Quantum mechanical bootstrap without inequalities: SYK bilinear spectrum

This paper introduces a "direct bootstrap" method that utilizes fractional powers of operators to derive constraint equations, successfully determining the bilinear spectrum of the Sachdev-Ye-Kitaev model without relying on the standard positivity conditions that fail to distinguish its specific boundary data.

The Gist

Imagine you are trying to solve a complex puzzle where the pieces are the possible energy levels of a tiny, quantum mechanical system. Usually, physicists solve these puzzles using a method called the "bootstrap." Think of the bootstrap like a strict bouncer at a club. The bouncer has a list of rules (mathematical inequalities) that say, "If you don't fit these rules, you can't be an energy level." By checking every possible energy against these rules, the bouncer eventually narrows down the list until only the correct, real energy levels remain.

This paper, titled "Quantum mechanical bootstrap without inequalities," by Kok Hong Thong and David Vegh, describes a situation where the usual "bouncer" fails.

The Problem: The Bouncer is Confused

The authors are studying a specific quantum system that mimics a famous model in physics called the SYK model (Sachdev–Ye–Kitaev). This model is famous for being chaotic and hard to solve, but it has a very specific set of energy levels (a spectrum) that physicists want to find.

In most quantum systems, the standard bootstrap method works perfectly. The "positivity" rules (the bouncer's list) are so strict that they eliminate all the wrong answers, leaving only the true energy levels.

However, for this specific SYK-like system, the authors found that the standard bouncer is degenerate. This means the rules are too loose. The "bouncer" allows a whole continuous range of wrong answers to pass through because the standard rules can't tell the difference between the correct boundary conditions (the specific way the system is "tied down" at its edges) and the wrong ones. It's like a bouncer who can't tell the difference between a VIP guest and a random tourist; everyone gets in, and you can't find the VIP.

The Solution: A New Kind of Key

To fix this, the authors invented a new tool they call the "Direct Bootstrap." Instead of relying on the bouncer's "no-entry" rules (inequalities), they decided to ask the system direct questions that force it to give a specific answer.

Here is how they did it, using a simple analogy:

1. Fractional Powers as Special Keys:
   Usually, physicists use standard "keys" (operators) made of whole numbers, like $Z$, $Z^2$, $Z^3$. The authors realized they needed "fractional keys," like $Z^{0.5}$ or $Z^{1.3}$.

   • Analogy: Imagine trying to open a lock with a standard key. It doesn't fit. But if you use a key with a slightly jagged, fractional shape, it fits perfectly. These fractional keys allow the authors to probe the "edges" of the system in a way standard keys cannot.

2. The "Anomaly" as a Whisper:
   When they used these fractional keys, they noticed something strange happening at the boundaries of the system. In physics, this is called an "anomaly."

   • Analogy: Imagine a room with soundproof walls. If you shout in the middle, you hear nothing. But if you whisper right against the wall, the wall vibrates in a specific way that tells you exactly how the wall is built. The "anomaly" is that vibration. It carries secret information about the boundary conditions that the standard rules missed.

3. Connecting the Dots (Taylor Expansion):
   The authors found that these fractional keys created three different "families" of equations. Each family gave them a clue about the boundary, but each clue was slightly "degenerate" (confusing) on its own.

   • Analogy: Imagine you have three different maps of a city. Map A says the treasure is "somewhere north." Map B says "somewhere east." Map C says "somewhere south." Alone, none of them help. But if you overlay them (using a mathematical technique called Taylor expansion), the lines cross at a single, precise point.

The Result: Solving Without Guessing

By combining these three families of clues, the authors created a system of three equations with three unknowns.

• Old Way: "Is this energy level allowed? Yes/No." (Result: Too many "Yes" answers).
• New Way (Direct Bootstrap): "If the energy is $X$, then the boundary must be $Y$, and the correlation must be $Z$." (Result: Only one specific set of numbers works).

