Anomalous Hall Effect in Chiral Superconductors

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1 Anomalous Hall Effect in Chiral Superconductors Vudtiwat Ngampruetikorn & James Sauls Supported by NSF & Northwestern University NSF DMR

2 Where are chiral superconductors? Chiral SC are interesting Majorana state, edge currents, exotic vortices Where are they? 3 He-A The only well-understood chiral superfluid in theory & experiments? UPt3 Strong evidence of chirality (Polar Kerr effect) What are the nature of the chiral state / order parameter??? Sr2RuO4 Some signs of chirality

3 (no external B-field) Anomalous Hall Effect for studying Chiral SC Hall effect requires 1. Broken time-reversal symmetry (TRS) 2. Broken mirror symmetry Also broken in Chiral pairing states Hall experiments could Verify Chiral pairing Identify Order parameter (winding number & gap structure) 2D (Sr2RuO4) 3D (UPt3) z y x

4 Edge and Bulk Hall Effects Edge Hall Effect Bulk Hall Effect Thermal Hall conductance for Chiral p-wave states [Read&Green Induced by impurity scattering in the bulk (2000] K edge xy = 1 2 π2 k 2 B T/6π Often dominant when present Could be sensitive to surface quality or worse disconnected from the bulk Both indicate Broken TRS & Broken Mirror Symmetry

5 Heat flows from hot to cold Hot Cold Longitudinal Heat Currents

6 Anomalous Thermal Hall Effect Heat currents Thermal gradient Transverse (Hall) Heat Currents (in the bulk) Hot Cold Cold Chiral SC TRS & Mirror Symm. broken

7 Obtaining Thermal Hall currents from Quasiclassical Linear Response Theory propagators (encoding spectrum) order parameter effects of impurities Quasiclassical Transport Equations + Gap Equation + T-matrix Equations Solve Self-consistent Equilibrium states Input Linearize & Solve Keldysh response diagonal piece occupied spectral Heat Currents: j ( ) =2N f d 4 i vˆp g K (ˆp; ) ˆp function

8 Randomly distributed point-like impurities Standard model for studying impurity effects scattering in s- wave channel only Bulk Hall Effect in Chiral p-wave states only Too restrictive not useful for studying Chiral SC Point-like means kfr 1 It s never true! Must Consider Finite-size Impurity

9 Problems with point-like impurities Standard model for studying impurity effects scattering in s- wave channel only Bulk Hall Effect in Chiral p-wave states only Too restrictive to be useful for studying Chiral SC Point-like means kfr 1 no! It s never true! Consider Finite-size Impurity (but keep homogeneity)

10 Small impurity Hall effect in p x +ip y only Chiral p-wave (2D) Chiral d-wave (2D) Thermal Hall 0.03 k f R 0.2 hard-disc radius conductivity 0.02 suppressed Hall effect apple xy (T ) in d-wave states apple xx (T c ) 0.01 for small impurity T/T c T/T c Impurity density chosen so that normal-state conductivity is fixed

11 Bigger impurity Hall effect beyond p x +ip y Chiral p-wave (2D) Chiral d-wave (2D) Thermal Hall conductivity apple xy (T ) k f R hard-disc radius k f R =0.2 apple xx (T c ) T/T c T/T c Hall current very sensitive to chiral winding Good probe for chirality Impurity density chosen so that normal-state conductivity is fixed

12 Thermal Hall transport at zero T Bulk effect shows up only with enough impurities Zero-T Chiral p-wave (2D) Chiral d-wave (2D) Thermal Hall Conductance 40 k f R 0.2 K bulk xy 2 k 2 B T/6 20 Too clean no sub-gap states at zero energy Too dirty No Chiral SC No Hall effect 0 conservative estimate k f 0 = 100 n imp 2 0 n imp = v f 2 k B T c0 (Kxy bulk k f 0 ) Edge effect dominates even in impure systems

13 Thermal Hall transport at zero T When present, Bulk effect tends to dominate Zero-T Thermal Hall Conductance bulk Kxy 2 k 2 T /6 B conservative estimate kf 0 Chiral p-wave (2D) 40 at optimal bulk edge Kxy 100Kxy nimp 0 = 100 vf 0 = 2 kb Tc0 bulk (Kxy kf 0 ) Edge Hall conductance (T edge Kxy Chiral d-wave (2D) = π k T /6π! B kf R nimp 0 0) (p-wave) Bulk effect dominates (when present)

14 Conclusions Thermal Hall Effect can be used for verifying and studying Chiral SC no! Point-like impurity model [kfr 1] spuriously rules out Hall Effect in all but chiral p-wave states Finite-size impurity needed Bulk Hall Effect is often dominant but requires low-energy states If no low-energy states exist Edge Effect dominates even in dirty samples We also have a full set of results for 3D gap models proposed for UPt3 NSF DMR Thank you!