Resurrecting the coherent state variational algorithm for large $N$ gauge theories

This paper reassesses the feasibility of using coherent state variational methods for large $N$ gauge theories by introducing a new implementation applicable to SU($N$) lattice gauge theories with or without fermions, and presents initial results for Hamiltonian Yang-Mills theory on an infinite two-dimensional spatial lattice.

The Gist

Imagine you are trying to solve a massive, incredibly complex puzzle. This puzzle represents the fundamental forces that hold the universe together (specifically, the strong force that binds quarks inside protons and neutrons). The puzzle is so huge that it has an infinite number of pieces, and trying to solve it piece-by-piece with a computer is like trying to drink the ocean with a spoon.

For decades, physicists have used a method called "Monte Carlo simulation" to solve this. Imagine this as a blindfolded hiker stumbling around a mountain, taking random steps and hoping to eventually find the lowest valley (the ground state of the theory). It works, but it's slow, and it gets very messy when you try to look at the mountain from a distance (the "large N" limit, where the complexity of the puzzle becomes infinite).

The New Approach: The "Coherent State" Map

This paper, written by Laurence G. Yaffe, proposes a different way to solve the puzzle. Instead of stumbling around randomly, the author suggests using a "map" based on a mathematical concept called coherent states.

Think of the puzzle not as a chaotic mess, but as a smooth landscape. In the "large N" limit (where the puzzle becomes infinitely complex), the quantum weirdness fades away, and the landscape becomes "classical." It's like the difference between a foggy, chaotic night (quantum) and a clear, sunny day (classical).

The author's method is to find the absolute lowest point (the minimum) on this smooth landscape. Once you find the bottom of the valley, you can easily figure out the shape of the hills around it. This allows physicists to calculate things like the mass of particles (glueballs) and how they bounce off each other, which is very hard to do with the old "stumbling" method.

The Tool: "Gordion"

To do this, the author built a new computer program named "Gordion." The name is a clever reference to the legend of Alexander the Great, who faced a tangled knot (the Gordian Knot) that no one could untie. Instead of trying to untie it thread by thread, Alexander simply sliced through it with his sword.

Similarly, the "Gordion" program doesn't try to untangle every single thread of the infinite puzzle. Instead, it uses a "loop-list" strategy. It focuses on the most important loops (paths the particles take) and ignores the rest, effectively "cutting through" the complexity.

What Did They Find?

The author tested this new method on several scenarios:

1. Simple Test Cases: They started with tiny, simple puzzles (one "plaquette" or square loop). The program worked perfectly, matching the known exact answers. This proved the "sword" was sharp and the map was accurate.
2. 2D Grid (Flat World): They applied it to a two-dimensional grid. Even without simplifying the math too much, the program got very close to the correct answers, even in areas where the puzzle is usually very hard (weak coupling).
3. 3D Grid (Real World Simulation): They tried it on a 2+1 dimensional grid (two space dimensions plus time). This is much harder. The program worked well for strong interactions but started to struggle as the interactions got weaker.

The Limitations: The "Truncation" Problem

The main challenge is that the program has to ignore some pieces of the puzzle to run on a normal desktop computer. This is called "truncation."

• The Analogy: Imagine trying to describe a complex painting by only listing the colors of the biggest brushstrokes. At first, this works great. But as you zoom in (or as the physics gets more subtle), you miss the fine details.
• The Result: The program works beautifully when the "paint" is thick and bold (strong coupling). But as the paint gets thinner and more detailed (weak coupling), the approximation starts to drift. The program sometimes produces results that are physically impossible (like a probability greater than 100%), signaling that it has run out of useful pieces to work with.

The "Factorization" Attempt

The author tried a clever trick to fix the missing pieces. They guessed that if a large loop is made of two smaller loops, the value of the big loop is just the product of the two small ones. They called this "factorization."

However, the results were disappointing. Sometimes this guess helped, but often it made things worse or didn't change anything. It's like trying to guess the flavor of a complex soup by just multiplying the flavors of two ingredients; it doesn't always capture the full taste.

Conclusion

The paper concludes that this "coherent state" approach is a powerful new way to look at these infinite puzzles. It allows physicists to work directly with the "infinite" version of the theory, avoiding the statistical noise of random simulations.

While the current version (running on a standard desktop computer) hasn't solved the hardest parts of the 3D puzzle yet, it has proven the concept works. The author suggests that with better computers (supercomputers) and smarter ways to handle the missing pieces, this method could eventually solve problems that are currently impossible, such as calculating exactly how particles scatter and decay in a way that is much more direct than current methods.

In short: The author has sharpened a new sword (Gordion) and shown it can slice through the simplest knots perfectly. It's starting to cut through the bigger knots, but it needs a bigger hand (more computing power) and a sharper edge (better approximations) to finish the job.

Technical Summary

Technical Summary: Resurrecting the Coherent State Variational Algorithm for Large N Gauge Theories

Problem Statement
The paper addresses the challenge of numerically solving non-Abelian gauge theories (specifically SU(N) and U(N)) in the large $N$ limit. While the large $N$ limit reduces quantum field theory dynamics to a classical minimization problem on a phase space of coherent states, solving this problem for non-trivial lattice gauge theories remains computationally difficult. Existing methods, such as Euclidean Monte Carlo simulations, face limitations in extracting real-time properties (like decay widths and scattering amplitudes) and require stochastic sampling. Furthermore, standard Hamiltonian truncation methods struggle with the exponential growth of the Hilbert space in 2+1 and 3+1 dimensions. The author reassesses the feasibility of using coherent state variational methods to directly access the $N=\infty$ limit, aiming to compute ground state properties, excitation spectra, and scattering amplitudes without finite volume effects or statistical sampling variance.

