Exchange coupling in a one-dimensional Wigner crystal

Abstract

We consider a long quantum wire at low electron densities. In this strong interaction regime a Wigner crystal may form, in which electrons comprise an antiferromagnetic Heisenberg spin chain. The coupling constant J is exponentially small, as it originates from tunneling of two neighboring electrons through the segregating potential barrier. We study this exponential dependence, properly accounting for the many-body effects and the finite width of the wire.PACS numbers: 73.21. Hb,75.10.Pq,75.30.Et,71.70.Gm Quantum wires exhibit a plethora of interesting phenomena. In particular, quantization of conductance, which is a fundamental manifestation of the quantum nature of the electron, has been actively studied ever since its first observation in quantum point contacts [1]. The phenomenon presents itself as very flat plateaus of linear conductance G at integer multiples of G 0 = 2e 2 /h, as a function of gate voltage which tunes the electron density in the wire. Since the first observation of the phenomenon, its various facets have been studied by measurements of thermal transport [2,3], noise [4], and experiments on systems with superconducting elements [5].A new generation of experiments in quantum wires has revealed an unexpected structure at low electron densities: a plateau at about 0.7 G 0 for short [6,7,8,9, 10] and at 0.5 G 0 for long quantum wires [11,12,13,14]. These new features have generated much interest as they are likely caused by electron correlation effects. The origin of the new plateau has not yet been established; however, the experiments [6,7,8] point to the important role played by the electron spins.In recent experiments of a different kind, involving tunneling between two parallel ballistic quantum wires, mapping of the spectrum of spin and charge excitations was achieved [15]. In addition, unexpected behavior indicating electron localization was observed at low densities. Of particular interest is the concurrence of the localization with the drops in the conductance steps, which indicates [15] a possible connection with the 0.7 structure [6,7,8,9,10].Recent theoretical work suggests that qualitatively new transport properties of quantum wires at low electron densities may be due to the formation of a Wigner crystal state of electrons. The latter is expected to occur when the density is low enough for the potential energy of the Coulomb repulsion to overwhelm the kinetic energy of the electrons in the wire. In the crystalline configuration, the electron spins form an antiferromagnetic Heisenberg spin chain, with the exchange coupling constant J emerging as a new energy scale. The Heisenberg exchange can be viewed as arising from tunneling of two neighboring electrons through the potential barrier created by their mutual repulsion, and by the repulsion from all other electrons of the wire. As a result, J is expected to be small compared to the Fermi energy E F , and qualitatively new transport properties of the quantum wires, such as the plateau at 0.5 G 0 , are expected [16,17] at tempera...