Relativistic Dirac fermions on one-dimensional lattice

Transcription

1 Niodem Szpa DUE, Relativistic Dirac fermions on one-dimensional lattice Niodem Szpa Universität Duisburg-Essen & Ralf Schützhold Plan: 2 Jan 211 Discretized relativistic Dirac fermions (in an external electric potential) Cold fermions on the lattice Realization of fermions with strongly repulsive bosons on the lattice Relativistic QED effects: spontaneous pair creation in strong external fields Physical parameters and approximations in the model (Wor in progress...) 1

2 Niodem Szpa DUE, Discretization of the Dirac equation in 1+1 dimensions Representation of the Clifford algebra {γ µ, γ ν } = 2η µν can be chosen to be γ = σ 3, γ 1 = iσ 1. The one-particle Hamiltonian reads H = α i (p i + A i ) + mβ + A = iσ 2 x + mσ 3 + A + σ 2 A 1 No magnetic field in one spatial dimension gauge in which A = φ and A 1 =. No spin one spatial dimension Ψ(t) H = L 2 (R) 2 has only two components. We discretize the space dimension and introduce lattice: x = a and R d := {x }. Discretized Hilbert space H d := L 2 (R d ) 2 discretized wave f s Ψ α (t) := Ψα (t, x ), α = 1, 2. Discretized derivative: x Ψ α (t, x) x=x n Ψ α n+1 (t) Ψα n 1 (t)/(2a) The discretized one-particle Hamiltonian H d Ψ n = i 2a σ 2(Ψ n+1 Ψ n 1 ) + mσ 3 Ψ n + φψ n. 2

3 Niodem Szpa DUE, Discretized relativistic Dirac fermions The corresponding many-particle Hamiltonian (a self-adjoint operator in the Foc space F) Ĥ = ˆΨ (Hf ) ˆΨ(f ) with ˆΨ α = ˆΨ α (f ) taes the Fermi-Hubbard form Ĥ = 1 2a + m ˆΨ2 ˆΨ 1 +1 ˆΨ 2 ˆΨ h.c. ˆΨ1 ˆΨ1 ˆΨ 2 ˆΨ2 + φ (t) ˆΨ1 ˆΨ1 + ˆΨ 2 ˆΨ2. (1) First term: jumping between the neighbouring sites Remaining two terms: can be treated as combination of effective external potentials Jumping terms split the lattice into into two dynamically independent (non-interacting) sublattices: (A) containing ˆΨ 1 2 and ˆΨ and (B) containing ˆΨ and ˆΨ

4 Niodem Szpa DUE, It is sufficient to consider only one copy of them, say A. Ĥ A = 1 2a + m ˆΨ2 2+1 ˆΨ1 2 ˆΨ ˆΨ1 2 + h.c. ˆΨ1 2 ˆΨ1 2 ˆΨ ˆΨ φ 2 ˆΨ1 2 ˆΨ1 2 + φ ˆΨ ˆΨ (2) With the following identifications, which only modify local phases, ˆΨ 1 2 = ( 1) â, φ 2 = φ, ˆΨ = ( 1)ˆb, φ 2+1 = φ (3) we get Ĥ A = 1 2a ˆb â + ˆb 1â + h.c. + m â â ˆb ˆb + φ â â + φ ˆb ˆb. (4) 4

5 Niodem Szpa DUE, Spectrum of the discretized Dirac Hamiltonian The free Hamiltonian ĤA, with φ =, can be diagonalized. Discrete Fourier transform on the lattice â = 1 â(p)e ip dp ˆb = 1 ˆb(p)e ip dp (5) with anti-commutation relations where {â, â l } = δ,l Ĥ = 1 {â(p), â(q)} = δ(p q) we obtain m (â(p) â(p) ˆb(p) ˆb(p) ) + 1 ( 2a sin(p) â(p) ˆb(p) + ˆb(p) â(p)) dp. (6) This Hamiltonian can be diagonalized via (ã(p) b(p) ) = U (â(p) ), U = 1 ( E+m E ˆb(p) 2 E m E E m E E+m E ), E = m a 2 sin2 (p) (7) what leads to Ĥ = 1 m a 2 sin2 (p) ã(p) ã(p) b(p) b(p) dp. (8) The spectrum of Ĥ consists therefore of two intervals m a 2, m m, m a 2. Additional localized (!) potential φ can only introduce isolated eigenvalues bound states. 5

6 Niodem Szpa DUE, Cold fermions on the lattice 6

7 Niodem Szpa DUE, Fermi-Hubbard model in a 2-periodic potential Set up an optical lattice with cold fermionic atoms described by the discretized Dirac Hamiltonian Ĥ A = 1 2a ˆb â + ˆb 1â + h.c. + m â â ˆb ˆb + φ â â + φ ˆb ˆb. It should be sufficient to consider a standard Fermi-Hubbard Hamiltonian for nonrelativistic fermions on one-dimensional lattice Ĥ = J ĉ +1ĉ + ĉ ĉ+1 + V ĉ ĉ (9) with an additional (slowly varying) external potential V = W + φ and W = ( 1) m. However, this derivation is not quite correct. It contains some intrinsic inconsistency: W is quicly varying (on the length scale of the lattice) and modifies the energy spectrum (bands and gaps) and the Wannier functions used to derive the Fermi-Hubbard Hamiltonian from an underlying continuous physical model. 7

