scholarly journals Imaging the electronic Wigner crystal in one dimension

Science ◽  
2019 ◽  
Vol 364 (6443) ◽  
pp. 870-875 ◽  
Author(s):  
I. Shapir ◽  
A. Hamo ◽  
S. Pecker ◽  
C. P. Moca ◽  
Ö. Legeza ◽  
...  
Keyword(s):  
Real Space ◽  
Wigner Crystal ◽  
One Dimension ◽  
Many Body ◽  
Quantum Crystal ◽  
Electronic Density ◽  
Eugene Wigner ◽  
Minimal Invasiveness ◽  
Wigner Crystals ◽  
The Many

The quantum crystal of electrons, predicted more than 80 years ago by Eugene Wigner, remains one of the most elusive states of matter. In this study, we observed the one-dimensional Wigner crystal directly by imaging its charge density in real space. To image, with minimal invasiveness, the many-body electronic density of a carbon nanotube, we used another nanotube as a scanning-charge perturbation. The images we obtained of a few electrons confined in one dimension match the theoretical predictions for strongly interacting crystals. The quantum nature of the crystal emerges in the observed collective tunneling through a potential barrier. These experiments provide the direct evidence for the formation of small Wigner crystals and open the way for studying other fragile interacting states by imaging their many-body density in real space.

scholarly journals Imaging Generalized Wigner Crystal States in a WSe2/WS2 Moiré Superlattice

2021 ◽  
Author(s):  
Feng Wang ◽  
Hongyuan Li ◽  
Shaowei Li ◽  
Emma Regan ◽  
Danqing Wang ◽  
...  
Keyword(s):  
Electrical Transport ◽  
Nearest Neighbor ◽  
Real Space ◽  
Wigner Crystal ◽  
Scanning Tunneling ◽  
Honeycomb Lattice ◽  
Transition Metal Dichalcogenide ◽  
Tunneling Microscopy ◽  
Eugene Wigner ◽  
Wigner Crystals

Abstract The Wigner crystal state, first predicted by Eugene Wigner in 19341, has fascinated condensed matter physicists for nearly 90 years2-10. Studies of two-dimensional (2D) electron gases first revealed signatures of the Wigner crystal in electrical transport measurements at high magnetic fields2-4. More recently optical spectroscopy has provided evidence of generalized Wigner crystal states in transition metal dichalcogenide (TMDC) moiré superlattices6-9. Direct observation of the 2D Wigner crystal lattice in real space, however, has remained an outstanding challenge. Scanning tunneling microscopy (STM) in principle has sufficient spatial resolution to image the Wigner crystal, but conventional STM measurements can potentially alter fragile Wigner crystal states in the process of measurement. Here we demonstrate real-space imaging of 2D Wigner crystals in WSe2/WS2 moiré heterostructures using a novel non-invasive STM spectroscopy technique. We employ a graphene sensing layer in close proximity to the WSe2/WS2 moiré superlattice for Wigner crystal imaging, where local STM tunneling current into the graphene sensing layer is modulated by the underlying electron lattice of the Wigner crystal in the WSe2/WS2 heterostructure. Our measurement directly visualizes different lattice configurations associated with Wigner crystal states at fractional electron fillings of n = 1/3, 1/2, and 2/3, where n is the electron number per site. The n=1/3 and n=2/3 Wigner crystals are observed to exhibit a triangle and a honeycomb lattice, respectively, in order to minimize nearest-neighbor occupations. The n = 1/2 state, on the other hand, spontaneously breaks the original C3 symmetry and forms a stripe structure in real space. Our study lays a solid foundation toward the fundamental understanding of rich Wigner crystal states in WSe2/WS2 moiré heterostructures. Furthermore, this new STM technique is generally applicable to imaging novel correlated electron lattices in different van der Waals moiré heterostructures.

scholarly journals The bulk-corner correspondence of time-reversal symmetric insulators

2021 ◽  
Vol 6 (1) ◽  
Author(s):  
Sander Kooi ◽  
Guido van Miert ◽  
Carmine Ortix
Keyword(s):  
Time Reversal ◽  
Real Space ◽  
Topological Phases ◽  
Spin Orbit Coupling ◽  
Topological Properties ◽  
Many Body ◽  
Boundary Correspondence ◽  
Berry Phases ◽  
The Many ◽  
Do So

AbstractThe topology of insulators is usually revealed through the presence of gapless boundary modes: this is the so-called bulk-boundary correspondence. However, the many-body wavefunction of a crystalline insulator is endowed with additional topological properties that do not yield surface spectral features, but manifest themselves as (fractional) quantized electronic charges localized at the crystal boundaries. Here, we formulate such bulk-corner correspondence for the physical relevant case of materials with time-reversal symmetry and spin-orbit coupling. To do so we develop partial real-space invariants that can be neither expressed in terms of Berry phases nor using symmetry-based indicators. These previously unknown crystalline invariants govern the (fractional) quantized corner charges both of isolated material structures and of heterostructures without gapless interface modes. We also show that the partial real-space invariants are able to detect all time-reversal symmetric topological phases of the recently discovered fragile type.

scholarly journals A Simple Molecular Orbital Picture of RIXS Distilled from Many-Body Damped Response Theory

2020 ◽  
Author(s):  
Kaushik Nanda ◽  
Anna I. Krylov
Keyword(s):  
Molecular Orbital ◽  
Cross Sections ◽  
Transition Density ◽  
Wave Functions ◽  
Many Body ◽  
Response Theory ◽  
Electronic Density ◽  
Near Resonance ◽  
Alternative Approach ◽  
The Many

<div>Ab initio calculations of resonant inelastic X-ray scattering (RIXS) rely on the damped response theory, which prevents the divergence of response solutions in the resonant regime. Within the damped response theory formalism, RIXS moments are expressed as sum over all electronic states of the system (SOS expressions). By invoking resonance arguments, these expressions can be reduced to a few terms, an approximation commonly exploited for interpretation of the computed cross sections. We present an alternative approach: a rigorous formalism for deriving a simple molecular orbital picture of the RIXS process from the many-body calculations using damped</div><div>response theory. In practical implementations, the SOS expressions of RIXS moments are recast in terms of matrix elements between the zero-order wave functions and first-order frequency-dependent response wave functions of the initial and final states, such that the RIXS moments can be evaluated using complex response</div><div>one-particle transition density matrices (1PTDMs). Visualization of these 1PTDMs connects the RIXS process with the changes in electronic density. We demonstrate that the real and imaginary components of the response 1PTDMs can be interpreted as contributions of the undamped off-resonance and damped near-resonance SOS terms, respectively. By analyzing these 1PTDMs in terms of natural transition orbitals, we derive a rigorous, black-box mapping of the RIXS process into a molecular orbital picture. We illustrate the utility of the new tool by analyzing RIXS transitions in the OH radical, benzene, para-nitroaniline, and</div><div> 4-amino-4'-nitrostilbene. These examples highlight the significance of both near-resonance and off-resonance channels. </div>

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