Supersymmetric Quantum Mechanical Systems Research Articles

Late time dynamics in SUSY saddle-dominated scrambling through higher-point OTOC

In this article, we propose higher-point out-of-time-order correlators (OTOCs) as a tool to differentiate chaotic from saddle-dominated dynamics in late times. As a model, we study the scrambling dynamics in supersymmetric quantum mechanical systems. Using the eigenstate representation, we define the 2N-point OTOC using two formalisms, namely the ’Tensor Product formalism’ and the ’Partner Hamiltonian formalism’. We analytically find that the 2N-point OTOC for the supersymmetric 1D harmonic oscillator is in exact agreement with that of the 1D bosonic harmonic oscillator system. We show that the higher-point OTOC is a more sensitive measure of scrambling than the usual 4-point OTOC. To demonstrate this, we analyze a supersymmetric sextic 1D oscillator, for which the bosonic partner system has an unstable saddle in the phase space, while the saddle is absent in the fermionic counterpart. For such a system, we show that the saddle-dominated scrambling, higher anharmonic potential effects, and the supersymmetric OTOC exhibit similar dynamics due to supersymmetry constraints. Finally, we illustrate that the late-time dynamics of the higher-point OTOC become oscillatory after the peak for saddle-dominated scrambling and anharmonic oscillator systems. We propose the higher-point OTOC as a probe of late-time dynamics in non-chaotic systems that exhibit fast early-time scrambling.

Supersymmetric quantum mechanical system for locating the Riemann zeros

Supersymmetric quantum mechanical system for locating the Riemann zeros

Polarisability and magnetisation of electrically K-mouflaged objects: the Born-Infeld ModMax case study

We consider a family of non-linear theories of electromagnetism that interpolate between Born-Infeld at small distances and the recently introduced ModMax at large distances. These models are duality invariant and feature a K-mouflage screening in the Born-Infeld regime. We focus on computing the static perturbations around a point-like screened charge in terms of two decoupled scalar potentials describing the polar and the axial sectors respectively. Duality invariance imposes that the propagation speed of the odd perturbations goes to zero as fast as the effective screened charge of the object, potentially leading to strong coupling and an obstruction to the viability of the EFT below the screened radius. We then consider the linear response to external fields and compute the electric polarisability and the magnetic susceptibility. Imposing regularity of the perturbations at the position of the particle, we find that the polarisability for the odd multipoles vanishes whilst for the magnetisation Born-Infeld emerges as the only theory with vanishing susceptibility for even multipoles. The perturbation equations factorise in terms of ladder operators connecting different multipoles. There are two such ladder structures for the even sector: one that acts as an automorphism between the first four multipoles and another one that connects multipoles separated by four units. When requiring a similar ladder structure for the odd sector, Born-Infeld arises again as the unique theory. We use this ladder structure to relate the vanishing of the polarisability and the susceptibility to the values of conserved charges. Finally the perturbation equations correspond to a supersymmetric quantum mechanical system such that the polar sector can be described in terms of Schrödinger’s equations with four generalised hyperbolic Pösch-Teller potentials whose eigenfunctions are in correspondence with the multipoles.

Witten index for weak supersymmetric systems: Invariance under deformations

When a 4D supersymmetric theory is placed on [Formula: see text], the supersymmetric algebra is necessarily modified to [Formula: see text] and we are dealing with a weak supersymmetric system. For such systems, the excited states of the Hamiltonian are not all paired. As a result, the Witten index [Formula: see text] is no longer an integer number, but a [Formula: see text]-dependent function. However, this function stays invariant under deformations of the theory that keep the supersymmetry algebra intact. Based on the Hilbert space analysis, we give a simple general proof of this fact. We then show how this invariance works for two simplest weak supersymmetric quantum mechanical systems involving a real or a complex bosonic degree of freedom.

Weak supersymmetric su(N|1) quantum systems

In this paper, we present several examples of supersymmetric quantum mechanical systems with weak superalgebra [Formula: see text]. One of them is the weak [Formula: see text] oscillator. It has a singlet ground state, [Formula: see text] degenerate states at the first excited level, etc. Starting from the level [Formula: see text], the system has complete supersymmetric multiplets at each level involving [Formula: see text] degenerate states. Due to the fact that the supermultiplets are not complete for [Formula: see text], the Witten index represents a nontrivial function of [Formula: see text]. This system can be deformed with keeping the algebra intact. The index is invariant under such deformation. The deformed system is not exactly solved, but the invariance of the index implies that the energies of the states at the first [Formula: see text] levels of the spectrum are not shifted, and we are dealing with a quasi-exactly solvable system. Another system represents a weak generalization of the superconformal mechanics with [Formula: see text] complex supercharges. Also in this case, starting from a certain energy, the spectrum involves only complete supersymmetric [Formula: see text]-plets. (There also exist normalizable states with lower energies, but they do not have normalizable superpartners. To keep supersymmetry, we have to eliminate these states.)

