1. Transport of heat and charge at a z=2 Quantum Critical Point: Violation of Wiedemann-Franz Law and other non-fermi-liquid properties. A.Vishwanath UC Berkeley Cond-Mat/0510597

2. Acnowledgements In collaboration with: –Daniel Podolsky (UC Berkeley) –J. Moore (UC Berkeley) –S. Sachdev (Harvard) Special thanks to –T. Senthil (IISc, MIT)

3. Introduction Metallic transport of heat and charge – best known example: Fermi liquid with impurities. Expect: JQJQ J el E Where is electrical (thermal) conductivity. And thermopower: P Break particle-hole (P) symmetry: S≠0. T Break time reversal (T) symmetry: PT Break both (P and T) symmetries: B

4. Introduction Metallic transport of heat and charge – best known example: Fermi liquid with impurities. Expect: JQJQ J el E Where is electrical (thermal) conductivity. And thermopower: Fermi Liquid: As And (Wiedemann-Franz Law) while (vanishes at low T)

5. Metallic Transport without Fermi Liquids Non-Fermi liquid phases with metallic transport? –Experiments on amorphous superconducting films in a field (eg. Steiner and Kapitulnik). Metallic transport at critical points? –Expt: Quantum criticality in heavy fermion systems – non-Fermi liquid metallic transport. –Theory: Metallic conductivity at the XY (superfluid-Mott insulator transition of lattice bosons at integer filling) critical point. However, thermal conductivity infinite & particle hole symmetric. –Here, tractable quantum critical point with nontrivial thermoelectric and thermal conductivity

6. Pair-breaking transition out of a 2D superconductor T=0 Transition between a superconductor and a diffusive Fermi liquid. –Clean FL, transition occurs at infinitesimal attractive coupling. –S-wave superconductors, adding nonmagnetic impurities does not change this (Anderson) –BUT non-s-wave Sc, nontrivial critical point. S-wave D-wave

7. Physical Realizations Layered unconventional superconductors: –Cuprates (overdoped transition) –Sr 2 RuO 4 (p+ip Sc.) –Organic materials (eg. (TMTSF) 2 PF 6 – p-wave)

8. Landau theory of pair-breaking transition I Begin with fermions in a random potential with an attractive interaction that favors an unconventional pairing state. Decouple using Hubbard Stratanovich field Φ. Hubbard Stratanovitch: + h.c.

9. Landau Theory of pair-breaking transition II Integrate out fermions to obtain effective action in terms of Φ (Hertz-Millis, Herbut). Non-s-wave character simplifies calculation Where g=UN(0)=coupling constant, and and δ=> p-h symm. breaking

10. Landau Theory of pair breaking transition III Thus, we obtain is a z=2 theory: –Also considered in the context of dissipative JJA (Dalidovich and Philips) Only role of disorder is to render fermions diffusive. Other effects suppressed: 2D weak localization irrelevant if Disorder effect on Bosons, only if Quartic term marginally irrelevant d=2,z=2. Line of fixed points (η)

11. Calculating Transport Coefficients Define electrical, energy and heat currents: Use Kubo formula to calculate transport coefficients: Contributions from diffusive fermions also present: etc.

12. At criticality two limits: ω/T →∞ (T=0, easier ) ω/T →0 (dc limit, more relevant to experiments). Transport Quantities in the Free Theory divergent metallic Why divergence? Relaxation rate (E) at low energies – thermal still finite (dominated by higher E). For finite T transport, interactions dangerously irrelevant.

13. Effect of Interactions I: HF Approximation Regularizes transport by generating a finite ‘r eff ’ Within a self consistent Hartree- Fock approximation: This serves as a low frequency cutoff (ω→ r eff )

14. Effect of Interactions II: Results from HF The α coefficient is weakly temperature dependent (unlike a Fermi liquid, where it is vanishes as T/E F ). Widermann-Franz law violation – bare violation is large, but adding to fermionic component, and since violation is a <1% effect; can be in either direction.

15. Effect of Interactions III: Beyond HF Hartree-Fock treatment strictly valid (Millis,Fisher Hoenberg) onlly if: –Otherwise interactions importantant To get around this restrictive condition, alternate approach – derive an effective classical Langevin dynamics for the quantum critical regime, and numerically simulate dynamics (Sachdev). Proceedure: –Integrate out all ω n ≠0 fields to obtain effective classical action for –Where If Log is large, justifies proceedure

16. Effect of Interactions III: Classical Action Define the renormalized Mass (R) –R cutoff independent, (only ultraviolet divergence of S c now accounted for). Therefore correlations functions cutoff independent. –Note, R,U set by position (r-r c ), T –If U/R << 1, then Hartree Fock holds.

17. Effect of Interactions:Langevin Dynamics For dynamics (eg. transport) obtain an effective Langevin equation in the usual manner Similarly for the other transport coefficients. Need to subtract U/R=0 result.

18. Fluctuation conductivity near finite temperature supeconductor transition – Aslamazov-Larkin contribution from Langevin dynamics with R~(T-Tc), U=0. (eg. σ~T/R) Other contributions to conductivity – Maki-Thompson + DOS equally singular for s-wave. However, not singular for d-wave (pairbreaking) (Yip). Contribution to Thermal (Niven and Smith) and Nenst (Ussishkin) coefficients not singular even for s-wave. Safe to ignore these contributions here. Relation to Fluctuation Effects at Classical Transitions

19. Transport in a field and 3D One can use results on classical Langevin dynamics (Ussishkin) in a field to obtain 3D, quartic term irrelevant at QCP: Corrections to diagonal coefficients vanish as T->0.

20. Dual Description? Vortex Dissipation In vortex variables:

21. Summary Considered a tractable quantum critical point without particle-hole or Lorentz/Gallilean symmetry. All transport quantities allowed to be nontrivial. Electrical, thermal and thermoelectric conductivities calculated in T>0 d.c. limit using HF and going beyond by mapping to a classical Langevin equation. Non-Fermi liquid transport obtains (Wiedemann-Franz law violation and non vanishing theroelectric coefficient). Could be seen at d-sc to diffusive metal transition – experimental signature reduced by fermionic background.