Bootstrapping Matrix Quantum Mechanics

Large $N$ matrix quantum mechanics is central to holographic duality but not solvable in the most interesting cases. We show that the spectrum and simple expectation values in these theories can be obtained numerically via a “bootstrap” methodology. In this approach, operator expectation values are related by symmetries—such as time translation and $SU(N)$ gauge invariance—and then bounded with certain positivity constraints. We first demonstrate how this method efficiently solves the conventional quantum anharmonic oscillator. We then reproduce the known solution of large $N$ single matrix quantum mechanics. Finally, we present new results on the ground state of large $N$ two matrix quantum mechanics.

Figure 1

Bootstrap allowed region (shaded) for the anharmonic oscillator (1) with $g=1$. Upper plot: the allowed region for $(E,⟨x^2⟩)$ near the ground state solution (marked by the red cross) for different sizes of the bootstrap matrix $K=7$, 8, 9; lower plot: the allowed region near the first excited state.

Figure 2

One matrix quantum mechanics bootstrap for the Hamiltonian (8). $L$ is the maximal length of trial operators. Upper: The markers show the minimal energies allowed by the bootstrap constraints, in comparison with the exact ground state solution. Lower: the expectation values of $\mathrm{tr}{X}^2$, for the minimal energy parameters found in the upper plot.

Figure 3

Minimal energy configuration in the bootstrap allowed region for $L=3$, 4. The gray dashed curves are rigorous lower and upper bounds of the ground state energy from the Born-Oppenheimer approximation. In the plots we have set $m=1$.