Generalized Wigner Crystal Research Articles

Quantum Melting of Generalized Wigner Crystals in Transition Metal Dichalcogenide Moiré Systems.

The generalized Wigner crystal (GWC) is a novel quantum phase of matter driven by further-range interaction at fractional fillings of a lattice. The role of further-range interaction as the driver for the incompressible state is akin to the Wigner crystal. On the other hand, the significant role of commensurate filling is akin to the Mott insulator. Recent progress in simulator platforms presents unprecedented opportunities to investigate quantum melting in the strongly interacting regime through synergy between theory and experiments. However, the earlier theory literature presents diverging predictions. We study the quantum freezing of GWC through large-scale density matrix renormalization group simulations of a triangular lattice extended Hubbard model. We find a single first-order phase transition between the Fermi liquid and the sqrt[3]×sqrt[3] GWC state. The GWC state shows long-range antiferromagnetic 120° Néel order. Our results present the simplest answers to the question of the quantum phase transition into the GWC phase and the properties of the GWC phase.

Mapping charge excitations in generalized Wigner crystals.

Transition metal dichalcogenide-based moiré superlattices exhibit strong electron-electron correlations, thus giving rise to strongly correlated quantum phenomena such as generalized Wigner crystal states. Evidence of Wigner crystals in transition metal dichalcogenide moire superlattices has been widely reported from various optical spectroscopy and electrical conductivity measurements, while their microscopic nature has been limited to the basic lattice structure. Theoretical studies predict that unusual quasiparticle excitations across the correlated gap between upper and lower Hubbard bands can arise due to long-range Coulomb interactions in generalized Wigner crystal states. However, the microscopic proof of such quasiparticle excitations is challenging because of the low excitation energy of the Wigner crystal. Here we describe a scanning single-electron charging spectroscopy technique with nanometre spatial resolution and single-electron charge resolution that enables us to directly image electron and hole wavefunctions and to determine the thermodynamic gap of generalized Wigner crystal states in twisted WS2 moiré heterostructures. High-resolution scanning single-electron charging spectroscopy combines scanning tunnelling microscopy with a monolayer graphene sensing layer, thus enabling the generation of individual electron and hole quasiparticles in generalized Wigner crystals. We show that electron and hole quasiparticles have complementary wavefunction distributions and that thermodynamic gaps of ∼50 meV exist for the 1/3 and 2/3 generalized Wigner crystal states in twisted WS2.

Dielectric catastrophe at the Wigner-Mott transition in a moiré superlattice

The bandwidth-tuned Wigner-Mott transition is an interaction-driven phase transition from a generalized Wigner crystal to a Fermi liquid. Because the transition is generally accompanied by both magnetic and charge-order instabilities, it remains unclear if a continuous Wigner-Mott transition exists. Here, we demonstrate bandwidth-tuned metal-insulator transitions at fixed fractional fillings of a MoSe2/WS2 moiré superlattice. The bandwidth is controlled by an out-of-plane electric field. The dielectric response is probed optically with the 2s exciton in a remote WSe2 sensor layer. The exciton spectral weight is negligible for the metallic state with a large negative dielectric constant. It continuously vanishes when the transition is approached from the insulating side, corresponding to a diverging dielectric constant or a ‘dielectric catastrophe’ driven by the critical charge dynamics near the transition. Our results support the scenario of continuous Wigner-Mott transitions in two-dimensional triangular lattices and stimulate future explorations of exotic quantum phases in their vicinities.

Mott and generalized Wigner crystal states in WSe2/WS2 moiré superlattices.

Moiré superlattices can be used to engineer strongly correlated electronic states in two-dimensional van der Waals heterostructures, as recently demonstrated in the correlated insulating and superconducting states observed in magic-angle twisted-bilayer graphene and ABC trilayer graphene/boron nitride moiré superlattices1-4. Transition metal dichalcogenide moiré heterostructures provide another model system for the study of correlated quantum phenomena5 because of their strong light-matter interactions and large spin-orbit coupling. However, experimental observation of correlated insulating states in this system is challenging with traditional transport techniques. Here we report the optical detection of strongly correlated phases in semiconducting WSe2/WS2 moiré superlattices. We use a sensitive optical detection technique and reveal a Mott insulator state at one hole per superlattice site and surprising insulating phases at 1/3 and 2/3 filling of the superlattice, which we assign to generalized Wigner crystallization on the underlying lattice6-11. Furthermore, the spin-valley optical selection rules12-14 of transition metal dichalcogenide heterostructures allow us to optically create and investigate low-energy excited spin states in the Mott insulator. We measure a very long spin relaxation lifetime of many microseconds in the Mott insulating state, orders of magnitude longer than that of charge excitations. Our studies highlight the value of using moiré superlattices beyond graphene to explore correlated physics.

