Wigner Crystal Phase Research Articles

On some structural phase transitions in coupled quantum wires at finite temperature

In this paper, we explore some structural phase transitions in GaAs-based coupled electron-electron (e-e) and electron-hole (e-h) quantum wires at finite temperature. To this endeavour, the intra- and inter-wire static-structure factors, pair-correlation functions and static (charge) density susceptibilities are calculated over a wide range of temperature T, particle number density parameter r se and some selected values of inter-wire spacing d. The particle exchange-correlations (xc) are included using the dynamic version of self-consistent mean-field theory of Singwi et al (the qSTLS theory), and the results have been compared with the static STLS model. It has been found that in the e-h system, the inclusion of dynamic nature of xc leads to the formation of Wigner crystal (WC) state in the close proximity of two wires at wave-vector q ∼ 3.5k Fe . However, a charge-density-wave (CDW) instability is observed at q ∼ 2k Fe when the xc are treated statically (k Fe being the electron’s Fermi wave vector). On the other hand, the e-e system shows comparatively small signatures of the WC phase when wires are kept sufficiently far apart, but, a long-wavelength instability is encountered in close vicinity of the wires. Interestingly, the CDW phase is completely missing in the e-e system at the investigated parameters. Expectedly, the quantum phase transitions are predicted to occur in the strongly correlated regime i.e. at sufficiently small T and high r se .

Quasi-localized charge approximation approach for the nonlinear structures in strongly coupled Yukawa systems

Strongly coupled systems occupying the transitional range between the Wigner crystal and fluid phases are the most dynamic constituents of the nature. Highly localized but strongly interacting elements in this phase possess enough thermal energy to trigger the transition between a variety of short to long range order phases. Nonlinear excitations are often carriers of proliferating structural modifications in the strongly coupled Yukawa systems. Well represented by laboratory dusty plasmas, these systems show explicit propagation of nonlinear shocks and solitary structures both in experiments and first principles simulations. The shorter scale length contributions remain absent at strong screening in the present approximate models, which nevertheless prescribe nonlinear solitary solutions that consequently lose their coherence in a numerical evolution of the system under the special implementation of a quasi-localized charge approximation (QLCA) formulation. The stable coherent structures self-consistently emerge following an initial transient in the numerical evolution that adapts QLCA approach to spatiotemporal domain for accessing the nonlinear excitations in the strong screening limit. The present κ∼1 limit of the existing Yukawa fluid models to show agreement with the experiment and molecular dynamical simulations has, therefore, been overcome, and the coherent nonlinear excitations have become characterizable up to κ∼2.7, before they become computationally challenging in the present implementation.

Thermal melting of a quantum electron solid in the presence of strong disorder: Anderson localization versus the Wigner crystal

We consider the temperature-induced melting of a Wigner solid in one-dimensional (1D) and two-dimensional (2D) lattices of electrons interacting via the long-range Coulomb interaction in the presence of strong disorder arising from charged impurities in the system. The system simulates semiconductor-based 2D electron layers where Wigner crystallization is often claimed to be observed experimentally. Using exact diagonalization and utilizing the inverse participation ratio as well as conductance to distinguish between the localized insulating solid phase and the extended metallic liquid phase, we find that the effective melting temperature may be strongly enhanced by disorder since the disordered crystal typically could be in a localized glassy state incorporating the combined nonperturbative physics of both Anderson localization and Wigner crystallization. This disorder-induced enhancement of the melting temperature may explain why experiments often manage to observe insulating disorder-pinned Wigner solids in spite of the experimental temperature being decisively far above the theoretical melting temperature of the pristine Wigner crystal phase in many cases.

Damping of Pseudo-Goldstone Fields.

Approximate symmetries abound in nature. If these symmetries are also spontaneously broken, the would-be Goldstone modes acquire a small mass, or inverse correlation length, and are referred to as pseudo-Goldstones. At nonzero temperature, the effects of dissipation can be captured by hydrodynamics at sufficiently long scales compared to the local equilibrium. Here, we show that, in the limit of weak explicit breaking, locality of hydrodynamics implies that the damping of pseudo-Goldstones is completely determined by their mass and diffusive transport coefficients. We present many applications: superfluids, QCD in the chiral limit, Wigner crystal and density wave phases in the presence of an external magnetic field or not, nematic phases, and (anti)ferromagnets. For electronic density wave phases, pseudo-Goldstone damping generates a contribution to the resistivity independent of the strength of disorder, which can have a linear temperature dependence provided the associated diffusivity saturates a bound. This is reminiscent of the phenomenology of strange metal high-T_{c} superconductors, where charge density waves are observed across the phase diagram.

