matthew s. foster, 1 maxim dzero, 2 victor gurarie, 3 and emil a. yuzbashyan 4 1 rice...
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Quantum quench in p+ip superfluids : Winding numbers and topological states far from equilibrium. 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 - PowerPoint PPT PresentationTRANSCRIPT
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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
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Spin-polarized (spinless) fermions in 2D: P-wave BCS Hamiltonian
P-wave superconductivity in 2D
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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
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Spin-polarized (spinless) fermions in 2D: P-wave BCS Hamiltonian
Anderson pseudospins
P-wave superconductivity in 2D
{k,-k} vacant
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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
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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
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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
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Quantum quench protocol1. Prepare initial state2. “Quench” the Hamiltonian:
Non-adiabatic perturbation
Quantum Quench: Coherent many-body evolution
1. 2.
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Quantum quench protocol1. Prepare initial state2. “Quench” the Hamiltonian:
Non-adiabatic perturbation3. Exotic excited state,
coherent evolution
Quantum Quench: Coherent many-body evolution
1. 2.
3.
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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
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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)
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P-wave Quantum Quench
BCS BEC
• Initial p+ip BCS or BEC state:
• Post-quench Hamiltonian:
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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
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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)
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Pseudospin winding number Q: Dynamics
Pseudospin winding number Q
Chiral p-wave model:
Spins along arcs evolvecollectively:
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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)
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Pseudospin winding number Q: Unchanged by quench!
Foster, Dzero, Gurarie, Yuzbashyan (unpublished)
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Pseudospin winding number Q: Unchanged by quench!
“Topological”Gapless State
Foster, Dzero, Gurarie, Yuzbashyan (unpublished)
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“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)
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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
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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
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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)
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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
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Pseudospin winding Q
Ret GF winding W
Bulk signature? “Cooper pair” distribution
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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
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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