anew

_posts/2024-09-12-dark_states.md

@@ -10,7 +10,7 @@ related_posts: false
 toc: true
 ---
 
-- A slightly different way of looking at dark bands in the BZ. See [Chung et al](https://doi.org/10.1038/s41567-024-02586-x)
+- A slightly different way of looking at dark bands in the Brillouin Zone (BZ). See [Chung et al.](https://doi.org/10.1038/s41567-024-02586-x).
 
 This review discusses the discovery of **condensed-matter dark states** in the material **Palladium Diselenide (PdSe₂)**, where entire bands of quantum states in the Brillouin zone remain undetectable via angle-resolved photoemission spectroscopy (ARPES). The dark states arise due to interference effects between sublattices, a novel feature in quantum materials.
 
@@ -20,7 +20,7 @@ This review discusses the discovery of **condensed-matter dark states** in the m
 
 In quantum mechanics, a **dark state** refers to a state that cannot absorb or emit photons and is thus undetectable through typical spectroscopic methods. These states are typically well-known in atomic and molecular systems where they arise due to **quantum interference** or **conservation of angular momentum**.
 
-In the condensed matter context, dark states have been less explored, especially when caused by interference between **sublattices**. This paper expands the concept to solid-state systems, where dark states emerge due to destructive interference within the crystal’s sublattices. These states remain hidden from ARPES measurements because their transition matrix elements vanish.
+In the condensed-matter context, dark states have been less explored, especially when caused by interference between **sublattices**. This paper expands the concept to solid-state systems, where dark states emerge due to destructive interference within the crystal’s sublattices. These states remain hidden from ARPES measurements because their transition matrix elements vanish.
 
 ### Key Definitions:
 
@@ -31,21 +31,21 @@ In the condensed matter context, dark states have been less explored, especially
 
 ## 2. Crystal Structure and Sublattice Symmetry
 
-PdSe₂ is chosen for its crystal structure, which consists of two pairs of **sublattices** labeled A, B, C, and D. These sublattices are related by **glide-mirror symmetries**, which leads to specific **quantum phases** that control the interference patterns of electronic wavefunctions in the Brillouin zone.
+PdSe₂ is chosen for its crystal structure, which consists of two pairs of **sublattices** labeled \( A \), \( B \), \( C \), and \( D \). These sublattices are related by **glide-mirror symmetries**, which leads to specific **quantum phases** that control the interference patterns of electronic wavefunctions in the Brillouin zone.
 
-Mathematically, the electronic structure of PdSe₂ is described using the **tight-binding Hamiltonian** model. The dominant states arise from **Pd 4d orbitals**, and the relative phases $ \varphi*{AB}, \varphi*{AC}, \varphi\_{AD} $ between sublattices dictate whether the interference is constructive or destructive.
+Mathematically, the electronic structure of PdSe₂ is described using the **tight-binding Hamiltonian** model. The dominant states arise from **Pd 4d orbitals**, and the relative phases \( \varphi*{AB}, \varphi*{AC}, \varphi\_{AD} \) between sublattices dictate whether the interference is constructive or destructive:
 
-$$
-H_{PdSe_2} =
+\[
+H*{PdSe_2} =
 \begin{pmatrix}
-f_{AA} & f_{AB} & f_{AC} & f_{AD} \\
-f_{AB} & f_{AA} & f_{AC} & f_{AD} \\
-f_{AC} & f_{AD} & f_{AA} & f_{AB} \\
-f_{AD} & f_{AC} & f_{AB} & f_{AA}
+f*{AA} & f*{AB} & f*{AC} & f*{AD} \\
+f*{AB} & f*{AA} & f*{AC} & f*{AD} \\
+f*{AC} & f*{AD} & f*{AA} & f*{AB} \\
+f*{AD} & f*{AC} & f*{AB} & f\_{AA}
 \end{pmatrix}
-$$
+\]
 
-The key discovery here is that PdSe₂ has a unique sublattice arrangement where **multiple glide-mirror symmetries** connect these sublattices, resulting in **double destructive interference** under certain conditions, leading to the appearance of **dark states**.
+The key discovery is that PdSe₂ has a unique sublattice arrangement where **multiple glide-mirror symmetries** connect these sublattices, resulting in **double destructive interference** under certain conditions, leading to the appearance of **dark states**.
 
 ### Key Definitions:
 
@@ -55,109 +55,31 @@ The key discovery here is that PdSe₂ has a unique sublattice arrangement where
 
 ---
 
-## 3. Angle-Resolved Photoemission Spectroscopy (ARPES), Light Polarization, and Dark States
+## 3. ARPES, Light Polarization, and Dark States
 
 The experimental technique used in the paper, **ARPES**, allows researchers to probe the electronic band structure of a material. However, in PdSe₂, an entire band of electronic states in the Brillouin zone is **invisible** regardless of the photon energy or light polarization used. This is a clear indicator of **dark states** resulting from **sublattice interference**.
 
