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

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

Problem Statement
The real-time quantum dynamics of many-body systems driven through continuous phase transitions is a fundamental problem in nonequilibrium physics. While the Kibble-Zurek (KZ) mechanism and the generalized Full Finite-Time Scaling (FTS) theory have been rigorously tested in one spatial dimension (1D) using exactly solvable models and advanced numerical techniques, their application in two and higher dimensions remains significantly underdeveloped. This gap exists because real-time simulations of strongly correlated quantum systems in dimensions $d > 1$ face severe computational obstacles: Quantum Monte Carlo methods operate in imaginary time and require analytic continuation to access real-time dynamics, while conventional tensor network methods are limited by rapid entanglement growth. Consequently, controlled tests of KZ theory beyond 1D have been scarce.

Methodology
The authors address this challenge by applying the recently developed fuzzy sphere regularization scheme to the paradigmatic 2D transverse-field Ising model. This approach maps the system onto the surface of a sphere with a magnetic monopole at the center, forming Landau levels. By projecting the Hamiltonian onto the Lowest Landau Level (LLL), the model realizes an effective Ising transition where the spin degrees of freedom dominate.

Key methodological features include:

• Symmetry Preservation: The construction maintains exact $SO(3)$ rotational invariance at the microscopic level. This symmetry restricts the dynamics to a specific angular momentum sector (matching the initial state), significantly reducing the effective Hilbert space and enabling controlled tensor-network evolution to longer times than possible on conventional lattices.
• Quench Protocol: The system is driven from the paramagnetic phase to criticality by linearly ramping the transverse field $h(t) = h_c - vt$ at a rate $v$.
• Numerical Algorithms: The authors employ the Time-Dependent Variational Principle (TDVP) algorithm augmented by a controlled bond expansion technique, implemented via the FuzzifiED package and ITensor. Exact Diagonalization (ED) is used for smaller system sizes to validate results and explore the slow-quench limit.
• Observables: Three key quantities are computed at the critical point: the squared order parameter $\langle m_z^2 \rangle$, the excitation energy density $Q$, and the two-point correlation function $C(\theta)$.

Key Results
The study establishes the following numerical findings for the 2D quantum Ising transition on the fuzzy sphere:

1. Squared Order Parameter ($\langle m_z^2 \rangle$):

   • At intermediate quench rates, $\langle m_z^2 \rangle$ exhibits clean data collapse when scaled using the known critical exponents of the 3D Ising universality class ($\nu \approx 0.63, z=1$).
   • Extrapolation to the thermodynamic limit confirms the predicted KZ power-law scaling, $\langle m_z^2 \rangle \sim v^{2\Delta\nu/(1+z\nu)}$, where $\Delta$ is the scaling dimension of the order parameter.
   • At fast quench rates, the system crosses over to a regime where $\langle m_z^2 \rangle$ becomes independent of system size and quench rate, consistent with fast quench scaling on conventional lattices.

2. Excitation Energy Density ($Q$):

   • Unlike $\langle m_z^2 \rangle$, the excitation energy density $Q$ fails to reach the universal scaling regime for the available system sizes.
   • The authors attribute this failure to symmetry-enforced level sparsity. Because the initial state, Hamiltonian, and quench protocol all preserve specific symmetries (including $SO(3)$, particle-hole symmetry, and Ising $Z_2$), the excited states accessible during the non-equilibrium process are restricted to the same symmetry sector as the ground state.
   • This restriction results in a sparse energy spectrum and a significantly larger effective finite-size gap at criticality, preventing the system from reaching the scaling limit within the studied sizes.
   • However, in the slow-quench limit (accessible via ED), the system recovers the expected adiabatic scaling $Q \sim v^2$.

3. Two-Point Correlation Function ($C(\theta)$):

   • The correlation function exhibits the expected exponential decay in the non-equilibrium regime.
   • The data collapses onto a universal curve when scaled according to FTS theory.
   • The decay length of the collapsed curve allows for the estimation of the non-universal scaling coefficient in the freeze-out time/length relation ($\hat{\xi}_v \propto v^{-\nu/(1+z\nu)}$). This coefficient is found to be approximately 5.6, providing a quantitative link between the quench rate and the characteristic domain size.

Significance and Claims
The paper claims to establish the fuzzy sphere as a quantitative laboratory for studying nonequilibrium quantum critical phenomena in $(2+1)$ dimensions. By overcoming the computational barriers of conventional lattice approaches, this work provides the first controlled real-time dynamical test of KZ theory and FTS for a 2D quantum critical point.

The authors emphasize that while the fuzzy sphere has previously been confined to equilibrium ground-state physics, this work demonstrates its viability for real-time dynamics. The methodology is not limited to the Ising model; since the fuzzy sphere construction can realize a broad family of $(2+1)d$ Conformal Field Theories (CFTs), the authors posit that their approach offers a general route to studying the real-time critical dynamics of strongly coupled CFTs that are otherwise computationally intractable. The results confirm that if the equilibrium critical point is faithfully captured by the geometry, the universal dynamic scaling laws (controlled by equilibrium critical theory and the quench protocol) manifest naturally in the quench dynamics.