Methodology
The paper describes a new implementation of the coherent state variational algorithm, originally formulated in the 1980s, utilizing a unified C++ program named "Gordion." The methodology relies on the following core components:

1. State Representation: The algorithm uses a "loop-list" truncation scheme. The state is represented by a finite set of expectation values of Wilson loops (and fermion bilinears if present). These observables are selected based on their "strong-coupling order," defined by the order in a strong-coupling expansion at which omitting the observable would first introduce an error.
2. Variational Parameters: Instead of using Wilson loop expectation values directly as variational parameters (which violates unitarity constraints and leads to complex minimization boundaries), the algorithm employs Riemann normal coordinates. These are the coefficients of generators from the infinite-dimensional coherence group $G$ (exponentials of anti-Hermitian combinations of loops and electric fields). Deformations of the state are generated by acting with these group elements, ensuring the state remains within the physical domain.
3. Algorithmic Steps:
   • Symbolic Generation: The program symbolically evaluates commutators between retained observables and coherence group generators to define geodesic equations on the large $N$ phase space.
   • Energy/Free Energy Minimization: It computes the gradient and curvature (Hessian) of the large $N$ Hamiltonian (or Euclidean free energy) with respect to the Riemann normal coordinates.
   • Newton Iteration: The minimum is found via Newton iteration, solving linear equations to predict the location of the minimum and integrating the geodesic equations to update expectation values.
   • Spectrum Extraction: For Hamiltonian theories, the small oscillation frequencies around the minimum are determined by solving a generalized eigensystem involving the curvature matrix and the Lagrange bracket (inverse Poisson bracket), yielding the glueball or meson mass spectrum.
4. Observable Approximation: An initial attempt is made to reduce truncation errors by approximating the expectation values of non-retained observables using a factorization scheme ($W_{\Gamma} \approx W_{\Gamma_1} W_{\Gamma_2}$ for self-intersecting loops), though this is found to be of limited utility in the current implementation.

Key Contributions

• Implementation of "Gordion": The author presents a modern, efficient C++ implementation capable of handling U(N) gauge theories on cubic lattices up to 4 dimensions, with or without fundamental fermions, in both Hamiltonian and Euclidean formulations.
• Benchmarking: The code is validated against exactly solvable one-plaquette models in both Euclidean and Hamiltonian formulations, demonstrating convergence to exact results as the number of variational parameters (generators) increases.
• 2D Euclidean Yang-Mills: Initial results are presented for 2D Euclidean Yang-Mills on an infinite lattice without utilizing the non-local reduction to independent plaquette variables. The method successfully reproduces exact results for free energy and various loop expectation values down into the weak coupling regime ($\lambda \approx 0.7$).
• 2+1D Hamiltonian Yang-Mills: The paper presents the first application of this method to 2+1 dimensional Hamiltonian Yang-Mills theory on an infinite lattice. It calculates the ground state energy and the masses of the lowest glueball states in various symmetry channels ($A_1, A_2, B_1, B_2, E$).

Results

• Convergence: In one-plaquette models, the variational results converge rapidly to exact solutions. Three to five generators are sufficient to reproduce the exact free energy and loop expectations within visual indistinguishability, even near the third-order phase transition.
• 2D Euclidean Theory: Using generators up to order 6 (involving up to three plaquette operators) and observables truncated at order 28, the algorithm reproduces exact free energy and loop expectation values with high accuracy for $\lambda > 0.7$. Truncation errors become noticeable as $\lambda$ decreases further.
• 2+1D Hamiltonian Theory:
  • The method successfully computes the ground state energy and expectation values of small loops.
  • Glueball masses are extracted for multiple symmetry channels. The results show the expected strong-coupling behavior (masses scaling linearly with $\lambda$).
  • However, convergence in the weak coupling regime is slower than in the 2D Euclidean case. The sixth-order generator results do not yet clearly exhibit the linear approach to the origin required for a continuum limit extrapolation.
  • Limitations: The calculations are limited by the rapid growth of the number of observables and the size of the geodesic equations. For 2+1D theory, calculations with order 6 generators failed to converge below $\lambda \approx 1.4$ due to the emergence of unphysical loop expectations (violating $|W_\Gamma| \le 1$) and complex curvature eigenvalues, signaling the breakdown of the current truncation scheme.
• Observable Approximation: The factorization-based approximation for non-retained observables was tested but found to be inconclusive. It significantly increased computational storage requirements without consistently improving accuracy or mimicking higher-order truncation results.

Significance and Claims
The paper claims that the coherent state variational approach is a viable, deterministic method for studying large $N$ gauge theories directly at $N=\infty$. Its primary significance lies in its ability to:

1. Work directly in infinite volume, avoiding finite-size effects inherent in standard lattice simulations.
2. Provide direct access to the excitation spectrum (glueball/meson masses) and, in principle, decay widths and scattering amplitudes via higher-order derivatives of the classical Hamiltonian, without the need for analytic continuation from Euclidean time.
3. Handle dynamical fermions without the complications of the Dirac determinant.

The author modestly concludes that while the current results for 2+1D Hamiltonian theory are a "promising initial effort," they are limited by truncation errors and computational resources (the work was performed on a desktop computer). The paper does not claim to have solved 3+1D QCD or reached the continuum limit of 2+1D Yang-Mills. Instead, it establishes the feasibility of the approach and identifies specific technical challenges—such as the need for better observable selection criteria, improved approximation schemes for truncated observables, and the potential utility of finite-dimensional master field representations—that must be addressed to extend the method to physically relevant weak-coupling regimes. The author emphasizes that this classical minimization problem, while hard, is likely more tractable than direct Hamiltonian truncation for finite $N$ or real-time evolution on quantum computers.