Francisco Correa

Self-isospectrality, special supersymmetry, and their effect on the band structure.

We study a planar model of a nonrelativistic electron in periodic magnetic and electric fields that produce a 1D crystal for two spin components separated by a half-period spacing. We fit the fields to create a self-isospectral pair of finite-gap associated Lamé equations shifted for a half-period, and show that the system obtained is characterized by a new type of supersymmetry. It is a special nonlinear supersymmetry generated by three commuting integrals of motion, related to the parity-odd operator of the associated Lax pair, that coherently reflects the band structure and all its peculiarities. In the infinite-period limit it provides an unusual picture of supersymmetry breaking.

Aharonov–Bohm effect on AdS2 and nonlinear supersymmetry of reflectionless Pöschl–Teller system

Abstract We explain the origin and the nature of a special nonlinear supersymmetry of a reflectionless Poschl–Teller system by the Aharonov–Bohm effect for a non-relativistic particle on the AdS 2 . A key role in the supersymmetric structure appearing after reduction by a compact generator of the AdS 2 isometry is shown to be played by the discrete symmetries related to the space and time reflections in the ambient Minkowski space. We also observe that a correspondence between the two quantum non-relativistic systems is somewhat of the AdS/CFT holography nature.

Finite-gap systems, tri-supersymmetry and self-isospectrality

We show that an n-gap periodic quantum system with parity-even smooth potential admits 2n − 1 isospectral super-extensions. Each is described by a tri-supersymmetry that originates from a higher-order differential operator of the Lax pair and two-term nonsingular decompositions of it; its local part corresponds to a spontaneously partially broken centrally extended nonlinear N = 4 supersymmetry. We conjecture that any finite-gap system having antiperiodic singlet states admits a self-isospectral tri-supersymmetric extension with the partner potential to be the original one translated for a half-period. Applying the theory to a broad class of finite-gap elliptic systems described by a two-parametric associated Lame equation, our conjecture is supported by the explicit construction of the self-isospectral tri-supersymmetric pairs. We find that the spontaneously broken tri-supersymmetry of the self-isospectral periodic system is recovered in the infinite-period limit.

Hidden supersymmetry in quantum bosonic systems

Abstract We show that some simple well-studied quantum mechanical systems without fermion (spin) degrees of freedom display, surprisingly, a hidden supersymmetry. The list includes the bound state Aharonov–Bohm, the Dirac delta and the Poschl–Teller potential problems, in which the unbroken and broken Nxa0=xa02 supersymmetry of linear and nonlinear (polynomial) forms is revealed.

The Bogoliubov-de Gennes system, the AKNS hierarchy, and nonlinear quantum mechanical supersymmetry

Abstract We show that the Ginzburg–Landau expansion of the grand potential for the Bogoliubov–de Gennes Hamiltonian is determined by the integrable nonlinear equations of the AKNS hierarchy, and that this provides the natural mathematical framework for a hidden nonlinear quantum mechanical supersymmetry underlying the dynamics.

Hidden nonlinear su(2|2) superunitary symmetry of N=2 superextended 1D Dirac delta potential problem

Abstract We show that the N = 2 superextended 1D quantum Dirac delta potential problem is characterized by the hidden nonlinear su ( 2 | 2 ) superunitary symmetry. The unexpected feature of this simple supersymmetric system is that it admits three different Z 2 -gradings, which produce a separation of 16 integrals of motion into three different sets of 8 bosonic and 8 fermionic operators. These three different graded sets of integrals generate two different nonlinear, deformed forms of su ( 2 | 2 ) , in which the Hamiltonian plays a role of a multiplicative central charge. On the ground state, the nonlinear superalgebra is reduced to the two distinct 2D Euclidean analogs of a superextended Poincare algebra used earlier in the literature for investigation of spontaneous supersymmetry breaking. We indicate that the observed exotic supersymmetric structure with three different Z 2 -gradings can be useful for the search of hidden symmetries in some other quantum systems, in particular, related to the Lame equation.

On Hidden broken nonlinear superconformal symmetry of conformal mechanics and nature of double nonlinear superconformal symmetry

Abstract We show that for positive integer values l of the parameter in the conformal mechanics model the system possesses a hidden nonlinear superconformal symmetry, in which reflection plays a role of the grading operator. In addition to the even so ( 1 , 2 ) ⊕ u ( 1 ) -generators, the superalgebra includes 2 l + 1 odd integrals, which form the pair of spin- ( l + 1 2 ) representations of the bosonic subalgebra and anticommute for order 2 l + 1 polynomials of the even generators. This hidden symmetry, however, is broken at the level of the states in such a way that the action of the odd generators violates the boundary condition at the origin. In the earlier observed double nonlinear superconformal symmetry, arising in the superconformal mechanics for certain values of the boson–fermion coupling constant, the higher order symmetry is of the same, broken nature.

Hidden nonlinear supersymmetry of finite-gap Lamé equation

A bosonized nonlinear (polynomial) supersymmetry is revealed as a hidden symmetry of the finite-gap Lame equation. This gives a natural explanation for peculiar properties of the periodic quantum system underlying diverse models and mechanisms in field theory, nonlinear wave physics, cosmology and condensed matter physics.

Peculiarities of the hidden nonlinear supersymmetry of the Pöschl–Teller system in the light of the Lamé equation

A hidden nonlinear bosonized supersymmetry was revealed recently in the Poschl–Teller and finite-gap Lame systems. In spite of the intimate relationship between the two quantum models, the hidden supersymmetry in them displays essential differences. In particular, the kernel of the supercharges of the Poschl–Teller system, unlike the case of the Lame equation, includes nonphysical states. By means of the Lame equation, we clarify the nature of these peculiar states, and show that they encode essential information not only on the original hyperbolic Poschl–Teller system, but also on its singular hyperbolic and trigonometric modifications, and reflect the intimate relation of the model to a free-particle system.

Hidden superconformal symmetry of the spinless Aharonov-Bohm system

A hidden supersymmetry is revealed in the spinless Aharonov–Bohm problem. The intrinsic supersymmetric structure is shown to be intimately related to the scale symmetry. As a result, a bosonized superconformal symmetry is identified in the system. Different self-adjoint extensions of the Aharonov–Bohm problem are studied in the light of this superconformal structure and interacting anyons. The scattering problem of the original Aharonov–Bohm model is discussed in the context of the revealed supersymmetry.