2D Wigner Crystal Research Articles

A site-resolved two-dimensional quantum simulator with hundreds of trapped ions.

A large qubit capacity and an individual readout capability are two crucial requirements for large-scale quantum computing and simulation1. As one of the leading physical platforms for quantum information processing, the ion trap has achieved a quantum simulation of tens of ions with site-resolved readout in a one-dimensional Paul trap2-4 and of hundreds of ions with global observables in a two-dimensional (2D) Penning trap5,6. However, integrating these two features into a single system is still very challenging. Here we report the stable trapping of 512 ions in a 2D Wigner crystal and the sideband cooling of their transverse motion. We demonstrate the quantum simulation of long-range quantum Ising models with tunable coupling strengths and patterns, with or without frustration, using 300 ions. Enabled by the site resolution in the single-shot measurement, we observe rich spatial correlation patterns in the quasi-adiabatically prepared ground states, which allows us to verify quantum simulation results by comparingthe measured two-spin correlations with the calculated collective phonon modes and with classical simulated annealing. We further probe the quench dynamics of the Ising model in a transverse field to demonstrate quantum sampling tasks. Our work paves the way for simulating classically intractable quantum dynamics and for running noisy intermediate-scale quantum algorithms7,8 using 2D ion trap quantum simulators.

Experimental observations of Wigner crystals

In 1934, Eugene Wigner, who was studying at Princeton University, predicted the existence of electronic crystals. Electrons have both kinetic energy and potential energy of interaction. When the density of electronic states satisfies certain conditions, due to the repulsion between electrons, electrons will tend to arrange themselves in a regular lattice structure, forming electron crystals, which is also known as Wigner crystals. For nearly 90 years, Wigner crystals have fascinated condensed matter physicists. Physicists have designed many ingenious semiconductor heterojunctions to obtain lower electron densities and added magnetic fields to achieve larger effective mass of electron. In 1979, experiments revealed the existence of a phase transition from an electron liquid phase to an electron crystal on the surface of liquid helium, and subsequent experiments observed the characteristics of two-dimensional(2D) Wigner crystals in 2D electron gas under high magnetic fields. However, direct observation of 2D Wigner lattices in real space remains a formidable challenge. Through the graphene sensing layer of <inline-formula><tex-math id="M3">\begin{document}$ {WSe}_{2} $\end{document}</tex-math></inline-formula>/<inline-formula><tex-math id="M4">\begin{document}$ {WS}_{2} $\end{document}</tex-math></inline-formula>moiré superlattice, Hongyuan Li, Wang Feng et al. observed the real-space morphologies of Wigner crystals in their experiments. And in a recent study, researchers used high-resolution scanning tunneling microscopy to directly image magnetic field-induced Wigner crystals in bernal stacking bilayer graphene and investigated their structural properties as a function of electron density, magnetic field, and temperature. In this paper, we will introduce some interesting things about Wigner crystals through four representative researches briefly.

Possible fractional quantum Hall effect and Wigner crystal states in two-dimensional Coulomb gases as a result of anyon correlations

Recently, we have derived an approximate analytic formula for the ground state energy of (2D) charged anyon gas. This result allows us in the present paper to estimate the exchangecorrelation energy, which is as it is well known from a literature the difference of the ground state energy and the Hartree-Fock energy, for fermions with the anyon parameter ν=1, and to formulate the anyon correlation energy for other ν<1. The minima of the last one for all fixed values of ν as well as the minima of the ground state energy at the same ν are only the ν dependent, which means they are only the anyon statistics dependent with no effect of the Coulomb interaction. From this, we conclude that there should appear the localized real space macroscopic structure for any fixed ν from ν<1. For instance, since the Landau level fillingfactor (LLFF) of the fractional quantum Hall effect (FQHE) quasi-particles is equal to the anyon parameter then these quasi-particles should condense in the some real space structure at this LLFF, forming the FQHE plateau and showing that the FQHE is a result of anyoncorrelations. On the other hand, expressing the above mentioned minima function of ν through the density parameter rs of the Coulomb gas, we obtain the values of rs, for which the 2D Wigner crystal (WC) can be observed. It is interesting that the simultaneous observation of the FQHE and WC states in accord with our consideration.

