From e0d4ea9975dcbdc17399d97ad27db0b3475732e5 Mon Sep 17 00:00:00 2001 From: Utkarsh Singh Date: Wed, 15 Jan 2025 19:32:32 +0100 Subject: [PATCH] anew --- _posts/2023-08-01-mace.md | 45 ++++++++--- _posts/2024-04-18-kan.md | 28 +++---- _posts/2024-04-21-gnn_kan.md | 123 ++++++++++++++++--------------- _posts/2024-05-01-cognn.md | 103 +++++++++++++------------- _posts/2024-06-13-renorm.md | 47 ++++++++---- _posts/2024-09-12-dark_states.md | 35 +++++---- package-lock.json | 6 +- package.json | 4 +- 8 files changed, 224 insertions(+), 167 deletions(-) diff --git a/_posts/2023-08-01-mace.md b/_posts/2023-08-01-mace.md index 8d43eaa2..249db2d1 100644 --- a/_posts/2023-08-01-mace.md +++ b/_posts/2023-08-01-mace.md @@ -10,9 +10,11 @@ related_posts: false --- ### **Introduction** + MACE (Message Passing Atomic Cluster Expansion) is an equivariant message-passing neural network that uses higher-order messages to enhance the accuracy and efficiency of force fields in computational chemistry. ### **Node Representation** + Each node \(\large{i}\) is represented by: \[ @@ -22,38 +24,43 @@ Each node \(\large{i}\) is represented by: where \(r_i \in \mathbb{R}^3\) is the position, \(\large{z_i}\) is the chemical element, and \(\large{h_i^{(t)}}\) are the learnable features at layer \(\large{t}\). ### **Message Construction** + Messages are constructed hierarchically using a body order expansion: \[ -m_i^{(t)} = \sum_j u_1(\sigma_i^{(t)}, \sigma_j^{(t)}) + \sum_{j_1, j_2} u_2(\sigma_i^{(t)}, \sigma_{j_1}^{(t)}, \sigma_{j_2}^{(t)}) + \cdots + \sum_{j_1, \ldots, j_\nu} u_\nu(\sigma_i^{(t)}, \sigma_{j_1}^{(t)}, \ldots, \sigma_{j_\nu}^{(t)}) +m*i^{(t)} = \sum_j u_1(\sigma_i^{(t)}, \sigma_j^{(t)}) + \sum*{j*1, j_2} u_2(\sigma_i^{(t)}, \sigma*{j*1}^{(t)}, \sigma*{j*2}^{(t)}) + \cdots + \sum*{j*1, \ldots, j*\nu} u*\nu(\sigma_i^{(t)}, \sigma*{j*1}^{(t)}, \ldots, \sigma*{j\_\nu}^{(t)}) \] ### **Two-body Message Construction** + For two-body interactions, the message \(m_i^{(t)}\) is: \[ -A_i^{(t)} = \sum_{j \in N(i)} R_{kl_1l_2l_3}^{(t)}(r_{ij}) Y_{l_1}^{m_1}(\hat{r}_{ij}) W_{kk_2l_2}^{(t)} h_{j,k_2l_2m_2}^{(t)} +A*i^{(t)} = \sum*{j \in N(i)} R*{kl_1l_2l_3}^{(t)}(r*{ij}) Y*{l_1}^{m_1}(\hat{r}*{ij}) W*{kk_2l_2}^{(t)} h*{j,k_2l_2m_2}^{(t)} \] where \(\large{R}\) is a learnable radial basis, \(\large{Y}\) are spherical harmonics, and \(\large{W}\) are learnable weights. \(\large{C}\) are Clebsch-Gordan coefficients ensuring equivariance. ### **Higher-order Feature Construction** + Higher-order features are constructed using tensor products and symmetrization: \[ -\large{B_{i, \eta \nu k LM}^{(t)} = \sum_{lm} C_{LM \eta \nu, lm} \prod_{\xi=1}^\nu \sum_{k_\xi} w_{kk_\xi l_\xi}^{(t)} A_{i, k_\xi l_\xi m_\xi}^{(t)}} +\large{B*{i, \eta \nu k LM}^{(t)} = \sum*{lm} C*{LM \eta \nu, lm} \prod*{\xi=1}^\nu \sum*{k*\xi} w*{kk*\xi l*\xi}^{(t)} A*{i, k*\xi l*\xi m\_\xi}^{(t)}} \] where \(\large{C}\) are generalized Clebsch-Gordan coefficients. ### **Message Passing** + The message passing updates the node features by aggregating messages: \[ -\large{h_i^{(t+1)} = U_{kL}^{(t)}(\sigma_i^{(t)}, m_i^{(t)}) = \sum_{k'} W_{kL, k'}^{(t)} m_{i, k' LM} + \sum_{k'} W_{z_i kL, k'}^{(t)} h_{i, k' LM}^{(t)}} +\large{h*i^{(t+1)} = U*{kL}^{(t)}(\sigma*i^{(t)}, m_i^{(t)}) = \sum*{k'} W*{kL, k'}^{(t)} m*{i, k' LM} + \sum*{k'} W*{z*i kL, k'}^{(t)} h*{i, k' LM}^{(t)}} \] ### **Readout Phase** + In the readout phase, invariant features are mapped to site energies: \[ @@ -63,35 +70,39 @@ In the readout phase, invariant features are mapped to site energies: where: \[ -\large{E_i^{(t)} = R_t(h_i^{(t)}) = \sum_{k'} W_{\text{readout}, k'}^{(t)} h_{i, k' 00}^{(t)} \quad \text{for } t < T} +\large{E*i^{(t)} = R_t(h_i^{(t)}) = \sum*{k'} W*{\text{readout}, k'}^{(t)} h*{i, k' 00}^{(t)} \quad \text{for } t < T} \] \[ -\large{E_i^{(T)} = \text{MLP}_{\text{readout}}^{(t)}(\{h_{i, k 00}^{(t)}\})} +\large{E*i^{(T)} = \text{MLP}*{\text{readout}}^{(t)}(\{h\_{i, k 00}^{(t)}\})} \] ### **Equivariance** + The model ensures equivariance under rotation \(\large{Q \in O(3)}\): \[ \large{h_i^{(t)}(Q \cdot (r_1, \ldots, r_N)) = D(Q) h_i^{(t)}(r_1, \ldots, r_N)} \] -where \(\large{D(Q)}\) is a Wigner D-matrix. For feature \(\large{h_{i, k LM}^{(t)}}\), it transforms as: +where \(\large{D(Q)}\) is a Wigner D-matrix. For feature \(\large{h\_{i, k LM}^{(t)}}\), it transforms as: \[ -\large{h_{i, k LM}^{(t)}(Q \cdot (r_1, \ldots, r_N)) = \sum_{M'} D_L(Q)_{M'M} h_{i, k LM'}^{(t)}(r_1, \ldots, r_N)} +\large{h*{i, k LM}^{(t)}(Q \cdot (r_1, \ldots, r_N)) = \sum*{M'} D*L(Q)*{M'M} h\_{i, k LM'}^{(t)}(r_1, \ldots, r_N)} \] ## Properties and Computational Efficiency 1. **Body Order Expansion**: + - MACE constructs messages using higher body order expansions, enabling rich representations of atomic environments. 