From d77474fc2374a741d3ea065a6ef89688f9b18699 Mon Sep 17 00:00:00 2001 From: Angela Riva <62027430+angelariva@users.noreply.github.com> Date: Thu, 2 May 2024 14:40:08 +0200 Subject: [PATCH] Update anderson-model.md --- docs/src/examples/anderson-model.md | 5 +++-- 1 file changed, 3 insertions(+), 2 deletions(-) diff --git a/docs/src/examples/anderson-model.md b/docs/src/examples/anderson-model.md index 795e735..96a71d3 100644 --- a/docs/src/examples/anderson-model.md +++ b/docs/src/examples/anderson-model.md @@ -2,14 +2,15 @@ In these two examples, we use the fermionic chain mapping proposed in [^khon_efficient_2021] to perform tensor network simulations of the Single Impurity Anderson Model (SIAM). The SIAM Hamiltonian is defined as: -$$ +``math \hat H^\text{SIAM} = \hat H_\text{loc} + \hat H_\text{hyb} + \hat H_\text{cond} = \overbrace{\epsilon_d \hat d^\dagger \hat d}^{\hat H_\text{loc}} + \underbrace{\sum_{k} V_k \Big( \hat d^\dagger \hat c_k + \hat c_k^\dagger \hat d \Big)}_{H_\text{hyb}} + \underbrace{\sum_k \epsilon_k \hat c_k^\dagger \hat c_k}_{H_I^\text{chain}}. -$$ +`` All of the operators obey to the usual fermionic anti-commutation relations: $\{\hat c_i, \hat c_j^\dagger \} = \delta_{ij}$, $\{\hat c_i, \hat c_j \} =\{\hat c_i^\dagger, \hat c_j^\dagger \} =0$ $\forall i,j$. The chain mapping is based on a thermofield-like transformation [2], performed with fermions: ancillary fermionic operators $\hat c_{2k}$ are defined, one for each of the original fermionic modes $\hat c_{1k}$. A Bogoliubov transformation is then applied, so that two new fermionic modes $\hat f_{1k}$ and $\hat f_{2k}$ are defined as a linear combination of $\hat c_{1k}$ and $\hat c_{2k}$. Two chains are defined: the chain labelled $1$ for the empty modes, the chain labelled $2$ for the filled modes. The following relations are used to define the functions equivalent to the spectral density of the bosonic case, one for each chain: $$ + V_{1k} = V_{k} \sin \theta_k = \sqrt{\frac{1}{e^{\beta \epsilon_k}+1}} \\ V_{2k} = V_{k} \cos \theta_k = \sqrt{\frac{1}{e^{-\beta \epsilon_k}+1}}, $$