Four-site System Research Articles

Conserved quantities in parity-time symmetric systems

Conserved quantities such as energy or the electric charge of a closed system, or the Runge-Lenz vector in Kepler dynamics are determined by its global, local, or accidental symmetries. They were instrumental to advances such as the prediction of neutrinos in the (inverse) beta decay process and the development of self-consistent approximate methods for isolated or thermal many-body systems. In contrast, little is known about conservation laws and their consequences in open systems. Recently, a special class of these systems, called parity-time ($\mathcal{PT}$) symmetric systems, has been intensely explored for their remarkable properties that are absent in their closed counterparts. A complete characterization and observation of conserved quantities in these systems and their consequences is still lacking. Here we present a complete set of conserved observables for a broad class of $\mathcal{PT}$-symmetric Hamiltonians and experimentally demonstrate their properties using single-photon linear optical circuit. By simulating the dynamics of a four-site system across a fourth-order exceptional point, we measure its four conserved quantities and demonstrate their consequences. Our results spell out non-local conservation laws in non-unitary dynamics and provide key elements that will underpin self-consistent analysis of non-Hermitian quantum many-body systems that are forthcoming.

The Holstein polaron problem revisited

The Holstein Hamiltonian was proposed half a century ago; since then, decades of research have come up empty handed in the pursuit of a closed-form solution. An exact solution to the two-site Holstein model is presented in this paper. The obtained results provide a clear image of the Hamiltonian structure and allow for the investigation of the symmetry, energy level crossings and polaronic characteristics of the system. The main mathematical tool is a three-term recurrence relation between the wave function amplitudes, which was obtained using the properties of a family of orthogonal functions, namely the Poisson–Charlier polynomials. It is shown that, with the appropriate choice of basis, the eigenfunctions of the problem naturally fall into two families (parities) associated with the discrete symmetry of the Hamiltonian. The asymptotic solution to the recurrence relation is found by using the Birkhoff expansion. The asymptotic sets the truncation criterion for the wave function, which ensures the accurate calculation of the energy levels for any strength of electron–phonon interaction. The level crossing of states with different parities is discussed and the exact points of broken symmetry are found analytically. The results are used as the building blocks for studying a four-site system. The inherited symmetries lead to the formation of a sparse matrix that is convenient for numerical calculations.

From strong chaos via weak chaos to regular behaviour: Optimal interplay between chaos and order

We investigate the interplay between chaotic and integrable Hamiltonian systems. In detail, a fully connected four-site lattice system associated with the discrete nonlinear Schrödinger equation is studied. On an embedded two-site segment (dimer) of the four-site system (tetramer) the coupling element between its two sites is time-periodically modified by an external driving term rendering the dimer dynamics chaotic, along with delocalisation of initially single-site excitations. Starting from an isolated dimer system the strength of the coupling to the remaining two sites of the tetramer is treated as a control parameter. It is striking that when the dimer interacts globally with the remaining two sites, thus constituting a fully connected tetramer, a non-trivial dependence of the degree of localisation on the strength of the coupling is found. There even exist ranges of optimal coupling strengths for which the driven tetramer dynamics becomes not only regular but also restores complete single-site localisation. We relate the re-establishment of complete localisation with transitions from permanent chaos via regular transients to permanent stable motion on a torus in the higher-dimensional phase space. In conclusion, increasing the dimension of a system can have profound effects on the character of the dynamics in higher-dimensional mixed phase spaces such that even full stabilisation of motion can be accomplished.

Entropy production of a bound nonequilibrium interface

We study the entropy production of a microscopic model for nonequilibrium wetting. We show that, in contrast to the equilibrium case, a bound interface in a nonequilibrium steady state produces entropy. Interestingly, in some regions of the phase diagram a bound interface produces more entropy than a free interface. Moreover, by solving exactly a four-site system, we find that the first derivative of the entropy production with respect to the control parameter displays a discontinuity at the critical point of the wetting transition.

Structural Change of a Tetrahedral Four-Site System with Arbitrary Electron Occupancy

Structural Change of a Tetrahedral Four-Site System with Arbitrary Electron Occupancy

Structural Change of a Tetrahedral Four-Site System with Arbitrary Electron Occupancy

We have performed a comprehensive theoretical study on the structural change of a tetrahedral (T d ) four-site system, which is the most conceivable simple model for various point defects in covalent semiconductors, for all possible electron (hole) occupancies. The system is characterized by the electron transfer energy T , on-site Coulomb repulsion energy U and the lattice relaxation energy S . Possible stable structural configurations and their stabilities are determined exactly for the electronic ground and excited states. The symmetry of the stable structures are found to be T d , C 3v , C 2v , C s or C depending on the electron number n (=0∼8), the electron spin S σ and the relative strengths of T , U and S . The minimum of the ground state energy of the lowest spin S σ state is lower than those of higher spin states except for n =5 and 6. The obtained results are applied to discuss the structural change of point defects in group IV and III-V semiconductors. Because of the absence of the electron-hole symmetry, the off-center instability of a substitutional deep-impurity is more favorable for acceptor than for donor.