A counterexample to Fermi isospectral rigidity for two dimensional discrete periodic Schrödinger operators

This paper uses numerical certification to construct a counterexample proving that two-dimensional discrete periodic Schrödinger operators are not Fermi isospectrally rigid, thereby disproving a long-standing conjecture regarding the irreducibility of Fermi varieties for nontrivial real-valued potentials.

The Gist

Here is an explanation of the paper "A Counterexample to Fermi Isospectral Rigidity," translated into simple, everyday language with creative analogies.

The Big Picture: The "Silent" Drum

Imagine you have a giant, infinite drum made of a grid of tiny points (like a pixelated drumhead). This drum can vibrate. The way it vibrates depends on two things:

1. The Shape of the Drum: How the grid is connected (this is the "Laplacian" or the standard grid).
2. The Weight on the Drum: Imagine placing tiny weights on some of the grid points. This is the Potential ($V$).

In physics, the "sound" of this drum is described by its energy levels. If you tap the drum, it produces specific notes (frequencies).

The Big Question:
If you listen to the drum and hear a specific set of notes (a specific "energy spectrum"), can you figure out exactly where the weights are placed?

• The Old Belief (Rigidity): For a long time, mathematicians believed that if the drum sounds exactly like a perfectly flat, weightless drum, then it must be perfectly flat. In other words, "If it sounds like nothing, it is nothing."
• The New Discovery: This paper proves that belief is wrong for 2-dimensional drums. You can build a drum with a complex pattern of weights that sounds exactly the same as a flat drum, at least for a specific range of notes.

The Key Concepts (Translated)

1. The "Fermi Variety" (The Drum's Signature)

Think of the "Fermi variety" as the drum's fingerprint. It's a mathematical map that shows all the possible ways the drum can vibrate at a specific energy level.

• The Goal: The authors wanted to find a pattern of weights (a "potential") that creates the exact same fingerprint as a drum with no weights at all.

2. The "Rigidity" Problem

Imagine you are a detective. You walk into a room and hear a sound.

• Scenario A (Rigidity): You hear a pure tone. You conclude, "There are no obstacles in the room."
• Scenario B (The Counterexample): You hear the exact same pure tone, but when you look around, you see a complex maze of furniture (weights) arranged in a very specific, hidden pattern. The furniture is there, but it's "invisible" to the sound.

The paper proves that Scenario B is possible in a 2D world.

3. The "Magic" Grid (3x5)

The authors didn't just guess; they built a specific example. They chose a repeating pattern of weights on a grid that repeats every 3 steps horizontally and 5 steps vertically.

• Think of this like a wallpaper pattern. The weights repeat every 3 inches across and 5 inches down.
• They calculated the exact weight for every single spot in that 3x5 block.

How They Did It (The "Mathematical Detective Work")

This wasn't a simple "aha!" moment; it was a massive computational feat.

Step 1: Turning Physics into Algebra
They translated the physics of the vibrating drum into a giant system of polynomial equations (think of these as a complex recipe with 14 different ingredients and 15 variables).

• They needed to find a set of numbers (the weights) that made all 14 equations equal zero simultaneously.
• If you find such numbers, you've found a "ghost drum" that sounds like a flat one.

Step 2: The Needle in the Haystack
The math showed there are billions of possible solutions, but most are imaginary or complex. They needed a real solution (real weights you could actually build).

• Finding this specific solution was like finding a single needle in a haystack the size of a city.
• They used a computer technique called Homotopy Continuation. Imagine you are walking through a dark maze. Instead of trying to map the whole maze, you just keep walking forward until you stumble upon the exit. They tracked thousands of paths until one led to a real solution.

Step 3: The "Certification" (The Gold Standard)
Here is the most important part. Computers make rounding errors. If you tell a computer, "The answer is 3.587996," the computer might actually be off by a tiny fraction. In math, "close enough" isn't good enough.

