Graphene, with its massless linearly-dispersing carriers, in the quantum Hall regime provides an instructive comparison with conventional two-dimensional (2D) systems in which carriers have a nonzero band mass and quadratic dispersion. We investigate the influence of Landau level mixing in graphene on Wigner crystal states in the $n^\mathrm{th}$ Landau level obtained using single Landau level approximation. We show that the Landau level mixing does not qualitatively change the phase diagram as a function of partial filling factor $ν$ in the $n^\mathrm{th}$ level. We find that the inter-Landau level mixing, quantified by relative occupations of the two Landau levels, $ρ_{n+1}/ρ_{n}$, oscillates around 2% and, in general, remains small ($< 4%$) irrespective of the Landau level index $n$. Our results show that the single Landau level approximation is applicable in high Landau levels, even though the energy gap between the adjacent Landau levels vanishes.
Wigner crystallization can be induced in a quantum dot by increasing the effective electron-electron interaction through a decrease of the electron density or by the application of a strong magnetic field. We show that the ground state in both cases is very similar but the energy scales are very different and therefore also the dynamics.
Wigner crystallization in mesoscopic quantum dots containing only few ($N < 50$) electrons exhibits a number of interesting peculiarities: (i) there exist two distinct crystal phases, and (ii) the phase boundary sensitively depends on the precise particle number. In this paper we demonstrate that this behavior can be used to control the {\em collective} transport properties by adding or removing a {\em single electron}.
We study the competition between the Wigner crystal and the Laughlin liquid states in an ultracold quasi two-dimensional rapidly rotating polarized fermionic dipolar gas, and find that the Wigner crystal has a lower energy below a critical filling factor. We examine the quantum crystal to liquid transition for different confinements in the third direction. Our analysis of the phonon spectra of the Wigner crystal with the account of phonon-phonon interactions also shows the stability of the Wigner crystal for sufficiently low filling factors (ν<1/7).
We provide a quantitative determination of the crystallization onset for two electrons in a parabolic two-dimensional confinement. This system is shown to be well described by a roto-vibrational model, Wigner crystallization occurring when the rotational motion gets decoupled from the vibrational one. The Wigner molecule thus formed is characterized by its moment of inertia and by the corresponding sequence of rotational excited states. The role of a vertical magnetic field is also considered. Additional support to the analysis is given by the Hartree-Fock phase diagram for the ground state an…
Using many-body configuration interaction techniques we show that Wigner crystallization occurs at the zigzag edges of graphene at surprisingly high electronic densities up to $0.8$ $\mbox{nm}^{-1}$. In contrast with one-dimensional electron gas, the flat-band structure of the edge states makes the system interaction dominated, facilitating the electronic localization. The resulting Wigner crystal manifests itself in pair-correlation functions, and evolves smoothly as the edge electron density is lowered. We also show that the crystallization affects the magnetization of the edges. While the e…
We explore the theory of electrons confined by one dimensional power law potentials. We calculate the density profile in the high density electron gas, the low density Wigner crystal, and the intermediate regime. We extract the momentum space wavefunction of the electron at the Fermi surface, which can be measured in experiments on tunneling between parallel wires. The onset of localization leads to a dramatic broadening of the momentum space wavefunction together with pronounced sharpening (in energy) of the tunneling spectrum.
Magnetic-Field-Induced Wigner Crystallization of Charged Interlayer Excitons in van der Waals Heterostructures
North Carolina Central University（北卡罗来纳中央大学）；Institute of Spectroscopy, Russian Academy of Sciences（俄罗斯科学院光谱学研究所）；National Research University ”Higher School of Economics”（国立研究大学高等经济大学）
We use Density Functional Theory to study interacting spinless electrons on a one-dimensional quantum ring in the density range where the system undergoes Wigner crystallization. The Wigner transition leads to a drastic ``collective'' electron localization due to the Wigner crystal pinning, provided a weak impurity potential is applied. To reveal this localization we examine a persistent current in a ring penetrated by a magnetic flux. Using the DFT-OEP method we calculated the current as a function of the interaction parameter r_S. We find that in the limit of vanishing impurity potential the…
Wigner crystallization of electrons in a 2D quantum dots is reported. It proceeds in two stages: I) via radial ordering of electrons on shells and II) freezing of the inter-shell rotation. The phase boundary of the crystal is computed in the whole temperature--density plane, and the influence of quantum effects and of the particle number is analyzed.
