Jump-Diffusion Stochastic Quantization for Euclidean Lattice Field Theories

Technical Summary: Jump-Diffusion Stochastic Quantization for Euclidean Lattice Field Theories

Problem Statement
The paper addresses a fundamental failure mode in the numerical simulation of Euclidean quantum field theories (QFTs) using stochastic quantization: the loss of ergodicity due to action barriers. In conventional stochastic quantization, field configurations evolve via a continuous diffusion process (Langevin dynamics) governed by a stochastic partial differential equation. While efficient for local observables, this approach struggles when the action landscape $S[\phi]$ contains high barriers separating distinct sectors (e.g., topological sectors or phase coexistence regions).
In such scenarios, the probability of a continuous path crossing a barrier scales as $\exp[-\Delta S_{\text{barrier}}]$. As the system approaches the continuum limit or critical points, these barriers become insurmountable within feasible simulation times, leading to "topological freezing." The sampler becomes trapped in a single sector, yielding incorrect ensemble averages despite appearing converged. Existing solutions, such as tempering, multicanonical approaches, or geometric strategies (open boundaries), often require significant system modifications or complex reweighting.

Methodology
The author proposes a natural generalization of stochastic quantization by extending the underlying stochastic process from pure diffusion to jump-diffusion. This framework incorporates non-continuous paths, specifically Lévy flights, allowing the field $\phi$ to make finite jumps in addition to infinitesimal diffusive steps.

The theoretical foundation is built upon the Courrège theorem and the Lévy-Itô decomposition, which state that the most general continuous-time Markov process consists of three components: deterministic drift, Gaussian diffusion, and jumps. The evolution of the probability distribution $P(\phi, t)$ is described by a generalized Chapman-Kolmogorov (CK) equation, or a generalized Fokker-Planck equation:
$$\partial_t P = \underbrace{-\partial_\phi [b P] + \partial_\phi^2 P}_{\text{Diffusion}} + \underbrace{\int d\phi' [W(\phi|\phi')P(\phi') - W(\phi'|\phi)P(\phi)]}_{\text{Jumps}}$$
where $W(\phi'|\phi)$ is the transition rate density for jumps. To ensure convergence to the target distribution $e^{-S[\phi]}$, the jump term must satisfy a detailed balance condition.

The paper outlines three specific strategies for constructing these jump updates:

1. Additive Proposals (Class A): Utilizing a fixed, symmetric proposal density $q_0(\eta)$ combined with a Metropolis acceptance step. This is analogous to standard Metropolis-Hastings but applied as a jump within the stochastic quantization framework.
2. Invertible Maps (Class B): Employing deterministic, invertible transformations $T$ (e.g., global shifts or rotations) that preserve the measure (Jacobian $J_T = 1$). These are accepted with a Metropolis factor based on the action change $\Delta S$.
3. State-Informed Families of Maps (Class C): Drawing inspiration from discrete system literature, this strategy involves a family of invertible maps $\{T_m\}$ parameterized by $m$ (e.g., location or size). Instead of a uniform selection, the parameter $m$ is chosen with a probability $p(m|\phi)$ that depends on the current state. The paper derives a "locally balanced" weight where the selection probability is proportional to $e^{-\Delta S_m/2}$. This approach effectively distributes the action cost between the forward and reverse moves, significantly improving acceptance rates for difficult transitions.

The efficiency of these jumps is analyzed by deriving an acceptance estimate. For measure-preserving symmetric jumps, the acceptance rate is shown to be $erfc(\sigma/ \sqrt{8})$, where $\sigma$ is the standard deviation of the action cost $\Delta S$. For state-informed jumps, the relevant cost variable becomes $\Delta S_{\text{eff}} = \ln(Z'/Z)$, where $Z$ is a normalization factor over the family of maps.

Key Results
The methodology was tested on two benchmark systems:

1. Tilted Double Well:

   • Setup: A 1D system with a potential $S(x) = \gamma(x^2-1)^2 + cx$, creating two local minima separated by a barrier.
   • Outcome: Standard Langevin dynamics initialized in one well remained trapped, failing to sample the correct global average ($\langle x \rangle_{\text{diff}} \approx 0.985$ vs. true $\approx -0.765$). The jump-diffusion approach, using a "flip" map with added Gaussian noise, successfully restored ergodicity, crossing the barrier frequently and recovering the correct expectation value ($\langle x \rangle_{\text{jumps}} \approx -0.755$).

2. 2D $U(1)$ Gauge Theory:

   • Setup: Compact $U(1)$ gauge theory on a torus, known for topological freezing near the continuum limit (large $\beta$). The topological charge $Q$ changes via barriers of cost $\sim 2\beta$.
   • Global Updates: A "constant flux" jump was constructed to shift the topological charge by $Q=\pm 1$ by spreading the flux uniformly across the lattice. This update successfully maintained transitions between topological sectors even at $\beta=12.5$, where standard Langevin dynamics were completely frozen (zero transitions). The jump-diffusion sampler reproduced exact analytical results for the topological susceptibility $\chi_t$ and plaquette averages.
   • Local (Winding) Updates: To address realistic theories where global updates are costly due to matter fields, "winding" updates (local gauge transformations with a $2\pi$ twist) were tested.
     • Uninformed: Randomly placed winding updates suffered from low acceptance rates as the winding size $L_w$ decreased, leading to freezing for small $L_w$.
     • State-Informed: Using the Class C strategy (selecting the winding position based on local action cost), the acceptance rate was dramatically improved. Remarkably, the smallest winding ($L_w=2$) with state-informed selection achieved higher efficiency than uninformed global updates and restored correct topological sampling.

Significance and Claims
The paper claims to provide a complete, Markovian generalization of stochastic quantization that naturally incorporates jumps to overcome ergodicity breaking.

• Restoration of Ergodicity: The primary contribution is demonstrating that jump-diffusion processes can efficiently explore action landscapes with barriers, restoring ergodicity in regimes where traditional diffusion fails (e.g., topological freezing).
• Theoretical Framework: It establishes a rigorous theoretical basis for jump updates via the generalized Fokker-Planck equation, offering a systematic way to design updates that satisfy detailed balance.
• Efficiency via State-Informed Selection: A key insight is that by treating jump parameters as a state-dependent selection problem (Class C), one can mitigate the high cost of individual jump maps. This allows for efficient sampling even with localized updates, which are crucial for theories with matter fields.
• Future Directions: The author positions this work as a stepping stone for more complex theories. The framework is intended to be applied to the $CP(N-1)$ model (incorporating gauge and matter fields) and eventually to full QCD. The paper also notes the potential application to complex Langevin simulations (to avoid singularities in the complex plane) and discrete systems where diffusion is undefined.

The work does not claim to solve critical slowing down in all geometries immediately but suggests that combining jumps with kernel freedom (e.g., Fourier acceleration) is a promising future direction. The results are presented as a proof-of-concept for the efficacy of jump-diffusion in restoring ergodicity in lattice field theories.