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Matthew S. Foster, 1 Maxim Dzero, 2 Victor Gurarie, 3 and Emil A. Yuzbashyan 4 1 Rice University, 2 Kent State Un

Quantum quench in p+ip superfluids : Winding numbers and topological states far from equilibrium. Matthew S. Foster, 1 Maxim Dzero, 2 Victor Gurarie, 3 and Emil A. Yuzbashyan 4 1 Rice University, 2 Kent State University , 3 University of Colorado at Boulder, 4 Rutgers University

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Matthew S. Foster, 1 Maxim Dzero, 2 Victor Gurarie, 3 and Emil A. Yuzbashyan 4 1 Rice University, 2 Kent State Un

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  1. Quantum quench in p+ipsuperfluids: Winding numbers and topological states far from equilibrium Matthew S. Foster,1Maxim Dzero,2 Victor Gurarie,3 and Emil A. Yuzbashyan4 1 Rice University, 2 Kent State University, 3 University of Colorado at Boulder, 4 Rutgers University April 23rd, 2013

  2. P-wave superconductivity in 2D Spin-polarized (spinless) fermions in 2D: P-wave BCS Hamiltonian

  3. P-wave superconductivity in 2D Spin-polarized (spinless) fermions in 2D: P-wave BCS Hamiltonian “P + i p” superconducting state: • At fixed density n: •  is a monotonically decreasing function of 0 BCS BEC

  4. P-wave superconductivity in 2D Spin-polarized(spinless) fermions in 2D: P-wave BCS Hamiltonian Anderson pseudospins {k,-k} vacant

  5. Topological superconductivity in 2D Pseudospin winding numberQ : BCS G. E. Volovik1988; Read and Green 2000 • 2D Topological superconductor • Fully gapped when   0 • Weak-pairing BCS state topologically non-trivial • Strong-pairing BEC state topologically trivial BEC

  6. Topological superconductivity in 2D Pseudospin winding numberQ : G. E. Volovik1988; Read and Green 2000 • Retarded GF winding number W : • W = Q in ground state Niu, Thouless, and Wu 1985 G. E. Volovik1988

  7. Topological superconductivity in 2D Topological signatures: Majorana fermions Chiral 1D Majorana edge states quantized thermal Hall conductance Isolated Majorana zero modes in type II vortices J. Moore • Realizations? • 3He-A thin films, Sr2RuO4(?) • 5/2 FQHE: Composite fermion Pfaffian • Cold atoms • Polar molecules • S-wave proximity-induced SC on surface of 3D Z2 Top. Insulator Volovik 1988, Rice and Sigrist 1995 Moore and Read 1991, Read and Green 2000 Gurarie, Radzihovsky, Andreev 2005; Gurarie and Radzihovsky 2007 Zhang, Tewari, Lutchyn, Das Sarma 2008; Sato, Takahashi, Fujimoto 2009; Y. Nisida 2009 Cooper and Shlyapnikov2009; Levinsen, Cooper, and Shlyapnikov 2011 Fu and Kane 2008

  8. Quantum Quench: Coherent many-body evolution 1. 2. Quantum quench protocol Prepare initial state “Quench” the Hamiltonian: Non-adiabatic perturbation

  9. Quantum Quench: Coherent many-body evolution 1. 2. Quantum quench protocol Prepare initial state “Quench” the Hamiltonian: Non-adiabatic perturbation Exotic excited state, coherent evolution 3.

  10. Quantum Quench: Coherent many-body evolution Experimental Example: Quantum Newton’s Cradle for trapped 1D 87Rb Bose Gas Dynamics of a topological many-body system: Need a global perturbation! Topological “Rigidity” vs Quantum Quench (Fight!) Kinoshita, Wenger, and Weiss 2006

  11. P-wave superconductivity in 2D: Dynamics • Ibañez, Links, Sierra, and Zhao (2009):Chiral 2D P-wave BCS Hamiltonian • Same p+ip ground state, non-trivial (trivial) BCS (BEC) phase • Integrable (hyperbolic Richardson model) • Method: Self-consistent non-equilibrium mean field theory • (Exact solution to nonlinear classical spin dynamics via integrability, Lax construction) • For p+ip initial state, dynamics are identical to “real” p-wave Hamiltonian • Exact in thermodynamic limit if pair-breaking neglected Richardson (2002) Dunning, Ibanez, Links, Sierra, and Zhao (2010) Rombouts, Dukelsky, and Ortiz (2010)