Topological magnetic textures in magnetic topological insulators

The surfaces of intrinsic magnetic topological insulators (TIs) host magnetic moments exchange-coupled to Dirac electrons. We study the magnetic phases arising from tuning the electron density using variational and exact diagonalization approaches. In the dilute limit, we find that magnetic skyrmions are formed, which bind to electrons, leading to a skyrmion Wigner crystal phase while at higher densities spin spirals accompanied by chiral one-dimensional channels of electrons are formed. The binding of electrons to textures raises the possibility of manipulating textures with electrostatic gating. We determine the phase diagram capturing the competition of intrinsic spin-spin interactions and carrier density and comment on the possible application to experiments in magnetic TIs and spintronic devices such as skyrmion-based memory.

Interaction-driven transition between the Wigner crystal and the fractional Chern insulator in topological flat bands

We investigate an interaction-driven transition between crystalline and liquid states of strongly correlated spinless fermions within topological flat bands at low density (with filling factors $\ensuremath{\nu}=\frac{1}{5}, \frac{1}{7}, \frac{1}{9}$). Using exact diagonalization for finite-size systems with periodic boundary conditions, we distinguish different phases, whose stability depends on the interaction range, controlled by the screening parameter of the Coulomb interaction. The crystalline phases are identified by a crystallization strength, calculated from the Fourier transforms of pair correlation density, while the fractional Chern insulator (FCI) phases are characterized using momentum counting rules, entanglement spectrum, and overlaps with corresponding fractional quantum Hall states. The type of the phase depends on a particular single-particle model and its topological properties. We show that for $\ensuremath{\nu}=\frac{1}{7}$ and $\frac{1}{5}$ it is possible to tune between the Wigner crystal and fractional Chern insulator phase in the kagome lattice model with the band carrying the Chern number $C=1$. In contrast, in the $C=2$ models, the Wigner crystallization was absent at $\ensuremath{\nu}=\frac{1}{5}$, and appeared at $\ensuremath{\nu}=\frac{1}{9}$, suggesting that $C=2$ FCIs are more stable against the formation of crystalline order.

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.

Pair-correlation functions and density-modulated states in electron–hole bilayer: thermal and mass-asymmetry effects

We have studied the effect of temperature and electron–hole mass-asymmetry on pair-correlation functions (PCFs) of a coupled electron–hole bilayer, treating correlations dynamically using the dynamical self-consistent mean-field approach of Singwi, Tosi, Land, and Sjolander. Taking a fixed electron–hole effective mass ratio $$m_h^*/m_e^*=5$$ , both intra- and interlayer static PCFs are calculated for their dependence on carrier density $$r_{se}$$ , temperature $$\tau _e$$ , and interlayer spacing d. Besides, the static wavevector-dependent density susceptibility is obtained to probe the role of these effects on the hitherto existence of density-modulated phases in the bilayer. We find that thermal effects in overall make correlations weaker, but the mass-asymmetry results in stronger correlations both within and across the layers, with the effect being most pronounced in the holes layer. In the strong coupling regime (i.e., large $$r_{se}/\tau _e d$$ ), the intra- and interlayer PCFs show pronounced in-phase periodic oscillations, typical of a spatially localized phase in the bilayer. Correspondingly, the in-phase component of static density susceptibility is seen to exhibit in the close approach of two layers an apparently diverging peak at a wavevector coincident with the location of sharp peak in the static structure factor, implying the emergence of a density-modulated phase in the bilayer. Parallel to the zero-temperature study, the charge-density-wave (CDW) phase is found to dominate at higher densities, with a cross-over to the Wigner crystal (WC) phase below a critical density. As an interesting result, we find that while thermal effects tend to oppose the formation of density-modulated phase, the electron–hole mass-asymmetry boosted correlations favour it, with the two effects almost cancelling out at $$\tau _e=0.125$$ , thus resulting in the CDW-WC cross-over at nearly the same critical density as for a zero-temperature mass-symmetric electron–hole bilayer. Our prediction of coupled WC phase is found to be in qualitative agreement with path-integral Monte Carlo simulations.

