Bootstrapping supersymmetric (matrix) quantum mechanics

TL;DR

This work develops and applies a quantum-mechanics bootstrap framework to SUSY QM and the Marinari–Parisi matrix model, deriving rigorous ground-state bounds from positivity of moment matrices combined with Heisenberg, gauge, and zero-temperature constraints. For N=1 SUSY QM with a cubic potential, the bounds are tight and align with perturbative and semiclassical instanton analyses across coupling regimes, while in the quadratic case the method confirms a unique zero-energy ground state and SUSY pairing of excited states. In the SUSY matrix case, a 44×44 bootstrap at large N reproduces the expected E ∝ g^{2/3} scaling with a rigorous lower bound κ > 0.196, and reveals a spurious small-g kink attributed to truncation. The results demonstrate the bootstrap’s ability to extract nonperturbative information about SUSY systems and point toward improvements (larger matrices, additional constraints) to sharpen bounds and access further observables.

Abstract

We apply the quantum-mechanics bootstrap to supersymmetric quantum mechanics (SUSY QM) and to its matrix relative, the Marinari-Parisi model, which is conjectured to describe the worldvolume of unstable $D0$ branes. Using positivity of moment matrices together with Heisenberg, gauge, and (zero-temperature) thermal constraints, we obtain rigorous bounds on ground-state data. In the cases where SUSY is spontaneously broken, we find bounds that apply to the lowest-energy normalizable eigenstate. For $N = 1$ SUSY QM with a cubic superpotential, we obtain tight bounds that agree well with available approximation methods. At weak coupling they match well with the semiclassical instanton contribution to SUSY-breaking ground-state energy, while at strong coupling they exhibit the expected scaling and match well with Hamiltonian truncation. For the SUSY matrix QM, we construct a $44 \times 44$ bootstrap matrix and obtain bounds at large $N$. At strong coupling, we obtain the expected $E \sim κ\ g^{2/3}$ scaling of $E$ with $g$ and extract a lower bound on the coefficient $κ> .196$. At small coupling, the theory has a critical point $g_c$ where the two wells merge into one. We find a spurious kink at $g = \sqrt{2} g_c$. We attribute this to truncation error and solver limitations, and discuss possible improvements.

Figures (13)

• Figure 1: Quadratic superpotential with $\epsilon = -1$, $\omega = 1$. The matrix sizes are $K \times K$ with $K = 4$ (blue), $K = 8$ (orange), $K = 12$ (green) and $K = 16$ (red). The exact results are represented by black crosses. The results for $\epsilon = 1$ are the same as all of the energies shifted up by 1. Thus all states except the ground state are paired. These constraints use $\langle H \mathcal{O} \rangle = E \langle \mathcal{O} \rangle$
• Figure 2: (Left) Constraint region for E vs $\langle x \rangle$ for a bootstrap matrix of size K x K where K is 5 (blue) and 6 (dark blue). The $\langle x \rangle < 0$ portion of the figure shows the results when $\epsilon = 1$ and the $\langle x \rangle > 0$ portion of the figure shows the results when $\epsilon = -1$. (Right) Constraint region for E vs $\langle x^2 \rangle$ for a bootstrap matrix of size K x K where K is 4 (blue) and 5 (dark blue).
• Figure 3: Bounds on the minimum of the energy vs g (colored dots) compared to expectation from perturbation theory (solid blue line). The bounds were found using bootstrap matrices of the form shown in Equation \ref{['eq:bloc_M']}, where let the size $K$ of the $K$ x $K$ blocks $A$, $B$, $C$, and $D$ take the values 2 (purple dots), 3 (blue dots), 4 (green dots), 5 (yellow dots), and 6 (red dots).
• Figure 4: Ground state energy bounds for SUSY QM with $W = 1/2 x^2 + g/3 x^3$. The green dots are the upper bound and the purple are the lower bound. The blue line is the WKB approximation and the orange line is the approximation from the Rayleigh-Ritz method with a $60 \times 60$ matrix, using the leading $g^{2/3}$ part of the Hamiltonian \ref{['eq:Hleading']}.
• Figure 5: Ground state energy (left) and $x^2$ (right) lower bounds for SUSY QM with $W = 1/2 x^2 + g/3 x^3$ for level 7 (red), level 8 (purple), level 9 (blue) and level 10 (green). For smaller $g$ Mathematica's solver breaks down (see later figures).