Thermal Bootstrap of Matrix Quantum Mechanics

TL;DR

This work develops a thermal bootstrap framework for ungauged matrix quantum mechanics by combining stationary-state constraints, KMS-based convex inequalities, and semidefinite relaxations of the matrix logarithm to bound finite-temperature observables. The method is validated on the anharmonic oscillator and extended to the ungauged one-matrix QM, with finite-$N$ and planar-limit implementations that yield convergent bounds on the energy $E(\beta)$ across temperatures. Analytic high- and low-temperature expansions are used to benchmark the results, and the bootstrap successfully interpolates between them while enabling extraction of the adjoint-sector energy gap $\Delta_1$. Additionally, the approach provides insights into thermal metastability and a preliminary foray into the two-matrix case, highlighting both the potential and the challenges for applying thermal bootstrap to holographic matrix models such as BFSS at finite temperature.

Abstract

We implement a bootstrap method that combines stationary state conditions, thermal inequalities, and semidefinite relaxations of matrix logarithm in the ungauged one-matrix quantum mechanics, at finite rank N as well as in the large N limit, and determine finite temperature observables that interpolate between available analytic results in the low and high temperature limits respectively. We also obtain bootstrap bounds on thermal phase transition as well as preliminary results in the ungauged two-matrix quantum mechanics.

Figures (8)

• Figure 1: Bootstrap bounds on the thermal energy expectation value $E({\beta})$ of the quantum anharmonic oscillator (\ref{['hamanh']}) as a function of the temperature $T=1/{\beta}$, computed using different $(m,k)$-relaxations of the matrix logarithm, with maximal word length $L=10$. Note that the bounds with $(m,k)=(3,3)$ and $(4,4)$ are virtually indistinguishable. The result from numerically solving the Schrödinger equation is shown in solid red curve. From the $(m,k)=(4,4)$ lower bootstrap bound, one can extract the energy gap $\Delta = 2.748$, which is within $\sim$ 0.3% of the physical value $\Delta = 2.739$.
• Figure 2: Bootstrap bounds on $E({\beta})/(N^2-1)$ in the ungauged one-matrix QM at finite $N$ as well as at $N=\infty$, computed at coupling $g=2$ with maximal word length $L=8$, are shown in colors ranging from red to violet. For $N=2$, the Hamiltonian truncation results (shown in black dots) lie within the bootstrap bounds and differ from the lower bound by $\sim 10^{-4}$ at low temperatures to $\sim 10^{-2}$ at $T=0.5$.
• Figure 3: The bootstrap bounds on ${E({\beta})/N^2}$ at various values of coupling $g$, computed using MOSEK with maximal word length $L=6$ and $L=10$, and using SDPA-DD with $L=8$ for numerical stability. The upper and lower bounds at $L=10$, shown in black, are virtually indistinguishable in the plot; their difference is further exhibited in Figure \ref{['fig:convergence2']}.
• Figure 4: $\delta E/E$ as a function of the temperature, where $\delta E$ is the difference between the upper and lower bootstrap bounds on the energy expectation value $E$, is shown in logarithmic scale at various values of coupling $g$ and maximal word length $L$.
• Figure 5: Left: bootstrap results for ${\beta}(E({\beta})-E_0)/N^2$ in the ungauged one-matrix QM computed with maximal word length $L=10$, shown in solid black curve (where the distinction between the upper and lower bounds is invisible), compared to the leading terms in the low temperature expansion (\ref{['lowTExp']}) shown in dashed orange, and the leading terms in the high temperature expansion (\ref{['gN0highT']}) shown in dashed blue. Right: the upper and lower bootstrap bounds on $E({\beta})/N^2$ after subtracting off the low temperature approximation $E_{L.T.}({\beta})/N^2 = e_0 + \Delta_1 e^{-{\beta}\Delta_1}$.