Hermitian Interaction Research Articles

Hermitian and pseudo-Hermitian Hamiltonians of [formula omitted] system—Spectrum, exceptional point, quantum–classical correspondence

We in this paper study the relation of hermiticity and energy spectrum for Hamiltonians consisting of SU(1,1) generators. In contrast with the common belief, the transition from real to imaginary spectra can appear at an exceptional point for the Hermitian Hamiltonian of a conservative system. The imaginary domain of spectrum resulted from an inverted potential well unbounded from below. An outward force applies on the particle, which moves acceleratingly away from the central point similar to the minisuperspace model of expanding universe. The Hamiltonian remains Hermitian beyond the exceptional point in a price that the boson-operator realization of SU(1,1) generator Ŝz becomes non-Hermitian. Oppositely a non-Hermitian Hamiltonian (called pseudo-Hermitian) possesses real eigenvalues in the entire region of interaction constant, which increases the gradient of potential well. While the Hermitian interaction decreases the gradient continuously to zero, namely the exceptional point, where eigenstates are degenerate. We extend the non-Hermitian Hamiltonian to the periodically time-dependent and two-dimensional system. The probability density of wave function coincides with classical orbits, which are derived from the corresponding classical-counterpart of the non-Hermitian quantum Hamiltonian.

Effective protection of quantum coherence by a non-Hermitian driving potential

We investigate the effects of non-Hermitian driving potential on quantum coherence in a bipartite system. The results show that the dynamical localization will revive after being destroyed by the Hermitian interaction, which provides evidence of the restoration of quantum coherence by a non-Hermitian driving potential. Besides, the entanglement between the two subsystems also decays with the boosting of non-Hermitian driving strength, which gives more evidence that non-Hermitian driving potential will protect quantum coherence. The physics behind this phenomenon is the domination of the quasieigenstate with maximum imaginary value of the quasieigenvalue over the dynamics of the non-Hermitian system. Our discovery establishes a restoration mechanism of quantum coherence in interacting and dissipative quantum systems which is highly expectable in updated experiments from many-body physics to quantum information.

Universal conductance of a PT -symmetric Luttinger liquid after a quantum quench

We study the non-equilibrium dynamics and transport of a PT-symmetric Luttinger liquid (LL) after an interaction quench. The system is prepared in domain wall initial state. After a quantum quench to spatially homogeneous, PT-symmetric LL, the domain wall develops into a flat central region that spreads out ballistically faster than the conventional Lieb-Robinson maximal speed. By evaluating the current inside the regular lightcone, we find a universal conductance $e^2/h$, insensitive to the strength of the PT-symmetric interaction. On the other hand, by repeating the very same time evolution with a hermitian LL Hamiltonian, the conductance is heavily renormalized by the hermitian interaction as $e^2/hK$ with $K$ the LL parameter. Our analytical results are tested numerically, confirming the universality of the conductance in the non-hermitian realm.

Dissipation-Induced Luttinger Liquid Correlations in a One-Dimensional Fermi Gas.

We study a one-dimensional Fermi gas in the presence of dissipative coupling to environment through the Lindblad equation. The dissipation involves energy exchange with the environment and favours the relaxation of electrons to excitations. After switching on the dissipation, the system approaches a steady state, which is described by a generalized Gibbs ensemble. The fermionic single particle density matrix resembles deceivingly to that in a hermitian interaction quench. It decays inversely with the distance for short times due to the fermionic correlations in the initial state, which changes into a noninteger power law decay for late times, representing dissipation-induced Luttinger liquid behavior. However, the crossover between the two regions occurs due to dissipation-induced damping, and is unrelated to the propagation of excitations. The velocity of information spreading is set by the dissipative coupling, and differs significantly from the original sound velocity. The thermodynamic entropy grows as -t ln t initially, and saturates to an extensive value. Our results can be tested experimentally in one-dimensional Dirac systems.