2. **Computational Efficiency**: + - The use of higher-order messages reduces the required number of message-passing layers to two, enhancing computational efficiency and scalability. 3. **Receptive Field**: + - MACE maintains a small receptive field by decoupling correlation order increase from the number of message-passing iterations, facilitating parallelization. 4. **State-of-the-Art Performance**: @@ -106,12 +117,15 @@ For further details, refer to the [Batatia et al.](https://arxiv.org/abs/2206.07 ### 1. **Spherical Harmonics** **Concept**: + - Spherical harmonics \(Y^L_M\) are functions defined on the surface of a sphere. They are used in many areas of physics, including quantum mechanics and electrodynamics, to describe the angular part of a system. **Role in MACE**: + - Spherical harmonics are used to decompose the angular dependency of the atomic environment. This helps in capturing the rotational properties of the features in a systematic way. **Mathematically**: + - The spherical harmonics \(Y^L_M(\theta, \phi)\) are given by: \[ @@ -123,29 +137,35 @@ where \(P^M_L\) are the associated Legendre polynomials. ### 2. **Clebsch-Gordan Coefficients** **Concept**: + - Clebsch-Gordan coefficients are used in quantum mechanics to combine angular momenta. They arise in the coupling of two angular momentum states to form a new angular momentum state. **Role in MACE**: + - In MACE, Clebsch-Gordan coefficients are used to combine features from different atoms while maintaining rotational invariance. They ensure that the resulting features transform correctly under rotations, preserving the physical symmetry of the system. **Mathematically**: + - When combining two angular momentum states \(\vert l_1, m_1\rangle\) and \(\vert l_2, m_2\rangle\), the resulting state \(\vert L, M\rangle\) is given by: \[ -|L, M\rangle = \sum_{m_1, m_2} C_{L, M}^{l_1, m_1; l_2, m_2} |l_1, m_1\rangle |l_2, m_2\rangle +|L, M\rangle = \sum*{m_1, m_2} C*{L, M}^{l_1, m_1; l_2, m_2} |l_1, m_1\rangle |l_2, m_2\rangle \] -where \(C_{L, M}^{l_1, m_1; l_2, m_2}\) are the Clebsch-Gordan coefficients. +where \(C\_{L, M}^{l_1, m_1; l_2, m_2}\) are the Clebsch-Gordan coefficients. ### 3. **\(O(3)\) Rotations** **Concept**: + - The group \(O(3)\) consists of all rotations and reflections in three-dimensional space. It represents the symmetries of a 3D system, including operations that preserve the distance between points. **Role in MACE**: + - Ensuring that the neural network respects \(O(3)\) symmetry is crucial for modeling physical systems accurately. MACE achieves this by using operations that are invariant or equivariant under these rotations and reflections. **Mathematically**: + - A rotation in \(O(3)\) can be represented by a 3x3 orthogonal matrix \(Q\) such that: \[ @@ -157,14 +177,17 @@ where \(I\) is the identity matrix. ### 4. **Wigner D-matrix** **Concept**: + - The Wigner D-matrix \(D^L(Q)\) represents the action of a rotation \(Q\) on spherical harmonics. It provides a way to transform the components of a tensor under rotation. **Role in MACE**: + - Wigner D-matrices are used to ensure that the feature vectors in the neural network transform correctly under rotations. This is essential for maintaining the rotational equivariance of the model. **Mathematically**: + - For a rotation \(Q \in O(3)\) and a spherical harmonic of degree \(L\), the Wigner D-matrix \(D^L(Q)\) is a \((2L+1) \times (2L+1)\) matrix. If \(Y^L_M\) is a spherical harmonic, then under rotation \(Q\), it transforms as: \[ -Y^L_M(Q \cdot \mathbf{r}) = \sum_{M'=-L}^{L} D^L_{M'M}(Q) Y^L_{M'}(\mathbf{r}) +Y^L*M(Q \cdot \mathbf{r}) = \sum*{M'=-L}^{L} D^L*{M'M}(Q) Y^L*{M'}(\mathbf{r}) \] diff --git a/_posts/2024-04-18-kan.md b/_posts/2024-04-18-kan.md index 80251112..539be610 100644 --- a/_posts/2024-04-18-kan.md +++ b/_posts/2024-04-18-kan.md @@ -23,10 +23,10 @@ The motivation for KANs stems from the limitations of MLPs, such as fixed activa The Kolmogorov-Arnold representation theorem states: \[ -f(x) = \sum_{q=1}^{2n+1} \Phi_q \left( \sum_{p=1}^n \varphi_{q,p}(x_p) \right) +f(x) = \sum*{q=1}^{2n+1} \Phi_q \left( \sum*{p=1}^n \varphi\_{q,p}(x_p) \right) \] -where \(\varphi_{q,p} : [0, 1] \to \mathbb{R}\) and \(\Phi_q : \mathbb{R} \to \mathbb{R}\). +where \(\varphi\_{q,p} : [0, 1] \to \mathbb{R}\) and \(\Phi_q : \mathbb{R} \to \mathbb{R}\). ### 3. KAN Architecture @@ -34,16 +34,16 @@ KANs generalize the representation theorem to arbitrary depths and widths. Each #### 3.1. Mathematical Formulation of KANs -Define a KAN layer with \(n_{\text{in}}\)-dimensional inputs and \(n_{\text{out}}\)-dimensional outputs as a matrix of 1D functions: +Define a KAN layer with \(n*{\text{in}}\)-dimensional inputs and \(n*{\text{out}}\)-dimensional outputs as a matrix of 1D functions: \[ -\Phi = \{ \varphi_{q,p} \}, \quad p = 1, 2, \ldots, n_{\text{in}}, \quad q = 1, 2, \ldots, n_{\text{out}} +\Phi = \{ \varphi*{q,p} \}, \quad p = 1, 2, \ldots, n*{\text{in}}, \quad q = 1, 2, \ldots, n\_{\text{out}} \] -Activation function on edge \(\varphi_{l,j,i}\) between layer \(l\) and \(l+1\) is given by: +Activation function on edge \(\varphi\_{l,j,i}\) between layer \(l\) and \(l+1\) is given by: \[ -\varphi_{l,j,i}(x) = w \big(b(x) + \text{spline}(x)\big) +\varphi\_{l,j,i}(x) = w \big(b(x) + \text{spline}(x)\big) \] where \(b(x) = \text{silu}(x) = \frac{x}{1 + e^{-x}}\). @@ -51,13 +51,13 @@ where \(b(x) = \text{silu}(x) = \frac{x}{1 + e^{-x}}\). The output of each layer is computed as: \[ -x_{l+1, j} = \sum_{i=1}^{n_l} \varphi_{l,j,i}(x_{l,i}) +x*{l+1, j} = \sum*{i=1}^{n*l} \varphi*{l,j,i}(x\_{l,i}) \] in matrix form: \[ -x_{l+1} = \Phi_l x_l +x\_{l+1} = \Phi_l x_l \] where \(\Phi_l\) is the function matrix of layer \(l\). @@ -71,13 +71,13 @@ KANs can approximate functions by decomposing high-dimensional problems into sev Let \(f(x)\) be represented as: \[ -f = (\Phi_{L-1} \circ \Phi_{L-2} \circ \cdots \circ \Phi_1 \circ \Phi_0)x +f = (\Phi*{L-1} \circ \Phi*{L-2} \circ \cdots \circ \Phi_1 \circ \Phi_0)x \] -For each \(\Phi_{l,i,j}\), there exist \(k\)-th order B-spline functions \(\Phi_{l,i,j}^G\) such that: +For each \(\Phi*{l,i,j}\), there exist \(k\)-th order B-spline functions \(\Phi*{l,i,j}^G\) such that: \[ -\| f - (\Phi_{L-1}^G \circ \Phi_{L-2}^G \circ \cdots \circ \Phi_1^G \circ \Phi_0^G)x \|_{C^m} \leq C G^{-k-1+m} +\| f - (\Phi*{L-1}^G \circ \Phi*{L-2}^G \circ \cdots \circ \Phi*1^G \circ \Phi_0^G)x \|*{C^m} \leq C G^{-k-1+m} \] where \(G\) is the grid size and \(C\) depends on \(f\) and its representation. @@ -87,7 +87,7 @@ where \(G\) is the grid size and \(C\) depends on \(f\) and its representation. KANs can increase accuracy by refining the grid used in splines: \[ -\{c'_j\} = \arg\min_{\{c'_j\}} E_{x \sim p(x)} \left( \sum_{j=0}^{G2+k-1} c'_j B'_j(x) - \sum_{i=0}^{G1+k-1} c_i B_i(x) \right)^2 +\{c'_j\} = \arg\min_{\{c'_j\}} E_{x \sim p(x)} \left( \sum*{j=0}^{G2+k-1} c'\_j B'\_j(x) - \sum*{i=0}^{G1+k-1} c_i B_i(x) \right)^2 \] ### 6. Simplification Techniques @@ -101,10 +101,10 @@ KANs were shown to have better scaling laws than MLPs, achieving lower test loss #### Example Functions: 1. Bessel function: \(f(x) = J_0(20x)\) -2. High-dimensional function: +2. High-dimensional function: \[ -f(x_1, \ldots, x_{100}) = \exp\left( \frac{1}{100} \sum_{i=1}^{100} \sin^2\left(\frac{\pi x_i}{2}\right) \right) +f(x*1, \ldots, x*{100}) = \exp\left( \frac{1}{100} \sum\_{i=1}^{100} \sin^2\left(\frac{\pi x_i}{2}\right) \right) \] KANs can achieve near-theoretical scaling exponents \(\alpha = 4\), outperforming MLPs in accuracy and parameter efficiency. diff --git a/_posts/2024-04-21-gnn_kan.md b/_posts/2024-04-21-gnn_kan.md index df402715..51a0b7f7 100644 --- a/_posts/2024-04-21-gnn_kan.md +++ b/_posts/2024-04-21-gnn_kan.md @@ -16,99 +16,99 @@ related_posts: false 1. **Input Node and Edge Features**: - - Nodes: \(\mathbf{x}_i\) (node features) - - Edges: \(\mathbf{e}_{ij}\) (edge features) + - Nodes: \(\mathbf{x}\_i\) (node features) + - Edges: \(\mathbf{e}\_{ij}\) (edge features) 2. **Message Passing Layer** (per layer): - a. **Edge Feature Transformation**: + a. **Edge Feature Transformation**: - \[ - \mathbf{e}'_{ij} = f_e(\mathbf{e}_{ij}) - \] + \[ + \mathbf{e}'_{ij} = f_e(\mathbf{e}_{ij}) + \] - where \(f_e\) is a transformation function applied to edge features. + where \(f_e\) is a transformation function applied to edge features. - b. **Message Computation**: + b. **Message Computation**: - \[ - \mathbf{m}_{ij} = f_m(\mathbf{x}_i, \mathbf{x}_j, \mathbf{e}'_{ij}) - \] + \[ + \mathbf{m}_{ij} = f_m(\mathbf{x}\_i, \mathbf{x}\_j, \mathbf{e}'_{ij}) + \] - where \(f_m\) computes messages using node features \(\mathbf{x}_i\), \(\mathbf{x}_j\), and transformed edge features \(\mathbf{e}'_{ij}\). + where \(f*m\) computes messages using node features \(\mathbf{x}\_i\), \(\mathbf{x}\_j\), and transformed edge features \(\mathbf{e}'*{ij}\). - c. **Message