We study the possibility of Wigner crystallization in both single- and and bi-layer graphene using a real space tight binding model. In addition to verifying our earlier prediction for single layer graphene, we predict that the bilayer graphene can undergo a Wigner crystal transition at low enough carrier density thus offering a possibility of tunable and bipolar Wigner crystal. We find that aside from the Coulomb interaction of the carriers of two layers that stabilizes the charge ordered state, so does the inter layer coupling.
Optical probing of Wigner crystallization in monolayer WSe$_2$ via diffraction of longitudinal excitons
ITMO University（ITMO大学）；Moscow Institute of Physics and Technology (National Research University)（莫斯科物理科学与技术学院（国立研究大学））；Russian Quantum Center（俄罗斯量子中心）；Laboratory of Programmable Functional Materials, Center for Neurophysics and Neuromorphic Technologies（可编程功能材料实验室，神经物理与类脑技术中心）；Moscow Center for Advanced Studies（莫斯科高级研究中心）；Queen’s University（女王大学）
We consider a restricted three body problem, where two interacted particles are located in 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 (or holes) and direct excitons spatially separated in different parallel neighboring quantum wells that 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 much greater than the exci…
Competition between liquid and solid states in two-dimensional electron system is an intriguing problem in condensed matter physics. We have investigated competing Wigner crystal and fractional quantum Hall ( FQH ) liquid phases in atomically thin suspended graphene devices in Corbino geometry. Low temperature magnetoconductance and transconductance measurements along with $IV$ characteristics all indicate strong charge density dependent modulation of electron transport. Our results show unconventional FQH phases which do not fit the standard Jain's series for conventional FQH states, instead …
We investigate the effects of coupling between the two zigzag edges of graphene nanoribbons on the Wigner crystallization of electrons and holes using a combination of tight-binding, mean field Hubbard and many-body configuration interaction methods. We show that the thickness of the nanoribbon plays a crucial role in the formation of Wigner crystal. For ribbon widths smaller than 16 \mboxÅ, increased kinetic energy overcomes the long-range Coulomb repulsion and suppresses the Wigner crystallization. For wider ribbons up to 38 \mboxÅ wide, strong Wigner localization is observed for even number…
The quantum-classical crossover from the Fermi liquid towards the Wigner solid is numerically revisited, considering small square lattice models where electrons interact via a Coulomb $U/r$ potential. The studies of models without disorder and spin and including disorder and spin show that the electron solid is formed in two stages, giving rise to an intriguing solid-liquid regime at intermediate couplings
We consider the fate of the Wigner crystal state in a two dimensional system of massive Dirac electrons as the effective fine structure constant $α$ is increased. In a Dirac system, larger $α$ naively corresponds to stronger electron-electron interactions, but it also implies a stronger interband dielectric response that effectively renormalizes the electron charge. We calculate the critical density and critical temperature associated with quantum and thermal melting of the Wigner crystal state using two independent approaches. We show that at $α\gg 1$, the Wigner crystal state is best underst…
The ground state energy and the lowest excitations of a two dimensional Wigner crystal in a perpendicular magnetic field with one and two electrons per cell is investigated. In case of two electrons per lattice site, the interaction of the electrons {\em within} each cell is taken into account exactly (including exchange and correlation effects), and the interaction {\em between} the cells is in second order (dipole) van der Waals approximation. No further approximations are made, in particular Landau level mixing and {\em in}complete spin polarization are accounted for. Therefore, our calcula…
We study the ground states of cold atoms in the tight-binding bands built from p-orbitals on a two dimensional honeycomb optical lattice. The band structure includes two completely flat bands. Exact many-body ground states with on-site repulsion can be found at low particle densities, for both fermions and bosons. We find crystalline order at n=1/6 with a $\sqrt{3} \times \sqrt{3}$ structure breaking a number of discrete lattice symmetries. In fermionic systems, if the repulsion is strong enough, we find the bonding strength becomes \emph{dimerized} at n=1/2. Experimental signatures of crystal…