• The authors used a method called Krawczyk's Method.
• The Analogy: Imagine you found a treasure chest. To prove it's real, you don't just say, "It looks like gold." You put the chest in a box, seal it, and use a super-precise scanner to prove that inside that specific box, there is definitely a solid gold bar, and no other possibility exists.
• They used two different computer programs (Macaulay2 and Julia) to independently verify that their solution was mathematically proven to exist, not just a lucky guess.

Why This Matters

1. It Breaks a 30-Year-Old Rule: In the 1990s, famous mathematicians (Gieseker, Knörrer, and Trubowitz) guessed that this "ghost drum" phenomenon was impossible for real materials. This paper proves them wrong.
2. It Changes How We See Physics: It shows that in 2D, nature can hide complexity. You can have a material that looks and acts "flat" to certain waves, even if it's actually bumpy and complex underneath.
3. The Method is Powerful: The way they used computers to prove the existence of a solution (rather than just approximating it) is a major step forward for how mathematicians use computers in pure theory.

The Bottom Line

The authors found a specific, repeating pattern of weights on a 2D grid. When they "tapped" this grid, it produced the exact same sound signature as a grid with no weights at all.

Conclusion: Just because something sounds empty doesn't mean it is empty. The universe is more deceptive (and interesting) than we thought.

Technical Summary

Here is a detailed technical summary of the paper "A Counterexample to Fermi Isospectral Rigidity for Two Dimensional Discrete Periodic Schrödinger Operators" by Taylor Brysiewicz, Matthew Faust, and Wencai Liu.

1. Problem Statement and Context

The paper addresses two fundamental questions in the spectral theory of discrete periodic Schrödinger operators ($H = \Delta + V$) on the lattice $\mathbb{Z}^d$:

1. Fermi Isospectral Rigidity:

   • Context: Two potentials $X$ and $Y$ are Fermi isospectral at energy $\lambda_0$ if their Fermi varieties (the set of quasi-momenta $k$ where the operator has an eigenvalue $\lambda_0$) are identical: $F_{\lambda_0}(X) = F_{\lambda_0}(Y)$.
   • Known Results: For dimensions $d \geq 3$, it is a known theorem (Theorem 1.1) that if a real potential $V$ is Fermi isospectral to the zero potential ($V \equiv 0$) at any energy level, then $V$ must be identically zero. This is a "rigidity" result.
   • The Open Question: Does this rigidity hold in dimension $d=2$? Specifically, Question 1 (posed by the authors and others) asks: If $d=2$ and a real periodic potential $V$ is Fermi isospectral to 0, must $V=0$?

2. Irreducibility of Fermi Varieties:

   • Context: The Fermi variety $F_\lambda(V)$ is an algebraic variety. A major conjecture by Gieseker, Knörrer, and Trubowitz (1990s) stated that for any non-constant real-valued periodic potential in $d=2$, the Fermi variety is irreducible (cannot be decomposed into smaller algebraic components) at all energy levels.
   • The Connection: Theorem 1.3 establishes that if the Fermi variety $F_{[V]}(V)$ (where $[V]$ is the average potential) is reducible, then $V$ is Fermi isospectral to a constant potential. Thus, disproving rigidity implies disproving irreducibility.

2. Methodology

The authors employ a numerical certification approach to prove the existence of a counterexample. They do not rely solely on floating-point approximations but use rigorous interval arithmetic to guarantee the existence of an exact mathematical solution.

A. Mathematical Formulation

• Setup: They consider a $3\mathbb{Z} \times 5\mathbb{Z}$-periodic potential$V$on$\mathbb{Z}^2$. The potential is defined by 15 variables$v_{m,n}$(values of$V$ on a fundamental domain).
• Floquet Matrices: The spectral problem is reduced to analyzing the Floquet matrix $D_V(z)$, where $z = (z_1, z_2)$ are complex variables related to the quasi-momentum.
• Polynomial System: The condition that $V$ is Fermi isospectral to the zero potential at energy $\lambda=0$ is equivalent to the identity:
  $$\det(D_V(z)) = \det(D_0(z))$$
  By expanding this determinant and comparing coefficients of the monomials in $z_1$ and $z_2$, the authors construct a system of 14 polynomial equations ($f_1, \dots, f_{14}$) in 15 variables (the potential values).
• Goal: Find a non-zero real solution to this system $F(v) = 0$.