8 Niodem Szpa DUE, Re-derivation of the Fermi-Hubbard model in a 2-periodic potential Figure 1: Example of a 2-periodic potential: W (x) = A sin 2 (λx) + B sin 2 (2λx) easy experimental realization: superposing two lattice-generating laser fields with periods λ and 2λ leads to the Whittaer-Hill ODE for which the band spectrum has been studied by several authors We derive the Hubbard Hamiltonian from a continuous physical model in three steps: Schrödinger equation for nonrelativistic cold atoms in a 2-periodic potential two bands, two Wannier functions, two types of sites (even and odd) For a particular form of the potential: calculate parameters of the model (J, m) Add an additional slowly varying external potential. 8

9 Niodem Szpa DUE, Free particles in a 2-periodic potential. In the lowest energy band (after suitable rescaling of variables): the momentum p π, π and the energy E(p) E, E+ is periodic with a period. Then E(p) can be expanded into a Fourier-cosine-series E 1 (p) = n= E (n) 1 cos(np). (1) Transformation: momentum lattice representation: each cos(nx) δ, n + δ,+n (symmetric jumping by n sites to the right and left) The nearest neighbour approximation corresponds to the approximation E 1 (p) = E() 1 + E (1) 1 cos(p) = J cos(p) (we sete () 1 = ) (11) When a second 2-periodic potential is added (even perturbatively): potential becomes periodic (only) with the double period E 2 (p) periodic by π there appears a (small) gap in the middle of the lowest band (we are not interested in higher bands) So even if the form of E 1 (p) does not change significantly the lowest band splits into two separate bands. 9

10 Niodem Szpa DUE, Figure 2: Dispersion relations for two bands created by inserting a gap in the middle and doubling the period. The dashed (blue) line shows the unperturbed E 1 (p) while the solid and dotted (red) lines correspond to E 2± (p) = ± m 2 + J 2 cos 2 (p). 1

11 Niodem Szpa DUE, We can approximate such obtained dispersion relation by a simple formula E 2± (p) = ± m 2 + J 2 cos 2 (p). (12) For m J very good approximation of the exact dispersion relation E 2± (p) (can be qualitatively verified by using a version of the WKB method for periodic potentials). Two energy bands two Wannier functions on the lattice. Their linear combinations upper/lower states localized at even/odd sites. The Schrödinger Hamiltonian Ĥ1 = + W (x) with periodic W (x) diagonalized in the momentum space Ĥ 1 = 1 π E 1 (p) ˆψ (p) ˆψ(p)dp (13) π undergoes a transition E 1 (p) E 2 (p) after switching the 2-periodic potential on to Ĥ 2 = 1 π/2 π/2 ( ) ˆχ(p) ˆψ(p) ( ) ˆχ(p) M dp (14) ˆψ(p) with ˆχ(p) := ˆψ(p + π) and M = m 2 + J 2 cos 2 (p) m 2 + J. 2 cos 2 (p) (15) 11

12 Niodem Szpa DUE, The matrix M can be transformed via a similarity transformation M = U MU (U is defined above) to ( ) M m J cos(p) =. (16) J cos(p) m The new set of field operators (â(p) ˆb(p) ) = U ( ) ˆχ(p) ˆψ(p) (17) brings the Hamiltonian Ĥ2 to the form (6). Inverting the Fourier transformation (5) and going from the momentum to the site representation â(p) := e i2p â 2, ˆb(p) := e i(2+1)pˆb2+1 (18) where we finally obtain the Hubbard-type Hamiltonian (4) with 1/a = J and without the potential term containing φ. Ĥ A = 1 2a ˆb â + ˆb 1â + h.c. + m â â ˆb ˆb + φ â â + φ ˆb ˆb. (19) The odd sites (b 2+1 ) correspond to higher energy levels while the even sites (a 2 ) correspond to lower energy levels with the difference 2m between the levels. 12

13 Niodem Szpa DUE, The Wannier functions can be obtained from the inverse Fourier transformations a 2 = 1 π π e 2ip a(p)dp, b 2+1 = 1 π π e i(2+1)p b(p)dp (2) where now the a n, b n, a(p), b(p) (with no hat) refer to the corresponding wave functions. m/j we find that the even and odd Wannier functions In the limit a 2 = 1 2 ψ + 2 ψ 2, b 2+1 = 1 2 ψ ψ 2+1 (21) are build from the difference and sum, respectively, of the single-band Wannier functions for the lower and upper band ψ n := 1 π π/2 π/2 e inp ψ(p)dp, ψ + n := 1 π π/2 π/2 e inp ψ(p + π)dp. (22) Additional external potential If V (x) = W (x) + φ(x) where W (x) is 2-periodic and φ(x) is slowly varying (on the length scale λ) and vanishing at infinity the band structure of the spectrum remains untouched with only additional isolated eigenvalues possible. The above construction can be correspondingly modified. 13

