Magnetic evolution, phase transitions, and electronic band structure of the ferrotoroidic candidate <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>Ba</mml:mi> <mml:mn>6</mml:mn> </mml:msub> <mml:msub> <mml:mi>

Ferrotoroidic materials, which break both time- and space-reversal symmetries to enhance magnetoelectric responses, are of great interest for applications. <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mrow> <a:msub> <a:mi>Ba</a:mi> <a:mn>6</a:mn> </a:msub> <a:msub> <a:mi>Cr</a:mi> <a:mn>2</a:mn> </a:msub> <a:msub> <a:mi mathvariant="normal">S</a:mi> <a:mn>10</a:mn> </a:msub> </a:mrow> </a:math> , a recently synthesized quasi-one-dimensional ferrotoroidic candidate, combines polarization, magnetization, and toroidal moment, yet its electronic band structure, magnetic evolution, and phase transitions remain incompletely understood to date. Here, we investigate <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:mrow> <c:msub> <c:mi>Ba</c:mi> <c:mn>6</c:mn> </c:msub> <c:msub> <c:mi>Cr</c:mi> <c:mn>2</c:mn> </c:msub> <c:msub> <c:mi mathvariant="normal">S</c:mi> <c:mn>10</c:mn> </c:msub> </c:mrow> </c:math> , using electrical transport, direct current magnetization, alternating current susceptibility, specific heat, and muon spin rotation ( <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"> <e:mrow> <e:mi>μ</e:mi> <e:mi>SR</e:mi> </e:mrow> </e:math> ) measurements, along with various first-principles calculations. Electrical transport measurements and density functional theory confirm a narrow band gap ( <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"> <f:mrow> <f:mo>∼</f:mo> <f:mn>0.707</f:mn> <f:mspace width="0.16em"/> <f:mi>eV</f:mi> </f:mrow> </f:math> , experimentally; <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"> <h:mrow> <h:mo>∼</h:mo> <h:mn>0.680</h:mn> <h:mspace width="0.16em"/> <h:mi>eV</h:mi> </h:mrow> </h:math> , theoretically). <j:math xmlns:j="http://www.w3.org/1998/Math/MathML"> <j:mrow> <j:mi>μ</j:mi> <j:mi>SR</j:mi> </j:mrow> </j:math> measurements reveal that static magnetism dominates down to 2 K, with an ordered volume fraction of <k:math xmlns:k="http://www.w3.org/1998/Math/MathML"> <k:mrow> <k:mo>∼</k:mo> <k:mn>89</k:mn> <k:mo>%</k:mo> </k:mrow> </k:math> at this temperature. Two distinct transitions are identified: one around the Néel temperature <l:math xmlns:l="http://www.w3.org/1998/Math/MathML"> <l:mrow> <l:mo>[</l:mo> <l:msub> <l:mi>T</l:mi> <l:mi mathvariant="normal">N</l:mi> </l:msub> <l:mo>=</l:mo> <l:mrow> <l:mo>(</l:mo> <l:mn>8</l:mn> <l:mo>±</l:mo> <l:mn>1</l:mn> <l:mo>)</l:mo> </l:mrow> <l:mspace width="0.16em"/> <l:mi mathvariant="normal">K</l:mi> <l:mo>]</l:mo> </l:mrow> </l:math> and another corresponding to a phase transition near 30 K, as consistently indicated by our complementary experimental techniques. The absence of dynamic critical behaviors and the lack of a <p:math xmlns:p="http://www.w3.org/1998/Math/MathML"> <p:mi>λ</p:mi> </p:math> -type specific heat anomaly are consistent with a first-order-like transition at <q:math xmlns:q="http://www.w3.org/1998/Math/MathML"> <q:msub> <q:mi>T</q:mi> <q:mi mathvariant="normal">N</q:mi> </q:msub> </q:math> . Based on various complementary experimental results and theoretical calculations, we propose a hypothesis that the <s:math xmlns:s="http://www.w3.org/1998/Math/MathML"> <s:mrow> <s:mo>∼</s:mo> <s:mn>30</s:mn> <s:mspace width="0.16em"/> <s:mi mathvariant="normal">K</s:mi> </s:mrow> </s:math> broad feature arises from the complex interplay between crystal structure distortion and magnetic coupling, which is potentially directly linked to ferrotoroidicity. As the first study integrating <v:math xmlns:v="http://www.w3.org/1998/Math/MathML"> <v:mrow> <v:mi>μ</v:mi> <v:mi>SR</v:mi> </v:mrow> </v:math> with complementary experiments and first-principles calculations to explore ferrotoroidicity, this work provides key insights into the underlying physics of <w:math xmlns:w="http://www.w3.org/1998/Math/MathML"> <w:mrow> <w:msub> <w:mi>Ba</w:mi> <w:mn>6</w:mn> </w:msub> <w:msub> <w:mi>Cr</w:mi> <w:mn>2</w:mn> </w:msub> <w:msub> <w:mi mathvariant="normal">S</w:mi> <w:mn>10</w:mn> </w:msub> </w:mrow> </w:math> and related compounds in this area.