Competing Electronic Ground States in the Heavy-Fermion Superconductor <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi>CeRh</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:m

<a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mrow> <a:msub> <a:mrow> <a:mi>CeRh</a:mi> </a:mrow> <a:mn>2</a:mn> </a:msub> <a:msub> <a:mrow> <a:mi>As</a:mi> </a:mrow> <a:mn>2</a:mn> </a:msub> </a:mrow> </a:math> is rare among superconductors, in that the magnetic field tunes it between two distinct superconducting phases. Combined with a lack of local inversion symmetry and an upper critical field exceeding the Pauli paramagnetic limit, this excitingly suggests triplet multicomponent superconductivity. Preceding the superconducting onset, <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mi>f</c:mi> </c:math> -electron correlations cause long-range order, attributed both to local antiferromagnetism and itinerant (quadrupole) density waves. A magnetic field provides a significant perturbation of the <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mi>f</e:mi> </e:math> electrons and may reveal the nature of the many-body correlations. Therefore, we report comprehensive magnetization and magnetotransport studies on microstructured devices in fields of up to 73 T. Applied along the <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:mrow> <g:mi>c</g:mi> </g:mrow> </g:math> axis, the field causes a low-temperature change of majority (hole) carrier density at <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:mrow> <i:msub> <i:mi>μ</i:mi> <i:mn>0</i:mn> </i:msub> <i:mi>H</i:mi> <i:mo>≈</i:mo> <i:mn>2</i:mn> <i:mn>4</i:mn> <i:mtext> </i:mtext> <i:mtext> </i:mtext> <i:mi mathvariant="normal">T</i:mi> </i:mrow> </i:math> . By contrast, in-plane fields produce a cascade of phase transitions; the field-induced in-plane conductivity anisotropy and lack of accompanying magnetic features, plus the closed-dome nature of the overall phase boundary is consistent with a hierarchy of field-induced density-wave states.