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.

Ground States of Crystalline Caps: Generalized Jellium on Curved Space.

We study the ground states of crystals on spherical surfaces. These ground states consist of positive disclination defects in structures spanning from flat and weakly curved caps to closed shells. Comparing two continuum theories and one discrete-lattice simulation, we first investigate the transition between defect-free caps to single-disclination ground states and show it to be continuous and symmetry breaking. Further, we show that ground states adopt icosahedral subgroup symmetries across the full range of curvatures, even far from the closure of complete shells. While superficially similar to other models of 2D "jellium" (e.g., superconducting disks and 2D Wigner crystals), the interplay between the free edge of caps and the non-Euclidean geometry of its embedding leads to nontrivial ground state behavior that is without counterpart in planar jellium models.

2D Wigner crystals on silicon surface induced by nanosecond pulsed laser

During pulsed laser interaction on silicon surface, the transient electron lattices occur on plasma resonance induced from faster photons on surface in our experiment. Here, it is interesting that the electron crystals can be built on plasma surface induced by intensive faster photons on silicon, which is similar with Wigner crystal in intensive electric field or intensive magnetic field. In the two-dimensional (2D) Bravais lattices, the electron crystals of three-fold, four-fold and six-fold symmetry lattices can be observed on plasma surface by using the reflecting Talbot effect images. On these surface period lattices, the decay spectrum of faster emission on the 2D transient electron crystal was measured, which is related to the faster decay peak with lifetime of 100 ns. These new phenomena have a good application in preparing the PN junction of photo-electronic devices, where the surface electron gas on plasma surface can inject into the top layer to build N type region, and the positive ions under plasma can inject into the bottom layer for forming P type region on silicon with defects in vacuum.

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.

Study of Wigner crystal in n-GaAs/AlGaAs by surface acoustic waves

The surface acoustic wave (SAW) technique was used for studying a 65 nm wide GaAs quantum well containing a two dimensional electron system (2DES) with the density of 5·1010 cm-2 and the mobility of 8·106 cm2/V·s. A SAW propagated in proximity of the quantum well. Both SAW attenuation α and SAW velocity V were measured in perpendicular magnetic field of up to 18 T in the frequency f range 28.5–306 MHz at temperatures 40–380 mK. Then, both real and imaginary parts of the ac conductance σ(f) = σ1 (f) + i·σ2 (f) of the 2DES were calculated from α and V. Dependencies of α, V, σ1, and σ2 on magnetic fields showed a rich oscillation pattern, in particular, the fractional quantum Hall effect (FQHE). Thus, at low temperature, the FQHE state was observed at the filling factor ν = 0.20, whereas, in vicinity of this value, at ν = 0.19 and 0.21, the 2DES behaved as a 2D Wigner crystal weakly pinned by a disorder random potential. The calculated pinning frequencies at high and zero magnetic fields were 193 MHz and 12....

Signatures for Wigner crystal formation in the chemical potential of a two-dimensional electron system.

We investigate the evolution of the chemical potential of a two-dimensional electron system (2DES) as a function of density at a fixed magnetic field. By using a bilayer system, changes in the chemical potential of one 2DES are determined from the density variation induced in the second, nearby 2DES. At high magnetic fields around a filling factor of ν=1 or ν=2, the chemical potential jump associated with the condensation in a quantum Hall state exhibits two anomalies symmetrically located around these integer filling factors. They are attributed to the formation of a 2D Wigner crystal of quasiparticles.

