Thermal Bootstrap of Large-N Matrix Models via Conic Optimization

TL;DR

The paper addresses thermal bootstrap for large-$N$ matrix quantum mechanics by enforcing nonlinear KMS constraints via the Quantum Information Conic Solver, surpassing prior linear relaxations. It demonstrates stricter, rigorous bounds for the one-matrix anharmonic oscillator (including $L=12$) and provides meaningful, symmetry-facilitated bounds for a two-matrix truncation, linking results to the dual long-string description. By extracting $e_0$, $igl( ext{Delta}_1igr)$, and $h_{1111}$ from low-temperature data and comparing to the long-string effective theory, the work validates the nonlinear conic approach and its potential to illuminate nonperturbative sectors of matrix models. The study highlights both the practical gains and numerical challenges, arguing for higher-precision solvers to unlock further higher-excitation data and tighter couplings in more complex truncations. These advances broaden the applicability of thermal bootstrap to nonperturbative regimes relevant to quantum gravity and black hole physics.

Abstract

This paper is aimed at improving thermal bootstrapping methods for matrix quantum mechanics and quantum field theory. The thermal energies of the large-N one-matrix anharmonic oscillator and large-N two-matrix anharmonic oscillator were bounded without logarithmic relaxation using the Quantum Information Conic Solver. For the one-matrix anharmonic oscillator, dual to a theory of "long strings", stricter bounds yielded a value of the first long string excited energy within 0.001% of the physical value and the first estimation from symmetry and self-consistency equations alone of the first long string coupling coefficient.

Figures (6)

• Figure 1: Thermal energy bounds for $g=2$ found using MOSEK with $(3,3)$ logarithmic relaxation and QICS without relaxation for system sizes $L=8,10$.
• Figure 2: Bounds for system size $L=12$ found using QICS are stricter than what was previously achieved with thermal relaxation in Ref. cho2025thermalbootstrapmatrixquantum.
• Figure 3: Plot of the first four long string eigenvectors numerically computed from Eq. \ref{['finaleigenvalueequation']}.
• Figure 4: The difference between QICS bootstrap bounds from system size $L=12$ and the first order approximation with values in Ref. Marchesini:1978ud.
• Figure 5: QICS bounds compared to analytic approximations of the energy with values from Ref. Marchesini:1978ud and long string coupling coefficients from Table \ref{['longStringCouplingCoeffs']}.