In a certain regime of low carrier densities and strong correlations, electrons can crystallize into a periodic arrangement of charge known as Wigner crystal. Such phases are particularly interesting in one dimension (1D) as they display a variety of charge and spin ground states which may be harnessed in quantum devices as high-fidelity transmitters of spin information. Recent theoretical studies suggest that the strong Coulomb interactions in Mott insulators and other flat band systems, may provide an attractive higher temperature platform for Wigner crystallization, but due to materials and…
We devise a model to explain why twisted bi-layer graphene (TBLG) exhibits insulating behavior when $ν=2,3$ charges occupy a unit moiré cell, a feature attributed to Mottness, but not for $ν=1$, clearly inconsistent with Mott insulation. We compute $r_s=E_U/E_K$, where $E_U$ and $E_K$ are the potential and kinetic energies, respectively, and show that (i) the Mott criterion lies at a density $10^4$ higher than in the experiments and (ii) a transition to a series of Wigner crystalline states exists as a function of $ν$. We find, for $ν=1$, $r_s$ fails to cross the threshold ($r_s = 37$) for the…
When a strong magnetic field is applied perpendicularly (along z) to a sheet confining electrons to two dimensions (x-y), highly correlated states emerge as a result of the interplay between electron-electron interactions, confinement and disorder. These so-called fractional quantum Hall (FQH) liquids form a series of states which ultimately give way to a periodic electron solid that crystallizes at high magnetic fields. This quantum phase of electrons has been identified previously as a disorder-pinned two-dimensional Wigner crystal with broken translational symmetry in the x-y plane. Here, w…
Critical gate distance for Wigner crystallization in the two-dimensional electron gas
Center for Computational Quantum Physics, Flatiron Institute（Flatiron Institute 计算量子物理中心）；Department of Physics, Stanford University（斯坦福大学物理系）；Kavli Institute for Theoretical Physics, University of California, Santa Barbara（加州大学圣塔芭芭拉分校卡弗里理论物理研究所）；Department of Physics and Astronomy, Hofstra University（霍夫斯特拉大学物理与天文系）；Department of Physics, University of Wisconsin-Madison（威斯康星大学麦迪逊分校物理系）
The physics of flat band is novel and rich but difficult to access. In this regard, recently twisting of bilayer van der Waals (vdW)-bounded two-dimensional (2D) materials has attracted much attention, because the reduction of Brillouin zone will eventually lead to a diminishing kinetic energy. Alternatively, one may start with a 2D Kagome lattice, which already possesses flat bands at the Fermi level, but unfortunately these bands connect quadratically to other (dispersive) bands, leading to undesirable effects. Here, we propose, by first-principles calculation and tight-binding modeling, tha…
The competition between kinetic energy and Coulomb interactions in electronic systems can lead to complex many-body ground states with competing superconducting, charge density wave, and magnetic orders. Here we study the low temperature phases of a strongly interacting zinc-oxide-based high mobility two dimensional electron system that displays a tunable metal-insulator transition. Through a comprehensive analysis of the dependence of electronic transport on temperature, carrier density, in-plane and perpendicular magnetic fields, and voltage bias, we provide evidence for the existence of com…
We report on results of finite-size numerical studies of partially filled lowest Landau level at low electron filling factors. We find convincing evidence suggesting that electrons form Wigner Crystals at sufficiently low filling factors, and the critical filling factor is close to 1/7. At nu=1/7 we find the system undergoes a phase transition from Wigner Crystal to the incompressible Laughlin state when the short-range part of the Coulomb interaction is modified slightly. This transition is either continuous or very weakly first order.
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 {\em 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 $ν=1$ and $ν=2$ are presented. We interpret the resonance as the signature of Wigner crystal state around integer Landau levels.
