Matthew S. Foster,1 Maxim 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

Quantum quench in p+ip superfluids: Winding numbers and topological states

far from equilibrium

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

P-wave superconductivity in 2D

Spin-polarized (spinless) fermions in 2D: P-wave BCS Hamiltonian “P + i p” superconducting state:

P-wave superconductivity in 2D

At fixed density n: • µ is a monotonically

decreasing function of ∆0

BCS

BEC

Spin-polarized (spinless) 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

G. E. Volovik 1988; Read and Green 2000

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 state

Niu, Thouless, and Wu 1985 G. E. Volovik 1988

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. M

oore

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

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

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

Dynamics of a topological many-body system: Need a global perturbation!

Topological “Rigidity” vs Quantum Quench (Fight!)

Kinoshita, Wenger, and Weiss 2006

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

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

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

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)

Pseudospin winding number Q: Dynamics

Pseudospin winding number Q

Chiral p-wave model: Spins along arcs evolve collectively:

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)

Pseudospin winding number Q: Unchanged by quench!

Foster, Dzero, Gurarie, Yuzbashyan (unpublished)

Pseudospin winding number Q: Unchanged by quench!

“Topological” Gapless State

Foster, Dzero, Gurarie, Yuzbashyan (unpublished)

“Topological” gapless phase

Decay of gap (dephasing):

1) Initial state not at the QCP (|∆| ≠ ∆QCP, µ ≠ 0):

2) Initial state at the QCP (|∆| = ∆QCP, µ = 0):

Gapless Region A, Q = 0 Gapless

Region B, Q = 1

Foster, Dzero, Gurarie, Yuzbashyan (unpublished)

Foster, Dzero, Gurarie, Yuzbashyan (unpublished)

Retarded GF winding number W :

• Same as pseudospin winding Q in ground state

• Signals presence of chiral edge states in equilibrium

New to p-wave quenches: • Chemical potential µ(t)

also a dynamic variable!

• Phase II:

Niu, Thouless, and Wu 1985 G. E. Volovik 1988

Retarded GF winding number W: Dynamics

Retarded GF winding number W: Dynamics

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 state

• Signals presence of chiral edge states in equilibrium

Niu, Thouless, and Wu 1985 G. E. Volovik 1988

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)

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

Pseudospin winding Q Ret GF winding W

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)

Post-quench “Cooper pair” distribution: Gapped phase II

Gapped Region C,

Q = 0 W = 1

Gapped Region D,

Q = 1 W = 1

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

• 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 HBdG possesses/lacks edge state modes (Floquet)

• Parity of zeroes in Cooper pair distribution is odd whenever Q ≠ W