Aggregation**: + c. **Message Aggregation**: - \[ - \mathbf{m}_i = \sum_{j \in \mathcal{N}(i)} \mathbf{m}_{ij} - \] + \[ + \mathbf{m}_i = \sum_{j \in \mathcal{N}(i)} \mathbf{m}\_{ij} + \] - where \(\mathcal{N}(i)\) denotes the set of neighbors of node \(i\). + where \(\mathcal{N}(i)\) denotes the set of neighbors of node \(i\). - d. **Node Feature Update**: - - \[ - \mathbf{x}'_i = f_n(\mathbf{x}_i, \mathbf{m}_i) - \] + d. **Node Feature Update**: - where \(f_n\) updates node features using the aggregated messages \(\mathbf{m}_i\). + \[ + \mathbf{x}'\_i = f_n(\mathbf{x}\_i, \mathbf{m}\_i) + \] + + where \(f_n\) updates node features using the aggregated messages \(\mathbf{m}\_i\). 3. **Output Node and Edge Features**: - - - Nodes: \(\mathbf{x}'_i\) (updated node features) - - Edges: \(\mathbf{e}'_{ij}\) (updated edge features) + + - Nodes: \(\mathbf{x}'\_i\) (updated node features) + - Edges: \(\mathbf{e}'\_{ij}\) (updated edge features) ## E3-Equivariant GNN with Learnable Activation Functions on Edges 1. **Input Node and Edge Features**: - - - Nodes: \(\mathbf{x}_i\) (node features) - - Edges: \(\mathbf{e}_{ij}\) (edge features) + + - Nodes: \(\mathbf{x}\_i\) (node features) + - Edges: \(\mathbf{e}\_{ij}\) (edge features) 2. **Learnable Edge Feature Transformation**: - - - **Fourier-based Edge Transformation**: - - \[ - \mathbf{e}'_{ij} = \text{FourierTransform}(\mathbf{e}_{ij}) - \] - where the Fourier transformation is applied to edge features. Specifically, the transformation is defined as: + - **Fourier-based Edge Transformation**: + + \[ + \mathbf{e}'_{ij} = \text{FourierTransform}(\mathbf{e}_{ij}) + \] - \[ - \mathbf{e}'_{ij} = \sum_{k=1}^{K} a_{ij,k} \cos(k \mathbf{e}_{ij}) + b_{ij,k} \sin(k \mathbf{e}_{ij}) - \] + where the Fourier transformation is applied to edge features. Specifically, the transformation is defined as: - Here, \(a_{ij,k}\) and \(b_{ij,k}\) are learnable parameters, and \(K\) is the number of Fourier terms. + \[ + \mathbf{e}'_{ij} = \sum_{k=1}^{K} a*{ij,k} \cos(k \mathbf{e}*{ij}) + b*{ij,k} \sin(k \mathbf{e}*{ij}) + \] + + Here, \(a*{ij,k}\) and \(b*{ij,k}\) are learnable parameters, and \(K\) is the number of Fourier terms. 3. **Message Passing and Aggregation**: - + a. **Message Computation**: - \[ - \mathbf{m}_{ij} = \mathbf{e}'_{ij} \odot \mathbf{x}_j - \] - - where \(\odot\) denotes element-wise multiplication, combining the transformed edge features \(\mathbf{e}'_{ij}\) with the neighboring node features \(\mathbf{x}_j\). + \[ + \mathbf{m}_{ij} = \mathbf{e}'_{ij} \odot \mathbf{x}\_j + \] + + where \(\odot\) denotes element-wise multiplication, combining the transformed edge features \(\mathbf{e}'\_{ij}\) with the neighboring node features \(\mathbf{x}\_j\). b. **Message Aggregation**: - - \[ - \mathbf{m}_i = \sum_{j \in \mathcal{N}(i)} \mathbf{m}_{ij} - \] + + \[ + \mathbf{m}_i = \sum_{j \in \mathcal{N}(i)} \mathbf{m}\_{ij} + \] c. **Simple Node Feature Transformation**: - - \[ - \mathbf{x}'_i = \mathbf{W} (\mathbf{x}_i + \mathbf{m}_i) + \mathbf{b} - \] - - where \(\mathbf{W}\) is a learnable weight matrix and \(\mathbf{b}\) is a bias vector. + + \[ + \mathbf{x}'\_i = \mathbf{W} (\mathbf{x}\_i + \mathbf{m}\_i) + \mathbf{b} + \] + + where \(\mathbf{W}\) is a learnable weight matrix and \(\mathbf{b}\) is a bias vector. 4. **Output Node and Edge Features**: - - - Nodes: \(\mathbf{x}'_i\) (updated node features) - - Edges: \(\mathbf{e}'_{ij}\) (updated edge features) + + - Nodes: \(\mathbf{x}'\_i\) (updated node features) + - Edges: \(\mathbf{e}'\_{ij}\) (updated edge features) ## Full Implementation @@ -327,9 +327,11 @@ for epoch in range(num_epochs): print(f'Epoch {epoch + 1}, Train Loss: {train_loss:.4f}, Val Loss: {val_loss:.4f}') ``` + ## Detailed Explanation of Mathematical Formulations ### Learnable Edge Feature Transformation + For each edge $(i, j)$ with feature $\mathbf{e}_{ij}$: $$ @@ -339,6 +341,7 @@ $$ where $a_{ij,k}$ and $b_{ij,k}$ are learnable parameters, and $K$ is the number of terms. ### Message Computation + For each edge $(i, j)$: $$ @@ -348,6 +351,7 @@ $$ where $\odot$ denotes element-wise multiplication. ### Message Aggregation + For each node $i$: $$ @@ -357,6 +361,7 @@ $$ where $\mathcal{N}(i)$ denotes the set of neighbors of node $i$. ### Node Feature Update + For each node $i$: $$ diff --git a/_posts/2024-05-01-cognn.md b/_posts/2024-05-01-cognn.md index 96d4c6ac..87638b19 100644 --- a/_posts/2024-05-01-cognn.md +++ b/_posts/2024-05-01-cognn.md @@ -23,44 +23,47 @@ Co-GNNs introduce a novel, flexible message-passing mechanism where