Two-dimensional Wigner Crystal Research Articles

Sublattice Structure and Topology in Spontaneously Crystallized Electronic States.

The prediction and realization of the quantum anomalous Hall effect are often intimately connected to honeycomb lattices in which the sublattice degree of freedom plays a central role in the nontrivial topology. Two-dimensional Wigner crystals, on the other hand, form triangular lattices without sublattice degrees of freedom, resulting in a topologically trivial state. Here, we discuss the possibility of spontaneously formed honeycomb-lattice crystals that exhibit the quantum anomalous Hall effect. Starting from a single-band system with nontrivial quantum geometry, we derive the mean-field energy functional of a class of crystal states and express it as a model of sublattice pseudospins in momentum space. We find that nontrivial quantum geometry leads to extra terms in the pseudospin model that break an effective "time-reversal symmetry" and favor a topologically nontrivial pseudospin texture. When the effects of these extra terms dominate over the ferromagnetic exchange coupling between pseudospins, the anomalous Hall crystal state becomes energetically favorable over the trivial Wigner crystal state.

The Emergence of the Hexagonal Lattice in Two-Dimensional Wigner Fragments.

At very low density, the electrons in a uniform electron gas spontaneously break symmetry and form a crystalline lattice called a Wigner crystal. But which type of crystal will the electrons form? We report a numerical study of the density profiles of fragments of Wigner crystals from first principles. To simulate Wigner fragments, we use Clifford periodic boundary conditions and a renormalized distance in the Coulomb potential. Moreover, we show that high-spin restricted open-shell Hartree-Fock theory becomes exact in the low-density limit. We are thus able to accurately capture the localization in two-dimensional Wigner fragments with many electrons. No assumptions about the positions where the electrons will localize are made. The density profiles we obtain emerge naturally when we minimize the total energy of the system. We clearly observe the emergence of the hexagonal crystal structure, which has been predicted to be the ground-state structure of the two-dimensional Wigner crystal.

Interstitial-Induced Ferromagnetism in a Two-Dimensional Wigner Crystal.

The two-dimensional Wigner crystal (WC) occurs in the strongly interacting regime (r_{s}≫1) of the two-dimensional electron gas (2DEG). The magnetism of a pure WC is determined by tunneling processes that induce multispin ring-exchange interactions, resulting in fully polarized ferromagnetism for large enough r_{s}. Recently, Hossain etal. [Proc. Natl. Acad. Sci. U.S.A. 117, 32244 (2020)PNASA60027-842410.1073/pnas.2018248117] reported the occurrence of a fully polarized ferromagnetic insulator at r_{s}≳35 in an AlAs quantum well, but at temperatures orders of magnitude larger than the predicted exchange energies for the pure WC. Here, we analyze the large r_{s} dynamics of an interstitial defect in the WC, and show that it produces local ferromagnetism with much higher energy scales. Three hopping processes are dominant, which favor a large, fully polarized ferromagnetic polaron. Based on the above results, we speculate concerning the phenomenology of the magnetism near the metal-insulator transition of the 2DEG.

Imaging two-dimensional generalized Wigner crystals.

The Wigner crystal1 has fascinated condensed matter physicists for nearly 90 years2-14. Signatures of two-dimensional (2D) Wigner crystals were first observed in 2D electron gases under high magnetic field2-4, and recently reported in transition metal dichalcogenide moiré superlattices6-9. Direct observation of the 2D Wigner crystal lattice in real space, however, has remained an outstanding challenge. Conventional scanning tunnelling microscopy (STM) has sufficient spatial resolution but induces perturbations that can potentially alter this fragile state. Here we demonstrate real-space imaging of 2D Wigner crystals in WSe2/WS2 moiré heterostructures using a specially designed non-invasive STM spectroscopy technique. This employs a graphene sensing layer held close to the WSe2/WS2 moiré superlattice. Local STM tunnel current into the graphene layer is modulated by the underlying Wigner crystal electron lattice in the WSe2/WS2 heterostructure. Different Wigner crystal lattice configurations at fractional electron fillings of n = 1/3, 1/2 and 2/3, where n is the electron number per site, are directly visualized. The n = 1/3 and n = 2/3 Wigner crystals exhibit triangular and honeycomb lattices, respectively, to minimize nearest-neighbour occupations. The n = 1/2 state spontaneously breaks the original C3 symmetry and forms a stripe phase. Our study lays a solid foundation for understanding Wigner crystal states in WSe2/WS2 moiré heterostructures and provides an approach that is generally applicable for imaging novel correlated electron lattices in other systems.

