⚛️ high-energy theory

Thermal Bootstrap of Large-N Matrix Models via Conic Optimization

This paper improves thermal bootstrapping for large- $N$ matrix models by employing the Quantum Information Conic Solver to derive logarithmic-relaxation-free energy bounds, achieving high-precision estimates for long string excitation energies and coupling coefficients in the one-matrix anharmonic oscillator.

Original authors: Sophia Adams

Published 2026-02-17

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Sophia Adams

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to understand the behavior of a incredibly complex, chaotic dance party. The "guests" are tiny particles, and the "music" is the energy of the system. In the world of theoretical physics, this dance party is called a Matrix Model. It's a mathematical way to describe how particles interact, and it's so complex that even the smartest supercomputers struggle to predict exactly how the party will go, especially when the room gets hot (high temperature) or cold (low temperature).

For a long time, physicists have used a method called "The Bootstrap" to figure this out. Think of the Bootstrap like trying to guess the rules of a game just by watching people play, without ever being told the rulebook. You look at the patterns: "If Alice moves left, Bob usually moves right. If the music speeds up, everyone jumps higher." By applying strict rules of logic and symmetry (like "energy must be conserved"), you can narrow down the possibilities until you find the only answer that makes sense.

The Problem: The "Relaxation" Trap

In the past, when physicists tried to use the Bootstrap for hot systems (thermal states), they hit a wall. The math required to describe heat involves a tricky, non-linear rule called the KMS condition (a fancy way of saying "the system is in thermal equilibrium").

To solve this, previous methods used a shortcut called "Linear Relaxation."

The Analogy: Imagine you are trying to draw a perfect circle on a piece of paper, but your tools only let you draw straight lines. To get close, you draw a square, then an octagon, then a 100-sided polygon. It looks like a circle, but it's not exactly one.
The Issue: In physics, this "polygon" approximation meant that the results were never perfectly precise. As the systems got bigger (more particles), the errors piled up, and the math would eventually crash or give nonsense answers.

The Solution: The "Conic" Solver (QICS)

This paper introduces a new tool called QICS (Quantum Information Conic Solver).

The Analogy: Instead of trying to force a circle into a square, QICS is like a 3D printer that can mold the clay into the exact shape you need, including the curved parts. It handles the "non-linear" rules directly, without needing to approximate them with straight lines.

The author, Sophia Adams, used this new tool to "bootstrap" two specific types of matrix models:

The One-Matrix Model: A single giant matrix of interacting particles.
The Two-Matrix Model: Two giant matrices interacting with each other (this is a simplified version of a model used to describe Black Holes).

The Results: Cracking the Code

Here is what happened when she used the new "exact" tool:

1. Tighter Bounds (The Net Gets Smaller)
With the old method, the possible answers were like a wide net catching a fish. You knew the fish was somewhere in the net, but you didn't know exactly where. With QICS, the net shrank dramatically. The range of possible answers became so small that the result was almost identical to the "true" physical value.

Real-world impact: For the one-matrix model, she calculated the energy of the first excited state with an error of only 0.001%. That's like measuring the distance from New York to London and being off by less than the width of a human hair.

2. Discovering "Long Strings"
In the low-temperature limit (when the party cools down), the particles in these models behave like long, vibrating strings.

The Analogy: Imagine the chaotic dance floor suddenly freezing into a single, giant, vibrating guitar string.
The Discovery: Using her new bounds, Adams was able to calculate the vibration frequency (energy) and the tension (coupling coefficient) of this string for the first time, using only the rules of symmetry and self-consistency. She didn't need to guess; the math forced the answer out.

3. The Two-Matrix Challenge
The two-matrix model is much harder (like trying to solve a Rubik's cube while juggling). The old methods failed completely here due to numerical instability (the math got too messy). However, by using a clever trick—relabeling the variables to exploit a hidden symmetry (like realizing the dance floor is perfectly symmetrical)—she was able to get results for the two-matrix model that were previously impossible.

Why Does This Matter?

This paper is a major step forward for two reasons:

Better Tools: It proves that using "non-linear" solvers (like QICS) is superior to the old "linear relaxation" tricks. It allows physicists to tackle bigger, more complex systems without the math breaking down.
Black Hole Insights: These matrix models are simplified versions of theories that describe Black Holes and the fabric of space-time (M-theory). By getting more precise answers about how these models behave at different temperatures, we get closer to understanding how Black Holes work, how they store information, and what happens when they evaporate.

In a nutshell: Sophia Adams took a messy, hard-to-solve math problem about hot quantum particles, swapped out the old "approximate" tools for a brand-new "exact" tool, and managed to calculate the properties of a theoretical "long string" with incredible precision. It's like finally hearing the exact note a guitar string is playing, instead of just guessing the general pitch.

1. Problem Statement

The paper addresses the challenge of applying the thermal bootstrap method to Matrix Quantum Mechanics (MQM) in the large- $N$ limit. While the bootstrap method is powerful for deriving rigorous bounds on observables using symmetry and self-consistency, thermal states present a specific difficulty:

The KMS Constraint: Thermal equilibrium is characterized by the Kubo–Martin–Schwinger (KMS) condition, which is a nonlinear constraint on the thermal density matrix.
Limitations of Linear Methods: Previous studies (e.g., Ref. [5]) relied on Linear Semidefinite Programming (SDP). To handle the nonlinear KMS condition, these methods used logarithmic relaxations (approximating $\log(x)$ with linear functions). While rigorous, these approximations can fail to enforce the KMS condition exactly, leading to numerical instability and loose bounds, particularly as the system size (operator length $L$ ) increases.
Goal: The author aims to improve thermal bootstrapping by solving the KMS constraint exactly using nonlinear conic optimization, thereby achieving tighter bounds and accessing larger system sizes ( $L$ ) where linear methods fail.