each node in ## **Mathematical Formulation** 1. **Action Selection (Action Network \(\pi\))**: - - For each node \(v\), the action network predicts a probability distribution \(p^{(\ell)}_v\) over the actions \(\{S, L, B, I\}\) at layer \(\ell\): - - \[ - p^{(\ell)}_v = \pi \left( h^{(\ell)}_v, \{ h^{(\ell)}_u \mid u \in N_v \} \right) - \] - - - Actions are sampled using the Straight-through Gumbel-softmax estimator. + + - For each node \(v\), the action network predicts a probability distribution \(p^{(\ell)}\_v\) over the actions \(\{S, L, B, I\}\) at layer \(\ell\): + + \[ + p^{(\ell)}\_v = \pi \left( h^{(\ell)}\_v, \{ h^{(\ell)}\_u \mid u \in N_v \} \right) + \] + + - Actions are sampled using the Straight-through Gumbel-softmax estimator. 2. **State Update (Environment Network \(\eta\))**: - - The environment network updates the node states based on the selected actions: - - \[ - h^{(\ell+1)}_v = - \begin{cases} - \eta^{(\ell)} \left( h^{(\ell)}_v, \{ \} \right) & \text{if } a^{(\ell)}_v = \text{I or B} \\ - \eta^{(\ell)} \left( h^{(\ell)}_v, \{ h^{(\ell)}_u \mid u \in N_v, a^{(\ell)}_u = \text{S or B} \} \right) & \text{if } a^{(\ell)}_v = \text{L or S} - \end{cases} - \] + + - The environment network updates the node states based on the selected actions: + + \[ + h^{(\ell+1)}\_v = + \begin{cases} + \eta^{(\ell)} \left( h^{(\ell)}\_v, \{ \} \right) & \text{if } a^{(\ell)}\_v = \text{I or B} \\ + \eta^{(\ell)} \left( h^{(\ell)}\_v, \{ h^{(\ell)}\_u \mid u \in N_v, a^{(\ell)}\_u = \text{S or B} \} \right) & \text{if } a^{(\ell)}\_v = \text{L or S} + \end{cases} + \] 3. **Layer-wise Update**: - - A Co-GNN layer involves predicting actions, sampling them, and updating node states. - - Repeated for \(L\) layers to obtain final node representations \(h^{(L)}_v\). + - A Co-GNN layer involves predicting actions, sampling them, and updating node states. + - Repeated for \(L\) layers to obtain final node representations \(h^{(L)}\_v\). ### **Environment Network \(\eta\) Details** The environment network updates node states using a message-passing scheme based on the selected actions. Let’s consider the standard GCN layer and how it adapts to Co-GNN concepts: 1. **Message Aggregation**: - - For each node \(v\), aggregate messages from its neighbors \(u\) that are broadcasting or using the standard action: - \[ - m_v^{(\ell)} = \sum_{u \in N_v, a_u^{(\ell)} = \text{S or B}} h_u^{(\ell)} - \] + + - For each node \(v\), aggregate messages from its neighbors \(u\) that are broadcasting or using the standard action: + \[ + m*v^{(\ell)} = \sum*{u \in N_v, a_u^{(\ell)} = \text{S or B}} h_u^{(\ell)} + \] 2. **Node Update**: - - The node updates its state based on the aggregated messages and its current state: - \[ - h_v^{(\ell+1)} = \sigma \left( W^{(\ell)}_s h_v^{(\ell)} + W^{(\ell)}_n m_v^{(\ell)} \right) - \] + - The node updates its state based on the aggregated messages and its current state: + \[ + h_v^{(\ell+1)} = \sigma \left( W^{(\ell)}\_s h_v^{(\ell)} + W^{(\ell)}\_n m_v^{(\ell)} \right) + \] ### **Properties and Benefits** @@ -75,32 +78,32 @@ The environment network updates node states using a message-passing scheme based Consider a GCN (Graph Convolutional Network) adapted with Co-GNN concepts: 1. **GCN Layer (Traditional)**: - - \[ - h^{(\ell+1)}_v = \sigma \left( W^{(\ell)}_s h^{(\ell)}_v + W^{(\ell)}_n \sum_{u \in N_v} h^{(\ell)}_u \right) - \] + + \[ + h^{(\ell+1)}_v = \sigma \left( W^{(\ell)}\_s h^{(\ell)}\_v + W^{(\ell)}\_n \sum_{u \in N_v} h^{(\ell)}\_u \right) + \] 2. **Co-GNN Layer**: - - **Action Network**: Predicts action probabilities for each node. - - \[ - p^{(\ell)}_v = \text{Softmax} \left( W_a h^{(\ell)}_v + b_a \right) - \] - - - **Action Sampling**: Gumbel-softmax to select actions: - \[ - a^{(\ell)}_v \sim \text{Gumbel-Softmax}(p^{(\ell)}_v) - \] - - - **State Update (Environment Network)**: - - \[ - h^{(\ell+1)}_v = - \begin{cases} - \sigma \left( W^{(\ell)}_s h^{(\ell)}_v \right) & \text{if } a^{(\ell)}_v = \text{I or B} \\ - \sigma \left( W^{(\ell)}_s h^{(\ell)}_v + W^{(\ell)}_n \sum_{u \in N_v, a^{(\ell)}_u = \text{S or B}} h^{(\ell)}_u \right) & \text{if } a^{(\ell)}_v = \text{L or S} - \end{cases} - \] + + - **Action Network**: Predicts action probabilities for each node. + + \[ + p^{(\ell)}\_v = \text{Softmax} \left( W_a h^{(\ell)}\_v + b_a \right) + \] + + - **Action Sampling**: Gumbel-softmax to select actions: + \[ + a^{(\ell)}\_v \sim \text{Gumbel-Softmax}(p^{(\ell)}\_v) + \] + - **State Update (Environment Network)**: + + \[ + h^{(\ell+1)}_v = + \begin{cases} + \sigma \left( W^{(\ell)}\_s h^{(\ell)}\_v \right) & \text{if } a^{(\ell)}\_v = \text{I or B} \\ + \sigma \left( W^{(\ell)}\_s h^{(\ell)}\_v + W^{(\ell)}\_n \sum_{u \in N_v, a^{(\ell)}\_u = \text{S or B}} h^{(\ell)}\_u \right) & \text{if } a^{(\ell)}\_v = \text{L or S} + \end{cases} + \] ## Conclusion diff --git a/_posts/2024-06-13-renorm.md b/_posts/2024-06-13-renorm.md index 3b4255cf..0be83cbb 100644 --- a/_posts/2024-06-13-renorm.md +++ b/_posts/2024-06-13-renorm.md @@ -15,34 +15,39 @@ related_posts: false ## First, a one line (or two :D) introduction to key concepts **Magnetism in 2D Materials**: + - **2D Magnets** are materials with magnetic properties confined to two dimensions, influenced significantly by quantum effects. They are promising for applications in spintronics, where electronic spins are used to store, process, and transfer information. **Heisenberg Model**: + - Describes magnetic interactions through pairwise exchange interactions. - The Hamiltonian: - $$ H_0 = \sum_{i > j} J_{ij} \mathbf{S}_i \cdot \mathbf{S}_j $$ - where $$ J_{ij} $$ is the exchange interaction between spins $$ \mathbf{S}_i $$ and $$ \mathbf{S}_j $$. + $$ H*0 = \sum*{i > j} J*{ij} \mathbf{S}\_i \cdot \mathbf{S}\_j $$ + where $$ J*{ij} $$ is the exchange interaction between spins $$ \mathbf{S}\_i $$ and $$ \mathbf{S}\_j $$. **Electron-Phonon Coupling**: + - Refers to interactions between electrons and lattice vibrations (phonons). - These interactions affect various electronic properties, including magnetic exchange interactions. **Green’s Functions and Self-Energy**: -- **Green’s Functions** $$ G^{\sigma}_{ij}(i\omega_n) $$: Describe electron propagation with spin $$ \sigma $$. -- **Self-Energy** $$ \Sigma^{\sigma}_k(i\omega_n) $$: Represents interaction effects on electrons due to phonons and other electrons. + +- **Green’s Functions** $$ G^{\sigma}\_{ij}(i\omega_n) $$: Describe electron propagation with spin $$ \sigma $$. +- **Self-Energy** $$ \Sigma^{\sigma}\_k(i\omega_n) $$: Represents interaction effects on electrons due to phonons and other electrons. ## 1. Theory and Model -The paper extends the Heisenberg model to include electron-phonon interactions, recalculating the exchange interaction $$ J_{ij} $$ using the magnetic force theorem: +The paper extends the Heisenberg model to include electron-phonon interactions, recalculating the exchange interaction $$ J\_{ij} $$ using the magnetic force theorem: $$ J_{ij} = 2 \text{Tr}_{\omega L} \left[ \Delta_i G^{\uparrow}_{ij}(i\omega_n) \Delta_j G^{\downarrow}_{ji}(i\omega_n) \right] S^{-2} $$ where: + - $$ \Delta_i $$ is the exchange splitting at lattice site $$ i $$. -- $$ G^{\sigma}_{ij}(i\omega_n) $$ is the spin-polarized electron propagator. -- $$ \text{Tr}_{\omega L} $$ denotes the trace over Matsubara frequencies $$ i\omega_n $$ and orbital indices $$ L $$. +- $$ G^{\sigma}\_{ij}(i\omega_n) $$ is the spin-polarized electron propagator. +- $$ \text{Tr}\_{\omega L} $$ denotes the trace over Matsubara frequencies $$ i\omega_n $$ and orbital indices $$ L $$. Incorporating electron-phonon interactions, the Green’s function is renormalized using the Dyson equation: @@ -56,13 +61,13 @@ $$ \Delta \rightarrow \tilde{\Delta}_k(i\omega_n) = \Delta + \Sigma^{\uparrow}_k(i\omega_n) - \Sigma^{\downarrow}_k(i\omega_n) $$ -where the self-energy $$ \Sigma^{\sigma}_k(i\omega_n) $$ is given by: +where the self-energy $$ \Sigma^{\sigma}\_k(i\omega_n) $$ is given by: $$ \Sigma^{\sigma}_k(i\omega_n) = -T \sum_{k' \nu m} G^{\sigma}_{k'}(i\omega_n - i\omega_m) |g^{\nu \sigma}_{kk'}|^2 D_{k-k'}(i\omega_n - i\omega_m) $$ -Here, $$ g^{\nu \sigma}_{kk'} $$ is the electron-phonon coupling vertex, and $$ D_q(i\omega_n) $$ is the phonon propagator. +Here, $$ g^{\nu \sigma}\_{kk'} $$ is the electron-phonon coupling vertex, and $$ D_q(i\omega_n) $$ is the phonon propagator. **Context**: This theoretical framework allows the authors to predict how the electron-phonon interactions influence the magnetic properties of 2D materials by renormalizing the exchange interactions between spins. @@ -75,6 +80,7 @@ H = t \sum_{\langle ij \rangle \sigma} c^{\dagger}_{i\sigma} c_{j\sigma} + \frac $$ where: + - $$ t $$ is the nearest-neighbor hopping. - $$ \Delta $$ is the on-site exchange splitting. - $$ \omega_q $$ is the phonon frequency. @@ -101,7 +107,8 @@ $$ where $$ c $$ is a renormalization constant. **Derivation**: -- The linear temperature dependence arises from the self-energy correction, which modifies the exchange interaction strength $$ J_{ij} $$. + +- The linear temperature dependence