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Wigner crystal physics in quantum wires

Julia S. Meyer, К. А. Матвеев

Published Aug 4, 2013• 58 citations
OpenAlex

Abstract

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.

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Wigner crystal physics in quantum wires

Julia S Meyer, К. А. Матвеев

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.

arXiv:0808.2076•Journal of Physics Condensed Matter•Dec 10, 2008•126 citations

Spin coupling in zigzag Wigner crystals

A. D. Klironomos, Julia S. Meyer, Takashi Hikihara +1 more

We consider interacting electrons in a quantum wire in the case of a shallow confining potential and low electron density. In a certain range of densities, the electrons form a two-row (zigzag) Wigner crystal whose spin properties are determined by nearest and next-nearest neighbor exchange as well as by three- and four-particle ring exchange processes. The phase diagram of the resulting zigzag spin chain has regions of complete spin polarization and partial spin polarization in addition to a number of unpolarized phases, including antiferromagnetism and dimer order as well as a novel phase generated by the four-particle ring exchange.

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Transition from a One-Dimensional to a Quasi-One-Dimensional State in Interacting Quantum Wires

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Upon increasing the electron density in a quantum wire, the one-dimensional electron system undergoes a transition to a quasi-one-dimensional state. In the absence of interactions between electrons, this corresponds to filling up the second subband of transverse quantization, and there are two gapless excitation modes above the transition. On the other hand, strongly interacting one-dimensional electrons form a Wigner crystal, and the transition corresponds to it splitting into two chains (zigzag crystal). We show that the soft mode driving the transition to the zigzag state is gapped, and only one gapless mode exists above the transition. Furthermore, we establish that in the vicinity of the transition already arbitrarily weak interactions open a gap in the second mode. We then argue that only one gapless mode exists near the transition at any interaction strength.

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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.

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Conductance of a quantum wire at low electron density

К. А. Матвеев

We study the transport of electrons through a long quantum wire connecting two bulk leads. As the electron density in the wire is lowered, the Coulomb interactions lead to short-range crystalline ordering of electrons. In this Wigner crystal state the spins of electrons form an antiferromagnetic Heisenberg spin chain with exponentially small exchange coupling $J$. Inhomogeneity of the electron density due to the coupling of the wire to the leads results in violation of spin-charge separation in the device. As a result the spins affect the conductance of the wire. At zero temperature the low-energy spin excitations propagate freely through the wire, and its conductance remains $2{e}^{2}∕h$. Since the energy of the elementary excitations in the spin chain (spinons) cannot exceed $\ensuremath{\pi}J∕2$, the conductance of the wire acquires an exponentially small negative correction $\ensuremath{\delta}\phantom{\rule{0.2em}{0ex}}G\ensuremath{\propto}\ensuremath{-}\mathrm{exp}(\ensuremath{-}\ensuremath{\pi}J∕2T)$ at low temperatures $T⪡J$. At higher temperatures, $T⪢J$, most of the spin excitations in the leads are reflected by the wire, and the conductance levels off at a new universal value ${e}^{2}∕h$.

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Exchange coupling in a one-dimensional Wigner crystal

A. D. Klironomos, R. R. Ramazashvili, К. А. Матвеев

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.

Journal de Physique IV (Proceedings)•Dec 1, 2005•0 citations
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.
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.
We consider interacting electrons in a quantum wire in the case of a shallow confining potential and low electron density. In a certain range of densities, the electrons form a two-row (zigzag) Wigner crystal whose spin properties are determined by nearest and next-nearest neighbor exchange as well as by three- and four-particle ring exchange processes. The phase diagram of the resulting zigzag spin chain has regions of complete spin polarization and partial spin polarization in addition to a number of unpolarized phases, including antiferromagnetism and dimer order as well as a novel phase generated by the four-particle ring exchange.
Upon increasing the electron density in a quantum wire, the one-dimensional electron system undergoes a transition to a quasi-one-dimensional state. In the absence of interactions between electrons, this corresponds to filling up the second subband of transverse quantization, and there are two gapless excitation modes above the transition. On the other hand, strongly interacting one-dimensional electrons form a Wigner crystal, and the transition corresponds to it splitting into two chains (zigzag crystal). We show that the soft mode driving the transition to the zigzag state is gapped, and only one gapless mode exists above the transition. Furthermore, we establish that in the vicinity of the transition already arbitrarily weak interactions open a gap in the second mode. We then argue that only one gapless mode exists near the transition at any interaction strength.
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 JJJ 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.
We study the transport of electrons through a long quantum wire connecting two bulk leads. As the electron density in the wire is lowered, the Coulomb interactions lead to short-range crystalline ordering of electrons. In this Wigner crystal state the spins of electrons form an antiferromagnetic Heisenberg spin chain with exponentially small exchange coupling JJJ. Inhomogeneity of the electron density due to the coupling of the wire to the leads results in violation of spin-charge separation in the device. As a result the spins affect the conductance of the wire. At zero temperature the low-energy spin excitations propagate freely through the wire, and its conductance remains 2e2∕h2{e}^{2}∕h2e2∕h. Since the energy of the elementary excitations in the spin chain (spinons) cannot exceed $\ensuremath{\pi}J∕2$, the conductance of the wire acquires an exponentially small negative correction $\ensuremath{\delta}\phantom{\rule{0.2em}{0ex}}G\ensuremath{\propto}\ensuremath{-}\mathrm{exp}(\ensuremath{-}\ensuremath{\pi}J∕2T)$ at low temperatures T⪡JT⪡JT⪡J. At higher temperatures, T⪢JT⪢JT⪢J, most of the spin excitations in the leads are reflected by the wire, and the conductance levels off at a new universal value e2∕h{e}^{2}∕he2∕h.