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Zigzag Phase Transition in Quantum Wires and Localization in the Inhomogeneous One-Dimensional Electron Gas

Abhijit C. Mehta

Published Jan 1, 2013• 0 citations
OpenAlex

Abstract

In this work, we study two important themes in the physics of the interacting one-dimensional (1D) electron gas: the transition from one-dimensional to higher dimensional behavior, and the role of inhomogeneity. The interplay between interactions, reduced dimensionality, and inhomogeneity drives a rich variety of phenomena in mesoscopic physics. In 1D, interactions fundamentally alter the nature of the electron gas, and the homogeneous 1D electron gas is described by Luttinger Liquid theory. We use Quantum Monte Carlo methods to study two situations that are beyond Luttinger Liquid theory --- the quantum phase transition from a linear 1D electron system to a quasi-1D zigzag arrangement, and electron localization in quantum point contacts. Since the interacting electron gas has fundamentally different behavior in one dimension than in higher dimensions, the transition from 1D to higher dimensional behavior is of both practical and theoretical interest. We study the first stage in such a transition; the quantum phase transition from a 1D linear arrangement of electrons in a quantum wire to a quasi-1D zigzag configuration, and then to a liquid-like phase at higher densities. As the density increases from its lowest values, first, the electrons form a linear Wigner crystal; then, the symmetry about the axis of the wire is broken as the electrons order in a quasi-1D zigzag phase; and, finally, the electrons form a disordered liquid-like phase. We show that the linear to zigzag phase transition occurs even in narrow wires with strong quantum fluctuations, and that it has characteristics which are qualitatively different from the classical transition. Experiments in quantum point contacts (QPC's) show an unexplained feature in the conductance known as the ``0.7 Effect''. The presence of the 0.7 effect is an indication of the rich physics present in inhomogeneous systems, and we study electron localization in quantum point contacts to evaluate several different proposed mechanisms for the 0.7 effect. We show that electrons form a Wigner crystal in a 1D constriction; for sharp constriction potentials the localized electrons are separated from the leads by a gap in the density, while for smoother potentials, the Wigner crystal is smoothly connected to the leads. Isolated bound states can also form in smooth constrictions if they are sufficiently long. We thus show that localization can occur in QPC's for a variety of potential shapes and at a variety of electron densities. These results are consistent with the idea that the 0.7 effect and bound states observed in quantum point contacts are two distinct phenomena.