Boundary Supersymmetry of (1+1)D Fermionic Symmetry-Protected Topological Phases.

We prove that the boundaries of all nontrivial (1+1)-dimensional intrinsically fermionic symmetry-protected-topological phases, protected by finite on-site symmetries (unitary or antiunitary), are supersymmetric quantum mechanical systems. This supersymmetry does not require any fine-tuning of the underlying Hamiltonian, arises entirely as a consequence of the boundary 't Hooft anomaly that classifies the phase, and is related to a "Bose-Fermi" degeneracy different in nature from other well known degeneracies such as Kramers doublets.

Modular operators and entanglement in supersymmetric quantum mechanics

The modular operator approach of Tomita–Takesaki to von Neumann algebras is elucidated in the algebraic structure of certain supersymmetric (SUSY) quantum mechanical systems. A von Neumann algebra is constructed from the operators of the system. An explicit operator characterizing the dual infinite degeneracy structure of a SUSY two dimensional system is given by the modular conjugation operator. Furthermore, the entanglement of formation for these SUSY systems using concurrence is shown to be related to the expectation value of the modular conjugation operator in an entangled bi-partite supermultiplet state thus providing a direct physical meaning to this anti-unitary, anti-linear operator as a quantitative measure of entanglement. Finally, the theory is applied to the case of two-dimensional Dirac fermions, as is found in graphene, and a SUSY Jaynes Cummings model.

Supercanonical transformations and time-dependent Hamiltonians

Starting from general self-adjoint linear combinations of generators of the superalgebra a time-dependent Hamiltonian of a supersymmetric quantum mechanical system is defined by computing the supercommutator of the linear forms. The resulting Hamiltonian is given by the sum of two quadratic forms in odd, respectively even, generators of the superalgebra describing a class of systems containing bosons and fermions. Linear supercanonical transformations of wave vectors leave invariant a Heisenberg superalgebra and belong to the supergroup OSp. The equations of motion for the supercanonical transformations in the Heisenberg picture are shown to be systems of ordinary differential equations. The unitary time evolution operator is constructed using the adjoint map. For periodic Hamiltonians, it is shown that this is a procedure to obtain effective Floquet Hamiltonians. The examples show that the known superalgebras-based approaches for the nuclear shell model and the Jaynes-Cummings model are incorporated in the lowest-dimensional cases. The presented approach opens the possibility to study quantum control problems defined by linear combinations of superalgebra generators with Grassmannian coefficients.

Autocorrelation functions for quantum particles in supersymmetric Pöschl-Teller potentials

AbstractWe consider autocorrelation functions for supersymmetric quantum mechanical systems (consisting of a fermion and a boson) confined in trigonometric Pöschl–Teller partner potentials. We study the limit of rescaled autocorrelation functions (at random time) as the localization of the initial state goes to infinity. The limiting distribution can be described using pairs of Jacobi theta functions on a suitably defined homogeneous space, as a corollary of the work of Cellarosi and Marklof. A construction by Contreras-Astorga and Fernández provides large classes of Pöschl-Teller partner potentials to which our analysis applies.

Chaos and random matrices in supersymmetric SYK

We use random matrix theory to explore late-time chaos in supersymmetric quantum mechanical systems. Motivated by the recent study of supersymmetric SYK models and their random matrix classification, we consider the Wishart-Laguerre unitary ensemble and compute the spectral form factors and frame potentials to quantify chaos and randomness. Compared to the Gaussian ensembles, we observe the absence of a dip regime in the form factor and a slower approach to Haar-random dynamics. We find agreement between our random matrix analysis and predictions from the supersymmetric SYK model, and discuss the implications for supersymmetric chaotic systems.

A free supersymmetric system: Novel symmetries

We discuss a set of novel discrete symmetries of a free supersymmetric (SUSY) quantum-mechanical system which is the limiting case of a widely studied interacting SUSY model of a charged particle constrained to move on a sphere in the background of a Dirac magnetic monopole. The usual continuous symmetries of this model provide the physical realization of the de Rham cohomological operators of differential geometry. The interplay between the novel discrete symmetries and usual continuous symmetries leads to the physical realization of the relationship between the (co-)exterior derivatives of differential geometry. We have also exploited the supervariable approach to derive the nilpotent SUSY symmetries of the theory and provided the geometrical origin and interpretation for the nilpotency property. Ultimately, our present study (based on innate symmetries) proves that our free SUSY example is a tractable model for the Hodge theory.