Hartree-Fock theory predicted stripe or bubble phase in the third and higher Landau levels for two-dimensional electrons, and experimental evidences has been accumulated. In this paper theoretical confirmation of the stripe phase and bubble phase in higher Landau level is given by means of the density matrix renormalization group (DMRG) method, which can give essentially exact ground state for electron systems with up to 18 electrons. From the study of the pair correlation function, the stripe phase, bubble phase, and the Wigner crystal phase are identified, and phase diagram is obtained. The …
Recent experiments on the twisted transition metal dichalcogenide (TMD) material, $\rm WSe_2/WS_2$, have observed insulating states at fractional occupancy of the moiré bands. Such states were conceived as generalized Wigner crystals (GWCs). In this article, we investigate the problem of Wigner crystallization in the presence of an underlying (moiré) lattice. Based on the best estimates of the system parameters, we find a variety of homobilayer and heterobilayer TMDs to be excellent candidates for realizing GWCs. In particular, our analysis based on $r_{s}$ indicates that $\rm MoSe_{2}$ (among…
A system of confined charged electrons interacting via the long-range Coulomb force can form a Wigner crystal due to their mutual repulsion. This happens when the potential energy of the system dominates over its kinetic energy, i.e., at low temperatures for a classical system and at low densities for a quantum one. At $T=0$, the system is governed by quantum mechanics, and hence, the spatial density peaks associated with crystalline charge localization are sharpened for a lower average density. Conversely, in the classical limit of high temperatures, the crystalline spatial density peaks are …
We report the first experimental observation of a characteristic nonlinear threshold behavior from dc dynamical response as an evidence for a Wigner crystallization in high-purity GaAs 2D hole systems in zero magnetic field. The system under increasing current drive exhibits voltage oscillations with negative differential resistance. They confirm the coexistence of a moving crystal along with striped edge states as observed for electrons on helium surfaces. However, the threshold is well below the typical classical levels due to a different pinning and depinning mechanism that is possibly rela…
Magnetotransport measurements were performed in a ultra-high mobility GaAs/AlGaAs quantum well of density $\sim 3.0 \times 10^{11}$ $cm^{-2}$. The temperature dependence of the magnetoresistance $R_{xx}$ was studied in detail in the vicinity of $ν={9/2}$. In particular, we discovered new minima in $R_{xx}$ at filling factor $ν\simeq 4{1/5}$ and $4{4/5}$, but only at intermediate temperatures $80\lesssim T\lesssim 120$ mK. We interpret these as evidence for a fractional quantum Hall liquid forming in the N=2 Landau level and competing with bubble and Wigner crystal phases favored at lower tempe…
The ground state cohesive energies per electron of the composite fermion (CF) Fermi sea, the Laughlin state and the charge density wave (CDW) at higher Landau levels (LLs) are computed. It is shown that whereas for $n\geq 2$ LL, the CDW state is generally more energetically preferable than those of the CF liquid and the Laughlin liquid, the $ν=4+1/6$ CF liquid state unexpectedly has lower ground state energy than that of the CDW state. We suggest this CF liquid between the Wigner crystal and the bubble phase may lead to the crossover from the normal integer quantum Hall liquid to the novel re-…
Interactions between the constituents of a condensed matter system can drive it through a plethora of different phases due to many-body effects. A prominent platform for this type of behavior is a two-dimensional electron system in a magnetic field, which evolves intricately through various gaseous, liquid and solids phases governed by Coulomb interaction. Here we report on the experimental observation of a phase transition between the Laughlin liquid of composite fermions and the adjacent insulating phase of a magnetic field-induced Wigner solid. The experiments are performed in the lowest La…
We present a variational study of the 2D and 3D Wigner crystal phase of large polarons. The method generalizes that introduced by S. Fratini,P.\ Qu{é}merais [Mod. Phys. Lett. B {\bf 12} 1003 (1998)]. We take into account the Wigner crystal normal modes rather than a single mean frequency in the minimization procedure of the variational free energy. We calculate the renormalized modes of the crystal as well as the charge polarization correlation function and polaron radius. The solid phase boundaries are determined via a Lindemann criterion, suitably generalized to take into account the classic…
We investigate numerically different phases that can occur at half filling in the lowest and the first excited Landau levels in wide-well twodimensional electron systems exposed to a perpendicular magnetic field. Within a twocomponent model that takes into account only the two lowest electronic subbands of the quantum well, we derive a phase diagram that compares favorably with an experimental one by Shabani et al. [Phys. Rev. B 88, 245413 (2013)]. In addition to the compressible composite-fermion Fermi liquid in narrow wells with a substantial subband gap and the incompressible twocomponent (…
The existence of Wigner crystallization, one of the most significant hallmarks of strong electron correlations, has to date only been definitively observed in two-dimensional systems. In one-dimensional (1D) quantum wires Wigner crystals correspond to regularly spaced electrons; however, weakening the confinement and allowing the electrons to relax in a second dimension is predicted to lead to the formation of a new ground state constituting a zigzag chain with nontrivial spin phases and properties. Here we report the observation of such zigzag Wigner crystals by use of on-chip charge and spin…
