Nanoscale Defects as Probes of Time-Reversal Symmetry Breaking

Nanoscale defects such as nitrogen-vacancy (NV) centers can serve as sensitive and noninvasive probes of electromagnetic fields and fluctuations from materials, which in turn can be used to characterize these systems. Here we specifically discuss how NV centers can probe time-reversal symmetry breaking (TRSB) phenomena in low-dimensional electronic systems. We argue that the difference in relaxation rates <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:msub> <a:mi mathvariant="normal">Γ</a:mi> <a:mrow> <a:mo>±</a:mo> <a:mover accent="true"> <a:mi>z</a:mi> <a:mo stretchy="false">^</a:mo> </a:mover> </a:mrow> </a:msub> </a:math> of NV centers starting from <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline"> <f:mi>m</f:mi> <f:mo>=</f:mo> <f:mo>±</f:mo> <f:mn>1</f:mn> </f:math> spin states to the ground state with <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" display="inline"> <h:mi>m</h:mi> <h:mo>=</h:mo> <h:mn>0</h:mn> </h:math> directly probes TRSB. The effect arises from the difference in the fluctuation spectrum of left- and right-polarized electromagnetic fields emanating from such materials. In the quantum Hall setting, the NV center experiences (nearly zero) large additional contribution to its relaxation due to the presence of the material when its magnetic dipole (anti)aligns with the external field. More generally, the difference in the relaxation rates is sensitive to the imaginary part of the wave-vector-dependent Hall conductivity. We argue that this can be used to determine the Hall viscosity, which can potentially distinguish candidate fractional quantum Hall states and be used to infer the pairing angular momentum in TRSB superconductors. We compute the average relaxation rate <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline"> <j:mo stretchy="false">[</j:mo> <j:msub> <j:mi mathvariant="normal">Γ</j:mi> <j:mrow> <j:mo>+</j:mo> <j:mover accent="true"> <j:mi>z</j:mi> <j:mo stretchy="false">^</j:mo> </j:mover> </j:mrow> </j:msub> <j:mo>+</j:mo> <j:msub> <j:mi mathvariant="normal">Γ</j:mi> <j:mrow> <j:mo>−</j:mo> <j:mover accent="true"> <j:mi>z</j:mi> <j:mo stretchy="false">^</j:mo> </j:mover> </j:mrow> </j:msub> <j:mo stretchy="false">]</j:mo> </j:math> near thin film superconductors and find that it exhibits a Hebel-Slichter-like enhancement below <t:math xmlns:t="http://www.w3.org/1998/Math/MathML" display="inline"> <t:msub> <t:mi>T</t:mi> <t:mi>c</t:mi> </t:msub> </t:math> . The difference <v:math xmlns:v="http://www.w3.org/1998/Math/MathML" display="inline"> <v:msub> <v:mi mathvariant="normal">Γ</v:mi> <v:mrow> <v:mo>+</v:mo> <v:mover accent="true"> <v:mi>z</v:mi> <v:mo stretchy="false">^</v:mo> </v:mover> </v:mrow> </v:msub> <v:mo>−</v:mo> <v:msub> <v:mi mathvariant="normal">Γ</v:mi> <v:mrow> <v:mo>−</v:mo> <v:mover accent="true"> <v:mi>z</v:mi> <v:mo stretchy="false">^</v:mo> </v:mover> </v:mrow> </v:msub> </v:math> also inherits this peak but is only nonzero for <db:math xmlns:db="http://www.w3.org/1998/Math/MathML" display="inline"> <db:mi>T</db:mi> <db:mo>&lt;</db:mo> <db:msub> <db:mi>T</db:mi> <db:mi>c</db:mi> </db:msub> </db:math> and only if the superconductivity is chiral. We provide concrete estimates for observing this effect in stacked twisted bismuth strontium calcium copper oxide flakes.