TRIONS IN COUPLED QUANTUM WELLS AND WIGNER CRYSTALLIZATION

We consider a restricted three-body problem, where two interacted particles are located in a two-dimensional (2D) plane and interact with the third one located in the parallel spatially separated plane. The system of such type can be formed in the semiconductor coupled quantum wells, where the electrons (holes) and direct excitons spatially separated in different parallel neighboring quantum wells are sufficiently close to interact and form negative X- or positive X+ indirect trions. It is shown that at large interwell separations, when the interwell separation is much greater than the exciton Bohr radius, this problem can be solved analytically using the cluster approach. Analytical results for the energy spectrum and the wavefunctions of the spatially indirect trion are obtained, their dependencies on the interwell separations is analyzed and a conditional probability distribution is calculated. The formation of 2D Wigner crystal of trions at the low densities is predicted. It is shown that the critical density of the formation of the trion Wigner crystal is sufficiently greater than the critical density of the electron Wigner crystal in the same material.

DISTORTION OF THE 2D WIGNER CRYSTAL INTO A 'QUASI-3D' INSULATOR

Applying high magnetic fields parallel to the surface of a high-mobility 2D electron gas, we have recently observed a transition from the fractional quantum Hall liquid to an insulating phase, for filling factors as high as ν = 0.8 [B. A. Piot et al., Nature Phys. doi:10.1038/nphys1094]. This transition can be seen as a continuous distortion of the 2D Wigner crystal into a 'quasi-3D' insulating state. Here, we provide additional arguments confirming that carrier localization cannot be accounted for by disorder effects alone, and instead is most likely due to the stabilization of an electron solid by the strong parallel magnetic field in a 'quasi-3D' geometry.

Analytic approach to the ground-state energy of charged anyon gases

We derive an approximate analytic formula for the ground-state energy of the charged anyon gas. Our approach is based on the harmonically confined two-dimensional (2D) Coulomb anyon gas and a regularization procedure for vanishing confinement. To take into account the fractional statistics and Coulomb interaction we introduce a function, which depends on both the statistics and density parameters ($\ensuremath{\nu}$ and ${r}_{s}$, respectively). We determine this function by fitting to the ground-state energies of the classical electron crystal at very large ${r}_{s}$ (the 2D Wigner crystal), and to the Hartree-Fock (HF) energy of the spin-polarized 2D electron gas, and the dense 2D Coulomb Bose gas at very small ${r}_{s}$. The latter is calculated by use of the Bogoliubov approximation. Applied to the boson system $(\ensuremath{\nu}=0)$ our results are very close to recent results from Monte Carlo (MC) calculations. For spin-polarized electron systems $(\ensuremath{\nu}=1)$ our comparison leads to a critical judgment concerning the density range, to which the HF approximation and MC simulations apply. In dependence on $\ensuremath{\nu}$, our analytic formula yields ground-state energies, which monotonously increase from the bosonic to the fermionic side if ${r}_{s}>1$. For ${r}_{s}\ensuremath{\leqslant}1$ it shows a nonmonotonous behavior indicating a breakdown of the assumed continuous transformation of bosons into fermions by variation of the parameter $\ensuremath{\nu}$.

Ground-state energy and Wigner crystallization in thick two-dimensional electron systems

The ground-state energy of the two-dimensional (2D) Wigner crystal is determined as a function of the thickness of the electron layer and the crystal structure. The method of evaluating the exchange-correlation energy is tested using known results for the infinitely thin 2D system. Two methods, one based on the local-density approximation (LDA), and another based on the constant-density approximation (CDA) are established by comparing with quantum Monte Carlo (QMC) results. The LDA and CDA estimates for the Wigner transition of the perfect 2D fluid are at ${r}_{s}=38$ and 32, respectively, compared with ${r}_{s}=35\ifmmode\pm\else\textpm\fi{}5$ from QMC. For thick-2D layers as found in Hetero-junction-insulated-gate field-effect transistors, the LDA and CDA predictions of the Wigner transition are at ${r}_{s}=20.5$ and 15.5, respectively. Impurity effects are not considered here.

Universal Aspects of Coulomb-Frustrated Phase Separation

We study the consequences of Coulomb interactions on a system undergoing a putative first order phase transition. In two dimensions (2D), near the critical density, the system is universally unstable to the formation of new intermediate phases, which we call "electronic microemulsion phases," which consist of an intermediate scale mixture of regions of the two competing phases. A corollary is that there can be no direct transition as a function of density from a 2D Wigner crystal to a uniform electron liquid. In 3D, if the strength of the Coulomb interactions exceeds a critical value, no phase separation occurs, while for the weaker Coulomb strength electronic microemulsions are inevitable. This tendency is considerably more pronounced in anisotropic (quasi-2D or quasi-1D) systems, where a devil's staircase of transitions is possible.