They tested this on two specific cases ($\Delta = 1/4$ and $\Delta = 1/6$). As they added more terms to their mathematical "maps" (increasing the truncation order), their calculated energy levels converged rapidly to the exact values known from other methods.

Why It Matters (According to the Paper)

The paper claims a significant breakthrough: You don't need the "bouncer" (positivity/inequalities) to solve this problem.

Usually, the bootstrap method relies on saying "This is impossible" to narrow things down. This paper shows that for systems with tricky boundary conditions, you can instead say "This must be true" by using direct equations derived from the system's anomalies. The spectrum is determined by the equality of the constraints, not by the exclusion of the impossible.

In short, the authors found a way to solve a quantum puzzle that the standard rules couldn't crack, by using "fractional keys" to listen to the system's whispers at the edges and combining those whispers into a single, undeniable truth.

Technical Summary

1. Problem Statement

The authors address the challenge of applying the Quantum Mechanical (QM) Bootstrap method to a specific system where standard positivity-based constraints fail to uniquely determine the energy spectrum.

• The System: A quantum mechanical system defined on a finite interval $z \in [0, 1]$ with the Hamiltonian:
  $$H = S Z(1-Z)S + \left(\frac{1}{2} - \Delta\right)^2 \frac{1}{Z(1-Z)}$$
  where $S = i\partial_z$ and $Z=z$. This Hamiltonian arises as the Casimir operator of a folded string in $AdS_2$ and its spectrum coincides with the bilinear operator spectrum of the Sachdev–Ye–Kitaev (SYK) model for a specific self-adjoint extension (Robin boundary conditions).
• The Obstacle: In standard QM bootstrap applications, the spectrum is isolated by imposing Positive Semidefiniteness (PSD) constraints on a matrix of correlators. As the truncation order increases, the allowed region in parameter space shrinks to discrete points (the eigenvalues).
  However, for this specific system, the authors find that the standard PSD constraints are degenerate. The "positivity islands" do not collapse to isolated points but remain extended over a continuous range of energies. This occurs because the boundary conditions (which select the SYK spectrum) are not uniquely distinguished by the positivity constraints derived from integer-powered operators. The system requires a method that does not rely on inequalities to fix the spectrum.

2. Methodology: The "Direct Bootstrap"

The authors propose a novel approach called the Direct Bootstrap, which determines the spectrum using exact equality constraints derived from fractional powers of operators, bypassing the need for positivity bounds.

A. Operator Basis Extension

Instead of using only integer powers of position operators ($Z^n$), the authors extend the operator basis to include fractional powers:
$$O_{\sigma, \zeta, \xi} = S^\sigma Z^\zeta (1-Z)^\xi, \quad \sigma \in \mathbb{Z}_{\ge 0}, \quad \zeta, \xi \in \mathbb{R}$$
This extension is crucial because the Hamiltonian and the boundary conditions involve fractional powers (specifically related to the conformal dimension $\Delta$).

B. Anomalies and Recursion Relations

The bootstrap relies on commutator recursion relations $\langle [H, O] \rangle = \mathcal{A}(O)$.

• Dressed Anomalies: Due to the finite interval and the specific weight function $w = [z(1-z)]^{2\Delta-1}$, the commutators generate boundary anomalies (domain anomalies) when the operator $w^{-1}O$ does not preserve the domain of the Hamiltonian.
• Operator Families: The recursion relations preserve the fractional parts of the exponents $\zeta$ and $\xi$. Consequently, correlators split into distinct operator families labeled by a common fractional offset $\omega$.
• Anomaly-Free Cone: Within a specific family, one can construct an "anomaly-free" sector where boundary terms vanish. This sector closes on three unknowns: the energy $E_a$ and two seed correlators. However, this sector is insensitive to boundary conditions.

C. Extracting Constraints from the Anomalous Sector

To fix the boundary conditions and the spectrum, the authors analyze the anomalous sector (where boundary terms are non-zero).