  12. P-wave Quantum Quench • Initial p+ip BCS or BEC state: • Post-quench Hamiltonian: BCSBEC

  13. Exact quench phase diagram: Strong to weak, weak to strong quenches Gap dynamics similar to s-wave case Barankov, Levitov, and Spivak 2004, Warner and Leggett 2005 Yuzbashyan, Altshuler, Kuznetsov, and Enolskii, Yuzbashyan, Tsyplyatyev, and Altshuler 2005 Barankovand Levitov, Dzero and Yuzbashyan 2006 Phase I: Gap decays to zero. Phase II: Gap goes to a constant. Phase III: Gap oscillates. Foster, Dzero, Gurarie, Yuzbashyan (unpublished)

  14. Phase III weak to strong quench dynamics: Oscillating gap Initial parameters: Blue curve: classical spin dynamics (numerics 5024 spins) Red curve: solution to Eq. () * Gap dynamics for reduced 2-spin problem: Parameters completely determined by two isolated root pairs Foster, Dzero, Gurarie, Yuzbashyan (unpublished) *

  15. Pseudospin winding number Q: Dynamics Pseudospin winding numberQ Chiral p-wave model: Spins along arcs evolve collectively:

  16. Pseudospin winding number Q: Dynamics Pseudospin winding numberQ Winding number is again given by Well-defined, so long as spin distribution remains smooth (no Fermi steps)

  17. Pseudospin winding number Q: Unchanged by quench! Foster, Dzero, Gurarie, Yuzbashyan (unpublished)

  18. Pseudospin winding number Q: Unchanged by quench! “Topological” Gapless State Foster, Dzero, Gurarie, Yuzbashyan (unpublished)

  19. “Topological” gapless phase • Decay of gap (dephasing): • Initial state not at the QCP (|| QCP,  0): • Initial state at the QCP (|| = QCP,  = 0): Foster, Dzero, Gurarie, Yuzbashyan (unpublished) Q = 0 Q = 1 Gapless Region A, Gapless Region B,

  20. Retarded GF winding number W: Dynamics • Retarded GF • winding number W : • Same as pseudospinwinding Q in ground state • Signals presence of chiral edge states in equilibrium • New to p-wave quenches: • Chemical potential (t) also a dynamic variable! • Phase II: Niu, Thouless, and Wu 1985 G. E. Volovik 1988 Foster, Dzero, Gurarie, Yuzbashyan (unpublished)

  21. Retarded GF winding number W: Dynamics • Retarded GF • winding number W : • Same as pseudospin windingQin ground state • Signals presence of chiral edge states in equilibrium Niu, Thouless, and Wu 1985 G. E. Volovik 1988 Purple line: Quench extension of topological transition “winding/BCS” “non-winding/BEC” Foster, Dzero, Gurarie, Yuzbashyan (unpublished)

  22. Retarded GF winding number W: Dynamics • W  Q out of equilibrium! • Winding number W can change following quench across QCP • Result is nevertheless quantized as t   • Edge states can appear or disappear in mean field Hamiltonian spectrum (Floquet) Foster, Dzero, Gurarie, Yuzbashyan (unpublished)

  23. Retarded GF winding number W: Dynamics • W  Q out of equilibrium! • Winding number W can change following quench across QCP • Result is nevertheless quantized as t   • Edge states can appear or disappear in mean field Hamiltonian spectrum (Floquet) • …Does NOT tell us about occupation of edge or bulk states Foster, Dzero, Gurarie, Yuzbashyan (unpublished)

  24. Bulksignature? “Cooper pair” distribution Pseudospin winding Q Ret GF winding W

  25. Post-quench “Cooper pair” distribution: Gapped phase II As t  , spins precess around “effective ground state field” determined by the isolated roots. Gapped phase: Parity of distribution zeroes odd when Q (pseudospin)  W (Ret GF) Q = 0 W = 1 Q = 1 W = 1 Gapped Region C, Gapped Region D,

  26. Summary and open questions • Quantum quench in p-wave superconductor investigated • Dynamics in thermodynamic limit exactly solved via classical integrability • Quench phase diagram, exact asymptotic gap dynamics • 1) Gap goes to zero (pair fluctuations) • 2) Gap goes to non-zero constant • 3) Gap oscillates • same as s-wave case • Pseudospinwinding number Q is unchanged by the quench, leading to “gapless topological state” • Retarded GF winding number W can change under quench; asymptotic value is quantized. Corresponding HBdG possesses/lacks edge state modes (Floquet) • Parity of zeroes in Cooper pair distribution is odd whenever Q  W

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