Bilayer Wigner crystals in a transition metal dichalcogenide heterostructure.

One of the first theoretically predicted manifestations of strong interactions in many-electron systems was the Wigner crystal1-3, in which electrons crystallize into a regular lattice. The crystal can melt via either thermal or quantum fluctuations4. Quantum melting of the Wigner crystal is predicted to produce exotic intermediate phases5,6 and quantum magnetism7,8 because of the intricate interplay of Coulomb interactions and kinetic energy. However, studying two-dimensional Wigner crystals in the quantum regime has often required a strong magnetic field9-11 or a moiré superlattice potential12-15, thus limiting access to the full phase diagram of the interacting electron liquid. Here we report the observation of bilayer Wigner crystals without magnetic fields or moiré potentials in an atomically thin transition metal dichalcogenide heterostructure, which consists of two MoSe2 monolayers separated by hexagonal boron nitride. We observe optical signatures of robust correlated insulating states at symmetric (1:1) and asymmetric (3:1, 4:1 and 7:1) electron doping of the two MoSe2 layers at cryogenic temperatures. We attribute these features to bilayer Wigner crystals composed of two interlocked commensurate triangular electron lattices, stabilized by inter-layer interaction16. The Wigner crystal phases are remarkably stable, and undergo quantum and thermal melting transitions at electron densities of up to 6×1012 per square centimetre and at temperatures of up to about 40kelvin. Our results demonstrate that an atomically thin heterostructure is a highly tunable platform for realizing many-body electronic states and probing their liquid-solid and magnetic quantum phase transitions4-8,17.

Approach towards a quantum crystal structure of a two dimensional gas of particles with delta function interaction

Inspired by recent studies of the two-dimensional Wigner crystals for electrons, we study the time evolution of a gas of particles interacting with a Dirac pair potential. We find that the gas crystalizes into a square lattice for low free-particle quantum numbers.

Vacancions in the two-dimensional Wigner crystal

From a new point of view, the recombination of two-dimensional electrons enclosed in a MgZnO/ZnO heterojunction with localized holes in the valence band is considered. It is suggested to consider quasi-holes in the two-dimensional electron system as quasiparticles in the Wigner crystal. The quasiparticles corresponding to electrons removed from the Wigner crystal are vacancions. Because of quantum tunneling effect, they are not localized. The vacancion energies E(k) form a band of width D which is proportional to the vacancy tunneling probability. The width D corresponds to the photoluminescence band of the two-dimensional electron system. The shape of the photoluminescence band of the Wigner crystal is considered within the tight-binding approximation for dispersion E(k) and compared with the experimental results.

Hartree-Fock description of a Wigner crystal in two dimensions

Abstract We perform numerical studies of strongly interacting electrons on a two-dimensional lattice in the Hartree-Fock approximation. The results clearly show significant ordering of the positions of the electrons with decreasing temperature signaling the formation of a two-dimensional Wigner crystal. The degree of order as a function of temperature is in a good qualitative agreement with experimental results for the threshold voltage at which the depinning of the crystal occurs. In the limit of very strong electron-electron interactions, the ordering of the electrons becomes almost perfect.

Structural transitions and plasma oscillations in electron chains over liquid helium

A two-dimensional Wigner crystal formed by electrons located above a rectangular electrode with uniformly distributed positive charge has been studied by molecular dynamics methods. The authors have analyzed not only possible electron configurations over a narrow electrode region at a fixed electron density, depending on the width of the electrode, but also patterns of transitions from one configuration to another. In addition, this paper considers an electron configuration consisting of several chains. Using the Fourier transform for time dependencies of average electron coordinates during modeling, the authors have determined plasma oscillation spectra. Frequencies observed during modeling have been compared with theoretical frequencies for one and two chains.

Topological Phonon Modes in a Two-Dimensional Wigner Crystal**Supported by the National Basic Research Program of China under Grant No 2015CB921101, and the National Natural Science Foundation of China under Grant No 11325416.

