<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mmultiscripts> <mml:mi mathvariant="normal">V</mml:mi> <mml:mprescripts/> <mml:none/> <mml:mn>51</mml:mn> </mml:mmultiscripts> </mml:math> NMR evidence for interlayer-modulated charge ord

Charge order in the kagome superconductor <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mrow> <a:msub> <a:mi>CsV</a:mi> <a:mn>3</a:mn> </a:msub> <a:msub> <a:mi>Sb</a:mi> <a:mn>5</a:mn> </a:msub> </a:mrow> </a:math> exhibits a complex three-dimensional organization and intermediate-temperature anomalies whose bulk character has remained unsettled. We use orientation-dependent <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:mmultiscripts> <b:mi mathvariant="normal">V</b:mi> <b:mprescripts/> <b:none/> <b:mn>51</b:mn> </b:mmultiscripts> </b:math> NMR as a site-selective probe to determine the stacking of the charge density wave (CDW) state and its thermal evolution. Below <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"> <d:mrow> <d:msub> <d:mi>T</d:mi> <d:mi>CDW</d:mi> </d:msub> <d:mo>≈</d:mo> <d:mn>94</d:mn> <d:mspace width="0.28em"/> <d:mi mathvariant="normal">K</d:mi> </d:mrow> </d:math> , the field-linear splitting of the <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"> <g:mmultiscripts> <g:mi mathvariant="normal">V</g:mi> <g:mprescripts/> <g:none/> <g:mn>51</g:mn> </g:mmultiscripts> </g:math> central transition together with the anisotropy of the Knight shift tensor identify an interlayer-modulated <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"> <i:mrow> <i:mn>3</i:mn> <i:mi mathvariant="italic">q</i:mi> </i:mrow> </i:math> CDW whose local environments are consistent with a four-layer <k:math xmlns:k="http://www.w3.org/1998/Math/MathML"> <k:mrow> <k:mn>2</k:mn> <k:mo>×</k:mo> <k:mn>2</k:mn> <k:mo>×</k:mo> <k:mn>4</k:mn> </k:mrow> </k:math> stacking with mixed trihexagonal/Star-of-David distortions, in agreement with synchrotron x-ray determinations. For comparison, <l:math xmlns:l="http://www.w3.org/1998/Math/MathML"> <l:mrow> <l:msub> <l:mi>RbV</l:mi> <l:mn>3</l:mn> </l:msub> <l:msub> <l:mi>Sb</l:mi> <l:mn>5</l:mn> </l:msub> </l:mrow> </l:math> serves as a reference exhibiting a uniform trihexagonal <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mn>2</m:mn> <m:mo>×</m:mo> <m:mn>2</m:mn> <m:mo>×</m:mo> <m:mn>2</m:mn> </m:mrow> </m:math> stacking, allowing us to isolate features unique to the <n:math xmlns:n="http://www.w3.org/1998/Math/MathML"> <n:mrow> <n:mn>2</n:mn> <n:mo>×</n:mo> <n:mn>2</n:mn> <n:mo>×</n:mo> <n:mn>4</n:mn> </n:mrow> </n:math> state in <o:math xmlns:o="http://www.w3.org/1998/Math/MathML"> <o:mrow> <o:msub> <o:mi>CsV</o:mi> <o:mn>3</o:mn> </o:msub> <o:msub> <o:mi>Sb</o:mi> <o:mn>5</o:mn> </o:msub> </o:mrow> </o:math> . With <p:math xmlns:p="http://www.w3.org/1998/Math/MathML"> <p:mrow> <p:msub> <p:mi mathvariant="italic">H</p:mi> <p:mn>0</p:mn> </p:msub> <p:mo>∥</p:mo> <p:mi>c</p:mi> </p:mrow> </p:math> , the <r:math xmlns:r="http://www.w3.org/1998/Math/MathML"> <r:mmultiscripts> <r:mi mathvariant="normal">V</r:mi> <r:mprescripts/> <r:none/> <r:mn>51</r:mn> </r:mmultiscripts> </r:math> quadrupolar satellites through the intermediate temperature scale near <t:math xmlns:t="http://www.w3.org/1998/Math/MathML"> <t:mrow> <t:msub> <t:mi>T</t:mi> <t:mi>CO</t:mi> </t:msub> <t:mo>≈</t:mo> <t:mn>65</t:mn> <t:mspace width="0.28em"/> <t:mi mathvariant="normal">K</t:mi> </t:mrow> </t:math> reorganize into two well-resolved electric-field-gradient manifolds that coexist over a finite interval; their relative spectral weights interchange on cooling while the total integrated satellite intensity remains conserved and <w:math xmlns:w="http://www.w3.org/1998/Math/MathML"> <w:msub> <w:mi>ν</w:mi> <w:mi>Q</w:mi> </w:msub> </w:math> within each manifold is nearly temperature independent. The coexistence without critical broadening, together with conserved intensity, provides bulk evidence consistent with a first-order charge-order transition near <x:math xmlns:x="http://www.w3.org/1998/Math/MathML"> <x:msub> <x:mi>T</x:mi> <x:mi>CO</x:mi> </x:msub> </x:math> . Our measurements do not resolve whether this lower-temperature transition corresponds to a distinct in-plane order or a reorganization of the <y:math xmlns:y="http://www.w3.org/1998/Math/MathML"> <y:mrow> <y:mn>3</y:mn> <y:mi mathvariant="italic">q</y:mi> </y:mrow> </y:math> state; rather, they delimit this window and provide bulk, site-resolved constraints that connect prior reported anomalies to a thermodynamic first-order transition.