Causality and unitarity via the tree-loop duality relation

The tree-loop duality relation is used as a starting point to derive the constraints of causality and unitarity. Specifically, the Bogoliubov causality condition is ab initio derived at the individual graph level. It leads to a representation of a graph in terms of lower order cut graphs. Extracting the absorptive part gives then the general unitarity relation (Cutkosky rule). The derivation, being carried out directly in momentum space, holds for any local (polynomial) hermitian interaction vertices. This is in contrast to the technical difficulties arising from contact terms in the spacetime approach based on the largest time equation.

A generalization of the Fermi-Breit equation to non-Coulombic spatial interactions

Starting from the relativistic invariance properties at classical level, we generalize the Darwin equation to the case of non-Coulombic spatial interactions. The relativistic correction terms for vector interactions are derived from a given nonrelativistic potential. We show that, for a Coulombic potential, the results coincide with those obtained in the Coulomb gauge. The results are adapted to the quantum theory obtaining a generalization of the Fermi-Breit equation. An Hermitian interaction operator is constructed. A critical comparison with other possible treatments of the retardation terms is performed also discussing the usual choice of the Coulomb gauge. Special attention is devoted to the construction of a model for quark interaction.

Family of Hermitian low-momentum nucleon interactions with phase shift equivalence

Using a Schmidt orthogonalization transformation, a family of Hermitian low-momentum NN interactions is derived from the non-Hermitian Lee-Suzuki (LS) low-momentum NN interaction. As special cases, our transformation reproduces the Hermitian interactions for Okubo and Andreozzi. Aside from their common preservation of the deuteron binding energy, these Hermitian interactions are shown to be phase shift equivalent, all preserving the empirical phase shifts up to decimation scale Lambda. Employing a solvable matrix model, the Hermitian interactions given by different orthogonalization transformations are studied; the interactions can be very different from each other particularly when there is a strong intruder state influence. However, because the parent LS low-momentum NN interaction is only slightly non-Hermitian, the Hermitian low-momentum nucleon interactions given by our transformations, including the Okubo and Andreozzi ones, are all rather similar to each other. Shell model matrix elements given by the LS and several Hermitian low-momentum interactions are compared.

The 2 + doublet in 8Be: an example of accidental resonance degeneracy

By means of a numerical analysis of the 2 + doublet in 8Be, we show that accidental degeneracies of two resonant states mixed by a Hermitian interaction are associated with the occurrence of a simple and a double pole of the S-matrix in the complex energy plane. A true crossing may be brought about by the variation of just two parameters.

Degeneracy and crossing of resonance energy surfaces

We investigate the accidental degeneracy of resonances mixed by a Hermitian interaction. We give general expressions for the codimension in parameter space of any degeneracy of resonances. In the case of a degeneracy of two resonances which produces only one simple pole in the S matrix, the codimension is four for time reversal invariant interactions and six when the quantum system is not time reversal invariant. Close to this type of degeneracy the corresponding energy surfaces are two double cones lying in orthogonal subspaces with a common vertex at a double diabolic point. When the degeneracy of two resonances leads to one simple plus one double pole of S the codimension is two irrespective of the time reversal invariant character of the quantum system. Close to this type of degeneracy, the energy surfaces are a hyperbolic cone and a sphere which lie in orthogonal subspaces and touch one another at all points on a diabolic circle.

Valence-core self-consistency in theA=17system

We develop a systematic approach to the calculation of self-consistency effects in core plus valence nucleon systems. A detailed calculation of the effective one-body Hamiltonian for A=17 in an 11h..cap omega.. model space is readily extrapolated to a 20h..cap omega.. model space which is sufficient to develop accurate tails for the single-particle wave functions. The effective one-body interaction is found using a Brillouin-Wigner type perturbation theory which eliminates folded diagrams and leads to a manifestly Hermitian interaction. A renormalized Brueckner calculation is performed for A=16 and the results are employed in a shell-model study of A=17. The self-consistent results still have a weak dependence on the initial unperturbed Hamiltonian. However, we find a single unperturbed Hamiltonian which yields a reasonable binding energy for A=16 and, except for a weak spin-orbit splitting, reasonable results for the lowest states in the A=17 system. This agreement is important for continued valence s- d shell studies.

Dirac theory and Hermitian interaction terms

Dirac theory and Hermitian interaction terms