Bootstrapping ground state properties of classical frustrated magnets

This paper introduces a rigorous semidefinite programming method that adapts the Lasserre hierarchy to produce convergent two-sided bounds on ground state energy densities and correlation functions for translation-invariant classical frustrated magnets, overcoming limitations of prior analytical techniques by handling non-quadratic Hamiltonians and non-Bravais lattices with high efficiency.

The Gist

Imagine you are trying to find the perfect way to arrange a crowd of people in a giant, infinite stadium. Each person has a specific rule: they must stand exactly one meter away from the center of their own personal space (like a unit-length spin). However, they also have conflicting desires: some want to face their neighbors, while others want to face away. This is a "frustrated" system because you can't satisfy everyone's desires at once.

The goal is to find the arrangement that makes the crowd as calm (low energy) as possible. This is a classic problem in physics, but it's incredibly hard to solve because there are so many people and so many conflicting rules that the math gets messy and full of "dead ends."

Here is how the authors, Nisarga Paul and Gil Refael, solved this problem using a new method they call bootstrapping.

The Problem: A Maze with Many Dead Ends

Think of the traditional way of solving this as trying to find the lowest point in a massive, foggy mountain range. You might start walking down a hill, but you could easily get stuck in a small valley (a local minimum) thinking it's the bottom, when there is actually a much deeper valley nearby.

• The old way (Luttinger-Tisza): This was like looking at the mountain from a very high, blurry distance. It gave a good guess for simple mountains, but if the terrain was weird or the rules were complex, the guess was often wrong.
• The simulation way (Monte Carlo): This is like sending a robot to walk around the mountain. But in a frustrated system, the robot gets confused, spins in circles, and never finds the true bottom.

The Solution: The "Shadow" Method (Bootstrapping)

Instead of trying to find the exact arrangement of every single person (which is impossible), the authors decided to look at the shadows the crowd casts.

Imagine you don't know where the people are standing, but you know the rules of the game:

1. Positivity: If you ask, "How likely is it that two people are standing in a certain way?" the answer can't be negative.
2. Normalization: Every person must exist (the total probability is 1).
3. Geometry: The people are standing on a sphere (they can't stretch or shrink).

The authors created a mathematical "sieve" or a series of filters. They started with a very loose filter that only checked the basic rules. Then, they added more and more complex filters that checked deeper relationships between the people.

• The Analogy: Imagine trying to guess the shape of a hidden object by looking at its shadow.
  • Level 1: You see a shadow that looks like a circle. The object could be a ball, a plate, or a coin.
  • Level 2: You add a second light source. Now the shadow must match both angles. The object is now narrowed down to just a ball or a plate.
  • Level 3: You add a third light. Now the shadow must match three angles. The object is definitely a ball.

In this paper, the "shadows" are correlation functions (how one spin relates to another). The "lights" are mathematical constraints called Semidefinite Programming (SDP).

How It Works in Practice

The authors built a hierarchy of these filters:

1. The Setup: They defined a small patch of the infinite stadium (a few rows of seats).
2. The Constraints: They forced the math to obey the rules of probability and geometry within that patch.
3. The Result: The computer solves a "convex optimization" problem. This is a type of math problem that doesn't have dead ends; it always finds the best possible answer within the rules of that specific filter.

As they made the patch bigger and added more complex filters (higher levels of the hierarchy), the "shadow" became sharper and sharper.

• The Lower Bound: The method gives a guaranteed "floor" for how calm the crowd can be. It says, "The energy cannot be lower than X."
• The Upper Bound: They also used a standard simulation to find a specific arrangement and calculate its energy, giving a "ceiling." "The energy cannot be higher than Y."

The Magic of the Result

In many cases, the "floor" and the "ceiling" met almost perfectly.

• Precision: They found the exact energy of the ground state with incredible precision (accurate to 8 decimal places in some cases).
• No Guessing: Unlike other methods, this doesn't rely on guessing a starting point. It provides a rigorous proof that the answer is within a tiny range.
• Speed: Even though the math is complex, the computer could solve these problems in just a few seconds per setting.
• Visualizing the Crowd: Once they found the "shadow," they could reverse-engineer it to see what the actual arrangement of people (the spin texture) looked like. It matched the best guesses from other methods perfectly.

Why This Matters

This method is like having a super-accurate ruler for a world where everything is fuzzy.

• It works for any shape of stadium (not just simple grids).
• It works for any type of rule (even complex, non-linear ones).
• It works in the infinite limit (theoretically perfect), not just on a small computer simulation.

The authors showed that by looking at the "shadows" (correlations) and tightening the rules (the hierarchy), they could solve a problem that was previously considered too hard to solve with certainty. They didn't just guess the answer; they mathematically proved the range where the answer must live.

