NonEquilibrium Dynamics and Decoherence of Quantum Spin Chains

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Non-Equilibrium Dynamics and Decoherence of
Quantum Spin Chains

• Robert Cherng
• Harvard University

• Leonid Levitov (MIT)
• MIT UROP
• NDSEG

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Why quantum spin chains?

• Dynamics across quantum critical points
• BEC in optical lattices Greiner et al., Nature
  (2002)
• Fundamental issues
• Intrinsic dynamics, quantum criticality,
  decoherence
• Quantum spin chains
• Equilibrium properties well-known, microscopic
  approach, exact solutions

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Outline

• Solution of dynamic transverse XY model
• Jordan-Wigner fermionization, Landau-Zener
  tunneling
• Many-body decoherence
• Non-equilibrium steady state, mapping to
  equilibrium model at finite temperature
• Spin correlators
• Crossover behavior, conjecture on Toeplitz
  determinant

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Dynamic transverse XY model
Barouch and McCoy (PRA 3, 1971)
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Jordan-Wigner fermionization

• Transform hard-core bosons to fermions
• Fermionic Hamiltonian is quadratic

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Landau-Zener tunneling I.
Convenient Basis
Landau (Phys. Z, 1932) Zener (Proc. R. Soc. 1932)
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Landau-Zener tunneling II.
Brundobler (J. Phys. A 1993)
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Many-body decoherence

• Local observables are physically relevant
• Remove cutoffs in a physical way (n?? last)
• Does decoherence remain?

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Non-equilibrium steady state I.

• Evolve far past ground state into far future
• Decoherence appears at late times

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Non-equilibrium steady state II.

• Even/odd sublattices decouple dynamically
• Effective mixed state on each sublattice
• Dimensionless dispersion

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Effective dispersion
z/z100

• Full gap for zltz
• Gapless for zgtz
• Low-lying excitations for z?z

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1
0.01
0.1
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Mapping to isotropic XY model

• Low-lying excitation structure ? thermal
  isotropic XY (XX)
• Compare dispersions

Barouch and McCoy (PRA 3, 1971)
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Spin Correlators I.

• Full lattice and sublattice spin correlators
• Full lattice correlators factorize

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Spin Correlators II.

• Consider sublattice only and rewrite in terms of
  fermionic correlators
• Only one non-trivial fermionic correlator

Lieb et el (Ann. Phys. 1961)
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Transverse Correlators

• Magnetization is smooth
• gzn decays as n-2n

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Toeplitz Determinants

• Need asymptotics of large Toeplitz determinants
  for longitudinal correlators
• Notation and generating functions
• Limitations of known results Szëgo lemma,
  Fisher-Hartwig conjecture

Böttcher et. al., Analysis of Toeplitz Operators
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New Conjecture I.

• Representations of f(?) not unique Basor and
  Tracy (Phys. A 1991)
• Parameters related by

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New Conjecture II.

• Dont constrain ?p1 admits much larger class
  of generating functions
• Sum over all reps., drop subleading terms

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Longitudinal correlators I.

• Roots of the generating function
• Large n asymptotic expansion

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Longitudinal correlators II.
A

• Write in canonical form

??, ?? ?x,y? ?x,y0
?-1
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Small zz
A

• Low temperature, large effective field
• Some results on ?-1

?-1
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Crossover z?z
A

• g? analytic in z
• Near critical field

??, ??
?-1
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Large zz
A

• Linearized dispersions
• Low temperature, effective field near -1
• Some results on ?-1

?-1
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Full lattice correlators

• Recall relation between correlators
• ?x,y correlators on full lattice sensitive to
  crossover for ? correlators on sublattice

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Conclusion

• Exact dynamics of transverse XY model
• Landau-Zener tunneling and critical modes
• Intrinsic many-body decoherence
• Emergence of mixed density matrix for local
  observables at late times, description as thermal
  XX model
• Asymptotics of spin correlators
• New conjecture on Toeplitz determinant,
  non-trivial crossover behavior