Far from equilibrium topological p-wave superfluids Matthew S. Foster, 1 Victor Gurarie, 2 Maxim Dzero, 3 and Emil A. Yuzbashyan 4 1 Rice University, 2 University of Colorado at Boulder, 3 Kent State University, 4 Rutgers University March 28 th , 2014 PRB 88 , 104511 (2013); arXiv:1307.2256
P-wave superconductivity in 2D Spin-polarized fermions in 2D: P-wave Hamiltonian
P-wave superconductivity in 2D Spin-polarized fermions in 2D: P-wave Hamiltonian “P + i p” superconducting state: At fixed density n: • µ is a monotonically decreasing function of ∆ 0
P-wave superconductivity in 2D Spin-polarized fermions in 2D: P-wave BCS Hamiltonian Anderson pseudospins { k ,- k } vacant
Topological superconductivity in 2D Pseudospin winding number Q : BCS Volovik 88; Read and Green 00 2D Topological superconductor Fully gapped when µ ≠ 0 • BEC • Weak-pairing BCS state topologically non-trivial • Strong-pairing BEC state topologically trivial
Topological superconductivity in 2D Pseudospin winding number Q : “Topology of the state” Volovik 88; Read and Green 00 Retarded GF winding number W : “Topology of the effective single particle Hamiltonian” Niu, Thouless, and Wu 85 Volovik 88 • W = Q in equilibrium
Topological superconductivity in 2D Topological signatures: Majorana fermions 1. Chiral 1D Majorana edge states quantized thermal Hall conductance J. Moore 2. Isolated Majorana zero modes (vortices) Realizations? Cold atoms: 6 Li, 40 K • Gurarie, Radzihovsky, Andreev 05; Gurarie and Radzihovsky 07 3 He-A thin films, Sr 2 RuO 4 (?) • Volovik 88, Rice and Sigrist 95 • 5/2 FQHE: Composite fermion Pfaffian Moore and Read 91, Read and Green 00 • Polar molecules Cooper and Shlyapnikov 09; Levinsen, Cooper, and Shlyapnikov 11 • S-wave SC on surface of 3D Z2 Top. Insulator Fu and Kane 08
Topological superconductivity in 2D Topological signatures: Majorana fermions 1. Chiral 1D Majorana edge states quantized thermal Hall conductance J. Moore 2. Isolated Majorana zero modes (vortices) Realizations? Cold atoms: 6 Li, 40 K • Gurarie, Radzihovsky, Andreev 05; Gurarie and Radzihovsky 07 PROBLEM: Losses due to three-body processes Decay is too fast for adiabatic cooling Zhang et al. 04 Jona-Lasinio, Pricoupenko, Castin 08 Levinsen, Cooper, Gurarie 08 Gaebler, Stewart, Bohn, Jin 07 Fuchs et al. 08, Inada et al. 08
Topological superconductivity in 2D Topological signatures: Majorana fermions 1. Chiral 1D Majorana edge states quantized thermal Hall conductance J. Moore 2. Isolated Majorana zero modes (vortices) Realizations? Cold atoms: 6 Li, 40 K • Gurarie, Radzihovsky, Andreev 05; Gurarie and Radzihovsky 07 PROBLEM: Losses due to three-body processes Decay is too fast for adiabatic cooling SOLUTION: Forget equilibrium…?
Quantum Quench: Coherent many-body evolution 1. 2. Quantum quench protocol 1. Prepare initial state 2. “Quench” the Hamiltonian: Non-adiabatic perturbation
Quantum Quench: Coherent many-body evolution 1. 2. Quantum quench protocol 3. 1. Prepare initial state 2. “Quench” the Hamiltonian: Non-adiabatic perturbation 3. Exotic excited state, coherent evolution
Quantum Quench: Coherent many-body evolution Experimental Example: Quantum Newton’s Cradle for trapped 1D 87 Rb Bose Gas Kinoshita, Wenger, and Weiss 06
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 Richardson (2002) • Integrable (hyperbolic Richardson model) Dunning, Ibanez, Links, Sierra, and Zhao (2010) Rombouts, Dukelsky, and Ortiz (2010) Method: Self-consistent non-equilibrium mean field theory
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 Richardson (2002) • Integrable (hyperbolic Richardson model) Dunning, Ibanez, Links, Sierra, and Zhao (2010) Rombouts, Dukelsky, and Ortiz (2010) Method: Self-consistent non-equilibrium mean field theory • For p+ip initial state, dynamics are identical to “full” BCS p-wave • Exact analytical solution in thermodynamic limit if pair-breaking neglected
P-wave Quantum Quench • Initial p+ip BCS or BEC state: • Post-quench Hamiltonian: BCS BEC
Exact quench phase diagram: Strong to weak, weak to strong quenches Gap dynamics similar to s-wave case Barankov, Levitov, Spivak 04, Warner and Leggett 05 Yuzbashyan, Altshuler, Kuznetsov, Enolskii 05, Yuzbashyan, Tsyplyatyev, Altshuler 05 Barankov and Levitov, Dzero and Yuzbashyan 06 Dynamical phases: ∆ ( t →∞ ) Phase I: Gap decays to zero. Phase II: Gap goes to a constant. Phase III: Gap oscillates.