arises from the self-energy correction, which modifies the exchange interaction strength $$ J\_{ij} $$. - The renormalization constant $$ c $$ is determined by the specific electronic structure of the material and the strength of the electron-phonon coupling. ## 4. Application to $$ \mathrm{Fe_3GeTe_2} $$ @@ -109,6 +116,7 @@ where $$ c $$ is a renormalization constant. For the metallic 2D ferromagnet $$ \mathrm{Fe_3GeTe_2} $$, the authors use first-principles calculations to determine the electronic and phononic structures. The temperature dependence of the exchange interactions is calculated, showing a reduction of the Curie temperature by about 10% due to electron-phonon interactions. **First-Principles Calculation**: + - **Density Functional Theory (DFT)** is employed to calculate the electronic structure. - **Density Functional Perturbation Theory (DFPT)** is used for phonon calculations. - The electronic structure in the vicinity of the Fermi level is parameterized using maximally localized Wannier functions. @@ -118,6 +126,7 @@ For the metallic 2D ferromagnet $$ \mathrm{Fe_3GeTe_2} $$, the authors use first ## 5. Spin-Wave Renormalization **Spin-Wave Theory**: + - Spin waves, or magnons, are collective excitations in a magnetically ordered system. - The stability of magnetic order is influenced by spin-wave spectra, which can be calculated by diagonalizing the spin-wave Hamiltonian. @@ -130,14 +139,16 @@ $$ where $$ A $$ is the anisotropy parameter. **Magnon Eigenvectors and Spectra**: -- Magnon frequencies $$ \Omega_{q \nu} $$ are obtained by diagonalizing the spin-wave Hamiltonian: + +- Magnon frequencies $$ \Omega\_{q \nu} $$ are obtained by diagonalizing the spin-wave Hamiltonian: $$ H^{\text{SW}}_{\mu \nu}(q) = \left[ \delta_{\mu \nu} \left( 2A \Phi + \sum_{\chi} J_{\mu \chi}(0) \right) - J_{\mu \nu}(q) \right] \langle S^z \rangle $$ where: -- $$ J_{\mu \nu}(q) $$ are the Fourier transforms of the exchange interaction matrix. + +- $$ J\_{\mu \nu}(q) $$ are the Fourier transforms of the exchange interaction matrix. - $$ \Phi = 1 - \left( 1 - \langle S^2_z \rangle / 2 \right) $$ is the Anderson-Callen decoupling factor for $$ S = 1 $$. The magnon spectra exhibit optical and acoustic branches. Near the $$ \Gamma $$ point, the acoustic branch disperses quadratically: @@ -149,6 +160,7 @@ $$ where $$ D $$ is the spin-stiffness constant and $$ \Omega_0 $$ is the gap due to single-ion anisotropy. **Temperature-Dependent Magnetization**: + - Magnetization $$ \langle S^z \rangle $$ is determined by spin-wave excitations, using the Tyablikov decoupling (RPA): $$ @@ -166,10 +178,12 @@ By solving these equations self-consistently, the renormalized exchange interact The discussion highlights several key points: 1. **Adiabatic vs. Antiadiabatic Electron-Phonon Coupling**: + - Most systems can be treated adiabatically, where phonon energies are much smaller than electron energies. - For stronger renormalization effects, systems with narrow electron bands or high phonon energies relative to electron energies need to be considered. 2. **Out-of-Equilibrium Effects**: + - Non-equilibrium distributions, such as those induced by charge currents or laser fields, can enhance electron-phonon coupling. - This can lead to significant changes in exchange interactions and magnetic properties. @@ -181,22 +195,27 @@ The discussion highlights several key points: ### One (or two :D) line mathematical context **Magnetic Force Theorem**: -- The exchange interaction $$ J_{ij} $$ is derived from the magnetic force theorem, which involves calculating the energy cost of rotating spins $$ \mathbf{S}_i $$ and $$ \mathbf{S}_j $$. + +- The exchange interaction $$ J\_{ij} $$ is derived from the magnetic force theorem, which involves calculating the energy cost of rotating spins $$ \mathbf{S}\_i $$ and $$ \mathbf{S}\_j $$. - The exchange interaction is given by the integral over the Brillouin zone of the product of the spin-resolved Green's functions and exchange splitting. **Dyson Equation**: + - The renormalization of Green’s functions due to self-energy $$ \Sigma_k(i\omega_n) $$ is given by the Dyson equation. - This renormalizes both the propagators and the exchange splitting. **Electron-Phonon Self-Energy**: -- The self-energy $$ \Sigma^{\sigma}_k(i\omega_n) $$ represents the correction to the electron’s energy due to its interaction with phonons. + +- The self-energy $$ \Sigma^{\sigma}\_k(i\omega_n) $$ represents the correction to the electron’s energy due to its interaction with phonons. - The self-energy is calculated