2. Methodology

The core innovation is the application of the Quantum Information Conic Solver (QICS) to handle the operator relative entropy cone directly, rather than relaxing it into a linear SDP.

A. Models Studied

The paper analyzes two specific models in the 't Hooft large- $N$ limit:

One-Matrix Anharmonic Oscillator:
- Hamiltonian: $H = \text{Tr}(\frac{1}{2}P^2 + \frac{1}{2}X^2 + \frac{g}{N}X^4)$ .
- Context: The ungauged theory admits an effective description of "long strings" in the adjoint sector at low temperatures.
Two-Matrix Anharmonic Oscillator:
- Hamiltonian: $H = \text{Tr}(\frac{1}{2}(P_1^2 + P_2^2) + \frac{1}{2}(X_1^2 + X_2^2) - \frac{g}{N}[X_1, X_2]^2)$ .
- Context: A bosonic truncation of the BFSS matrix model (dual to M-theory/black D0-branes).

B. Constraints and Optimization

The bootstrap problem is formulated as a conic optimization problem with the following constraints:

Semidefinite Positivity: The moment matrix $M_{ij} = \langle \text{Tr}(O_i^\dagger O_j) \rangle_\beta \succeq 0$ .
Symmetries: Parity, Time-reversal, and Reality conditions.
Large- $N$ Factorization: Treating single-trace operators as independent variables.
KMS Condition (Thermal Constraint):
- Instead of linear relaxation, the author uses the operator relative entropy cone formulation:
  $\beta C \succeq A^{1/2} \log(A^{1/2} B^{-1} A^{1/2}) A^{1/2}$
  where $A, B$ are constructed from correlators and $C$ involves the commutator with the Hamiltonian.
- Solver: QICS is used, which implements the Skajaa–Ye algorithm to handle nonsymmetric cones (specifically the quantum relative entropy cone) directly without requiring conjugate barriers or linear approximations.

C. Symmetry Reduction (Two-Matrix)

For the two-matrix model, the author reformulates the problem using complex operators ( $X, \bar{X}, P, \bar{P}$ ) to manifest U(1) charge conservation. This significantly reduces the number of variables (e.g., from 220 to 81 for $L=6$ ), making the problem computationally tractable.

3. Key Contributions

Exact Thermal Constraints: The paper demonstrates that using QICS allows for the direct enforcement of the nonlinear KMS condition, eliminating the need for logarithmic relaxations used in previous linear SDP approaches.
Extended System Sizes:
- One-Matrix: Successfully computed rigorous bounds up to operator length $L=12$ . In contrast, linear solvers (MOSEK) with logarithmic relaxation became numerically unstable at $L=10$ and required relaxed tolerances to converge.
- Two-Matrix: Achieved bounds for $L=6$ by combining QICS with U(1) symmetry constraints, a regime where previous methods struggled with noise and instability.
Precision in Effective Theory Parameters: The method yielded high-precision estimates for the low-energy effective "long string" parameters (ground state energy, excited state energy, and coupling coefficients) directly from symmetry and self-consistency equations, without prior analytic input.

4. Results

One-Matrix Model ( $g=2$ )

Numerical Stability: QICS provided stable, rigorous bounds for $L=12$ (computation time $\sim 10-100$ minutes per optimization), whereas MOSEK failed to converge at $L=10$ without relaxing tolerances.
Long String Extraction: By fitting the bootstrap bounds to the low-temperature expansion of the effective long string theory, the author extracted:
- Ground State Energy ( $e_0$ ): $0.86545775 \pm 3 \times 10^{-7}$ (Matches analytic value to 7 decimal places).
- First Excited Energy ( $\Delta_1$ ): $2.1283360 \pm 2 \times 10^{-4}$ (Within 0.001% of the physical value).
- Coupling Coefficient ( $h_{1111}$ ): Estimated as $0.32731 \pm 0.07$ . This represents the first estimation of this coupling coefficient derived solely from bootstrap symmetry and self-consistency equations.
Fit Quality: The bootstrap bounds agreed excellently with the first-order long string approximation. Higher-order terms were necessary to match the bounds at higher temperatures.

Two-Matrix Model ( $g=0.1$ )

Symmetry Impact: Utilizing complex operators and U(1) charge conservation reduced the variable count by roughly 60%, enabling convergence at $L=6$ .
Challenges: Even with QICS, the two-matrix model remained numerically challenging at $L=6$ , with QICS failing to find bounds past temperature $T=0.56$ . This highlights the need for arbitrary-precision solvers for larger matrix models.

5. Significance and Discussion

Beyond Linear Relaxations: The work proves that nonlinear conic optimization is superior to linear relaxations for thermal bootstrapping. It provides a framework to enforce the KMS condition exactly, leading to tighter bounds and access to larger truncation levels ( $L$ ).
Bridge to Quantum Gravity: By successfully bootstrapping the two-matrix model (a proxy for the BFSS model), the paper takes a step toward extracting observables in matrix models dual to black hole dynamics and M-theory.
Future Directions: The primary limitation identified is numerical instability in double precision for large operator bases. The author suggests that future work should employ arbitrary-precision solvers to handle the rapid growth of the operator basis in multi-matrix models.
Code Availability: The code is open-source, facilitating further development in the field of thermal matrix model bootstrapping.

In summary, this paper represents a significant methodological leap in thermal quantum field theory bootstrapping, moving from approximate linear relaxations to exact nonlinear conic optimization, thereby enabling more precise extraction of physical parameters in large- $N$ matrix models.