1. Three Special Families: They identify three specific operator families with offsets $\omega \in \{0, 2\Delta, 4\Delta-1\}$.
2. Degeneracy in Boundary Data:
   • The $\omega=0$ family yields a constraint independent of the Robin parameter $r$.
   • The $\omega=2\Delta$ family depends on $c_A^2 r$.
   • The $\omega=4\Delta-1$ family depends on $c_A^2 r^2$.
   • Individually, these constraints are degenerate regarding the boundary parameter $r$ and the normalization $c_A$.
3. Cross-Family Taylor Expansion: The key innovation is to relate these three families via Taylor expansions. By expanding the fractional-power correlators of the $\omega \neq 0$ families in terms of the integer-family correlators ($f_{0,0,0}$ and $f_{0,1,1}$), the authors generate a system of equations linking the different sectors.

D. Solving the System

The combination of the three exact constraint equations (derived from the anomalous sectors) and the Taylor expansion relations yields a closed system of three equations for three unknowns ($E_a$, $f_{0,0,0}$, $f_{0,1,1}$).

• Result: The spectrum is determined directly by solving these algebraic equations. No positivity input (inequalities) is required.

3. Key Contributions

1. Direct Bootstrap Framework: Introduction of a bootstrap method that determines spectra via equality constraints derived from fractional operators and anomalies, rather than relying on PSD bounds.
2. Resolution of Degeneracy: Demonstration that for systems with fractional Robin boundary conditions, standard positivity bounds are insufficient. The degeneracy is lifted by exploiting the structural relations (Taylor expansions) between different fractional operator families.
3. SYK Spectrum Recovery: Successful recovery of the exact SYK bilinear spectrum (both fermionic and bosonic sectors) without solving the Schrödinger equation or using the analytic form of the wavefunctions.
4. Handling Domain Anomalies: A rigorous treatment of dressed anomalies arising from non-self-adjoint operators on bounded domains, separating bulk corrections from boundary data.

4. Results

The method was tested for two specific values of the conformal dimension: $\Delta = 1/4$ and $\Delta = 1/6$.

• Convergence: The calculated eigenvalues converge to the exact analytical values as the Taylor expansion truncation order $N$ increases.
  • For $\Delta = 1/4$, the convergence rate for the first few eigenvalues scales as $O(N^{-3.6})$ to $O(N^{-3.8})$, which is faster than the theoretical upper bound derived from the expansion of the lowest fractional power.
  • For $\Delta = 1/6$, similar power-law convergence was observed.
• Exact Reproduction: The lowest fermionic state ($h_2 = 2$) was reproduced exactly even at lower truncation orders.
• General Boundary Conditions: The method successfully determined the spectrum for arbitrary Robin parameters $r$, not just the specific SYK value. This confirms that the fractional-power constraints provide the necessary information to fix the boundary conditions without positivity.

5. Significance and Implications

• Beyond Positivity: This work challenges the assumption that the QM bootstrap requires positivity constraints to isolate spectra. It shows that for systems with specific boundary structures, exact algebraic constraints derived from anomalies and operator mixing are sufficient and potentially more powerful.
• New Tool for SYK: It provides a non-perturbative, numerical-analytic tool to study the SYK model's bilinear spectrum, which is central to understanding quantum chaos and holography.
• Future Directions: The authors suggest that this "Direct Bootstrap" could be applied to other systems with fractional boundary conditions (e.g., Heun's equations). They also propose that for more complex systems where exact relations are insufficient, a hybrid approach (combining exact anomalous constraints with PSD bounds) may be the optimal strategy.
• Higher Dimensions: The paper raises the question of whether similar "from bounds to equations" mechanisms exist in higher-dimensional Conformal Field Theory (CFT) bootstrap programs.

In summary, the paper presents a breakthrough in quantum mechanical bootstrap techniques by utilizing fractional operators and anomaly analysis to solve a system where traditional positivity methods fail, successfully recovering the SYK spectrum through direct algebraic constraints.