The ground state structure of a one-dimensional generalized Wigner crystal on a weakly disordered host-lattice

A study of the ground state of a one-dimensional electron system on a weakly disordered host-lattice (epitaxial substrate). It is shown that the long-range correlations in the electron system are destroyed at the position of the host-lattice sites in arbitrarily weak disorder. We have obtained an analytical dependence that links the value of the host-lattice disorder at which the destruction of long-range order occurs and the typical size of the electron system.

Ground state of one-dimension repulsing particles on disordered lattice

The ground state (GS) of interacting particles on a disordered one-dimensional (1D) host-lattice is studied by a new numerical method. It is shown that if the concentration of particles is small, then even a weak disorder of the host-lattice breaks the long-range order of Generalized Wigner Crystal (GWC), replacing it by the sequence of blocks (domains) of particles with random lengths. The mean domains length as a function of the host-lattice disorder parameter is also found. It is shown that the domain structure can be detected by a weak random field, whose form is similar to that of the ground state but has fluctuating domain walls positions. This is because the generalized magnetization corresponding to the field has a sufficiently sharp peak as a function of the amplitude of fluctuations for small amplitudes.

Low-temperature thermodynamics of two-dimensional electron gas on disordered host lattice

The low-temperature thermodynamic properties of two-dimensional electron ensemble on ordered and disordered host lattice are studied numerically by the transfer-matrix method. It is shown that at low temperatures and weak disordering of host lattice site positions a change in chemical potential leads to successive transitions of the system from ordered phases (like the generalized Wigner crystal) to disordered states. The ranges of stability of these crystals as functions of temperature, chemical potential and disorder parameter are established.

LOW-TEMPERATURE THERMODYNAMIC PROPERTIES OF A ONE-DIMENSIONAL GENERALIZED WIGNER CRYSTAL

The low-temperature thermodynamic properties of a one-dimensional generalized Wigner crystal at arbitrary values of electron density and arbitrary number of interacting electrons are studied. The modified transfer-matrixes method is applied. It is shown that increasing the number of interacting electrons leads to the appearance of more and more fine "stairs" in low-temperature dependence of chemical potential against electron density. An influence of the disorder in host-lattice site positions on thermodynamic characteristics of the system is considered. It is established that the disorder destroys the "stairs".

One-dimensional electron lattice system with a long-range interelectron repulsion on a disordered host lattice

We study what happens to a generalized Wigner crystal, GWC (a regular structure formed by narrow-band electrons on a one-dimensional periodic host lattice), when the host lattice suffers a random distortion that does not break its long-range order. We show that an arbitrarily weak distortion of this kind gives rise to soliton-like GWC defects (discrete solitons, DS) in the ground state, and thereby converts the ordered GWC into a new disordered macroscopic state—lattice Wigner glass (LWG). The ground-state DS concentration is found to be proportional to λ4 (λ is the typical host-lattice strain). We show that the low-temperature LWG thermodynamics and kinetics are fully described in DS terms. A new phenomenon of a super-slow logarithmic relaxation in the LWG is revealed. Its time turns out to be tens orders of magnitude greater than the microscopic ones. Analytical dependences of LWG thermodynamic quantities on temperature and λ are obtained for an arbitrary relationship between the relevant Coulomb energies and the electron bandwidth.