A relation between Z3-graded symmetry and shape invariant supersymmetric systems

We study a notable property of shape invariant supersymmetric quantum mechanical systems. Particularly, we demonstrate that each shape invariant supersymmetric system can constitute a Z3-graded symmetric algebra. The latter is known from the literature to provide topological invariants which are generalizations of the Witten index. In addition, we relate the Z3-graded algebra to the generators of a deformed Lie algebra underlying each shape invariant system. We generalize the results to the case of sequential shape invariant systems, in which case we find a sequence of Z3-graded algebras. Finally, we present an example of shape invariant supersymmetric quantum system for which we express the elements of the Z3 in terms of the operators that constitute the supersymmetric algebra of the quantum system. In view of the fact that the shape invariance condition is somewhat an additional algebraic condition to supersymmetric quantum systems, with no origin to some concrete algebraic structure, our results might be useful towards this line of research.

${\mathcal N}$-FOLD SUPERSYMMETRIC QUANTUM MECHANICS WITH REFLECTIONS

We formulate [Formula: see text]-fold supersymmetry in quantum mechanical systems with reflection operators. As in the cases of other systems, they possess the two significant characters of [Formula: see text]-fold supersymmetry, namely, almost isospectrality and weak quasi-solvability. We construct explicitly the most general one- and two-fold supersymmetric quantum mechanical systems with reflections. In the case of [Formula: see text], we find that there are seven inequivalent such systems, three of which are characterized by three arbitrary functions having definite parity while the other four characterized by two arbitrary functions. In addition, four of the seven inequivalent systems do not reduce to ordinary quantum systems without reflections. Furthermore, in certain particular cases, they are essentially equivalent to the most general two-by-two Hermitian matrix two-fold supersymmetric quantum systems obtained previously by us.

$\cN$-FOLD SUPERSYMMETRY IN QUANTUM MECHANICAL MATRIX MODELS

We formulate [Formula: see text]-fold supersymmetry in quantum mechanical matrix models. As an example, we construct general two-by-two Hermitian matrix two-fold supersymmetric quantum mechanical systems. We find that there are two inequivalent such systems, both of which are characterized by two arbitrary scalar functions, and one of which does not reduce to the scalar system. The obtained systems are all weakly quasi-solvable.

Factorization of supersymmetric Hamiltonians in curvilinear coordinates

Planar supersymmetric quantum mechanical systems with separable spectral problem in curvilinear coordinates are analyzed in full generality. We explicitly construct the supersymmetric extension of the Euler/Pauli Hamiltonian describing the motion of a light particle in the field of two heavy fixed Coulombian centers. We shall also show how the SUSY Kepler/Coulomb problem arises in two different limits of this problem: either, the two centers collapse in one center - a problem separable in polar coordinates -, or, one of the two centers flies to infinity - to meet the Coulomb problem separable in parabolic coordinates.

Supersymmetric Proof of the Hirzebruch-Riemann-Roch Theorem for Non-Kähler Manifolds

We present the proof of the HRR theorem for a generic complex compact manifold by evaluating the functional integral for the Witten index of the appropriate supersymmetric quantum mechanical system.

SQM with non-abelian self-dual fields: harmonic superspace description

We present a Lagrangian formulation for N=4 supersymmetric quantum-mechanical systems describing the motion in external non-Abelian self-dual gauge fields. For any such system, one can write a component supersymmetric Lagrangian by introducing extra bosonic variables with topological Chern-Simons type interaction. For a special class of such system when the fields are expressed in the `t Hooft ansatz form, it is possible to give a superfield description using harmonic superspace formalism. As a new explicit example, the N=4 mechanics with Yang monopole is constructed.

On the geometry of supersymmetric quantum mechanical systems

We consider some simple examples of supersymmetric quantum mechanical systems and explore their possible geometric interpretation with the help of geometric aspects of real Clifford algebras. This leads to natural extensions of the considered systems to higher dimensions and more complicated potentials.

Supersymmetry versus ghosts

We consider the simplest nontrivial supersymmetric quantum mechanical system involving higher derivatives. We unravel the existence of additional bosonic and fermionic integrals of motion forming a nontrivial algebra. This allows one to obtain the exact solution both in the classical and quantum cases. The supercharges Q and Q¯ are not Hermitially conjugate to each other anymore, which allows for the presence of negative energies in the spectrum. We show that the spectrum of the Hamiltonian is unbounded from below. It is discrete and infinitely degenerate in the free oscillatorlike case and becomes continuous running from −∞ to ∞ when interactions are added. Notwithstanding the absence of the ground state, there is no collapse, which suggests that a unitary evolution operator may be defined.

Supersymmetry in 5D gravity

We study a 5d gravity theory with a warped metric and show that two N=2 supersymmetric quantum-mechanical systems are hidden in the 4d spectrum. The supersymmetry can be regarded as a remnant of higher-dimensional general coordinate invariance and turns out to become a powerful tool to determine the physical 4d spectrum and the allowed boundary conditions. Possible extensions of the N=2 supersymmetry are briefly discussed.