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Universal conductance of nanowires near the superconductor-metal quantum transition Subir Sachdev (Harvard) Philipp Werner (ETH) Matthias Troyer (ETH) Physical Review Letters 92, 237003 (2004) Talk online at http://sachdev.physics.harvard.edu Why study quantum phase transitions ? T Quantum-critical gc • Theory for a quantum system with strong correlations: describe phases on either side of gc by expanding in deviation from the quantum critical point. 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. Important property of ground state at g=gc : temporal and spatial scale invariance; characteristic energy scale at other values of g: ~ g g c z 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 I. Quantum Ising Chain I. Quantum Ising Chain Degrees of freedom: j 1 N qubits, N "large" , or j 1 j 2 j j j , j 1 j 2 Hamiltonian of decoupled qubits: H 0 Jg xj j j j 2Jg j Coupling between qubits: H1 J zj zj 1 j j j j 1 j 1 Prefers neighboring qubits are either Full Hamiltonian j j 1 or j j 1 (not entangled) H H 0 H1 J g xj zj zj 1 j leads to entangled states at g of order unity Experimental realization LiHoF4 Weakly-coupled qubits g 1 Ground state: G 1 2g Lowest excited states: j j Coupling between qubits creates “flipped-spin” quasiparticle states at momentum p p e p ipx j j j pa 1 Excitation energy p 4 J sin 2 O g 2 Excitation gap 2 gJ 2 J O g 1 a p Entire spectrum can be constructed out of multi-quasiparticle states a Dynamic Structure Factor S ( p, ) : Weakly-coupled qubits g Cross-section to flip a to a (or vice versa) while transferring energy and momentum p S p, Z p Quasiparticle pole Three quasiparticle continuum ~3 Structure holds to all orders in 1/g At T 0, collisions between quasiparticles broaden pole to a Lorentzian of width 1 where the phase coherence time is given by 1 2k B T e kBT S. Sachdev and A.P. Young, Phys. Rev. Lett. 78, 2220 (1997) 1 Ground states: G Strongly-coupled qubits g 1 g 2 Ferromagnetic moment N0 G z G 0 Second state G obtained by G and G mix only at order g N Lowest excited states: domain walls dj j Coupling between qubits creates new “domainwall” quasiparticle states at momentum p p e ipx j p dj j pa 2 Excitation energy p 4 Jg sin 2 O g 2 Excitation gap 2 J 2 gJ O g 2 a p a Dynamic Structure Factor S ( p, ) : Strongly-coupled qubits g 1 Cross-section to flip a to a (or vice versa) while transferring energy and momentum p S p, N 02 2 p 2 Two domain-wall continuum ~2 Structure holds to all orders in g At T 0, motion of domain walls leads to a finite phase coherence time , and broadens coherent peak to a width 1 where 1 S. Sachdev and A.P. Young, Phys. Rev. Lett. 78, 2220 (1997) 2k B T e kBT Entangled states at g of order unity Z ~ g gc 1/ 4 “Flipped-spin” Quasiparticle weight Z A.V. Chubukov, S. Sachdev, and J.Ye, Phys. Rev. B 49, 11919 (1994) gc g N0 ~ gc g 1/ 8 Ferromagnetic moment N0 P. Pfeuty Annals of Physics, 57, 79 (1970) gc g Excitation energy gap ~ g gc gc g Dynamic Structure Factor S ( p, ) : Critical coupling g gc Cross-section to flip a to a (or vice versa) while transferring energy and momentum p S p, ~ c p c p 2 2 2 7 / 8 No quasiparticles --- dissipative critical continuum H I J g ix iz iz1 i Quasiclassical dynamics Quasiclassical dynamics ( ) i dt k zj t , kz 0 ei t 0 A T 7/4 1 i / R ... k T R 2 tan B 16 S. Sachdev and J. Ye, Phys. Rev. Lett. 69, 2411 (1992). S. Sachdev and A.P. Young, Phys. Rev. Lett. 78, 2220 (1997). z j z k ~ 1 jk P. Pfeuty Annals of Physics, 57, 79 (1970) 1/ 4 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 II. Landau-Ginzburg-Wilson theory Mean field theory and the evolution of the excitation spectrum 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 III. Superfluid-insulator transition Boson Hubbard model at integer filling Bosons at density f 1 Weak interactions: superfluidity Strong interactions: Mott insulator which preserves all lattice symmetries LGW theory: continuous quantum transitions between these states M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature 415, 39 (2002). I. The Superfluid-Insulator transition Boson Hubbard model Degrees of freedom: Bosons, b†j , hopping between the sites, j , of a lattice, with short-range repulsive interactions. U † H t bi b j - n j n j ( n j 1) 2 j ij j nj b b † j j M.PA. Fisher, P.B. Weichmann, G. Grinstein, and D.S. Fisher Phys. Rev. B 40, 546 (1989). For small U/t, ground state is a superfluid BEC with superfluid density density of bosons 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) What is the ground state for large U/t ? Incompressible, insulating ground states, with zero superfluid density, appear at special commensurate densities nj 3 nj 7 / 2 t U Ground state has “density wave” order, which spontaneously breaks lattice symmetries Excitations of the insulator: infinitely long-lived, finite energy quasiparticles and quasiholes p2 Energy of quasi-particles/holes: p ,h p p ,h 2m*p ,h Excitations of the insulator: infinitely long-lived, finite energy quasiparticles and quasiholes p2 Energy of quasi-particles/holes: p ,h p p ,h 2m*p ,h Excitations of the insulator: infinitely long-lived, finite energy quasiparticles and quasiholes p2 Energy of quasi-particles/holes: p ,h p p ,h 2m*p ,h Boson Green's function G ( p, ) : Insulating ground state Cross-section to add a boson while transferring energy and momentum p G p, Z p Quasiparticle pole ~3 Continuum of two quasiparticles + one quasihole Similar result for quasi-hole excitations obtained by removing a boson Entangled states at g t / U of order unity Z ~ gc g Quasiparticle weight Z A.V. Chubukov, S. Sachdev, and J.Ye, Phys. Rev. B 49, 11919 (1994) gc g p ,h ~ gc g for g gc Excitation energy gap p ,h =0 for g gc g rs ~ g gc Superfluid density rs ggcc g ( d z 2) Crossovers at nonzero temperature Quasiclassical dynamics Quasiclassical dynamics Relaxational dynamics ("Bose molasses") with phase coherence/relaxation time given by 1 Universal number k BT Q2 Conductivity (in d=2) = h kBT 1 K 20.9kHz S. Sachdev and J. Ye, Phys. Rev. Lett. 69, 2411 (1992). K. Damle and S. Sachdev Phys. Rev. B 56, 8714 (1997). universal function 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). 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 IV. Superconductor-metal transition in nanowires T=0 Superconductor-metal transition Repulsive BCS interaction Attractive BCS interaction R Superconductor Metal Rc R 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). T=0 Superconductor-metal transition yi R Continuum theory for quantum critical point Consequences of hyperscaling T No transition for T>0 in d=1 Normal Superconductor Rc Sharp transition at T=0 R Consequences of hyperscaling Quantum Critical Region T No transition for T>0 in d=1 Normal Superconductor Rc Sharp transition at T=0 R Consequences of hyperscaling Quantum Critical Region Consequences of hyperscaling Quantum Critical Region Effect of the leads Large n computation of conductance Quantum Monte Carlo and large n computation of d.c. conductance 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.