A novel planar back-gate design to control the carrier concentrations in GaAs-based double quantum wells

The precise control of a bilayer system consisting of two adjacent two-dimensional electron gases (2DEG) is demonstrated by using a novel planar back-gate approach based on ion implantation. This technique overcomes some common problems of the traditional design like the poor 2DEG mobility and leakage currents between the gate and the quantum well. Both bilayers with and without separate contacts have been prepared and tested. Tuning the electron density in one layer while keeping the second 2DEG at fixed density, one observes a dramatic increase of the carrier concentration. This tunneling resonance, which occurs at equal densities of both layers, demonstrates the separated contacts to each individual layer. In another sample with a smaller tunneling barrier and parallel contacted 2DEGs, the transition from a single 2DEG to a bilayer system is investigated at 50 mK in magnetic fields up to 12 T, showing the gate stability in high magnetic fields and very low temperatures. Transitions into an insulating (Wigner crystal) phase are observed in the individual layers in high fields at filling factors below 1/3. The absence of a fractional quantum Hall liquid at filling factor 1/5 in our structure seems to be a consequence of confining the electrons in quantum wells rather than at interfaces. The observed metal-insulator transitions appear to be nearly unaffected by the presence of the second layer separated by a barrier which is only 3 nm thick. We believe that this planar back-gate design holds great promise to produce controllable bilayers suitable to investigate the exotic (non-abelian) properties of correlated states.

Entanglement and correlations in a one-dimensional quantum spin- 12 chain with anisotropic power-law long-range interactions

The correlations, entanglement entropy, and fidelity susceptibility are calculated for a one-dimensional spin-1/2 XXZ chain with anisotropic power-law long range interactions by employing the density matrix renormalization group method. In particular, this long-range interaction is assigned to ferromagnetic for transversal components, while it can be either ferro- or antiferromagnetic for the longitudinal spin component. Two ground-state phase diagrams are established versus the anisotropy of the interactions which not only changes the phase boundaries of the counterparts with short-range interactions, but also leads to the emergence of exotic phases. We found that the long-range interactions of the z-component results in a Wigner crystal phase, whereas the transversal one may break a continuous symmetry, resulting in a continuous symmetry breaking phase.

Instantons, colloids and convergence of the 1/N expansion for the homogeneous electron gas

We investigate non-perturbative corrections to the large N expansion of the homogeneous electron gas. These are associated with instanton solutions to the effective action of the plasmon field. We show that, although the large field behavior of that action dominates the quadratic bare Coulomb term, there are no solutions at large field, and consequently none at large density. We argue that solutions would exist at low density if the large N theory had a Wigner crystal (WC) phase. However, we argue that this is not the case. Together with the implied convergence of the large N expansion, this implies that the homogeneous electron gas with N component spins and a Coulomb interaction scaling like 1∕N can only have a WC phase below a curve in the plane of N and density, which asymptotes to zero density at infinite N. We argue that for systems with a semi-classical expansion for order parameter dynamics, and a first order quantum transition between fluid and crystal phases, there are instantons associated with the decays of meta-stable fluid and crystal phases in the appropriate regions of the phase diagram. We argue that the crystal will decay into one or more colloidal or bubble phases (Kivelson and Spivak, 2006) rather than directly into the fluid. It is possible that the bubble phases remain stable all the way to the density where the crystal solution disappears. The bubbles of fluid will expand as the density is raised and gradually convert the system into a fluid filled with chunks of crystal. The transition to a translationally invariant phase is likely to be second order. Unfortunately, the HEG does not have a crystal phase at large N, where these semi-classical ideas could be examined in detail. We suggest that the evidence for negative dielectric function at intermediate densities for N=2 is an indicator of this second order transition. While the infinite N limit does not have a negative dielectric function at any density, it is possible that the closed large N equation for the plasmon two point function, derived in Banks (2018) might capture at least the qualitative features of the second order transition.