14 Niodem Szpa DUE, Realization of fermions with strongly repulsive bosons on a 2-periodic lattice Bose-Hubbard model in a 2-periodic potential For nonrelativistic bosonic atoms on the a 2-periodic lattice Bose-Hubbard Hamiltonian ĉ +1ĉ + ĉ ĉ+1 ĉ 2+1ĉ2+1 ĉ 2ĉ2 Ĥ = J + φ ĉ ĉ + U + m ˆn (ˆn 1) (23) with an additional inter-atomic interaction of strength U where ˆn := ĉ ĉ. Big repulsion on-site: boson-fermion mapping U 1 (in fact U J, m, φ ) effective Pauli blocing two states: and 1. Jordan-Wigner transformation: commutating ĉ, ĉ ĉ := exp ˆd ˆd l l ˆd, iπ l< anti-commutating ˆd, ˆd ĉ := ˆd exp iπ l< ˆd l ˆd l (24) The above Hamiltonian taes the form Ĥ = J ˆd ˆd +1 + ˆd ˆd +1 + m ˆd ˆd ˆd ˆd φ ˆd ˆd (25) 14

15 Niodem Szpa DUE, Relativistic QED effects: spontaneous pair creation in strong external fields Supercriticality for the Dirac equation Attractive Woods-Saxon potential (Dirac equation analytically solvable in terms of hypergeometric f s) V (x) = W 1 + ea( x L), W <, (26) Figure 3: Sum of the 2-periodic lattice potential and the slowly varying external Woods-Saxon potential. Time-dependent supercritical phenomena bound state resonance pair creation (nown for the continuous theory!) 15

16 Niodem Szpa DUE, Realization of the time-dependent supercritical potential and pair production on the lattice Preparation of the initial state Big ±m, cool down, trivial Fermi-level (all particles down) adiabatic to small ±m The potential adiabatically switch on the potential wait (pair creation) adiabatic switch off Detection of the created pair adiabatically to big ±m 1 particle up+1 hole down Physical detection: Rabi oscillations in the atoms up? (diff. freq. than down?) 16

17 Niodem Szpa DUE, Physical parameters and approximations in the model Model parameters from physical parameters E, J and m are determined by the parameters of the standing laser wave A, B and λ. The atom-atom repulsion parameter U depends on the type and of atoms used in the experiment. Approximations in the model for Bose-Hubbard we need E U so that there is effectively only one state at each site ( E means the energy difference between the ground and first excited localized state) U J to have only up to one particle pro site actually here one can relax the previous condition to E, U J 1/a J m in order to obtain a good approximation by discretizing the Dirac eq. (compared to the continuous eq.) m T The first two conditions must be experimentally nown. The conditions on the mass J m T are posed by us and it must be checed how good can they be satisfied. 17

18 Niodem Szpa DUE, Appendices 18

19 Niodem Szpa DUE, Transformation bac to the lattice representation One can now as the question what is the spatial representation of this Hamiltonian. By the inverse Fourier transformation from the momentum space bac to the lattice ã(p) = e ip ã, d(p) = e ip d (27) we find H =,l H l ã ã l + d dl (28) where H l = 1 m 2 + sin 2 (p)e i( l)p dp (29) We don t now a closed formula for H l but expect it to correspond to the discretized operator m 2 d on the lattice. We can argue for this by calculating an approximation for m 1 where the integrand becomes m 2 + sin 2 (p) = m + sin2 (p) 2m + O ( m 3) (3) what gives H l = m δ,l 1 8m (δ,l+2 2δ,l + δ,l 2 ) + O and the second term is just d /(2m). ( m 3) (31) 19

20 Niodem Szpa DUE, Analogously, we can find relation between the new ã, d and old ˆΨ 1, ˆΨ 2 lattice operators ã n = ˆΨ 1 1 C aa (p)e i(n )p dp + ˆΨ 2 1 C ab (p)e i(n )p dp (32) d n = ˆΨ 1 1 C ba (p)e i(n )p dp + ˆΨ 2 1 C bb (p)e i(n )p dp (33) where C xy := δ x(p)/δy(p) with x, y {a, b} being the coefficients in (7) matrix elements of U. Another argument comes from comparison with the Bose field for which the coefficients calculated exactly H Bose l can be H Bose l = 1 m 2 + sin 2 (p)e i( l)p dp = m 2 δ,l 1 4 (δ,l+2 2δ,l + δ,l 2 ) = (m 2 I d ) l (34) with particles jumping however only between every second site (either only even or only odd)! 2