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Adiabatic Theorem and Quantum Hall Effect

Tracy Zhang 03/06/09. Adiabatic Theorem and Quantum Hall Effect. Outline. What is the Adiabatic Theorem Basic Formulation Degenerate Eigenstates The Quantum Hall Effect Apply the Adiabatic Theorem The Integer Quantum Hall Effect. Adiabatic Theorem. concerns time-dependent Hamiltonian

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Adiabatic Theorem and Quantum Hall Effect

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  1. Tracy Zhang 03/06/09 Adiabatic Theorem and Quantum Hall Effect

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

  3. 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

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

  5. Statement • If dH/dt is small, then C_n(t) = C_n(0) *e^(i\gamma)‏ • If Φ(t) starts out as the mth eigenstate, then C_n(0) = delta_nm, hence C_n(t)=0 unless n=m • Φ(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

  6. Degenerate Eigenstates • For each eigenvalue, associate eigenspace of dimension=multiplicity • Define projection operator P_n onto nth eigenspace • Φ(t)=sum_n {P_n(t) Φ(t)} • P^2=1 • change variable t -> s, s=t / tau • adiabatic condition tau -> infinity

  7. Physical Revolution • operator U(s), • Φ(0) -> U(s)Φ(0) = Φ(t)‏ • U(s) satisfies i hbar ∂s U(s) = tau H(s) U(s)‏ U(0) =1

  8. 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)‏

  9. 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 Φ(0) lies in nth eigenspace. Then U_A(s,P)P(0)Φ(0)=U_A(s,P)Φ(0)‏ =P(s)U_A(s,P)Φ(0)‏ • U_A(s,P)Φ(0) remains in nth eigenspace

  10. 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

  11. Quantum Hall Effect • Phenomenon observed in 2DEG • Induced potential difference transverse to applied potential difference and B field

  12. Hall Conductance • sigma = I/V_x • If electron electron interraction is weak, sigma equals integer multiples of 1/2Pi (in units e=1, hbar=1)‏ • called Integer Quantum Hall Effect • Arises from quantization of cyclotron orbits in the 2DEG

  13. 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 -> fewer levels occupied

  14. Apply the Adiabatic Theorem • a different picture

  15. Set up the Problem • H(Phi_1, Phi_2) depends on two parameters, of which Phi_1 is a function of time. • At s=0, Phi_1=0 and Phi_2=0 • As s goes to 1, Phi_1 changes by one magnetic flux quantum (pi*hbar/e =pi)‏ • Want to show that the average charge transported over loop2 is an integer

  16. 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(s=0, Phi_2=phi)‏

  17. Result • Average charge transport (averaged over phi)‏ <Q>=1/2Pi int_0^2Pi { i<f(1,phi), ∂phi f(1,phi)>} • By geometric formulation of the problem, this is the First Chern Character associated with the vector bundle of groundstates, hence an integer.

  18. 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.

  19. The End.

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