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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 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

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Page 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

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

Page 2: 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

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

P-wave superconductivity in 2D

Page 3: 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

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

BCS BEC

Page 4: 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

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

Anderson pseudospins

P-wave superconductivity in 2D

{k,-k} vacant

Page 5: 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

Pseudospin winding number Q :

Topological superconductivity in 2D

BCS

BEC

2D Topological superconductor

• Fully gapped when 0

• Weak-pairing BCS state topologically non-trivial

• Strong-pairing BEC state topologically trivial

G. E. Volovik 1988; Read and Green 2000

Page 6: 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

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

Page 7: 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

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

Page 8: 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

Quantum quench protocol

1. Prepare initial state

2. “Quench” the Hamiltonian: Non-adiabatic perturbation

Quantum Quench: Coherent many-body evolution

1. 2.

Page 9: 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

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.

Page 10: 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

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

Page 11: 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

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)

Page 12: 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

P-wave Quantum Quench

BCS BEC

• Initial p+ip BCS or BEC state:

• Post-quench Hamiltonian:

Page 13: 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

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 2005Barankov and Levitov, Dzero and Yuzbashyan 2006

Page 14: 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

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)

Page 15: 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

Pseudospin winding number Q: Dynamics

Pseudospin winding number Q

Chiral p-wave model:

Spins along arcs evolvecollectively:

Page 16: 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

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)

Page 17: 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

Pseudospin winding number Q: Unchanged by quench!

Foster, Dzero, Gurarie, Yuzbashyan (unpublished)

Page 18: 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

Pseudospin winding number Q: Unchanged by quench!

“Topological”Gapless State

Foster, Dzero, Gurarie, Yuzbashyan (unpublished)

Page 19: 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

“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 Gapless

Region B, Q = 1

Foster, Dzero, Gurarie, Yuzbashyan (unpublished)

Page 20: 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

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 (t) also a dynamic variable!

• Phase II:

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

Retarded GF winding number W: Dynamics

Page 21: 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

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

Page 22: 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

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)

Page 23: 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

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

Page 24: 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

Pseudospin winding Q

Ret GF winding W

Bulk signature? “Cooper pair” distribution

Page 25: 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

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 = 0W = 1

Gapped Region D,

Q = 1W = 1

Page 26: 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

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 dynamics1) Gap goes to zero (pair fluctuations)2) Gap goes to non-zero constant3) 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


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