Spin and pair density waves in two-dimensional altermagnetic metals

Altermagnetism, a recently proposed and experimentally confirmed class of magnetic order, features collinear compensated magnetism with unconventional spin-split bands. Here, we show that in a metallic 2D <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mi>d</a:mi> </a:math> -wave altermagnet with <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:mrow> <b:mo>[</b:mo> <b:msub> <b:mi>C</b:mi> <b:mn>2</b:mn> </b:msub> <b:mo>|</b:mo> <b:mo>|</b:mo> <b:msub> <b:mi>C</b:mi> <b:mn>4</b:mn> </b:msub> <b:mo>]</b:mo> </b:mrow> </b:math> symmetry, secondary instabilities can arise. Using an unbiased functional renormalization group approach, we analyze the weak-coupling instabilities of a 2D Hubbard model with a preexisting altermagnetic order inspired by our electronic structure calculations of realistic material candidates from <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:mrow> <c:msub> <c:mi mathvariant="normal">V</c:mi> <c:mn>2</c:mn> </c:msub> <c:msub> <c:mi>X</c:mi> <c:mn>2</c:mn> </c:msub> <c:mspace width="0.16em"/> <c:mi mathvariant="normal">O</c:mi> </c:mrow> </c:math> ( <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"> <g:mrow> <g:mi>X</g:mi> </g:mrow> </g:math> = Te, Se) family. We identify two distinct spin-density-wave (SDW) states that break the underlying altermagnetic <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"> <h:mrow> <h:mo>[</h:mo> <h:msub> <h:mi>C</h:mi> <h:mn>2</h:mn> </h:msub> <h:mo>|</h:mo> <h:mo>|</h:mo> <h:msub> <h:mi>C</h:mi> <h:mn>4</h:mn> </h:msub> <h:mo>]</h:mo> </h:mrow> </h:math> symmetry. Additionally, we find spin-fluctuation-induced instabilities leading to a singlet <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"> <i:mi>d</i:mi> </i:math> -wave superconducting state and an unconventional commensurate pair-density-wave (PDW) state with extended <j:math xmlns:j="http://www.w3.org/1998/Math/MathML"> <j:mi>s</j:mi> </j:math> -wave and spin-triplet symmetry. We analyze the pairing mechanism and characterize the excitation spectrum, which exhibits Bogoliubov Fermi surfaces or nodal points depending on the gap size.