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European Journal Of Physics Education•Jan 1, 2013•0 citations
In this work, we study two important themes in the physics of the interacting one-dimensional (1D) electron gas: the transition from one-dimensional to higher dimensional behavior, and the role of inhomogeneity. The interplay between interactions, reduced dimensionality, and inhomogeneity drives a rich variety of phenomena in mesoscopic physics. In 1D, interactions fundamentally alter the nature of the electron gas, and the homogeneous 1D electron gas is described by Luttinger Liquid theory. We use Quantum Monte Carlo methods to study two situations that are beyond Luttinger Liquid theory --- the quantum phase transition from a linear 1D electron system to a quasi-1D zigzag arrangement, and electron localization in quantum point contacts. Since the interacting electron gas has fundamentally different behavior in one dimension than in higher dimensions, the transition from 1D to higher dimensional behavior is of both practical and theoretical interest. We study the first stage in such a transition; the quantum phase transition from a 1D linear arrangement of electrons in a quantum wire to a quasi-1D zigzag configuration, and then to a liquid-like phase at higher densities. As the density increases from its lowest values, first, the electrons form a linear Wigner crystal; then, the symmetry about the axis of the wire is broken as the electrons order in a quasi-1D zigzag phase; and, finally, the electrons form a disordered liquid-like phase. We show that the linear to zigzag phase transition occurs even in narrow wires with strong quantum fluctuations, and that it has characteristics which are qualitatively different from the classical transition. Experiments in quantum point contacts (QPC's) show an unexplained feature in the conductance known as the ``0.7 Effect''. The presence of the 0.7 effect is an indication of the rich physics present in inhomogeneous systems, and we study electron localization in quantum point contacts to evaluate several different proposed mechanisms for the 0.7 effect. We show that electrons form a Wigner crystal in a 1D constriction; for sharp constriction potentials the localized electrons are separated from the leads by a gap in the density, while for smoother potentials, the Wigner crystal is smoothly connected to the leads. Isolated bound states can also form in smooth constrictions if they are sufficiently long. We thus show that localization can occur in QPC's for a variety of potential shapes and at a variety of electron densities. These results are consistent with the idea that the 0.7 effect and bound states observed in quantum point contacts are two distinct phenomena.
We study the quantum phase transition of interacting electrons in quantum wires from a one-dimensional (1D) linear configuration to a quasi-1D zigzag arrangement using quantum Monte Carlo methods. As the density increases from its lowest values, first, the electrons form a linear Wigner crystal, then, the symmetry about the axis of the wire is broken as the electrons order in a quasi-1D zigzag phase, and, finally, the electrons form a disordered liquidlike phase. We show that the linear to zigzag phase transition is not destroyed by the strong quantum fluctuations present in narrow wires; it has characteristics which are qualitatively different from the classical transition.
The physics of interacting quantum wires has attracted a lot of attention recently. When the density of electrons in the wire is very low, the strong repulsion between electrons leads to the formation of a Wigner crystal. We review the rich spin and orbital properties of the Wigner crystal, in both the one-dimensional and the quasi-one-dimensional regimes. In the one-dimensional Wigner crystal the electron spins form an antiferromagnetic Heisenberg chain with exponentially small exchange coupling. In the presence of leads, the resulting inhomogeneity of the electron density causes a violation of spin-charge separation. As a consequence the spin degrees of freedom affect the conductance of the wire. Upon increasing the electron density, the Wigner crystal starts deviating from the strictly one-dimensional geometry, forming a zigzag structure instead. Spin interactions in this regime are dominated by ring exchanges, and the phase diagram of the resulting zigzag spin chain has a number of unpolarized phases as well as regions of complete and partial spin polarization. Finally we address the orbital properties in the vicinity of the transition from a one-dimensional to a quasi-one-dimensional state. Due to the locking between chains in the zigzag Wigner crystal, only one gapless mode exists. Manifestations of Wigner crystal physics at weak interactions are explored by studying the fate of the additional gapped low-energy mode as a function of interaction strength.
Abstract. The physics of interacting quantum wires has attracted a lot of attention recently. When the density of electrons in the wire is very low, the strong repulsion between electrons leads to the formation of a Wigner crystal. We review the rich spin and orbital properties of the Wigner crystal, both in the one-dimensional and quasi-one-dimensional regime. In the one-dimensional Wigner crystal the electron spins form an antiferromagnetic Heisenberg chain with exponentially small exchange coupling. In the presence of leads the resulting inhomogeneity of the electron density causes a violation of spin-charge separation. As a consequence the spin degrees of freedom affect the conductance of the wire. Upon increasing the electron density, the Wigner crystal starts deviating from the strictly one-dimensional geometry, forming a zigzag structure instead. Spin interactions in this regime are dominated by ring exchanges, and the phase diagram of the resulting zigzag spin chain has a number of unpolarized phases as well as regions of complete and partial spin polarization. Finally we address the orbital properties in the vicinity of the transition from a onedimensional to a quasi-one-dimensional state. Due to the locking between chains in the zigzag Wigner crystal, only one gapless mode exists. Manifestations of Wigner crystal physics at weak interactions are explored by studying the fate of the additional gapped low-energy mode as a function of interaction strength. PACS numbers: 71.10.Pm2 1.
We explore the role of electron correlation in quasi-one-dimensional quantum wires as the range of the interaction potential is changed and their thickness is varied by performing exact quantum Monte Carlo simulations at various electronic densities. In the case of unscreened interactions with a long-range 1/x1/x1/x tail there is a crossover from a liquid to a quasi-Wigner crystal state as the density decreases. When this interaction is screened, quasi-long-range order is prevented from forming, although a significant correlation with 4kF4{k}_{F}4kF​ periodicity is still present at low densities. At even lower electron concentration, exchange is suppressed and the electrons behave like spinless fermions. Finally, we study the effect of electron correlations in the double quantum wire experiment [Steinberg et al., Phys. Rev. B 73, 113307 (2006)] by introducing an accurate model for the screening in the experiment and explicitly including the finite length of the system in our simulations. We find that decreasing the electron density continuously drives the system from a liquid to a state with quite strong 4kF4{k}_{F}4kF​ correlations. This crossover takes place around $22\text{ }\ensuremath{\mu}{\text{m}}^{\ensuremath{-}1}$, near the density where the electron localization occurs in the experiment. The charge and spin velocities are also in good agreement with the experimental findings in the proximity of the crossover. We argue that correlation effects play an important role at the onset of the localization transition.
We consider electrons in a quantum wire interacting via a long-range Coulomb potential screened by a nearby gate. We focus on the quantum phase transition from a strictly one-dimensional to a quasi-one-dimensional electron liquid that is controlled by the dimensionless parameter nx0n{x}_{0}nx0​, where nnn is the electron density and x0{x}_{0}x0​ is the characteristic length of the transverse confining potential. If this transition occurs in the low-density limit, it can be understood as the deformation of the one-dimensional Wigner crystal to a zigzag arrangement of the electrons described by an Ising order parameter. The critical properties are governed by the charge degrees of freedom and the spin sector remains essentially decoupled. At large densities, on the other hand, the transition is triggered by the filling of a second one-dimensional subband of transverse quantization. Electrons at the bottom of the second subband interact strongly due to the diverging density of states and become impenetrable. We argue that this stabilizes the electron liquid as it suppresses pair-tunneling processes between the subbands that would otherwise lead to an instability. However, the impenetrable electrons in the second band are screened by the excitations of the first subband, so that the transition is identified as a Lifshitz transition of impenetrable polarons. We discuss the resulting phase diagram as a function of nx0n{x}_{0}nx0​.
Due to progress in nanotechnology high-quality quantum wires can nowadays be fabricated. The behavior of particles in one dimension differs significantly from that in three-dimensional (3D) systems, yet the physics of such low-dimensional systems is generally not very well represented in standard undergraduate or graduate curricula. For instance, the Fermi liquid paradigm, working well for electrons in 3D systems, breaks down in 1D and has to be replaced by the Tomonaga-Luttinger liquid (TLL) description. However, most of the introductory papers on the TLL restrict themselves to a summary of results to be compared with experiments, while review papers are often fraught with technical details only accessible to the theorist. The present paper provides an elementary discussion of some phenomena distinguishing quantum wires and quasi-one-dimensional systems from their 3D counterparts targeting experimentalists and graduate students. It aims to convey the basic ideas of TLL theory and to relate it to alternative phases, such as charge and spin density waves. The important role of fluctuations in quasi-1D systems is pointed out and the connection is made to the problem of high-Tc superconductivity. The discussion is kept on a level, which should be accessible with a basic knowledge of quantum mechanics.