Phase III 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 determined by two isolated Lax root pairs
Pseudospin winding number Q : Unchanged by quench! Smooth evolution of pseudospin texture: Topology of the state Q doesn’t change. Not what we are interested in!
Retarded GF winding number W Retarded GF winding number W : • Topology of the Bogoliubov-de Gennes Hamiltonian • Signals presence of Majorana edge modes • Chemical potential µ ( t ) = phase of ∆ ( t ) also a dynamic variable! • Phase II:
Retarded GF winding number W Retarded GF winding number W : • Topology of the Bogoliubov-de Gennes Hamiltonian • Signals presence of Majorana edge modes “winding”: Majorana edge modes “non-winding”
To wind, or not to wind far from equilibrium W : Retarded Green’s function • Can change after a quench Encoded in | ∆ ( t )| and µ ( t ) • • Determines spectrum of “nearby” states including edges (probe) • “Topology of the spectrum can change” Q : Pseudospin winding number • Conserved • “Topology of the state does not change” How to reconcile W , Q ? Occupation of modes (Cooper pair distribution function)
Phase III: Quench-induced Floquet superconductor • Weak-to-strong quench in III: ∆ ∞ ( t + T ) = ∆ ∞ ( t ) • Asymptotic GF same as for an externally driven Floquet system Floquet can be topological (edge states)! Lindner, Refael, Galitski 11 Kitagawa, Oka, Brataas, Fu, Demler 11 Gu, Fertig, Arovas, Auerbach 11 Rudner, Lindner, Berg, Levin 13 • Analyze winding of Floquet operator G ( T ) • Diagonalize G ( T ) in strip geometry
Phase III: Quench-induced Floquet superconductor • Weak-to-strong quench in III: ∆ ∞ ( t + T ) = ∆ ∞ ( t ) • Asymptotic GF same as for an externally driven Floquet system Phase III most relevant for cold atom experiments • Prepare initial state with very weak p+ip order 3-body losses negligible away • from Feshbach resonance Jona-Lasinio, Pricoupenko, Castin 08 Quench to strong pairing •
Phase III: Quench-induced Floquet topological superconductor • Weak-to-strong quench in III: ∆ ∞ ( t + T ) = ∆ ∞ ( t ) • Asymptotic GF same as for an externally driven Floquet system
Bulk signature? “Cooper pair” distribution
Bulk signature? “Cooper pair” distribution 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 Q = 1 Gapped Gapped Region C, Region D, W = 1 W = 1
Bulk signature? “Cooper pair” distribution Appears in the bulk RF spectroscopy amplitude Q = 0 Q = 1 Gapped Gapped Region C, Region D, W = 1 W = 1
Summary and open questions • Pseudospin winding number Q is unchanged by the quench • Retarded GF winding number W can change under quench. Effective H BdG possesses/lacks Majorana edge state modes • Weak-to-stong pairing quench: Floquet topological superfluid without external driving • Parity of zeroes in Cooper pair distribution is odd whenever Q ≠ W
Summary and open questions 1. State versus spectrum winding for other (e.g. driven Floquet) non-equilibrium systems? 2. Bulk distribution winding for other systems? 3. Quantized energy transport at the edge? 4. Solid state SF (“Higgs”) dynamics via pump-probe? • “Higgs Amplitude Mode in the BCS Superconductors Nb 1-x Ti x N Induced by Terahertz Pulse Excitation” Matsunaga, Hamada, Makise, Uzawa, Terai, Wang, Shimano PRL (2013)
Chiral P-wave BCS: Dynamics Heisenberg spin equations of motion: Self-consistent mean field theory (thermodynamic limit) Identical to self-consistent (time-dependent) Bogoliubov-de Gennes
Chiral P-wave BCS: Lax construction Lax vector components, “norm”
Chiral P-wave BCS: Lax construction Lax vector components, “norm” Generalized Gaudin algebra M. Gaudin 1972, 1976, 1983
Chiral P-wave BCS: Lax construction Lax vector components, “norm” Lax vector norm: Generates integrals of motion From Gaudin algebra: BCS Hamiltonian:
Chiral P-wave BCS: Lax construction Lax vector components, “norm” Yuzbashyan, Altshuler, Conserved spectral polynomial: Kuznetsov, and Enolskii (2005) Key to understanding ground state and quench dynamics
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