1. Subir Sachdev (Harvard) Philipp Werner (ETH) Matthias Troyer (ETH) Universal conductance of nanowires near the superconductor-metal quantum transition Talk online at http://sachdev.physics.harvard.edu Physical Review Letters 92, 237003 (2004)

2. T Quantum-critical Why study quantum phase transitions ? g gcgc Theory for a quantum system with strong correlations: describe phases on either side of g c by expanding in deviation from the quantum critical point. Important property of ground state at g=g c : temporal and spatial scale invariance; characteristic energy scale at other values of g: Critical point is a novel state of matter without quasiparticle excitations Critical excitations control dynamics in the wide quantum-critical region at non-zero temperatures.

3. Outline I.Quantum Ising Chain II.Landau-Ginzburg-Wilson theory Mean field theory and the evolution of the excitation spectrum. III.Superfluid-insulator transition Boson Hubbard model at integer filling. IV.Superconductor-metal transition in nanowires Universal conductance and sensitivity to leads

4. I. Quantum Ising Chain

5. 2Jg

6. Full Hamiltonian leads to entangled states at g of order unity

7. LiHoF 4 Experimental realization

8. Weakly-coupled qubits Ground state: Lowest excited states: Coupling between qubits creates “flipped-spin” quasiparticle states at momentum p Entire spectrum can be constructed out of multi-quasiparticle states p

9. Three quasiparticle continuum Quasiparticle pole Structure holds to all orders in 1/g S. Sachdev and A.P. Young, Phys. Rev. Lett. 78, 2220 (1997) ~3  Weakly-coupled qubits

10. Ground states: Lowest excited states: domain walls Coupling between qubits creates new “domain- wall” quasiparticle states at momentum p p Strongly-coupled qubits

11. Two domain-wall continuum Structure holds to all orders in g S. Sachdev and A.P. Young, Phys. Rev. Lett. 78, 2220 (1997) ~2  Strongly-coupled qubits

12. Entangled states at g of order unity g gcgc “Flipped-spin” Quasiparticle weight Z A.V. Chubukov, S. Sachdev, and J.Ye, Phys. Rev. B 49, 11919 (1994) g gcgc Ferromagnetic moment N 0 P. Pfeuty Annals of Physics, 57, 79 (1970) g gcgc Excitation energy gap 

13. Critical coupling No quasiparticles --- dissipative critical continuum

14. Quasiclassical dynamics P. Pfeuty Annals of Physics, 57, 79 (1970) S. Sachdev and J. Ye, Phys. Rev. Lett. 69, 2411 (1992). S. Sachdev and A.P. Young, Phys. Rev. Lett. 78, 2220 (1997).

15. Outline I.Quantum Ising Chain II.Landau-Ginzburg-Wilson theory Mean field theory and the evolution of the excitation spectrum. III.Superfluid-insulator transition Boson Hubbard model at integer filling. IV.Superconductor-metal transition in nanowires Universal conductance and sensitivity to leads

16. II. Landau-Ginzburg-Wilson theory Mean field theory and the evolution of the excitation spectrum

19. Outline I.Quantum Ising Chain II.Landau-Ginzburg-Wilson theory Mean field theory and the evolution of the excitation spectrum. III.Superfluid-insulator transition Boson Hubbard model at integer filling. IV.Superconductor-metal transition in nanowires Universal conductance and sensitivity to leads

20. III. Superfluid-insulator transition Boson Hubbard model at integer filling

21. Bosons at density f  M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature 415, 39 (2002). Weak interactions: superfluidity Strong interactions: Mott insulator which preserves all lattice symmetries LGW theory: continuous quantum transitions between these states

22. I. The Superfluid-Insulator transition Boson Hubbard model For small U/t, ground state is a superfluid BEC with superfluid density density of bosons M.PA. Fisher, P.B. Weichmann, G. Grinstein, and D.S. Fisher Phys. Rev. B 40, 546 (1989).

23. What is the ground state for large U/t ? Typically, the ground state remains a superfluid, but with superfluid density density of bosons The superfluid density evolves smoothly from large values at small U/t, to small values at large U/t, and there is no quantum phase transition at any intermediate value of U/t. (In systems with Galilean invariance and at zero temperature, superfluid density=density of bosons always, independent of the strength of the interactions)

24. What is the ground state for large U/t ? Incompressible, insulating ground states, with zero superfluid density, appear at special commensurate densities Ground state has “density wave” order, which spontaneously breaks lattice symmetries

25. Excitations of the insulator: infinitely long-lived, finite energy quasiparticles and quasiholes

30. Continuum of two quasiparticles + one quasihole Quasiparticle pole ~3  Insulating ground state Similar result for quasi-hole excitations obtained by removing a boson

31. Entangled states at of order unity g gcgc Quasiparticle weight Z g gcgc Superfluid density  s gcgc g Excitation energy gap  A.V. Chubukov, S. Sachdev, and J.Ye, Phys. Rev. B 49, 11919 (1994)

32. Quasiclassical dynamics S. Sachdev and J. Ye, Phys. Rev. Lett. 69, 2411 (1992). K. Damle and S. Sachdev Phys. Rev. B 56, 8714 (1997). Crossovers at nonzero temperature M.P.A. Fisher, G. Girvin, and G. Grinstein, Phys. Rev. Lett. 64, 587 (1990). K. Damle and S. Sachdev Phys. Rev. B 56, 8714 (1997).

33. Outline I.Quantum Ising Chain II.Landau-Ginzburg-Wilson theory Mean field theory and the evolution of the excitation spectrum. III.Superfluid-insulator transition Boson Hubbard model at integer filling. IV.Superconductor-metal transition in nanowires Universal conductance and sensitivity to leads

34. IV. Superconductor-metal transition in nanowires

35. T=0 Superconductor-metal transition Attractive BCS interaction R Repulsive BCS interaction RRcRc Metal Superconductor M.V. Feigel'man and A.I. Larkin, Chem. Phys. 235, 107 (1998) B. Spivak, A. Zyuzin, and M. Hruska, Phys. Rev. B 64, 132502 (2001).

36. T=0 Superconductor-metal transition R ii

37. Continuum theory for quantum critical point

38. Consequences of hyperscaling R Superconductor Normal RcRc Sharp transition at T=0 T No transition for T>0 in d=1

39. Consequences of hyperscaling R Superconductor Normal RcRc Sharp transition at T=0 T Quantum Critical Region No transition for T>0 in d=1

40. Consequences of hyperscaling Quantum Critical Region

41. Consequences of hyperscaling Quantum Critical Region

42. Effect of the leads

43. Large n computation of conductance

44. Quantum Monte Carlo and large n computation of d.c. conductance

45. Conclusions Universal transport in wires near the superconductor-metal transition Theory includes contributions from thermal and quantum phase slips ---- reduces to the classical LAMH theory at high temperatures Sensitivity to leads should be a generic feature of the ``coherent’’ transport regime of quantum critical points. Conclusions Universal transport in wires near the superconductor-metal transition Theory includes contributions from thermal and quantum phase slips ---- reduces to the classical LAMH theory at high temperatures Sensitivity to leads should be a generic feature of the ``coherent’’ transport regime of quantum critical points.