Theoretical proposal for dual transformation between the Josephson effect and quantum phase slip in single junction systems and nanowires

The dual transformation has been known to be a useful tool in various physical systems. In particular, in quantum field theoryand statistical mechanics, many studied cases incorporating duality are known. Similarly, electric circuits arranged in series and parallel within classical electrical engineering exhibit similar adding laws, which are satisfied by interchanging the role of resistance and conductance, inductance and capacitance, current and voltage. This rule is known as the duality principle of electrical circuits-. In recent years, numerous experiments and theoretical discussions have been conducted on the potential of quantum phase-slip (QPS) existing as a dual system in the Josephson junction (JJ) system using nanowires-. However, a deterministic experimental fact showing the existence of quantum phase-slip completely has not been found yet. Also, the exact theory of the dual transformation between the JJ system and the QPS junction (QPSJ) system has not been completed yet-. In this paper, we introduce two Hamiltonian, which are dual to each other, and propose a general theory to construct a dual system by applying the dual condition between current and voltage in an electric circuit. This method was named the dual Hamiltonian (DH) method. By using this method, the Hamiltonians of the QPS system and the JJ system are proved to be equivalent to each other by dual transformation and also prove to be an exact dual system. The remainder of this paper is organized as follows. In the next section, the DH method is applied to build a quantum LC circuit as a simple example.In section 2, as an application of the preceding section, the relationship between the QPS system and the JJ system in a single junction is introduced. In section 3, self-duality in various quantum junction circuits is briefly proved. In the section 4, superconductors and superinsulators are discussed from the standpoint of quantum phase transition. In section 5, duality is examined for the partition function of a single junction that incorporates quantum effects using path integration. In section 6, the derivation of the anisotropic XY (AXY ) model and dual anisotropic XY (DAXY ) model are described in the classical 1 + 1 dimensional system equivalent to the JJ and QPSJ in a nanowire, which is a quantum one-dimensional system. In section 7, the duality between the AXY model and DAXY model is proved by the Villain approximation. In the last section, the summary, discussion, and conclusions are presented.