1. 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 3 University of Colorado at Boulder, 4 Rutgers University April 23 rd, 2013 Quantum quench in p+ip superfluids: Winding numbers and topological states far from equilibrium

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

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

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

5. Pseudospin winding number Q : Topological superconductivity in 2D BCS BEC 2D Topological superconductor Fully gapped when   0Fully gapped when   0 Weak-pairing BCS state topologically non-trivialWeak-pairing BCS state topologically non-trivial Strong-pairing BEC state topologically trivialStrong-pairing BEC state topologically trivial G. E. Volovik 1988; Read and Green 2000

6. Pseudospin winding number Q : Topological superconductivity in 2D G. E. Volovik 1988; Read and Green 2000 Retarded GF winding number W : W = Q in ground stateW = Q in ground state Niu, Thouless, and Wu 1985 G. E. Volovik 1988

7. Topological superconductivity in 2D Topological signatures: Majorana fermions 1.Chiral 1D Majorana edge states quantized thermal Hall conductance 2.Isolated Majorana zero modes in type II vortices J. Moore Realizations? 3 He-A thin films, Sr 2 RuO 4 (?) 3 He-A thin films, Sr 2 RuO 4 (?) 5/2 FQHE: Composite fermion Pfaffian5/2 FQHE: Composite fermion Pfaffian Cold atomsCold atoms Polar moleculesPolar molecules S-wave proximity-induced SC on surface of 3D Z2 Top. InsulatorS-wave proximity-induced SC on surface of 3D Z2 Top. Insulator Moore and Read 1991, Read and Green 2000 Fu and Kane 2008 Gurarie, Radzihovsky, Andreev 2005; Gurarie and Radzihovsky 2007 Zhang, Tewari, Lutchyn, Das Sarma 2008; Sato, Takahashi, Fujimoto 2009; Y. Nisida 2009 Cooper and Shlyapnikov 2009; Levinsen, Cooper, and Shlyapnikov 2011 Volovik 1988, Rice and Sigrist 1995

8. Quantum quench protocol 1.Prepare initial state 2.“Quench” the Hamiltonian: Non-adiabatic perturbation Quantum Quench: Coherent many-body evolution 1.2.

9. Quantum quench protocol 1.Prepare initial state 2.“Quench” the Hamiltonian: Non-adiabatic perturbation 3.Exotic excited state, coherent evolution Quantum Quench: Coherent many-body evolution 1.2. 3.

10. Experimental Example: Quantum Newton’s Cradle for trapped 1D 87 Rb 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. Chiral 2D P-wave BCS Hamiltonian Ibañez, Links, Sierra, and Zhao (2009): Chiral 2D P-wave BCS Hamiltonian Same p+ip ground state, non-trivial (trivial) BCS (BEC) phaseSame p+ip ground state, non-trivial (trivial) BCS (BEC) phase Integrable (hyperbolic Richardson model)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 P-wave superconductivity in 2D: Dynamics Richardson (2002) Dunning, Ibanez, Links, Sierra, and Zhao (2010) Rombouts, Dukelsky, and Ortiz (2010)

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

13. Exact quench phase diagram: Strong to weak, weak to strong quenches Phase I: Gap decays to zero. Phase II: Gap goes to a constant. Phase III: Gap oscillates. Foster, Dzero, Gurarie, Yuzbashyan (unpublished) 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 Barankov and Levitov, Dzero and Yuzbashyan 2006

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

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

16. Pseudospin winding number Q: Dynamics Pseudospin winding number Q  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): 1)Initial state not at the QCP (|  |   QCP,   0): 2)Initial state at the QCP (|  | =  QCP,  = 0): Gapless Region A, Q = 0 Q = 0 Gapless Region B, Q = 1 Q = 1 Foster, Dzero, Gurarie, Yuzbashyan (unpublished)

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

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

22. Foster, Dzero, Gurarie, Yuzbashyan (unpublished) 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)

23.  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) Retarded GF winding number W: Dynamics

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

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

26. Summary and open questions Quantum quench in p-wave superconductor investigatedQuantum quench in p-wave superconductor investigated Dynamics in thermodynamic limit exactly solved via classical integrabilityDynamics in thermodynamic limit exactly solved via classical integrability Quench phase diagram, exact asymptotic gap dynamicsQuench 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 same as s-wave case Pseudospin winding number Q is unchanged by the quench, leading to “gapless topological state”Pseudospin winding 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 H BdG possesses/lacks edge state modes (Floquet)Retarded GF winding number W can change under quench; asymptotic value is quantized. Corresponding H BdG possesses/lacks edge state modes (Floquet) Parity of zeroes in Cooper pair distribution is odd whenever Q  WParity of zeroes in Cooper pair distribution is odd whenever Q  W