Low‐temperature thermodynamic properties of one‐dimensional disordered electron lattice system

AbstractThe ground state and low‐temperature thermodynamic properties of a one‐dimensional Generalized Wigner crystal on disordered host‐lattice have been studied. It is established that the spectrum of elementary excitations has a gapless structure. An instability of the ground state with respect to arbitrary small disordering of host‐lattice is revealed. This instability leads to long‐range correlations breaking. An influence of long‐range action of the inter‐particle repulsion potential on thermodynamic properties is studied. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

NUMERICAL STUDIES OF ONE-DIMENSIONAL GENERALIZED WIGNER CRYSTAL ON DISORDERED HOST-LATTICE

New numerical methods of the ground state and thermodynamic properties calculations of one-dimensional Generalized Wigner crystal on disordered host-lattice are proposed. Unlike computer simulation methods (for instance, Monte Carlo) these methods bring the exact results. Another attractive feature of the proposed methods is their speed: it is possible to study the systems with length about 104–105 nodes even on a personal computer. This is especially important in the case of weakly disordered systems and the long-range correlations. The gapless structure of low-energy excitation and breaking long-range correlations at arbitrary small disordering are established.

Low-temperature thermodynamic properties of a one-dimensional disordered electron lattice system

A new method of low-temperature thermodynamic calculation of a one-dimensional generalized Wigner crystal on a disordered host-lattice is proposed. This method is based on the system statistical sum presentation in terms of modified transfer-matrixes. A gapless structure of the elementary excitations spectrum of the system under consideration is found.

Thermodynamics of a one-dimensional generalized Wigner crystal

The thermodynamic characteristics of a one-dimensional generalized Wigner crystal are investigated for arbitrary temperature, particle density, and character of the interparticle repulsive pair potential. A method is proposed which permits calculating the partition function of the system with allowance for the interaction between an arbitrary number of neighboring particles. It is shown that increasing this number leads to the appearance of increasingly fine steps in the low-temperature dependence of the particle density on the chemical potential. These steps are remnants of a dependence of the “devil’s staircase” type that is characteristic for this sort of system at zero temperature.

Charge and spin order in one-dimensional electron systems with long-range Coulomb interactions

We study a system of electrons interacting through long--range Coulomb forces on a one--dimensional lattice, by means of a variational ansatz which is the strong--coupling counterpart of the Gutzwiller wave function. Our aim is to describe the quantum analogue of Hubbard's classical ``generalized Wigner crystal''. We first analyse charge ordering in a system of spinless fermions, with particular attention to the effects of lattice commensurability. We argue that for a general (rational) number of electrons per site $n$ there are three regimes, depending on the relative strength $V$ of the long--range Coulomb interaction (as compared to the hopping amplitude $t$). For very large $V$ the quantum ground state differs little from Hubbard's classical solution, for intermediate to large values of $V$ we recover essentially the Wigner crystal of the continuum model, and for small $V$ the charge modulation amounts to a small--amplitude charge--density wave. We then include the spin degrees of freedom and show that in the Wigner crystal regimes (i.e. for large $V$) they are coupled by an antiferromagnetic kinetic exchange $J$, which turns out to be smaller than the energy scale governing the charge degrees of freedom. Our results shed new light on the insulating phases of organic quasi--1D compounds where the long--range part of the interaction is unscreened, and magnetic and charge orderings coexist at low temperatures.

The instability of a 2D electron gas near the critical density for a Wigner polaron crystal giving the quantum state of cuprate superconductors

Abstract Experimental evidence that the metallic phase in the Bi2212 high Tc cuprate superconductors results from an instability of a 2D-electron gas with density lower or larger than the critical density ϱc for the formation of a generalized Wigner polaronic crystal is reported. The critical density ϱc is obtained from ϱ c = 1 S p where Sp is the measured area of condensed polarons. We have found that the 2D-electron gas at charge densities ϱ≠ϱc in the range ϱ c 2 c and ϱc T T ∗ and 2) a Fermi liquid with local density ϱi≠ϱc. The polaron pair filling factor ν=2ϱSp is shown to be the good variable to describe the phase diagram of the 2D-electron gas in the CuO2 plane as function of doping in several families of cuprate superconductors. At integer values of ν we find the insulating phases of a 2D-polaron gas: the generalized Wigner CDW at ν=1, and the generalized Wigner crystal at ν=2. At non integer values 1 δ i > l > near 1 (near 2), where is the average hole doping for the Fermi liquid and for the polaron gas. The sharp drop of the isotope exponent from ∼0.5 for ν 2 supports the proposed model for the pairing mechanism.

Generalized Wigner crystal in quasi 1-D platinum chain compounds

Generalized Wigner crystal in quasi 1-D platinum chain compounds