 The transition probability in ARPES is governed by **Fermi’s Golden Rule**:
 
-$$
-M_k = \int \psi_f^* (\mathbf{A} \cdot \mathbf{p}) \psi_i \, dV
-$$
+\[
+M_k = \int \psi_f^\* (\mathbf{A} \cdot \mathbf{p}) \psi_i \, dV
+\]
 
 Where:
 
-- $ \mathbf{A} $ is the electromagnetic vector potential.
-- $ \mathbf{p} $ is the momentum operator.
-- $ \psi_i $ and $ \psi_f $ are the initial and final electronic states.
+- \( \mathbf{A} \) is the electromagnetic vector potential.
+- \( \mathbf{p} \) is the momentum operator.
+- \( \psi_i \) and \( \psi_f \) are the initial and final electronic states.
 
-The critical point in PdSe₂ is that the **interference between sublattices** can lead to **destructive interference** when certain relative quantum phases $ \varphi*{AB}, \varphi*{AC}, \varphi\_{AD} $ cancel out the matrix elements $ M_k $. This makes some states completely **undetectable by ARPES**.
-
-The experimental data shows that with **p-polarized light**, only one of the **nine cuboidal Brillouin zones** exhibits detectable valence bands (centered at Γ₆). However, when using **s-polarized light**, even this valence band disappears, as shown in the data taken under identical experimental conditions.
-
-In summary:
-
-- With **p-polarized light**, constructive interference allows detection of the 000 state.
-- With **s-polarized light**, all pseudospin states vanish due to destructive interference.
-- Bands centered at $ \Gamma*{106}, \Gamma*{101}, \Gamma\_{016} $ in the kx and ky directions are **not observed** regardless of the polarization.
-
-This clearly indicates the existence of **dark states** in PdSe₂, which are **undetectable** at any photon energy or light polarization due to **double destructive interference** in these sublattices.
-
-### Key Definitions:
-
-- **ARPES**: A technique used to observe the energy and momentum distribution of electrons in a material, providing insights into its electronic structure.
-- **Fermi’s Golden Rule**: A formula that calculates the transition probability per unit time for a quantum system interacting with an external perturbation.
-- **p-polarized light**: Light in which the electric field oscillates parallel to the plane of incidence.
-- **s-polarized light**: Light in which the electric field oscillates perpendicular to the plane of incidence.
+The critical point in PdSe₂ is that the **interference between sublattices** can lead to **destructive interference** when certain relative quantum phases \( \varphi*{AB}, \varphi*{AC}, \varphi\_{AD} \) cancel out the matrix elements \( M_k \). This makes some states completely **undetectable by ARPES**.
 
 ---
 
-## 4. Phase Polarization and Quantum States in the Brillouin Zone
+## 4. Phase Polarization and Brillouin Zone Quantum States
 
-A major finding in this paper is the identification of **phase polarization** in the Brillouin zone of PdSe₂. The electronic wavefunctions in PdSe₂ are fully polarized to one of four possible states: **000, 0ππ, π0π, ππ0**, depending on the relative quantum phases $ \varphi*{AB}, \varphi*{AC}, \varphi\_{AD} $.
+A major finding in this paper is the identification of **phase polarization** in the Brillouin zone of PdSe₂. The electronic wavefunctions in PdSe₂ are fully polarized to one of four possible states: \( 000, 0\pi\pi, \pi0\pi, \pi\pi0 \), depending on the relative quantum phases \( \varphi*{AB}, \varphi*{AC}, \varphi\_{AD} \).
 