Real-space observation of charge ordering in epitaxial La2\u2212xSrxCuO4 films

The cuprate superconductors exhibit ubiquitous instabilities toward charge-ordered states. These unusual electronic states break the spatial symmetries of the host crystal, and have been widely appreciated as essential ingredients for constructing a theory for high-temperature superconductivity in cuprates. Here, we report real-space imaging of the doping-dependent charge orders in the epitaxial thin films of a canonical cuprate compound La2–xSrxCuO4 using scanning tunneling microscopy. As the films are moderately doped, we observe a crossover from incommensurate to commensurate (4a0, where a0 is the Cu–O–Cu distance) stripes. Furthermore, at lower and higher doping levels, the charge orders occur in the form of distorted Wigner crystal and grid phase of crossed vertical and horizontal stripes. We discuss how the charge orders are stabilized, and their interplay with superconductivity.

Probing the Melting of a Two-Dimensional Quantum Wigner Crystal via its Screening Efficiency.

One of the most fundamental and yet elusive collective phases of an interacting electron system is the quantum Wigner crystal (WC), an ordered array of electrons expected to form when the electrons' Coulomb repulsion energy eclipses their kinetic (Fermi) energy. In low-disorder, two-dimensional (2D) electron systems, the quantum WC is known to be favored at very low temperatures (T) and small Landau level filling factors (ν), near the termination of the fractional quantum Hall states. This WC phase exhibits an insulating behavior, reflecting its pinning by the small but finite disorder potential. An experimental determination of a T vs ν phase diagram for the melting of the WC, however, has proved to be challenging. Here we use capacitance measurements to probe the 2D WC through its effective screening as a function of T and ν. We find that, as expected, the screening efficiency of the pinned WC is very poor at very low T and improves at higher T once the WC melts. Surprisingly, however, rather than monotonically changing with increasing T, the screening efficiency shows a well-defined maximum at a T that is close to the previously reported melting temperature of the WC. Our experimental results suggest a new method to map out a T vs ν phase diagram of the magnetic-field-induced WC precisely.

Fractional Quantum Hall Phases of Bosons with Tunable Interactions: From the Laughlin Liquid to a Fractional Wigner Crystal.

Highly tunable platforms for realizing topological phases of matter are emerging from atomic and photonic systems and offer the prospect of designing interactions between particles. The shape of the potential, besides playing an important role in the competition between different fractional quantum Hall phases, can also trigger the transition to symmetry-broken phases, or even to phases where topological and symmetry-breaking order coexist. Here, we explore the phase diagram of an interacting bosonic model in the lowest Landau level at half filling as two-body interactions are tuned. Apart from the well-known Laughlin liquid, Wigner crystal, stripe, and bubble phases, we also find evidence of a phase that exhibits crystalline order at fractional filling per crystal site. The Laughlin liquid transits into this phase when pairs of bosons strongly repel each other at relative angular momentum 4ℏ. We show that such interactions can be achieved by dressing ground-state cold atoms with multiple different-parity Rydberg states.

Unconventional fractional quantum Hall states and Wigner crystallization in suspended Corbino graphene

Competition between liquid and solid states in two-dimensional electron systems 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 they appear to originate from residual interactions of composite fermions in partially filled Landau levels. Also at very low charge density with filling factors nu ,lesssim, 1/5, electrons crystallize into an ordered Wigner solid which eventually transforms into an incompressible Hall liquid at filling factors around ν ≤ 1/7. Building on the unique Corbino sample structure, our experiments pave the way for enhanced understanding of the ordered phases of interacting electrons.

Phase diagram of a symmetric electron–hole bilayer system: a variational Monte Carlo study

We study the phase diagram of a symmetric electron–hole bilayer system at absolute zero temperature and in zero magnetic field within the quantum Monte Carlo approach. In particular, we conduct variational Monte Carlo simulations for various phases, i.e. the paramagnetic fluid phase, the ferromagnetic fluid phase, the anti-ferromagnetic Wigner crystal phase, the ferromagnetic Wigner crystal phase and the excitonic phase, to estimate the ground-state energy at different values of in-layer density and inter-layer spacing. Slater–Jastrow style trial wave functions, with single-particle orbitals appropriate for different phases, are used to construct the phase diagram in the (rs, d) plane by finding the relative stability of trial wave functions. At very small layer separations, we find that the fluid phases are stable, with the paramagnetic fluid phase being particularly stable at and the ferromagnetic fluid phase being particularly stable at . As the layer spacing increases, we first find that there is a phase transition from the ferromagnetic fluid phase to the ferromagnetic Wigner crystal phase when d reaches 0.4 a.u. at rs = 20, and before there is a return to the ferromagnetic fluid phase when d approaches 1 a.u. However, for rs < 20 and a.u., the excitonic phase is found to be stable. We do not find that the anti-ferromagnetic Wigner crystal is stable over the considered range of rs and d. We also find that as rs increases, the critical layer separations for Wigner crystallization increase.