B. Computational Strategy

1. Candidate Generation:

   • Solving the system directly via homotopy continuation is computationally prohibitive (estimated 25 years for a full solution set).
   • The authors use a lazy-evaluation homotopy iterator (via HomotopyContinuation.jl) to track paths one by one until a real solution is found. This took approximately 400 minutes and found a solution on the 142,295th path.
   • This yielded an approximate candidate solution $V^*$.

2. Numerical Certification (Krawczyk's Method):

   • To prove the existence of an exact real solution near $V^*$, they apply Krawczyk's method.
   • The System: Since the original system has 14 equations and 15 variables, they add a linear constraint ($v_{1,1} = 61/17$) to create a square system (15 equations, 15 variables).
   • The Proof: Using interval arithmetic, they define a bounding box $B$ around the approximate solution $V^*$. They prove that the Newton operator associated with the system maps $B$ strictly into its interior ($N_F(B) \subset \text{int}(B)$).
   • Implication: By Banach's Fixed Point Theorem, this guarantees a unique, non-singular real root $\tilde{V}$ exists within the box $B$.
   • Verification: The certification was performed independently using two software packages:
     • NumericalCertification in Macaulay2.
     • HomotopyContinuation.jl in Julia.
   • Both implementations confirmed the existence of a real solution with rigorous error bounds.

3. Key Contributions and Results

Theorem 1.2 (Main Result)

The authors prove the existence of a nontrivial (non-zero) real-valued $3\mathbb{Z} \times 5\mathbb{Z}$-periodic potential$V$ such that:
$$F_0(V) = F_0(0)$$
This provides a negative answer to Question 1, demonstrating that Fermi isospectral rigidity fails in dimension $d=2$.

Theorem 1.4 (Irreducibility Conjecture Disproof)

As a corollary, they disprove the Gieseker-Knörrer-Trubowitz conjecture. They show there exists a non-constant real periodic potential $V$ on $\mathbb{Z}^2$ such that the Fermi variety $F_\lambda(V)$ is reducible for some $\lambda$.

• Specifically, for their constructed potential $\tilde{V}$, the average $[\tilde{V}] = 0$.
• Since $F_0(\tilde{V}) = F_0(0)$ and $F_0(0)$ is known to have two irreducible components (Remark 1), the Fermi variety of the non-constant potential $\tilde{V}$ is reducible.

4. Significance

1. Resolution of a Long-Standing Open Problem: The paper settles the question of Fermi isospectral rigidity in 2D, showing that the behavior in $d=2$ is fundamentally different from $d \geq 3$.
2. Disproof of a Classic Conjecture: It overturns a widely held belief (the Gieseker-Knörrer-Trubowitz conjecture) regarding the irreducibility of Fermi varieties for real potentials in 2D.
3. Methodological Advancement: The paper serves as a benchmark for numerical certification in algebraic geometry and spectral theory. It demonstrates how to rigorously prove the existence of solutions to complex polynomial systems arising from physical models, bridging the gap between numerical approximation and mathematical proof.
4. Implications for Spectral Theory: The existence of non-trivial isospectral potentials in 2D suggests that the inverse spectral problem (recovering a potential from its spectrum) is ill-posed or has non-unique solutions in this dimension, even under strong constraints like real-valuedness and periodicity.

In summary, the authors constructed a specific, certified counterexample using advanced computational algebraic geometry techniques, proving that in two dimensions, the Fermi spectrum does not uniquely determine the periodic potential, and that Fermi varieties can be reducible for non-constant real potentials.