Supersymmetric Quantum Mechanics (SUSYQM)

• Supersymmetric Quantum Mechanics (SUSYQM) is an operator-theoretic framework that unites bosonic and fermionic sectors via graded symmetry and nilpotent supercharges.
• It employs Hamiltonian factorization and shape invariance to generate isospectral partner potentials, facilitating spectral design and offering a robust approach for nonperturbative quantum analysis.
• Advanced methods, including bootstrap and matrix formulations, extend SUSYQM to derive rigorous energy bounds and analyze complex systems such as the Marinari–Parisi model.

Supersymmetric Quantum Mechanics (SUSYQM) is an operator-theoretic framework developed to study quantum systems exhibiting a graded symmetry structure between bosonic and fermionic degrees of freedom. It formalizes and exploits the pairing of quantum Hamiltonians through nilpotent supercharges acting on a Hilbert space equipped with a $\mathbb{Z}_2$ grading. SUSYQM is both a toy model for field-theoretic supersymmetry and a powerful algebraic method for the factorization, spectral design, and nonperturbative analysis of quantum Hamiltonians.

1. Algebraic Structure and Fundamental Definitions

SUSYQM is typically defined for a quantum system with a real coordinate $x$, conjugate momentum $p$, and a set of fermionic operators $(\psi, \psi^\dagger)$ satisfying $\{\psi, \psi^\dagger\}=1$ and $\psi^2 = (\psi^\dagger)^2 = 0$. The construction introduces a real superpotential $W(x)$ and associated supercharges

$$Q = (p + i W'(x))\,\psi, \qquad Q^\dagger = (p - i W'(x))\,\psi^\dagger,$$

which realize the supersymmetry algebra

$$\{Q, Q^\dagger\} = 2H, \qquad Q^2 = (Q^\dagger)^2 = 0$$

with block-diagonal Hamiltonian

$$H = \frac{1}{2}p^2 + \frac{1}{2}[W'(x)]^2 + \frac{1}{2}[\psi^\dagger, \psi]\,W''(x).$$

The grading partitions the Hilbert space into bosonic and fermionic sectors; in each, the effective Hamiltonian is

$$H_\epsilon = \frac{1}{2}p^2 + \frac{1}{2}[W'(x)]^2 + \frac{1}{2}\epsilon\,W''(x), \qquad \epsilon = \pm 1.$$

Eigenstates of $H$ are paired unless a zero-energy ground state exists, in which case supersymmetry is unbroken (Laliberte et al., 1 Oct 2025, Ayad, 2019).

2. Factorization, Partner Hamiltonians, and Shape Invariance

A central principle in SUSYQM is Hamiltonian factorization through first-order differential (Darboux) operators: $$A = \frac{d}{dx} + W(x), \qquad A^\dagger = -\frac{d}{dx} + W(x).$$ This yields

$$A^\dagger A = H_- = -\frac{1}{2}\frac{d^2}{dx^2} + V_-(x), \qquad AA^\dagger = H_+ = -\frac{1}{2}\frac{d^2}{dx^2} + V_+(x),$$

with partner potentials

$V_\pm(x) = \frac{1}{2}W^2(x) \pm \frac{1}{2}W'(x).$

These partner Hamiltonians are isospectral except for the possible zero-mode of $A$ (Socorro et al., 2019, Marques, 2011).

Shape invariance is the existence of a parameter mapping $a_0\mapsto a_1$ and offset $R(a_0)$ such that

$V_+(x;a_0) = V_-(x;a_1) + R(a_0).$

This property allows the entire spectrum and set of eigenfunctions of $H_\pm$ to be generated algebraically by successive application of $A^\dagger$. Canonical shape-invariant systems include the harmonic oscillator ($W(x)=x$) and Coulomb potential (Sekhon, 2022).

3. Supersymmetry Breaking, Instantons, and Nonperturbative Effects

Supersymmetry can be unbroken (existence of a normalizable zero-energy ground state satisfying $Q|0\rangle=Q^\dagger|0\rangle=0$) or broken (no such state). Classical perturbation theory does not induce SUSY breaking: any deformation $W\rightarrow W+\delta W$ preserves $E_0=0$ to all finite orders in perturbation theory (Ayad, 2019). Instead, nonperturbative effects such as instanton-induced tunneling are responsible.

For systems with degenerate minima in $W(x)$ (e.g., cubic superpotential $W(x)=\frac{1}{2}x^2 + \frac{1}{3}g x^3$), quantum-mechanical instantons connect wells and generate a splitting $\Delta E$ in the ground-state energy: $$E_{\text{inst}}\sim \frac{1}{2\pi}e^{-2S_{\text{inst}}},\qquad S_{\text{inst}} = |W(x_1)-W(x_2)|.$$ The bootstrap quantum mechanics approach rigorously bounds $E_{\text{min}}(g)$, reproducing the instanton result at weak coupling and yielding $E_0\sim \kappa_0 g^{2/3}$ at strong coupling (Laliberte et al., 1 Oct 2025).

