Chasing Down Eigenvectors

One of the great challenges in quantum physics is computing the lowest eigenvalues and eigenvectors of a Hamiltonian matrix in a vector space so large that vectors cannot be easily stored in computer memory.  Eigenvectors are vectors that repeat themselves under matrix multiplication, up to an overall factor given by the eigenvalue.  In quantum mechanics, the eigenvalues of the Hamiltonian matrix correspond to the energy levels that are observable in experiments, and the eigenvectors correspond to the wave functions.  There are efficient methods developed for such quantum mechanical problems, but they generally fail when some parameter in the Hamiltonian matrix exceeds some threshold value.  In the recent paper published in Phys. Rev. Lett. on “Eigenvector continuation with subspace learning”, nuclear theorists from NSCL demonstrated a new technique, called eigenvector continuation, to find eigenvectors of interest.  The key insight is that while an eigenvector resides in a linear space with enormous dimensions, the eigenvector path generated by smooth changes of the Hamiltonian matrix is well approximated by a surface that bends in only a few directions.  

Eigenvector continuation is a general technique that has the potential to benefit numerous applications in quantum many-body theory. The eigenvector continuation method is now being applied to lattice simulations of the structure of atomic nuclei in difficult cases where the signal to noise ratio would otherwise be too small. It is also being developed to extend the reach of perturbation theory in nuclear systems. Perturbation theory is an approximation method that starts from a known solution and then systematically includes deviations ordered according to size.  The method will be used to compute pairing properties in atomic nuclei, neutron matter, and nuclear matter.    

Figure: The path of eigenvector