Temperature dependence of the coupled electron–ripplon resonance spectrum of electron crystal over liquid helium

Spectra of coupled electron–ripplon oscillation of two-dimensional (2D) Wigner crystal with the surface electron density n s=6.4×10 8 cm −2 (melting temperature of 0.58 K) are measured under conditions of both complete and noncomplete screening of the holding electric field in temperature range 0.08–0.4 K. The frequencies of the resonances are compared with the results of calculation in the frameworks of a self-consistent theory by Monarkha and Shikin. It is shown that the amplitude of the (0,1) resonance mode tends to 0 as temperature reaches the melting temperature.

EXCHANGE FREQUENCIES IN THE 2D WIGNER CRYSTAL

Path Integral Monte Carlo is used to calculate exchange frequencies as electrons undergo ring exchanges in a "clean" 2d Wigner crystal as a function of density. Agreement with WKB calculations is found at very low density, but the results show an enhanced increase with density near melting. Remarkably, the exchange Hamiltonian closely resembles the measured exchanges in 2d3 He . Using the resulting multi-spin exchange model we find the spin Hamiltonian for rs ≥ 175 ± 10 is a frustrated antiferromagnetic, with a spin liquid ground state. For lower density the ground state will be ferromagnetic. Some effects of a magnetic field are presented.

Microwave resonance of the 2D Wigner crystal around integer Landau fillings.

We have observed a resonance in the real part of the finite frequency diagonal conductivity using microwave absorption measurements in high quality 2D electron systems near integer fillings. The resonance exists in some neighborhood of filling factor around corresponding integers and is qualitatively similar to previously observed resonance of weakly pinned Wigner crystal in high B and very small filling factor regime. Data measured around both nu=1 and nu=2 are presented. We interpret the resonance as the signature of the Wigner crystal state around integer Landau levels.

Two-dimensional Wigner crystal in anisotropic semiconductors

We investigate the effect of mass anisotropy on the Wigner crystallization transition in a two-dimensional (2D) electron gas. The static and dynamical properties of a 2D Wigner crystal have been calculated for arbitrary 2D Bravais lattices in the presence of anisotropic mass, as may be obtainable in Si MOSFETs with (110) surface. By studying the stability of all possible lattices, we find significant change in the crystal structure and melting density of the electron lattice with the lowest ground state energy.

ELECTRON–ELECTRON INTERACTIONS IN THE 2D FERROMAGNETIC ELECTRON FLUID

Motivated by current interest in spin-polarized states of the electron fluid at strong coupling, we evaluate the effective screened potential V↑↑(r) between two electrons in the 2D ferromagnetic jellium model. Exchange and correlation are treated in local approximation, following the approach given by C. A. Kukkonen and A. W. Overhauser (Phys. Rev.B20, 550 (1979)) for the paramagnetic fluid in 3D. The main features of V↑↑(r) in the fluid near freezing are discussed with reference to Wigner crystallization and to paired interstitial defects in the 2D Wigner crystal.

Polarization vectors in a 2D Wigner crystal

We analyse the polarization vectors of two acoustic phonon modes propagating in a two-dimensional (2D) electron crystal formed above a dielectric layer (liquid He) on a metallic substrate. The electrons are described with a finite extent in the perpendicular direction, thus forming a quasi-2D Wigner lattice. We have found that the acoustic modes are strictly polarized in the longitudinal ( L) and transverse ( T) direction only close to the centre of the first Brillouin zone, whereas towards the zone edge the polarization can be shifted from the L and T directions (depending on the propagating wave direction) up to 30°. We have also shown that the shape of the electron wave function has no significant influence on the polarization vectors, i.e. similar results are derived for a delocalized as well as a strictly localized (confined to a plane) 2D Wigner crystal.