Hamiltonian time-reversal symmetry

Similar to closed chaotic billiards the idea was to adjust RMT to the effective Hamiltonian Heff = H — iT (see the pioneering work (J.J.M. Verbaarschot et.al., 1985) and (H.-J. Sommers et.al., 1999) for references). These matrices correspond to GUE with broken time-reversal symmetry. A next natural step was to assume that in the transition region between GOE and GUE, the eigenfunctions are complex and may be thought as columns of the unitary random matrix (G. Lenz et.al., 1992 E. Kanzieper et.al., 1996) S = S +ieS2, composed of two independent orthogonal matrices. The parameter... 

The time-reversal symmetry of the Hamiltonian dynamics, also called the microreversibility, is the property that if the phase-space trajectory... 

The time evolution of the probability density is induced by Hamiltonian dynamics so that it has its properties—in particular, the time-reversal symmetry. However, the solutions of Liouville s equation can also break this symmetry as it is the case for Newton s equations. This is the case if each trajectory (43) has a different probability weight than its time reversal (44) and that both are physically distinct (45). 

The subscript C/ in (4.1.45) stands for unitary since the system Hamiltonian is invariant under general unitary transformations in case all antiunitary symmetries are broken. The effects of time reversal symmetry breaking on the level statistics were recently investigated experimentally by So et al. (1995) and Stoffregen et al. (1995) by measuring the resonance spectrum of quasi-two-dimensional microwave cavities in the presence of time reversal breaking elements such as ferrites and directional couplers. These experiments are of considerable relevance for quantum mechanics since quasi-two-dimensional microwave cavities are generally considered to be excellent models for two-dimensional quantum billiards. 

The matrices and are defined in perfect analogy with (239). Once again we see the structure of the gradient vector e /, but with the reference Hamiltonian Hq replaced by the two-index transformed Hamiltonian. To determine the vector structure of with respect to hermiticity and time reversal symmetry it suffices to look at the corresponding two-index transformed Hamiltonian... 

If the system has time reversal symmetry, then the Hamiltonians H and K commute. If V > is an eigenstate then Kip > s also an eigenstate with the same energy. The functions ij and Kil are orthogonal. This is Kramers theorem. For a system with time reversal symmetry all states are at least doubly degenerate. For a crystal with the wavefunctions (10) we then have that K ( V fct >. nd K > belong to the wavevector —k,... 

In this section we show how the symmetry of the Hamiltonian can be used to simplify the relaxation equations. We also present several important theorems involving time-correlation functions and memory functions. We begin by discussing time reversal symmetry. 

A word of caution is necessary. In the presence of an external field the time-reversal symmetry of the Hamiltonian may be removed. For example, in a magnetic field B the spin-dipole interaction with the field is of the form(—J B), where J is an angular momentum and B is the magnetic field. J has odd time-reversal symmetry (like rxp) so that if B 0 none of the above theorems hold unless B —> — B is also imposed. Thus for example Theorem 3 would become... 

Interestingly the time-reversal symmetry provides analogous properties in some respects to the properties of Hamiltonian systems. For example, consider a linear Hamiltonian system dz/dl = JAz, with A a symmetric matrix. If A is an eigenvalue of JA then JAu = Xu for some eigenvector u 0. Because the matrix JA is real, we know that A will also be an eigenvalue. At the same time, we know that since A is an eigenvalue of JA it is also an eigenvalue of its transpose (JAY = A J = —AJ, thus... 

The coupling coefficient on the right-hand side of Eq- (6.57) restricts the symmetry of the nuclear displacements to the direct square of the irrep of the electronic wave-function. This selection rule is made even more stringent by time-reversal symmetry. The Hamiltonian is based on displacement of nuclear charges, and not on momenta, so as an operator it is time-even or real. For spatially-degenerate irreps, which are... 

We shall now examine the effects of these two kinds of time-reversal symmetry on quantum systems under time-even Hamiltonians, i.e., in the absence of external... 

In the two-component case, the setup of the relativistic one-electron Hamiltonian is much slower than in the scalar-relativistic case. The formal ratio of two-component to scalar-relativistic transformation is 32. The doubled dimension of matrices contributes a factor of eight and the multiplication of complex numbers contributes another factor of four. The actual ratios for different relativistic approaches match the formal ratio. The computation time of the X2C Hamiltonian is very close to that of DKH6. DKH2 is ten times faster than X2C in this two-component case. The computational cost for the calculation of the relativistic one-electron Hamiltonian shows a dramatic increase from scalar to two-component, whereas the SCF time is only slightly increased. This is due to several factors. The required primitive repulsion integrals are the same as in the SCF stage, the electron density is always real, and time-reversal symmetry had been used in the two-component SCF calculations. The two-component relativistic Hamiltonian construction is now the bottleneck of the whole calculation. If the DLU approximation is employed, the computation time of the relativistic transformation is again dramatically reduced. 

The time-reversal symmetry of the crystalline Hamiltonian introduces an additional energy-level degeneracy.Let the Hamiltonian operator H be real. The transition in the time-dependent Schrodinger equation to a complex-conjugate equation with simultaneous time-inversion substitution... 

Having discussed the symmetry of the spinors and products of spinors, we are now in a position to discuss the symmetries of the one- and two-electron integrals. For computational applications, the integrals over molecular orbitals are often divided into symmetry classes for convenient handling of symmetry. In addition, the consideration of symmetry may produce some simplification in the expressions for the many-electron Hamiltonian. In the relativistic case we must use both point-group and time-reversal symmetry. 

For the two-electron integrals, we want to divide the integrals into symmetry classes, as for the nonrelativistic integrals. We also want to divide the integrals into classes by time reversal symmetry, as we did for the one-electron integrals. Because of the structure of the Kramers-restricted Hamiltonian in terms of the one- and two-particle Kramers replacement operators, we hope to obtain a reduction in the expression for the Hamiltonian from time-reversal symmetry. The classification by time-reversal properties is also important for the construction of the many-electron Hamiltonian matrix, whose symmetry properties we consider in the next section. 

We have seen how time-reversal symmetry and double-group symmetry are intimately connected in the matrices of one- and two-electron operators. These two symmetries are just as intimately connected in the many-electron Hamiltonian matrix. 

Thus, apart from inversion, time-reversal symmetry provides for full symmetry blocking of the Hamiltonian matrix for an odd number of electrons in complex or real groups and for an even number of electrons in complex groups. 

Because

double excitation in the open-shell space, and because we left excitations within this space out of the excitation operators, the second part of the normalization term is zero, and the energy is given by the left side of the equation. This technique can be used for open-shell Kramers pairs belonging to complex or real irreps, but not to quaternion irreps. In the last case, there are four determinants that are composed of the open-shell spinors, and even though they occur in pairs related by time-reversal symmetry, the Hamiltonian operator connects all four. In the case of complex irreps, the absolute value of the off-diagonal matrix element must be taken, because it will in general be complex. 

From our experience to date with the transformations, we can immediately foresee a problem. If the transformation is some complicated function of the momentum, we might not be able to separate out the perturbation from the zeroth-order Hamiltonian. This would be unfortunate, because magnetic operators break Kramers symmetry and we would be forced to perform calculations without spin (or time-reversal) symmetry. We might also be forced to perform finite-field calculations. We will address this problem as it arises. 

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