Technical Summary

Technical Summary: Bootstrapping Ground State Properties of Classical Frustrated Magnets

Problem Statement
Determining the ground state spin configuration and energy density of classical frustrated magnetic systems is a formidable optimization problem. For Ising spins, this problem is known to be NP-hard. For Heisenberg spins with translation-invariant Hamiltonians, the standard analytical approach is the Luttinger–Tisza (LT) method, which minimizes energy over sums-of-spirals ansatzes by relaxing local spin normalization constraints. However, the LT method has limited validity, particularly for non-Bravais lattices and non-quadratic Hamiltonians. Numerical alternatives, such as classical Monte Carlo, often face severe equilibration challenges in frustrated systems due to energy landscapes with many closely competing local minima. Furthermore, variational and Monte Carlo methods typically operate on finite-size systems and lack rigorous guarantees regarding their proximity to the true thermodynamic-limit ground state.

Methodology
The authors introduce a bootstrap method based on semidefinite programming (SDP) to produce rigorous two-sided bounds on ground state energy densities and correlation functions for translation-invariant classical spin models on infinite lattices. The approach adapts the Lasserre hierarchy from polynomial optimization to the context of frustrated magnetism.

1. Formulation as Polynomial Optimization: The problem of finding the ground state is reduced to a polynomial optimization problem where the constraints are the unit-norm condition of spins ($|S_i|^2 = 1$).
2. Moment Hierarchy: Instead of optimizing over spin configurations directly (a non-convex problem), the method optimizes over the space of translation-invariant correlation functions (moments). The method defines a hierarchy of finite-size convex optimizations labeled by a real-space patch $P$ and a positive integer $r$ (the hierarchy level).
3. Variables and Constraints:
   • Variables: The optimization variables are the expectation values $y[I] = \langle m_I \rangle$ of spin monomials $m_I$ of degree up to $r$ within the patch $P$.
   • Positivity (Moment Matrix): A moment matrix $L$, indexed by pairs of monomials, is constructed. The condition that $L$ is positive semidefinite ($L \succeq 0$) ensures that the moments correspond to a valid probability distribution.
   • Normalization (Localizing Matrices): The unit-length constraints are enforced via localizing matrices $C_r$, which impose linear relations among moments derived from the identity $\sum_a (S_r^a)^2 - 1 = 0$.
4. Convergence: The authors prove that as the patch size $P$ and hierarchy level $r$ grow to infinity, the lower bounds on the energy density converge monotonically to the true thermodynamic-limit ground state energy density. This proof utilizes tools from measure theory (Kolmogorov's extension theorem) and real algebraic geometry.
5. Upper Bounds and Spin Textures: To obtain two-sided bounds, the method is combined with a variational upper bound ($\epsilon_{ub}$) obtained via iterative one-site minimization on large periodic clusters. Additionally, explicit spin textures can be extracted from the SDP solution by performing an eigendecomposition of the two-point correlation matrix and projecting onto the three leading eigenvectors, followed by normalization.

Key Contributions

• Rigorous Two-Sided Bounds: The method provides rigorous lower and upper bounds on ground state energy densities and arbitrary polynomial observables directly in the thermodynamic limit, eliminating the need for finite-size extrapolation.
• Generalization of Luttinger–Tisza: The approach subsumes the LT method as a special case (specifically at the lowest hierarchy level on Bravais lattices) but extends applicability to non-Bravais lattices and non-quadratic (e.g., quartic) Hamiltonians.
• Efficiency: The method is computationally efficient, with typical run times of seconds per parameter point using standard SDP solvers (e.g., MOSEK).
• Texture Extraction: The method allows for the extraction of explicit candidate ground-state spin textures that match variational results well, even in highly frustrated regimes.

Results
The method was applied to two two-dimensional frustrated spin models:

1. Centered-Square Lattice (Quadratic Heisenberg Model): A model with inequivalent sites and non-uniform coordination where the LT method fails. The SDP lower bounds and variational upper bounds nearly coincided across the parameter range, demonstrating the SDP's superiority over LT and its ability to certify variational results.
2. Triangular Lattice (Bilinear-Biquadratic Model): A highly frustrated quartic model known for extensive ground state degeneracy at intermediate couplings. The SDP provided tight energy bounds (brackets of $10^{-5}$ to $10^{-8}$ per site) across the phase diagram. In the highly frustrated regime, the bounds were wider, reflecting the intrinsic difficulty, but the SDP successfully certified variational results. The extracted spin textures matched variational configurations in ordered phases and selected representatives from the degenerate manifold in the frustrated phase.

Significance and Claims
The paper claims that this bootstrap approach serves as a practical, general-purpose tool for ground state problems in classical frustrated magnetism, complementary to existing heuristic approaches and Monte Carlo simulations. Its primary significance lies in providing rigorous guarantees in the thermodynamic limit, a feature absent in variational and Monte Carlo methods. The authors emphasize that the method addresses the limitations of prior analytical methods (like LT) regarding non-quadratic interactions and non-Bravais lattices.

The authors modestly frame the work as an introduction of the method and its application to specific 2D models. They identify future directions, such as relaxing translation invariance to study spin glasses, developing semiclassical bootstraps for large-spin quantum corrections, and applying the approach to classical dynamical processes, but do not claim these have been realized in the current work.