From Strong to Weak Correlations in Breathing-Mode Kagome van der Waals Materials: <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi>Nb</mml:mi> </mml:mrow> <mml:mrow> <mml:

By combining downfolding with cluster dynamical mean-field theory, we study the degree of correlations in monolayer, bilayer, and bulk breathing-mode kagome van der Waals materials <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mrow> <a:mrow> <a:msub> <a:mrow> <a:mi>Nb</a:mi> </a:mrow> <a:mrow> <a:mn>3</a:mn> </a:mrow> </a:msub> </a:mrow> <a:mo stretchy="false">(</a:mo> <a:mi mathvariant="normal">F</a:mi> <a:mo>,</a:mo> <a:mi>Cl</a:mi> <a:mo>,</a:mo> <a:mi>Br</a:mi> <a:mo>,</a:mo> <a:mi mathvariant="normal">I</a:mi> <a:msub> <a:mrow> <a:mo stretchy="false">)</a:mo> </a:mrow> <a:mrow> <a:mn>8</a:mn> </a:mrow> </a:msub> </a:mrow> </a:math> . Our new material-specific many-body model library shows that in low-temperature bulk structures the Coulomb correlation strength steadily increases from I to Br, Cl, and F, allowing us to identify <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:msub> <g:mi>Nb</g:mi> <g:mn>3</g:mn> </g:msub> <g:msub> <g:mi mathvariant="normal">I</g:mi> <g:mn>8</g:mn> </g:msub> </g:math> as a weakly correlated insulator whose gap is only mildly affected by the local Coulomb interaction. <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline"> <j:mrow> <j:msub> <j:mrow> <j:mi>Nb</j:mi> </j:mrow> <j:mrow> <j:mn>3</j:mn> </j:mrow> </j:msub> </j:mrow> <j:mrow> <j:msub> <j:mrow> <j:mi>Br</j:mi> </j:mrow> <j:mrow> <j:mn>8</j:mn> </j:mrow> </j:msub> </j:mrow> </j:math> and <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline"> <l:mrow> <l:msub> <l:mrow> <l:mi>Nb</l:mi> </l:mrow> <l:mrow> <l:mn>3</l:mn> </l:mrow> </l:msub> </l:mrow> <l:mrow> <l:msub> <l:mrow> <l:mi>Cl</l:mi> </l:mrow> <l:mrow> <l:mn>8</l:mn> </l:mrow> </l:msub> </l:mrow> </l:math> are strongly correlated insulators, whose gaps are significantly influenced by Coulomb-induced vertex corrections. <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline"> <n:mrow> <n:msub> <n:mrow> <n:mi>Nb</n:mi> </n:mrow> <n:mrow> <n:mn>3</n:mn> </n:mrow> </n:msub> <n:msub> <n:mrow> <n:mi mathvariant="normal">F</n:mi> </n:mrow> <n:mrow> <n:mn>8</n:mn> </n:mrow> </n:msub> </n:mrow> </n:math> is a prototypical bulk Mott insulator whose gap is initially opened by strong correlation effects. Angle-resolved photoemission spectroscopy measurements comparing <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"> <q:mrow> <q:msub> <q:mrow> <q:mi>Nb</q:mi> </q:mrow> <q:mrow> <q:mn>3</q:mn> </q:mrow> </q:msub> </q:mrow> <q:mrow> <q:msub> <q:mrow> <q:mi>Br</q:mi> </q:mrow> <q:mrow> <q:mn>8</q:mn> </q:mrow> </q:msub> </q:mrow> </q:math> and <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"> <s:msub> <s:mi>Nb</s:mi> <s:mn>3</s:mn> </s:msub> <s:msub> <s:mi mathvariant="normal">I</s:mi> <s:mn>8</s:mn> </s:msub> </s:math> allow us to experimentally confirm these findings by revealing spectroscopic footprints of the degree of correlation. Our calculations further uncover how the thickness and the stacking affect the degree of correlations and predict that the entire material family can be tuned into correlated charge transfer or Mott-insulating phases upon electron or hole doping. Our magnetic property analysis based on our model parameter library additionally confirms that interlayer magnetic interactions likely drive the lattice phase transition to the low-temperature structures. The accompanying bilayer hybridization through interlayer dimerization yields magnetic singlet-like ground states in the Cl, Br, and I compounds. We further prove that all low-temperature compounds are dynamically stable and that electron-phonon coupling to the low-energy subspace is suppressed. Our findings establish <v:math xmlns:v="http://www.w3.org/1998/Math/MathML" display="inline"> <v:msub> <v:mi>Nb</v:mi> <v:mn>3</v:mn> </v:msub> <v:msub> <v:mi mathvariant="normal">X</v:mi> <v:mn>8</v:mn> </v:msub> </v:math> as a robust, versatile, and tunable class for van der Waals-based Coulomb and Mott engineering with a rich phase diagram and allow us to speculate on the symmetry-breaking effects necessary for the recently observed Josephson diode effect in <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"> <y:mrow> <y:msub> <y:mrow> <y:mi>NbSe</y:mi> </y:mrow> <y:mrow> <y:mn>2</y:mn> </y:mrow> </y:msub> </y:mrow> <y:mo>/</y:mo> <y:msub> <y:mi>Nb</y:mi> <y:mn>3</y:mn> </y:msub> <y:mrow> <y:msub> <y:mrow> <y:mi>Br</y:mi> </y:mrow> <y:mrow> <y:mn>8</y:mn> </y:mrow> </y:msub> </y:mrow> <y:mo>/</y:mo> <y:msub> <y:mi>NbSe</y:mi> <y:mn>2</y:mn> </y:msub> </y:math> heterostructures.