4. Matrix SUSYQM and the Marinari–Parisi Model

SUSYQM extends to matrix degrees of freedom: for an $N\times N$ Hermitian matrix $X$, conjugate momentum $P$, and matrix-valued fermions $\Psi, \Psi^\dagger$, the Hamiltonian with superpotential $W(X)=\frac{1}{2}X^2+\frac{1}{3}gX^3$ reads

$$H = \frac{1}{2}\operatorname{Tr}[P^2 + W'(X)^2] + \frac{1}{2}\operatorname{Tr}\{[\Psi^\dagger, \Psi] W''(X)\}.$$

Physical states are constrained by the SU$(N)$ Gauss law $G = i[X,P] - \{\Psi, \Psi^\dagger\} + 2N I$, so $\langle G O\rangle = 0$ (Laliberte et al., 1 Oct 2025).

The bootstrap method for matrix SUSYQM uses positivity of large moment matrices, Heisenberg and gauge constraints, and thermal ground-state conditions to provide rigorous bounds on the ground-state energy. In the Marinari–Parisi model, the energy scales as $E_0/N^2 \geq \kappa g^{2/3}$ with explicit lower bounds on $\kappa$ at large $N$. Near the critical coupling $g_c$, numerical artifacts related to truncation become significant.

5. Quantum-Mechanics Bootstrap Methods and Rigorous Bounds

The quantum-mechanics bootstrap framework combines positivity constraints on moment matrices

$$M_{ij} = \langle E | O_i^\dagger O_j | E \rangle \succeq 0,$$

Heisenberg constraints $\langle[H, O]\rangle=0$, gauge constraints for symmetry generators $\langle G O\rangle=0$, and thermal positivity at zero temperature. These conditions together define a semidefinite program (SDP) that can be systematically improved by enlarging the operator basis (Laliberte et al., 1 Oct 2025).

This approach yields rigorous bounds on ground-state energies, applicable whether SUSY is broken or unbroken, and converges monotonically to the correct values as the SDP level increases. For systems with spontaneous SUSY breaking, such as 1D polynomial superpotentials of odd degree, the ground-state energy is strictly bounded away from zero.

6. Off-Shell Formalism, N = 2 SUSYQM, and Cohomology

The N=2 model is formulated on a $(0+1|2)$-dimensional supermanifold, introducing Grassmann coordinates $\theta,\bar\theta$ and superfield expansions. Two nilpotent off-shell SUSY transformations $s_1, s_2$ emerge, acting as translations along the Grassmann directions. The component Lagrangian is constructed from the superpotential and auxiliary field $A$ (Krishna et al., 2013).

Nilpotency $s_1^2=s_2^2=0$ is geometrically interpreted as vanishing under repeated shifts, and the entire algebraic structure mirrors the de Rham cohomology, providing deep links between SUSYQM and topological field theories.

7. Applications, Extensions, and Outlook

SUSYQM provides algorithms for phase-equivalent inverse scattering (e.g., neutron–proton S-wave inversion), explicit algebraic classification of quasi-exactly solvable potentials (via Bethe ansatz and hidden $sl(2)$ symmetry), and coherent state constructions in polynomial Heisenberg algebras (Bozet et al., 26 Aug 2025, Li et al., 26 Nov 2025, García-Muñoz et al., 2023). It has been generalized to multidimensional and matrix systems, as well as to noncommutative geometries (Jim et al., 2024).

The quantum-mechanics bootstrap formalism in SUSYQM delivers not only systematically improvable bounds but also rigorous results for strong- and weak-coupling regimes, validates semiclassical instanton effects, and enables new computational strategies via SDP techniques. Open research directions include higher-level operator extensions, non-convex factorization constraints at large $N$, and fusion with complementary variational or Monte Carlo methods for more complex models (Laliberte et al., 1 Oct 2025).

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References

• Bootstrapping supersymmetric (matrix) quantum mechanics (Laliberte et al., 1 Oct 2025)
• General N = 2 Supersymmetric Quantum Mechanical Model: Supervariable Approach to its Off-Shell Nilpotent Symmetries (Krishna et al., 2013)
• Supersymmetric Quantum Mechanics and Path Integrals (Ayad, 2019)
• Supersymmetric Quantum Mechanics: two factorization schemes, and quasi-exactly solvable potentials (Socorro et al., 2019)
• Supersymmetric Quantum Mechanics For Atomic Electronic Systems (Markovich et al., 2011)