Adiabatic Theorem and Quantum Hall Effect

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Adiabatic Theorem and Quantum Hall Effect

• Tracy Zhang
• 03/06/09

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Outline

• What is the Adiabatic Theorem
• Basic Formulation
• Degenerate Eigenstates
• The Quantum Hall Effect
• Apply the Adiabatic Theorem
• The Integer Quantum Hall Effect

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Adiabatic Theorem

• concerns time-dependent Hamiltonian
• where dH/dt is small
• typical conditions H is bounded eigenvalues are
  discreet
• general idea a system which starts in the nth
  eigenstate remains in the nth eigenstate

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Basic Formulation

• Condition the eigenvalues are discreet, and the
  eigenstates non-degenerate
• Let F(t) be the general solution to the
  Schrodinger equation
• i hbar ?t F(t) H(t) F(t)?
• As H chances with t, so do its eigenvalues and
  eigenstates
• At any instant t, F(t) can be written as the
  linear combination of eigenstates phi_n(t), with
  coefficients c_n(t)

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Statement

• If dH/dt is small, then C_n(t) C_n(0)
  e(i\gamma)?
• If F(t) starts out as the mth eigenstate, then
  C_n(0) delta_nm, hence C_n(t)0 unless nm
• F(t) remains the nth eigenstate
• e(i\gamma) called the geometri phase, related to
  parallel transport in the parameter space of the
  Hamiltonian
• Berry's phase

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Degenerate Eigenstates

• For each eigenvalue, associate eigenspace of
  dimensionmultiplicity
• Define projection operator P_n onto nth
  eigenspace
• F(t)sum_n P_n(t) F(t)
• P21
• change variable t -gt s, st / tau
• adiabatic condition tau -gt infinity

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Physical Revolution

• operator U(s),
• F(0) -gt U(s)F(0) F(t)?
• U(s) satisfies
• i hbar ?s U(s) tau H(s) U(s)?
• U(0) 1

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Adiabatic Revolution

• operators H_A(s, P), U_A(s, P),
• U_A(s, P) acts on the eigenspace associated with
  P
• satisfy
• i hbar ?s U_A tau H_A U_A
• U_A(0,P)1
• decoupling requirement
• U_A(s, P)P(0)P(s)U_A(s,P)?
• U_A(s, P)Q(0)P(s)U_A(s,P)?

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Why is it adiabatic

• de-coupling U_A(s,P)P(0) orthog. to
  U_A(s,P)Q(0)?
• Suppose P projects onto nth eigenspace, and F(0)
  lies in nth eigenspace. Then
• U_A(s,P)P(0)F(0)U_A(s,P)F(0)?
• P(s)U_A(s,P)F(0)?
• U_A(s,P)F(0) remains in nth eigenspace

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Statement

• H_A(s,P)-1/tau ?s U_A(s,P) U_A(s,P)
• U_tau (s) U_A(s,P) O(1/ tau)?
• If H is bounded, whose domain is
  time-indenpendent, is continuously
  differentiable, and has gaps in the spectrum,
  then H_A and U_A can always be found to
  approximate physical evolution
• Gap condition can be removed

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Quantum Hall Effect

• Phenomenon observed in 2DEG
• Induced potential difference transverse to
  applied potential difference and B field

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Hall Conductance

• sigma I/V_x
• If electron electron interraction is weak, sigma
  equals integer multiples of 1/2Pi (in units e1,
  hbar1)?
• called Integer Quantum Hall Effect
• Arises from quantization of cyclotron orbits in
  the 2DEG

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Landau Levels

• Hamiltonian of the electron has the form of 1D
  harmonic oscillator
• En hbar w_c (½ n) , called Landau levels
• Highly degenerate, degeneracy proportional to B
• Strong B field -gt fewer levels occupied

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Apply the Adiabatic Theorem

• a different picture

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Set up the Problem

• H(Phi_1, Phi_2) depends on two parameters, of
  which Phi_1 is a function of time.
• At s0, Phi_10 and Phi_20
• As s goes to 1, Phi_1 changes by one magnetic
  flux quantum (pihbar/e pi)?
• Want to show that the average charge transported
  over loop2 is an integer

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Time Evolution in Two Stages

• Initial state f0
• Step one
• Restrict the Hamiltonian to Phi_2 axis, define
  parallel transport U(phi) along the axis.
  U(phi)f0 takes system to the state where Phi_2
  phi
• Step two
• Fix phi, define the adiabatic evolution
  operators H_A(s), U_A(s).
• U_A(s,P)U(phi)f0 f(s0, Phi_2phi)?

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Result

• Average charge transport (averaged over phi)?
• ltQgt1/2Pi int_02Pi iltf(1,phi), ?phi f(1,phi)gt
• By geometric formulation of the problem, this is
  the First Chern Character associated with the
  vector bundle of groundstates, hence an integer.

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References

• Griffths. Introduction to Quantum Mechanics.
• Kato. On the Adiabatic Theorem of Quantum
  Mechnics. 1950.
• Avron et al. Adiabatic Theorems and Applications
  to the Quantum Hall Effect, 1987.

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The End.