Far from equilibrium topological p-wave superfluids Matthew S. - - PowerPoint PPT Presentation

Matthew S. Foster,1 Victor Gurarie,2 Maxim Dzero,3 and Emil A. Yuzbashyan4

1 Rice University, 2 University of Colorado at Boulder,

3 Kent State University, 4 Rutgers University March 28th, 2014 PRB 88, 104511 (2013); arXiv:1307.2256

Far from equilibrium topological p-wave superfluids

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:

P-wave superconductivity in 2D

At fixed density n:

• µ is a monotonically

decreasing function of ∆0

Spin-polarized fermions in 2D: P-wave BCS Hamiltonian Anderson pseudospins

P-wave superconductivity in 2D

{k,-k} vacant

Pseudospin winding number Q :

Topological superconductivity in 2D

BCS BEC

2D Topological superconductor

• Fully gapped when µ ≠ 0
• Weak-pairing BCS state

topologically non-trivial

• Strong-pairing BEC state

topologically trivial

Volovik 88; Read and Green 00

Pseudospin winding number Q : “Topology of the state”

Topological superconductivity in 2D

Volovik 88; Read and Green 00

Retarded GF winding number W : “Topology of the effective single particle Hamiltonian”

• W = Q in equilibrium

Niu, Thouless, and Wu 85 Volovik 88

Realizations?

• Cold atoms: 6Li, 40K
• 3He-A thin films, Sr2RuO4(?)
• 5/2 FQHE: Composite fermion Pfaffian
• Polar molecules
• S-wave SC on surface of 3D Z2 Top. Insulator

Topological superconductivity in 2D

Topological signatures: Majorana fermions

• 1. Chiral 1D Majorana edge states

quantized thermal Hall conductance

• 2. Isolated Majorana zero modes (vortices)
• J. Moore

Moore and Read 91, Read and Green 00 Fu and Kane 08 Gurarie, Radzihovsky, Andreev 05; Gurarie and Radzihovsky 07 Cooper and Shlyapnikov 09; Levinsen, Cooper, and Shlyapnikov 11 Volovik 88, Rice and Sigrist 95

Topological superconductivity in 2D

Topological signatures: Majorana fermions

• 1. Chiral 1D Majorana edge states

quantized thermal Hall conductance

• 2. Isolated Majorana zero modes (vortices)
• J. Moore

Realizations?

• Cold atoms: 6Li, 40K

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 Gurarie, Radzihovsky, Andreev 05; Gurarie and Radzihovsky 07

Topological superconductivity in 2D

Topological signatures: Majorana fermions

• 1. Chiral 1D Majorana edge states

quantized thermal Hall conductance

• 2. Isolated Majorana zero modes (vortices)
• J. Moore

Realizations?

• Cold atoms: 6Li, 40K

PROBLEM: Losses due to three-body processes Decay is too fast for adiabatic cooling

• SOLUTION: Forget equilibrium…?

Gurarie, Radzihovsky, Andreev 05; Gurarie and Radzihovsky 07

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 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.

Quantum Quench: Coherent many-body evolution

Experimental Example: Quantum Newton’s Cradle for trapped 1D 87Rb Bose Gas

Kinoshita, Wenger, and Weiss 06

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

P-wave superconductivity in 2D: Dynamics

Richardson (2002) Dunning, Ibanez, Links, Sierra, and Zhao (2010) Rombouts, Dukelsky, and Ortiz (2010)

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

• 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 superconductivity in 2D: Dynamics

Richardson (2002) Dunning, Ibanez, Links, Sierra, and Zhao (2010) Rombouts, Dukelsky, and Ortiz (2010)

P-wave Quantum Quench

BCS BEC

• Initial p+ip BCS or BEC state:
• Post-quench Hamiltonian:

Exact quench phase diagram: Strong to weak, weak to strong quenches

Dynamical phases: ∆(t →∞) Phase I: Gap decays to zero. Phase II: Gap goes to a constant. Phase III: Gap oscillates.

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

Gap dynamics for reduced 2-spin problem: Parameters determined by two isolated Lax root pairs Initial parameters:

Phase III quench dynamics: Oscillating gap

*

Blue curve: classical spin dynamics (numerics 5024 spins) Red curve: solution to Eq. ( )

*

Pseudospin winding number Q: Unchanged by quench!