 - The **000 state** (blue pseudospin) is **visible** in ARPES under **p-polarized light**, because constructive interference ensures a non-zero matrix element.
-- The other states, **0ππ, π0π, and ππ0** (red, yellow, and green pseudospins), are **dark states** because two of the three quantum phases are $ \pi $, leading to **double destructive interference**. These states are completely undetectable by ARPES under **any light polarization**.
-
-The phase polarization forms a **checkerboard pattern** in momentum space, where each region of the Brillouin zone is polarized to one of these four states. The **dark states** correspond to areas where double destructive interference occurs, making the electronic states invisible in ARPES measurements.
-
-### Key Definitions:
-
-- **Pseudospin**: An abstract concept used to describe two-level quantum systems, often associated with sublattices or quantum states.
-- **Brillouin Zone**: The fundamental region of reciprocal space in a crystal, within which the electronic wavefunctions are defined.
-
----
-
-## 5. Generalization to Other Materials
-
-The paper also generalizes the findings on dark states to other material systems with similar sublattice structures. This includes:
-
-- **Cuprates**: In high-temperature superconductors such as **Bi2201**, shadow bands have been observed in ARPES that cannot be explained by typical band theory. The paper demonstrates that these shadow bands are **dark states**, undetectable due to sublattice interference.
-  - For cuprates, the two nearly degenerate Fermi surfaces (FS1 and FS2) show that segments of the Fermi surface polarized to **000 states** are visible in ARPES with **p-polarized light**, while segments polarized to **0ππ states** are visible with **s-polarized light**.
-  - However, parts of the Fermi surface polarized to **π0π** and **ππ0** remain undetectable due to dark states.
-- **Lead Halide Perovskites**: In **CsPbBr₃**, ARPES measurements show that two distinct valence bands (VB1 and VB2) are observed depending on the photon energy used (kz = $ \Gamma_7 $ for VB1, kz = $ \Gamma_8 $ for VB2). However, only specific bands appear under **p-polarized light**, and both bands vanish under **s-polarized light**. This behavior is explained as a result of **dark states** in the perovskite's sublattice structure.
-
-These findings demonstrate that the phenomenon of **dark states** is not limited to PdSe₂ but is a **universal feature** in materials with two pairs of sublattices connected by **glide-mirror symmetries**.
-
-### Key Definitions:
-
-- **Shadow Bands**: Bands in the ARPES data of cuprates that appear with lower intensity or are completely undetectable due to their sublattice interference.
-- **Band Folding**: A phenomenon in ARPES where multiple bands appear due to the periodicity of the crystal lattice, often related to structural distortions or superlattice formations.
+- The other states, \( 0\pi\pi, \pi0\pi, \pi\pi0 \) (red, yellow, and green pseudospins), are **dark states** because two of the three quantum phases are \( \pi \), leading to **double destructive interference**. These states are completely undetectable by ARPES under **any light polarization**.
 
 ---
-
-## 6. Novel Contributions and Impact of the Paper
-
-The paper makes several important novel contributions to the field of condensed matter physics:
-
-1. **Discovery of Condensed-Matter Dark States**: This study is the first to report the observation of dark states in a solid-state material (PdSe₂). These dark states are completely undetectable in ARPES due to destructive interference between wavefunctions from different sublattices. This represents a new class of quantum states in condensed matter physics that had previously only been studied in atomic and molecular systems.
-
-2. **Mechanism of Sublattice Interference**: The authors introduce a rigorous theoretical framework that explains how the interference between sublattices, connected by **glide-mirror symmetries**, leads to the emergence of dark states. The framework is supported by first-principles calculations and the **tight-binding Hamiltonian** model, which accurately describes the band structure and phase relationships in PdSe₂.
-
-3. **Generalization to Other Systems**: The study extends the concept of dark states to other materials, such as **cuprates** and **lead halide perovskites**, providing a unified explanation for previously unexplained ARPES data in these systems. For example, the **shadow bands** in cuprates and the **vanishing bands** in perovskites can now be understood as the result of sublattice interference and dark states.
-
-4. **Light Polarization Effects**: One of the most striking aspects of the paper is the detailed discussion of how **light polarization** affects the visibility of electronic states in ARPES. The authors show that:
-   - **p-polarized light** can detect certain quantum states (e.g., the 000 state in PdSe₂).
-   - **s-polarized light** renders all states invisible due to complete destructive interference.
-     This effect has profound implications for future experimental studies using ARPES, as it highlights the need to carefully control light polarization to detect or suppress specific quantum states.
-
-### Why This Paper is Impressive
-
-This paper is highly impressive for several reasons:
-
-- **Novel Concept**: The generalization of **dark states** to condensed matter systems is a significant breakthrough. It opens up a new avenue of research for understanding hidden quantum states in materials that were previously inaccessible to experimental probes like ARPES.
-- **Comprehensive Approach**: The combination of sophisticated experimental techniques (e.g., ARPES) with detailed theoretical modeling (tight-binding calculations and symmetry analysis) makes this paper a comprehensive and authoritative study on the topic.
-- **Broad Implications**: The findings are not limited to PdSe₂ but extend to a wide range of materials with sublattice structures. This could have far-reaching implications for the study of high-temperature superconductors, optoelectronic materials, and other quantum systems.
-- **Resolution of Long-Standing Puzzles**: By explaining phenomena such as the shadow bands in cuprates and the vanishing bands in perovskites, the paper resolves several long-standing puzzles in the field of condensed matter physics.
-
-In summary, the discovery of condensed-matter dark states, the detailed analysis of sublattice interference, and the generalization to other material systems make this paper a landmark contribution to the study of quantum materials.
-
----
-
-### Key Definitions (Recap):
-
-- **Dark State**: A quantum state that does not interact with light, making it undetectable in spectroscopic experiments.
-- **Sublattice Interference**: The interaction between wavefunctions from different sublattices, leading to constructive or destructive interference.
-- **p-polarized light**: Light whose electric field oscillates parallel to the plane of incidence, often used to detect quantum states in ARPES.
-- **s-polarized light**: Light whose electric field oscillates perpendicular to the plane of incidence, which can cause destructive interference in certain quantum states.
-- **Fermi’s Golden Rule**: A quantum mechanical formula used to calculate the transition probability of electrons between states when interacting with light.