We investigate the spin–orbit coupling effect in a two-dimensional (2D) Wigner crystal. It is shown that sufficiently strong spin–orbit coupling and an appropriate sign of g-factor could transform the Wigner crystal to a topological phonon system. We demonstrate the existence of chiral phonon edge modes in finite size samples, as well as the robustness of the modes in the topological phase. We explore the possibility of realizing the topological phonon system in 2D Wigner crystals confined in semiconductor quantum wells/heterostructure. It is found that the spin–orbit coupling is too weak for driving a topological phase transition in these systems. It is argued that one may look for topological phonon systems in correlated Wigner crystals with emergent effective spin–orbit coupling.

Transport of a sliding Wigner crystal in the four flux composite fermion regime

In two-dimensional (2D) electron systems, Wigner crystals (WC) and fractional quantum Hall effect (FQHE) liquids are competing ground states under low temperatures $(T)$ and high magnetic fields $(B)$. Here we report differential conductivity results demonstrating the reentrant insulating phase around $\ensuremath{\nu}=1$/5 in a 2D hole system in AlGaAs/GaAs quantum wells and unexpected features in the solid-liquid phase transition between WC and FQHE liquids in ultrahigh magnetic fields up to 45 T. Remarkably, the electric field $(E)$ plays an equivalent role as the temperature does in our phase diagram. From the $E\ensuremath{-}T$ ``duality'' analysis, a characteristic length of 450 nm is derived, which can be understood as the phase-coherent domain size of WC. Moreover, evidence shows that with weak disorder the insulating phase and composite fermion liquid could be coexisting around $\ensuremath{\nu}=$ 1/5, pointing to the possibility that the insulating phase is the four flux quantum Wigner crystal, as proposed by theories.

The melting of the classical two-dimensional Wigner crystal

We report an extensive Monte Carlo study of the melting of the classical two-dimensional Wigner crystal for a system of point particles interacting via the 1/r-Coulomb potential. A hexatic phase is found in large enough systems. With the multiple histograms method and the finite-size scaling theory, we show that the fluid/hexatic phase transition is weakly first order. No set of critical exponents, consistent with a Kosterlitz-Thouless transition and the finite-size scaling analysis for this transition, has been found.

Critical field for dynamic transitions in two-dimensional electron crystals under liquid helium

Dynamic transitions in two-dimensional Wigner crystals over a liquid helium surface are studied under conditions of incomplete screening of the holding potential. The critical electric field that produces a transition is found to depend on the surface density of the electron layer and not on the holding field. It is assumed that the dynamic transition is associated primarily with the destruction of the spatial order of the system, which leads to delocalization of electrons and, thereby, to a change in the electro-ripplon interaction.

Nambu-Goldstone bosons with fractional-power dispersion relations

We pin down the origin of a peculiar dispersion relation of domain wall fluctuation, the so-called ripplon, in a superfluid-superfluid interface. A ripplon has a dispersion relation $\ensuremath{\omega}\ensuremath{\propto}{k}^{3/2}$ due to the nonlocality of the effective Lagrangian, which is mediated by gapless superfluid phonons in the bulk. We point out the analogy to the longitudinal phonon in the two-dimensional Wigner crystal.

Taking one charge off a two-dimensional Wigner crystal

A planar array of identical charges at vanishing temperature forms a Wigner crystal with hexagonal symmetry. We take off one (reference) charge in a perpendicular direction, hold it fixed, and search for the ground state of the whole system. The planar projection of the reference charge should then evolve from a sixfold coordination (centre of a hexagon) for small distances to a threefold arrangement (centre of a triangle), at large distances d from the plane. The aim of this paper is to describe the corresponding non-trivial lattice transformation. For that purpose, two numerical methods (direct energy minimisation and Monte Carlo simulations), together with an analytical treatment, are presented. Our results indicate that the d = 0 and d → ∞ limiting cases extend for finite values of d from the respective starting points into two sequences of stable states, with intersecting energies at some value dt; beyond this value the branches continue as metastable states.