Observation of Wigner crystal phase and ripplon-limited mobility behavior in monolayer CVD MoS2 with grain boundary

Two-dimensional electron gas (2DEG) is crucial in condensed matter physics and is present on the surface of liquid helium and at the interface of semiconductors. Monolayer MoS2 of 2D materials also contains 2DEG in an atomic layer as a field effect transistor (FET) ultrathin channel. In this study, we synthesized double triangular MoS2 through a chemical vapor deposition method to obtain grain boundaries for forming a ripple structure in the FET channel. When the temperature was higher than approximately 175 K, the temperature dependence of the electron mobility μ was consistent with those in previous experiments and theoretical predictions. When the temperature was lower than approximately 175 K, the mobility behavior decreased with the temperature; this finding was also consistent with that of the previous experiments. We are the first research group to explain the decreasing mobility behavior by using the Wigner crystal phase and to discover the temperature independence of ripplon-limited mobility behavior at lower temperatures. Although these mobility behaviors have been studied on the surface of liquid helium through theories and experiments, they have not been previously analyzed in 2D materials and semiconductors. We are the first research group to report the similar temperature-dependent mobility behavior of the surface of liquid helium and the monolayer MoS2.

Wigner crystallization in transition metal dichalcogenides: A new approach to correlation energy

We introduce a new approach for the correlation energy of one- and two-valley two-dimensional electron gas (2DEG) systems. Our approach is based on an interpolation between two limits, a random phase approximation at high densities and a classical approach at low densities which gives excellent agreement with available Quantum Monte Carlo (QMC) calculations. The two-valley 2DEG model is introduced to describe the electron correlations in monolayer transition metal dichalcogenides (TMDs). We study the zero-temperature transition from a Fermi liquid to a quantum Wigner crystal phase in monolayer TMDs. Consistent with QMC, we find that electrons crystallize at ${r}_{s}=31$ in one-valley 2DEG. For two valleys, we predict Wigner crystallization at ${r}_{s}=30$, implying that valley degeneracy has little effect on the critical ${r}_{s}$, in contrast to an earlier claim.

Wigner crystal phases in bilayer graphene

It is generally believed that a Wigner Crystal in single layer graphene can not form because the magnitudes of the Coulomb interaction and the kinetic energy scale similarly with decreasing electron density. However, this scaling argument does not hold for the low energy states in bilayer graphene. We consider the formation of a Wigner Crystal in weakly doped bilayer graphene with an energy gap opened by a perpendicular electric field. We argue that in this system the formation of the Wigner Crystal is not only possible, but different phases of the crystal with very peculiar properties may exist here depending on the system parameters.

Ordered phases in a bilayer system of dipolar fermions

The liquid-to-ordered phase transition in a bilayer system of fermions is studied within the context of a recently proposed density-functional theory [Phys. Rev. A {\bf 92}, 023614 (2015)]. In each two-dimensional layer, the fermions interact via a repulsive, isotropic dipolar interaction. The presence of a second layer introduces an attractive {\em interlayer} interaction, thereby allowing for inhomogeneous density phases which would otherwise be energetically unfavourable. For any fixed layer separation, we find an instability to a commensurate one-dimensional stripe phase in each layer, which always precedes the formation of a triangular Wigner crystal. However, at a certain {\em fixed} coupling, tuning the separation can lead to the system favoring a commensurate triangular Wigner crystal, or one-dimensional stripe phase, completely bypassing the Fermi liquid state. While other crystalline symmetries, with energies lower than the liquid phase can be found, they are never allowed to form owing to their high energetic cost relative to the triangular Wigner crystal and stripe phase.