What is the fate of the ground state of a two-dimensional electron system (2DES) at very low Landau level filling factors ($ν$) where interaction reigns supreme? An ordered array of electrons, the so-called Wigner crystal, has long been believed to be the answer. It was in fact the search for the elusive Wigner crystal that led to the discovery of an unexpected, incompressible liquid state, namely the fractional quantum Hall state at $ν=1/3$. Understanding the competition between the liquid and solid ground states has since remained an active field of fundamental research. Here we report exper…
Microwave spectroscopy within the Landau filling ($ν$) range of the integer quantum Hall effect (IQHE) has revealed pinning mode resonances signifying Wigner solids (WSs) composed of quasi-particles or -holes. We study pinning modes of WSs in wide quantum wells (WQWs) for $ 0.8\leν\le1.2$, varying the density, $n$, and tilting the sample by angle $θ$ in the magnetic field. Three distinct WS phases are accessed. One phase, S1, is phenomenologically the same as the WS observed in the IQHEs of narrow QWs. The second phase, S2, exists at $ν$ further from $ν=1$ than S1, and requires a sufficiently …
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.
Effect of periodic potential on the stripe phase realized in the higher Landau levels is investigated by the Hartree-Fock approximation. The period of the potential is chosen to be two to six times of the fundamental period of the stripe phase. It is found that the stripe aligns perpendicularly to the external potential in contrast to a naive expectation and hydrodynamic theory. Charge modulation towards the Wigner crystallization along the stripe is essential for the present unexpected new result.
For filling factors $ν$ in the range between 4.16 and 4.28, we simultaneously detect {\it two} resonances in the real diagonal microwave conductivity of a two--dimensional electron system (2DES) at low temperature $T \approx 35$ mK. We attribute the resonances to Wigner crystal and Bubble phases of the 2DES in higher Landau Levels. For $ν$ below and above this range, only single resonances are observed. The coexistence of both phases is taken as evidence of a first order phase transition. We estimate the transition point as $ν=4.22$.
We show that there can be no direct first order transition between a Fermi liquid and an insulating electronic (Wigner) crystalline phase in a clean two-dimensional electron gas in a metal-oxide-semiconductor field-effect transistor (MOSFET); rather, there must always exist intermediate ``micro-emulsion'' phases, and an accompanying sequence of continuous phase transitions. Among the intermediate phases which we find are a variety of electronic liquid crystalline phases, including stripe-related analogues of classical smectics and nematics. The existence of these phases can be established in t…
We study the fractional quantum Hall effect in the central Landau level of bilayer graphene. By tuning the external applied magnetic field and the electric bias between the two layers one can access a regime where there is a degeneracy between Landau levels with orbital characters corresponding to N=0 and N=1 Galilean Landau levels. While the Laughlin state is generically the ground state for filling $ν=1/3$ we find that it can be destroyed and replaced by a Wigner crystal at the same filling factor by tuning the bias and applied field. This competition does not take place at $ν=2/3$ where the…
We focused on the transverse AC magneto-conductance of a high mobility $p$-GaAs/AlGaAs quantum well ($p=1.2\times 10^{11}$~cm$^{-2}$) in the vicinity of two values of the Landau level filling factor $ν$: $ν=1$ (integer quantum Hall effect) and $ν=1/3$ (fractional quantum Hall effect). The complex transverse AC conductance, $σ_{xx}^{AC} (ω)$, was found from simultaneous measurements of attenuation and velocity of surface acoustic waves (SAWs) propagating along the interface between a piezoelectric crystal and the two-dimensional hole system under investigation. We analyzed both the real and ima…
Moiré superlattices are emerging as a new route for engineering strongly correlated electronic states in two-dimensional van der Waals heterostructures, as recently demonstrated in the correlated insulating and superconducting states in magic-angle twisted bilayer graphene and ABC trilayer graphene/boron nitride moiré superlattices. Transition metal dichalcogenide (TMDC) moiré heterostructures provide another exciting model system to explore correlated quantum phenomena, with the addition of strong light-matter interactions and large spin-orbital coupling. Here we report the optical detection …
Graphene, a single free-standing sheet of graphite with honeycomb lattice structure, is a semimetal with carriers that have linear dispersion. A consequence of this dispersion is the absence of Wigner crystallization in graphene, since the kinetic and potential energies both scale identically with the density of carriers. We study the ground state of graphene in the presence of a strong magnetic field focusing on states with broken translational symmetry. Our mean-field calculations show that at integer fillings a uniform state is preferred, whereas at non-integer filling factors Wigner crysta…
The ground state phase diagram of 2D electrons in a high Landau level (index N=2) is studied by the density matrix renormalization group method. Pair correlation functions are systematically calculated for various filling factors from v=1/8 to 1/2. It is shown that the ground state phase diagram consists of three different CDW states called stripe-phase, bubble-phase, and Wigner crystal. The boundary between the stripe and the bubble phases is determined to be v_c = 0.38, and that for the bubble phase and Wigner crystal is v_c = 0.24. Each transition is of first order.
