Periodic Orbits in Quantum Many-Body Systems · • Established method to compute classical orbits...
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FACULTY OF PHYSICS
Periodic Orbits inQuantum Many-Body Systems
Daniel Waltnerwith Maram Akila, Boris Gutkin, Petr Braun, Thomas Guhr
Quantum-Classical Transition in Many-Body Systems:Indistinguishability, Interference and Interactions,
MPI Dresden, 16. February 2017
Dresden, 16. Februar 2017
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Outline
• Semiclassical connection for the short-time behaviour of a quantum
many-body system
• Connection established for experimentally and theoretically topical
system of a spin chain
• Establish a quantum evolution of reduced dimension
• Impact of collective dynamics on the quantum spectrum
Dresden, 16. Februar 2017
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Motivation
Semiclassical connection for a single particle:
Gutzwiller trace formula:
ρ(E) =∑
n
δ(E − En)
︸ ︷︷ ︸∼ ρ(E) +
∑
γ
AγeiSγ/~
︸ ︷︷ ︸quantum level sum over classical
density orbits with action Sγand stability coefficient Aγ
Single-particle systems:
• Billiards: Sγ = ~klγ : Fourier-transform
with respect to k:
Spectrum of the classical orbits δ(l − lγ)
Stöckmann, Stein (1990)
• Kicked top: Fourier-transform with
respect to spin quantum number s Kus, Haake, Delande (1993)Dresden, 16. Februar 2017
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Kicked Top
Hamiltonian:H(t) =
4J (sz)2
(s+ 1/2)2+
2b · s
(s+ 1/2)
∞∑
n=−∞
δ(t− n)
Kick part of kicked top:
Quantum Classical
HK =2b · s
s+ 1/2
UK = exp(−i(s+ 1/2)HK
)
with n(t+ 1) = Rb(2|b|)n(t)
• magnetic field b = (bx, 0, bz)
• spin vector s = (sx, sy, sz)
• spin quantum number s
• unit vector n(t)
• rotation around b with angle
2|b|: Rb(2|b|)
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Kicked Top
“Ising” part of kicked top:
Quantum Classical
HI =4J(sz)
2
(s+ 1/2)2
UI = exp(−i(s+ 1/2)HI
)
with n(t+ 1) = Rz(8Jnz)n(t)
• “Ising” coupling J
• spin vector s = (sx, sy, sz)
• spin quantum number s
• unit vector n(t)
• rotation around z with angle
8Jnz: Rz(8Jnz)
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Kicked Top - Classical Dynamics
Combination of kick and Ising part: U = UI UK
Parameters: tan β = bx/bz, |b| = 1.27, J = 0.7
β = 0 β = 0.2 β = π/4
regular mixed chaotic
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Motivation
Many-particle system: Two limit parameters particle number N and spin
quantum number s
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Classical Motion
Many-particle systems: relative motion of particles provides additional
degree of freedom
Nuclear physics:Incoherent single
Coherent (collective) motion particle motion
Giant-Dipole Resonance:
Baldwin, Klaiber (1947)
Scissor Mode:Bohle, et al. (1984)
⇒ Description by effective models
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Previous Studies and AimsSemiclassics for many-particle systems:
• Propagator, trace formula for bosonic many-particle systems
Engl et al. (2014); Engl, Urbina, Richter (2015); Dubertrand, Müller (2016)
• Classical dynamics in spin chains for s≫ 1 with variable interaction
range Gessner, Bastidas, Brandes, Buchleitner (2016)
Aims:
• Quantum many-body systems: identify classical periodic orbits and
their impact on the quantum spectrum
• Consider no effective degrees of freedom: coordinates in a real
physical system (kicked spin-chain)
• Identify impact of high energy and
short time collective motion
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Kicked Spin Chain
Quantum System: kicked spin chain consisting of N coupled
spin-s-particles:
H(t) =N∑
n=1
4Jszn+1szn
(s+ 1/2)2
︸ ︷︷ ︸+
2
s+ 1/2
N∑
n=1
b · sn
︸ ︷︷ ︸
∞∑
τ=−∞
δ(t− τ)
next nearest local kick
neighbor Ising part HK
interaction HI
Periodic boundary conditions: sN+1 = s1
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Kicked Spin Chain
Motivation:
• Many-particle generalization of the kicked top
• Such systems are in the center of experimental studies:
◦ Ytterbium experiments with s = 5/2: Immanuel Bloch group
(Munich)
◦ ion traps with ≈ 10 spins: Christopher Monroe group (Maryland)
◦ ultracold fermionic atoms in optical traps: Selim Jochim group