using second-order perturbation theory in the electron-phonon coupling. **Spin-Wave Theory**: + - The spin-wave Hamiltonian is derived by expanding the Heisenberg Hamiltonian to second order in spin deviations. - Diagonalizing the resulting Hamiltonian gives the magnon eigenvalues (frequencies) and eigenvectors. **Temperature-Dependent Magnetization**: + - The magnetization $$ \langle S^z \rangle $$ is obtained using the Tyablikov decoupling method, which approximates the thermal averages of spin operators. - The self-consistent solution of the magnetization equations provides the temperature dependence of $$ \langle S^z \rangle $$ and $$ J(T) $$. diff --git a/_posts/2024-09-12-dark_states.md b/_posts/2024-09-12-dark_states.md index c6b45998..de63ca17 100644 --- a/_posts/2024-09-12-dark_states.md +++ b/_posts/2024-09-12-dark_states.md @@ -17,11 +17,12 @@ This review discusses the discovery of **condensed-matter dark states** in the m ## 1. Introduction to Dark States -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 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. ### Key Definitions: + - **Dark State**: A quantum state that does not interact with light and is therefore undetectable by traditional spectroscopic methods. - **Quantum Interference**: The phenomenon where the probability amplitudes of quantum states add or cancel out, affecting the visibility of quantum transitions. @@ -31,10 +32,10 @@ In the condensed matter context, dark states have been less explored, especially 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} \\ @@ -43,9 +44,10 @@ f_{AD} & f_{AC} & f_{AB} & f_{AA} \end{pmatrix} $$ -The key discovery here is that PdSe₂ has a unique sublattice arrangement where **multiple glide-mirror symmetries** connect these sublattices, resulting in **double destructive interference** under certain conditions, leading to the appearance of **dark states**. +The key discovery here is that PdSe₂ has a unique sublattice arrangement where **multiple glide-mirror symmetries** connect these sublattices, resulting in **double destructive interference** under certain conditions, leading to the appearance of **dark states**. ### Key Definitions: + - **Sublattice**: A subset of atoms within a crystal lattice that repeats in a regular pattern. - **Glide-Mirror Symmetry**: A symmetry operation combining a reflection with a translation. - **Tight-Binding Hamiltonian**: A mathematical model used to describe the movement of electrons in a material by considering the hopping between atoms. @@ -63,32 +65,35 @@ M_k = \int \psi_f^* (\mathbf{A} \cdot \mathbf{p}) \psi_i \, dV $$ 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. -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 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. +- 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. - + --- ## 4. Phase Polarization and Quantum States in the Brillouin Zone -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, 0ππ, π0π, ππ0**, 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**. @@ -96,6 +101,7 @@ A major finding in this paper is the identification of **phase polarization** in 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. @@ -104,15 +110,16 @@ The phase polarization forms a **checkerboard pattern** in momentum space, where ## 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. - + - 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. @@ -131,11 +138,12 @@ The paper makes several important novel contributions to the field of condensed 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. + 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. @@ -146,10 +154,9 @@ In summary, the discovery of condensed-matter dark states, the detailed analysis --- ### 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. - - diff --git a/package-lock.json b/package-lock.json index 4327cd9c..e9b4e8a5 100644 --- a/package-lock.json +++ b/package-lock.json @@ -1,12 +1,12 @@ { - "name": "al-folio", + "name": "utksi.github.io", "lockfileVersion": 3, "requires": true, "packages": { "": { "devDependencies": { - "@shopify/prettier-plugin-liquid": "1.4.0", - "prettier": "3.1.1" + "@shopify/prettier-plugin-liquid": "^1.4.0", + "prettier": "^3.1.1" } }, "node_modules/@shopify/liquid-html-parser": { diff --git a/package.json b/package.json index dc4c51ab..7374b3b8 100644 --- a/package.json +++ b/package.json @@ -1,6 +1,6 @@ { "devDependencies": { - "@shopify/prettier-plugin-liquid": "1.4.0", - "prettier": "3.1.1" + "@shopify/prettier-plugin-liquid": "^1.4.0", + "prettier": "^3.1.1" } }