We report a new correlated phase of two-dimensional charged carriers in high magnetic fields, manifested by an anisotropic insulating behavior at low temperatures. It appears near Landau level filling factor $ν=1/2$ in hole systems confined to wide GaAs quantum wells when the sample is tilted in magnetic field to an intermediate angle. The parallel field component ($B_{||}$) leads to a crossing of the lowest two Landau levels, and an elongated hole wavefunction in the direction of $B_{||}$. Under these conditions, the in-plane resistance exhibits an insulating behavior, with the resistance alo…
The $ν=2$ quantum Hall state at low Zeeman coupling is well-known to be a translationally invariant singlet if Landau level mixing is small. At zero Zeeman interaction, as Landau level mixing increases, the translationally invariant state becomes unstable to aninhomogeneous state. This is the first realistic example of a full Hall crystal, which shows the coexistence of quantum Hall order and density wave order. The full Hall crystal differs from the more familiar Wigner crystal by a topological property, which results in it having only linearly dispersing collective modes at small $q$, and no…
Measurements in very low disorder two-dimensional electrons confined to relatively wide GaAs quantum well samples with tunable density reveal reentrant $ν=1$ integer quantum Hall states in the lowest Landau level near filling factors $ν=4/5$ and 6/5. These states are not seen at low densities and become more prominent with increasing density and in wider wells. Our data suggest a close competition between different types of Wigner crystal states near these fillings. We also observe an intriguing disappearance and reemergence of the $ν=4/5$ fractional quantum Hall effect with increasing density…
We study two-dimensional interacting electrons in a weak perpendicular magnetic field with the filling factor $ν\gg 1$ and in the presence of a quenched disorder. In the framework of the Hartree-Fock approximation, we obtain the mean-field phase diagram for the partially filled highest Landau level. We find that the CDW state can exist if the Landau level broadening $1/2τ$ does not exceed the critical value $1/2τ_{c}=0.038ω_{H}$. Our analysis of weak crystallization corrections to the mean-field results shows that these corrections are of the order of $(1/ν)^{2/3}\ll 1$ and therefore can be ne…
When two-dimensional electrons are subjected to a very strong magnetic field, they are believed to form a triangular Wigner crystal. We demonstrate that, in the entire crystal phase, this crystal is very well represented by a composite-fermion-crystal wave function, revealing that it is not a simple Hartree-Fock crystal of electrons but an inherently quantum mechanical crystal characterized by a non-perturbative binding of quantized vortices to electrons, which establishes a long range quantum coherence in it. It is suggested that this has qualitative consequences for experiment.
A sufficiently large perpendicular magnetic field quenches the kinetic (Fermi) energy of an interacting two-dimensional (2D) system of fermions, making them susceptible to the formation of a Wigner solid (WS) phase in which the charged carriers organize themselves in a periodic array in order to minimize their Coulomb repulsion energy. In low-disorder 2D electron systems confined to modulation-doped GaAs heterostructures, signatures of a magnetic-field-induced WS appear at low temperatures and very small Landau level filling factors ($ν\simeq1/5$). In dilute GaAs 2D \textit{hole} systems, on t…
