Classical realization of two-site Fermi-Hubbard systems

Abstract

A classical wave optics realization of the two-site Hubbard model, describing the dynamics of interacting fermions in a double-well potential, is proposed based on light transport in evanescentlycoupled optical waveguides.PACS numbers: 71.10. Fd, 42.82.Et Quantum-classical analogies have been explored on many occasions to mimic and visualize in a purely classical setting the dynamical aspects embodied in a wide variety of quantum systems [1,2]. In particular, in the past two decades engineered photonic lattices have provided a useful model system to investigate wave optics analogous of solid state phenomena [2][3][4][5] [11], refer to single-particle phenomena and are based on the formal similarity between the paraxial optical wave equation in photonic lattices and the nonrelativistic Schrödinger equation of a single particle in periodic potentials [2]. However, much of the richer physics in condensed-matter comes from many-body phenomena and electron correlations. The simplest and paradigmatic model which describes correlation effects of electrons in a lattice, arising from the the competition among chemical bonding, Coulomb repulsion and Pauli exclusion principle, is perhaps provided by the Hubbard model (HM) [12]. This model is capable of capturing some many-body aspects of the electronic properties of condensed matter, such as metal-insulator transitions, itinerant magnetism, and electronic superconductivity (see, e.g., [13,14] and reference therein). In spite of the simplicity of its Hamiltonian structure, very few exact results are known for the HM, mainly for finite clusters or for the infinite onedimensional chain [13][14][15]. The simplest solvable and nontrivial system, which can still capture some of the main relevant properties of larger clusters and of the infinite chain, is provided by the two-site Hubbard Hamiltonian (see, for instance, [16]). The two-site HM, being exactly solvable, has been considered by several authors as a simplified theoretical model [16][17][18][19][20][21][22]. In particular, it is useful as a toy model for understanding the binding of molecules like H 2 [17][18][19], and it was proposed to model electron-molecular vibration coupling in organic charge-transfer salts [20] and the electronic structure in π systems [21]. Since photons are bosons and they do not interact when propagating in linear optical structures, one would expect that photonics is not a suited system to simulate in a classical setting the physics of interacting electrons in solids. In recent works [23], it has been pointed out that photonic structures could provide a noteworthy laboratory system to simulate the physics of few interacting bosons in the framework of the BoseHubbard model. In this Brief Report it is shown that light transport in suitably engineered coupled waveguide structures can mimic the dynamics of interacting fermions as well. In particular, an optical realization of the two-site HM is proposed, in which light propagation in four evanescently-coupled waveguides reproduces the temporal ...

Hubbard pair cluster in the external fields. Studies of the polarization and susceptibility

“…At the same time, it has been noticed that the exact solutions to the model can be obtained for small clusters, consisting of several lattice sites [48][49][50][51][52][53][54][55][56][57][58][59][60][61][62][63][64][65][66][67][69][70][71][72][73][74][75][76][77][78][79][80][81][82][83][84]. Intensive investigations of such systems have been carried out both from the point of view of static properties [48-50, 52, 58, 62, 64, 66-68, 71-73, 75-80, 82, 83, 85-88], as well as for dynamical description [54,59,61,70,81,84]. In case of very small atomic clusters, exact results for the Hubbard model have been obtained by analytical methods [48-50, 52, 54, 62, 66, 67, 71, 73-75, 79, 80, 82, 84, 85].…”

Hubbard pair cluster in the external fields. Studies of the chemical potential

“…The magnetic field has been taken into account in [22], however, the influence of the external electric field on the cluster properties has not been studied there.Motivated by such a possibility, we undertake the exact study of the Hubbard simplest cluster, namely the pair (or dimer), embedded simultaneously in the external magnetic and electric fields. We note that such system has already attracted some attention [28][29][30][31][32][33][34][35]. Also the effect of the electric field on the properties of the Hubbard model was studied [36,37].…”

Hubbard pair cluster with elastic interactions. Studies of thermal expansion, magnetostriction and electrostriction

“…Among such systems, a two-atomic cluster (dimer) plays a special role, being the smallest nanosystem where the Hubbard model can be adopted, and the exact analytical solutions can be found. As a consequence, numerous aspects of physics of Hubbard dimers focused considerable attention in the literature [30][31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46][47][48][49]. These facts serve as a sound motivation for further comprehensive study of the two-atomic Hubbard system aimed at its full and exact thermodynamic description.…”