- Re[Hmn]/Sqrt[EmEn] seen from the low-low-energy corner of matrix H, Dec 22, 2006
- Re[Hmn]/Sqrt[EmEn] seen from the low-high-energy corner of matrix H, Dec 22, 2006
- Im[Hmn]/Sqrt[EmEn] seen from the low-low-energy corner of matrix H, Dec 22, 2006
- Im[Hmn]/Sqrt[EmEn] seen from the low-high-energy corner of matrix H, Dec 22, 2006

Four short movies available below show three dimensional views of the similarity renormalization group evolution of real and imaginary parts of matrix elements of a model Hamiltonian from the paper "Limit cycles of effective theories" [Phys. Rev. D75, 025005 (2007); hep-th/0611015]. Familiarity with that work helps in understanding of what is visible in the movies. The films show a result of solving 1764 coupled nonlinear differential equations that determine the RG evolution of the entire Hamiltonian matrix.

There is a grid of points on the horizontal plane. Each and every grid point has coordinates m and n that are the subscripts of the matrix elements of the Hamiltonian H. The third coordinate, extending in the vertical direction from -1.5 to 2.5, is used to show the magnitude of the matrix elements. Namely, the vertical coordinate is equal to the ratio of the matrix element Hmn to the square root of the product of eigenvalues Em and En of the free Hamiltonian H0. The films show the RG evolution of the real and imaginary parts of the Hamiltonian matrix elements separately and only with subscripts between -1 and 16. Off-diagonal matrix elements with smaller or grater subscripts evolve in such a way that their visualization would require extention of the blue planes horizontally. Diagonal elements above 16 contain eigenvalues of H divided by eigenvalues of H0 and diagonal elements below -1 are equal 1-glambda, as a result of dividing diagonal matrix elements of H by the diagonal matrix elements of H0. The matrix elements with any subscript above 16 or below -1 are not shown because the goal is to produce a finite and as simple image as possible and focus viewers' attention on how the eigenvalues of the full Hamiltonian appear one after another on the diagonal in the cyclic sequence. The evolution shown in the films extends from lambda = 19.8 million down to lambda = 18.9, i.e., over 6 orders of magnitude. At the beginning and end of this range, the coupling constant glambda is zero. The natural logarithm of lambda varies from 16.8 to 2.94. The four videos show the same evolution as seen from two different points of view: from the low-low energy corner of the Hamiltonian matrix and from its low-high energy corner.