Spin dephasing as a probe of mode temperature, motional state distributions, and heating rates in a two-dimensional ion crystal

We employ spin-dependent optical dipole forces to characterize the transverse center-of-mass (COM) motional mode of a two-dimensional Wigner crystal of hundreds of $^9$Be$^+$. By comparing the measured spin dephasing produced by the spin-dependent force with the predictions of a semiclassical dephasing model, we obtain absolute mode temperatures in excellent agreement with both the Doppler laser cooling limit and measurements obtained from a previously published technique (B. C. Sawyer et al. Phys. Rev. Lett. \textbf{108}, 213003 (2012)). Furthermore, the structure of the dephasing histograms allows for discrimination between initial thermal and coherent states of motion. We also apply the techniques discussed here to measure, for the first time, the ambient heating rate of the COM mode of a 2D Coulomb crystal in a Penning trap. This measurement places an upper limit on the anomalous single-ion heating rate due to electric field noise from the trap electrode surfaces of $\frac{d\bar{n}}{dt}\sim 5$ s$^{-1}$ for our trap at a frequency of 795 kHz, where $\bar{n}$ is the mean occupation of quantized COM motion in the axial harmonic well.

Macroscopic Quantum System, Highly Correlated Electron State, and High-Temperature Superconductivity in Iron Pnictides

It is shown that the qualitative model of the high-temperature superconductivity suggested earlier for cuprates and doped picene and based on the idea that the valence electron state depends on the character of the chemical bonds they form and on the Coulomb interaction between the electrons is not only confirmed by the experimental data on iron pnictides but is also improved. From the chemical point of view, the high-temperature superconductivity is associated with additional $\pi$ bonding along chains of covalently bonded ions via a delocalized $\pi$ orbital, just like in cuprates. From the physical point of view, as the data on iron pnictides show, the superconductivity is associated with a FeAs layer transition into the state similar to a macroscopic quantum system characterized by a highly correlated electron state, formation of two-dimensional crystals of electron pairs with quantized energy levels, and a strong Coulomb interaction between these crystals. Superconductivity in such a system is accomplished by a two-dimensional Wigner crystal consisting of one-dimensional Wigner crystals formed by bosons, i.e., singlet electron pairs that are in the same quasi-one-dimensional state extending along the ion chain, which corresponds to a delocalized $\pi$ orbital in chemistry. The model applicability to three different classes of materials (cuprates, picene, iron pnictides) indicates that it can prove useful for development of the theory of superconductivity taking into consideraion the highly correlated state of valence electrons and strong Coulomb interactions between the electrons.

Wigner-crystal formulation of strong-coupling theory for counterions near planar charged interfaces

We present a new analytical approach to the strong electrostatic coupling regime (SC) that can be achieved equivalently at low temperatures, high charges, low dielectric permittivity, etc. Two geometries are analyzed in detail: one charged wall first, and then two parallel walls at small distances that can be likely or oppositely charged. In all cases, only one type of mobile counterions is present, and ensures electroneutrality (salt-free case). The method is based on a systematic expansion around the ground state formed by the two-dimensional Wigner crystal(s) of counterions at the plate(s). The leading SC order stems from a single-particle theory, and coincides with the virial SC approach that has been much studied in the last 10 years. The first correction has the functional form of the virial SC prediction, but the prefactor is different. The present theory is free of divergences and the obtained results, both for symmetrically and asymmetrically charged plates, are in excellent agreement with available data of Monte Carlo simulations under strong and intermediate Coulombic couplings. All results obtained represent relevant improvements over the virial SC estimates. The present SC theory starting from the Wigner crystal and therefore coined Wigner SC, sheds light on anomalous phenomena like the counterion mediated like-charge attraction, and the opposite-charge repulsion.

Counterions at Highly Charged Interfaces: From One Plate to Like-Charge Attraction

We present an analytical approach for similarly and highly charged planar interfaces in the presence of counterions. The procedure is physically transparent and based on an exact low temperature expansion around the ground state formed by the two-dimensional Wigner crystal of counterions. The one plate problem is worked out, together with the two plates situation. Unlike previous approaches, the expansion is free of divergences, and is shown to be in excellent agreement with available data of MonteCarlo simulations under strong Coulombic couplings. In the two plates case, the present results shed light on the like-charge attraction regime.

Diagrammatic quantum field formalism for localized electrons

We introduce a diagrammatic quantum field formalism for the evaluation of normalized expectation values of operators, and suitable for systems with localized electrons. It is used to develop a convergent series expansion for the energy in powers of overlap integrals of single-particle orbitals. This method gives intuitive and practical rules for writing down the expansion to arbitrary order of overlap, and can be applied to any spin configuration and to any dimension. Its applicability for systems with well localized electrons has been illustrated with examples, including the two-dimensional Wigner crystal and spin-singlets in the low-density electron gas.