(Heidelberg)
◦ Bose-Einstein condensate formed by two-level systems:
Markus Oberthaler’s talk
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Semiclassics
Identification of many-body periodic orbits:
Trace formula (s≫ 1):
TrUT =
∫da
⟨a∣∣UT
∣∣ a⟩∼
∑
γ(T )
AγeisSγ
for isolated periodic orbits γ(T ), stability prefactor Aγ , classical action
Sγ
for non isolated orbit Aγ diverges
Fourier-transform yields action spectrum:
ρ(S) ∝
scut∑
s=1
e−isSTrUT ∼∑
γ(T )
Aγδ(S − Sγ)
Waltner, Braun, Akila, Guhr (2017)
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Classical Dynamics
Classical state represented as unit vector nm(t) on the Bloch sphere for
spin m
Dynamics:
nm(t+ 1) = Rz(4Jχm)Rb(2|b|)nm(t)
χm = nzm−1 + nzm+1
Periodic orbits for T = 1:
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Classical Dynamics
Problems specific for many-body system:
ρ(S) ∝
scut∑
s=1
e−isSTrUT︸ ︷︷ ︸ ∼
∑
γ(T )
Aγδ(S − Sγ)
︸ ︷︷ ︸(2s+ 1)N
×(2s+ 1)N− number of orbits grows
dimensional grows exponentially with N
s = 100 N = 1 → dim UT = 201
s = 100 N = 2 → dim UT = 40401
s = 1/2 N = 14 → dim UT = 16384
s = 100 N = 14 → dim UT = 1.76 · 1032
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Duality Relation
Aim: Reduce dimension of UT
Time propagation:
|ψ(t+ 1)〉 = U |ψ(t)〉
Particle propagation:∣∣∣ψ(n+ 1)⟩= U
∣∣∣ψ(n)⟩
Duality:
TrUT = TrUN
Dimensions:
dimU = (2s+ 1)N × (2s+ 1)N ,
dimU = (2s+ 1)T × (2s+ 1)T
⇒ Analogy between time- and particle-propagation
Gutkin, Osipov (2016); Akila, Waltner, Gutkin, Guhr (2016)Dresden, 16. Februar 2017
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Action Spectrum T = 1
Identification of individual peaks in ρ(S) ∼∑
γ(T )Aγδ(S − Sγ)
Parameters: N = 7, J = 0.75, bx = bz = 0.9, scut = 800
Akila, Waltner, Gutkin, Braun, Guhr (2016) Dresden, 16. Februar 2017
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Action Spectrum T = 1
Larger N : number of orbits competes with resolution
Parameters: N = 19, J = 0.7, bx = bz = 0.9, scut = 4650
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Dominance of Collectivity
Parameters: T = 2, J = 0.7, bx = bz = 0.9, scut = 114
N = 3 N = 4
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✽✵✵✵
✶✷✵✵✵
S S
Peaks resulting from repetitions Large peaks dominatenot negligible the spectrum
Akila, Waltner, Gutkin, Braun, Guhr (2016)
Dresden, 16. Februar 2017
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Dominance of Collectivity
Observation generalizes to N = 4k (k ∈ N):
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Classical Collective Dynamics
4-dimensional manifold of periodic orbits for N = 4:
identified by χm = χ such that (Rz(4Jχ)Rb(2|b|))2 = 1
blue spins influenced bythe green and vice versa
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Scaling
Dominant for large N :
Aγ diverges for orbits forming manifold
⇒ Study scaling of the largest peak of ρ(S) with scut:
|ρ(Sγ)| ∝ (scut)α, α ∼ N/5
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→ Poster by M. Akila: “Semi-classics in many-body spin chains”Dresden, 16. Februar 2017
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Conclusions
• Established method to compute classical orbits in quantum
many-particle system and identified impact on quantum spectrum
for a spin chain
• Duality reduces dimension of UT by an exchange of N and T
• Collective dynamics dominates the quantum spectrum
0 1 2 3 4 5 6
1.×1043
2.×1043
3.×1043
Dresden, 16. Februar 2017
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Conclusions
• Established method to compute classical orbits in quantum
many-particle system and identify impact on quantum spectrum for
a spin chain
• Duality reduces dimension of UT by an exchange of N and T
• Collective dynamics dominates the quantum spectrum
Thank you for your attention!
Akila, Waltner, Gutkin, Braun, Guhr, arXiv 1611.05749
Dresden, 16. Februar 2017
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Dominance of Collectivity
Orbits on manifold dominate spectrum for specific s
TrUT ∼∑
γ(T )
AγeisSγ ≈ Amane
isSman
Difference of the phase:∆(s) = ImLogTrUT − sSman
⇒ TrUT dominated by a type of collective motion
Akila, Waltner, Gutkin, Braun, Guhr (2016)Dresden, 16. Februar 2017