Smooth evolution

• f pseudospin

texture: Topology of the state Q doesn’t change. Not what we are interested in!

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 Retarded GF winding number W “winding”: Majorana edge modes “non-winding”

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) To wind, or not to wind far from equilibrium

Phase III: Quench-induced Floquet superconductor Floquet can be topological (edge states)!

• Analyze winding of Floquet
• perator G(T)
• Diagonalize G(T) in strip

geometry

Lindner, Refael, Galitski 11 Kitagawa, Oka, Brataas, Fu, Demler 11 Gu, Fertig, Arovas, Auerbach 11 Rudner, Lindner, Berg, Levin 13

• Weak-to-strong quench in III: ∆∞( t + T ) = ∆∞ ( t )
• Asymptotic GF same as for an externally driven Floquet system

Phase III: Quench-induced Floquet superconductor 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

• Quench to strong pairing
• Weak-to-strong quench in III: ∆∞( t + T ) = ∆∞ ( t )
• Asymptotic GF same as for an externally driven Floquet system

Jona-Lasinio, Pricoupenko, Castin 08

• Weak-to-strong quench in III: ∆∞( t + T ) = ∆∞ ( t )
• Asymptotic GF same as for an externally driven Floquet system

Phase III: Quench-induced Floquet topological superconductor

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)

Bulk signature? “Cooper pair” distribution

Gapped Region C,

Q = 0 W = 1

Gapped Region D,

Q = 1 W = 1

Appears in the bulk RF spectroscopy amplitude

Bulk signature? “Cooper pair” distribution

Gapped Region C,

Q = 0 W = 1

Gapped Region D,

Q = 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 HBdG 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

Nb1-xTixN Induced by Terahertz Pulse Excitation”

Matsunaga, Hamada, Makise, Uzawa, Terai, Wang, Shimano PRL (2013)

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: Dynamics

Lax vector components, “norm”

Chiral P-wave BCS: Lax construction

Lax vector components, “norm” Generalized Gaudin algebra

Chiral P-wave BCS: Lax construction

• M. Gaudin 1972, 1976, 1983

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” Conserved spectral polynomial: Key to understanding ground state and quench dynamics

Chiral P-wave BCS: Lax construction

Yuzbashyan, Altshuler, Kuznetsov, and Enolskii (2005)

In the ground state (zero quench): One isolated pair of roots ; (N – 1) positive, real, doubly-degenerate roots Gap, chemical potential encoded in isolated roots

Ground state: Lax roots

P-wave quantum quench: Lax roots

Quench

• Initial p+ip BCS or BEC state:
• Post-quench Hamiltonian:
• Quench parameter:

Roots: Strong-to-weak quench #1 (β > 0)

P-wave quantum quench: Lax roots

Quench

100 spins (numerics)

1 isolated pair

• Initial p+ip BCS or BEC state:
• Post-quench Hamiltonian:
• Quench parameter:

Roots: Strong-to-weak quench #1 (β > 0)

Quench

100 spins (numerics)

No isolated pair

P-wave quantum quench: Lax roots

• Initial p+ip BCS or BEC state:
• Post-quench Hamiltonian:
• Quench parameter:

Roots: Strong-to-weak quench #2 (β > 0)

Quench

100 spins (numerics)

2 isolated pairs

P-wave quantum quench: Lax roots

• Initial p+ip BCS or BEC state:
• Post-quench Hamiltonian:
• Quench parameter:

Roots: Weak-to-strong quench #1 (β < 0)

Quench dynamics: Isolated roots determine phase diagram

Via continuum limit of the spectral polynomial, only possibilities: (same as s-wave) 1) No isolated pairs. Gap decays to zero. 2) One isolated pair. Effective one-spin spectral polynomial: Gap goes to a constant ; non-eqlm chemical potential 3) Two isolated pairs. Effective two-spin spectral polynomial: Gap oscillates according to elliptic EOM (2-spin problem).

Yuzbashyan, Altshuler, Kuznetsov, and Enolskii 2005 Yuzbashyan, Tsyplyatyev, and Altshuler 2005 Barankov and Levitov 2006 Dzero and Yuzbashyan 2006 Foster, Dzero, Gurarie, Yuzbashyan 2013