoes a quantum phase transition belonging to the 3D Ising universality class, and at the critical point we compute the scaling dimensions from the state-operator correspondence, which largely agrees with the prediction from the conformal field theory. These results pave the way to construct sign-problem-free models for QMC simulations on the fuzzy sphere, which could advance the future study on more sophisticated criticalities.
Conformal field theory (CFT) offers a modern viewpoint for understanding phase transitions. However, directly accessing the conformal algebra and microscopically uncovering the emergent conformal symmetry, especially in higher dimensions, remains a significant challenge. Motivated by recent advances in revealing CFT features via the fuzzy two-sphere, here we generalize this approach to higher dimensions and aim to expose the conformality at the (3+1)-D quantum critical point. We demonstrate this framework by investigating quantum phase transitions belonging to the Ising and Yang-Lee universality classes in a (3+1)-D quantum model (equivalent to a classical four-dimensional system), realized via Landau level projection on the fuzzy three-sphere. By computing the energy spectra at criticality, we explicitly verify the state-operator correspondence, a hallmark of conformal invariance. Together with prior advances, this work establishes a new pathway for the microscopic study of emergent conformality in higher-dimensional phase transitions.
The Kibble--Zurek mechanism captures universality when a system is driven through a continuous phase transition. Here we study the dynamical aspect of quantum phase transitions in the Ising Field Theory where the quantum critical point can be crossed in different directions in the two-dimensional coupling space leading to different scaling laws. Using the Truncated Conformal Space Approach, we investigate the microscopic details of the Kibble--Zurek mechanism in terms of instantaneous eigenstates in a genuinely interacting field theory. For different protocols, we demonstrate dynamical scaling in the non-adiabatic time window and provide analytic and numerical evidence for specific scaling properties of various quantities. In particular, we argue that the higher cumulants of the excess heat exhibit universal scaling in generic interacting models for a slow enough ramp.
The Kibble-Zurek mechanism captures universality when a system is driven through a continuous phase transition. Here we study the dynamical aspect of quantum phase transitions in the Ising Field Theory where the quantum critical point can be crossed in different directions in the two-dimensional coupling space leading to different scaling laws. Using the Truncated Conformal Space Approach, we investigate the microscopic details of the Kibble-Zurek mechanism in terms of instantaneous eigenstates in a genuinely interacting field theory. For different protocols, we demonstrate dynamical scaling in the non-adiabatic time window and provide analytic and numerical evidence for specific scaling properties of various quantities. In particular, we argue that the higher cumulants of the excess heat exhibit universal scaling in generic interacting models for a slow enough ramp.
Recent work has suggested that the Kibble-Zurek mechanism can be re-interpreted as a theory of critical scaling out of equilibrium. While this has been shown for some classical and integrable models, in this work we demonstrate one crucial aspect of critical scaling theory: universality. We solve for the time-dependent finite-size scaling functions of the 1D transversefield Ising chain during a linear-in-time ramp of the field through the quantum critical point. We then simulate Mott-insulating bosons in a tilted potential, an experimentally-studied system in the same equilibrium universality class, and demonstrate that universality holds for the dynamics as well. We find qualitatively athermal features of the scaling functions, such as negative spin correlations, and show that they should be robustly observable within present cold atom experiments. In addition, we discuss recent results in which the Ising model is extended by imbuing the magnetic field with dynamics. We predict using Kibble-Zurek scaling and numerically confirm that the magnetic field ramp is dynamically arrested at the quantum critical point. Extensions of this theory indicate that dynamic localization near critical points should be omnipresent in nature, and may have implications in particle physics as a possible mechanism for giving the Higgs boson a light mass.