Some recent results in supersymmetric quantum mechanics are presented. New semi-classical approximation formulas for Witten's realization of supersymmetric quantum mechanics are discussed. Implications of the supersymmetric structure of Pauli's Hamiltonian are also considered. In particular, the paramagnetisation of a non-interacting electron gas is related to a modified version of Witten's index. We also show that the supersymmetry in this system provides a new counter example for the paramagnetic conjecture.
This thesis constructs quantum mechanical theories of finite simple graphs for both labeled and unlabeled graphs.These theories may provide a description for discrete spacetime in a quantum gravity theory. Finite simple graphs and their properties are introduced. The first and second quantization of the standard quantum mechanics of particle systems are reviewed. A quantum mechanical theory of graphs similar to first quantization of standard quantum mechanics is developed via a correspondence of graph edges to particles in particle systems. An algorithm for constructing quantum states of graphs which is independent of graph vertex labeling is developed. It builds antisymmetric graphstates which describe fermiongraphs, the analog of systems of fermions for graphs. A quantum mechanical theory of graphs similar to second quantization of standard quantum mechanics is developed via the same correspondence of graph edges and particles. Edge creation and annihilation operators are defined which insert or delete edges between any pair of grap hvertices. The commutation and anticommutation relations of these operators are defined to produce states which are symmetricand antisymmetric upon any vertex label permutation. Operators for certain graph properties and constructions are defined in this theory. Dynamics are then incorporated into the second quantization approach of the quantum mechanical graph theory by developing two model Hamiltonians, in analogy with the Hamiltonians of the Ising and Heisenberg model sof ferromagnetism. From these Hamiltonians, statistical mechanics can be used to describe the dynamics ofgraphs.
Supersymmetry, originally proposed in high-energy physics, can emerge as a remarkable low-energy structure in condensed matter systems. While emergent supersymmetry at quantum critical points is widely discussed in models with continuous symmetries, real materials are constrained by microscopic discrete symmetries. To address this, we investigate (2+1)-dimensional Gross-Neveu-Yukawa theories coupling Dirac fermions to a complex order parameter with discrete
Zn anisotropy. Using the functional renormalization group, we find that for
n>3, the anisotropic perturbations are irrelevant at the fixed point, yielding a
N=2 Wess-Zumino supersymmetric critical point. In the ordered phase, this dangerously irrelevant anisotropy gives rise to a second characteristic length scale,
ξ′, alongside the usual correlation length,
ξ. By tracking mass thresholds along symmetry-broken renormalization group trajectories, we extract the exponents
ν′ and
ν without imposing prior scaling assumptions. For the
Z4,
Z5, and
Z6 models, our results support the scaling relation
ν′/ν=1+∣yn∣/p with
p=2 in the isotropic framework used here.
We discuss a supersymmetric generalization of the Sachdev-Ye-Kitaev (SYK) model. These are quantum mechanical models involving
N Majorana fermions. The supercharge is given by a polynomial expression in terms of the Majorana fermions with random coefficients. The Hamiltonian is the square of the supercharge. The
N=1 model with a single supercharge has unbroken supersymmetry at large
N, but nonperturbatively spontaneously broken supersymmetry in the exact theory. We analyze the model by looking at the large
N equation, and also by performing numerical computations for small values of
N. We also compute the large
N spectrum of ``singlet'' operators, where we find a structure qualitatively similar to the ordinary SYK model. We also discuss an
N=2 version. In this case, the model preserves supersymmetry in the exact theory and we can compute a suitably weighted Witten index to count the number of ground states, which agrees with the large
N computation of the entropy. In both cases, we discuss the supersymmetric generalizations of the Schwarzian action which give the dominant effects at low energies.
The interplay between supersymmetry and classical and quantum computation is discussed. First, it is shown that the problem of computing the Witten index of
N≤2 quantum mechanical systems is
#P-complete and therefore intractable. Then, the notions of supersymmetry in the space of qubits and supersymmetric quantum circuits are introduced and some of their properties discussed. In particular, it is shown that these define a nontrivial subclass of quantum algorithms with robustness properties typical of supersymmetric systems. Concrete examples, including the supersymmetric SYK model and fermion hard-core models are discussed. Some applications and open questions are suggested.
Supersymmetry (SUSY) helps solve the hierarchy problem in high-energy physics and provides a natural groundwork for unifying gravity with other fundamental interactions. While being one of the most promising frameworks for theories beyond the Standard Model, its direct experimental evidence in nature still remains to be discovered. Here we report experimental realization of a supersymmetric quantum mechanics (SUSY QM) model, a reduction of the SUSY quantum field theory for studying its fundamental properties, using a trapped ion quantum simulator. We demonstrate the energy degeneracy caused by SUSY in this model and the spontaneous SUSY breaking. By a partial quantum state tomography of the spin-phonon coupled system, we explicitly measure the supercharge of the degenerate ground states, which are superpositions of the bosonic and the fermionic states. Our work demonstrates the trapped-ion quantum simulator as an economic yet powerful platform to study versatile physics in a single well-controlled system.