Quantum non-Abelian hydrodynamics: Anyonic or spin–orbital entangled liquids, nonunitarity of scattering matrix and charge fractionalization

Abstract

In this article we develop an exact(non-adiabatic,non-perturbative) density matrix scattering theory for a two component quantum liquid which interacts or scatters off from a generic spin-dependent quantum potential.The generic spin dependent quantum potential[eq. (1)] is a matrix potential, hence, adiabaticity criterion is ill-defined. Therefore the full matrix potential should be treated non-adiabatically. We succeed in doing so using the notion of vectorial matrices which allows us to obtain an exact analytical expression for the scattered density matrix ,̺sc [eq.(30)]. We find that the number or charge density in scattered fluid,Tr(̺sc), expressions in eqs. (32) depends on non-trivial quantum interference coefficients, Q αβ 0ijk which arises due to quantum interference between spin-independent and spin-dependent scattering amplitudes and among spin-dependent scattering amplitudes. Further it is shown that Tr(̺sc) can be expressed in a compact form [eq.(39)] where the effect of quantum interference coefficients can be include using a vector Q αβ which allows us to define a vector order parameter Q. Since the number density is obtained using and exact scattered density matrix, therefore, we do not need to prove that Q is non-zero. However for sake of completeness we make detailed mathematical analysis for the conditions under which the vector order parameter Q would be zero or non-zero.We find that in presence of spin-dependent interaction the vector order parameter Q is necessarily non-zero and is related to the commutator and anti-commutator of scattering matrix S with its dagger S † [eq. (78)]. It is further shown that Q = 0, implies four physically equivalent conditions,i.e, spin-orbital entanglement is non-zero, NonAbelian scattering phase ,i.e, matrices, scattering matrix is NonUnitary and the broken time reversal symmetry for scattered density matrix. This also implies that quasi particle excitation are anyonic in nature, hence, charge fractionalization is a natural consequence. This aspect has also been discussed from the perspective of number or charge density conservation, which implies i.e, Tr(̺sc) = Tr(̺in). On the other hand Q = 0 turns out to be a mathematically forced unphysical solution in presence of spin-dependent potential or scattering which is equivalent 1 to Abelian hydrodynamics ,Unitary scattering matrix, absence of spinspace entanglement, and preserved time reversal symmetry.We have formulated the theory using mesoscopic language, specifically, we have considered two terminal systems connected to spin-dependent scattering region, which is equivalent to having two potential wells separated by a generic spin-dependent potential barrier. The formulation using mesoscopic language is practically useful because it leads directly to the measured quantities such as conductance and spin-polarization density in the leads, however, the presented formulation is not limited to the mesoscopic system only, its generality has been stressed at various places in this article.