Magnetic fields and lattice systems

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4 Magnetic fields and lattice systems Harper-Hofstadter Hamiltonian Landau gauge A = (0, B x, 0) (homogeneous B-field). Transition amplitude along x gains y-dependence: J x J x e i a2 B e y = J x e i Φy = J x e i 2παy H = J x e iφy a a x+1,y J y a a +1 +h.c. 4/18

5 Magnetic fields and lattice systems Harper-Hofstadter Hamiltonian Landau gauge A = (0, B x, 0) (homogeneous B-field). Transition amplitude along x gains y-dependence: J x J x e i a2 B e y = J x e i Φy = J x e i 2παy H = J x e iφy a a x+1,y J y a a +1 +h.c. What is the single particle spectrum of this Hamiltonian? For rational α = p/q we get a periodic Hamiltonian! 4/18

6 Magnetic fields and lattice systems The Hofstadter Butterfly Flux φ 0.5 Ε k Energy 3 0 2Π k q 2 p = 1 and q = 2 5/18

7 Magnetic fields and lattice systems The Hofstadter Butterfly Flux φ 0.5 Ε k Energy 3 0 2Π k q 3 p = 1 and q = 3 5/18

8 Magnetic fields and lattice systems The Hofstadter Butterfly Flux φ 0.5 Ε k Energy 3 0 2Π k q 4 p = 1 and q = 4 5/18

9 Magnetic fields and lattice systems The Hofstadter Butterfly Flux φ 0.5 Ε k Energy 3 0 2Π k q 8 p = 2 and q = 5 5/18

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12 Magnetic fields and lattice systems The Hofstadter Butterfly Ε k Π p = 2, q = 5 with open boundaries (cylinder) k For rational value α = p/q the Bloch-Band of HH-model breaks up into q distinct energy bands Nontrivial topological invariants for each band (Chern number) Open system exhibits (several) edge states Problem: in solid state systems α = a 2 B e 2π of order 1 requires fields of order 10 5 T (assuming a Å) Solution: Artificial lattice with larger lattice-distance 6/18

13 Effective models Engineering Magnetic fields with cold atoms D. Jaksch and P. Zoller, New J. Phys (2003) two 2D optical lattices for two different internal atomic states Rabi transition between internal states induces hopping phase accumulation by special geometry for the Raman beams 7/18

14 Effective models Realizing effective models in periodically driven systems General Hamiltonian with time dependent driving and a tilt in x direction (interactions play no role) H(t) = J x a a x+1,y J y a a +1 +h.c. + (x +V (t))n 8/18

15 Effective models Realizing effective models in periodically driven systems General Hamiltonian with time dependent driving and a tilt in x direction (interactions play no role) H(t) = J x a a x+1,y J y a a +1 +h.c. + (x +V (t))n Unitary transformation U(t) = e i (γt+χ(t) Θ)n with γ = xω and χ (t) = t 0 V (t )dt and Θ fixes the gauge 8/18

16 Effective models Realizing effective models in periodically driven systems General Hamiltonian with time dependent driving and a tilt in x direction (interactions play no role) H(t) = J x a a x+1,y J y a a +1 +h.c. + (x +V (t))n Unitary transformation U(t) = e i (γt+χ(t) Θ)n with γ = xω and χ (t) = t 0 V (t )dt and Θ fixes the gauge Hamiltonian in new frame H U HU iu U (choose ω = ) H(t) = J x e i(ṽ(t) Ṽx+1,y(t) ωt+ Θ ) a a x+1,y +h.c. J y e i(ṽ(t) Ṽ +1 (t)+ Θ ) a a +1 +h.c. 8/18

17 Effective models Realizing effective models in periodically driven systems General Hamiltonian with time dependent driving and a tilt in x direction (interactions play no role) H(t) = J x a a x+1,y J y a a +1 +h.c. + (x +V (t))n Unitary transformation U(t) = e i (γt+χ(t) Θ)n with γ = xω and χ (t) = t 0 V (t )dt and Θ fixes the gauge Hamiltonian in new frame H U HU iu U (choose ω = ) H(t) = J x e i(ṽ(t) Ṽx+1,y(t) ωt+ Θ ) a a x+1,y +h.c. J y e i(ṽ(t) Ṽ +1 (t)+ Θ ) a a +1 +h.c. for ω J we may time average to find effective Hamiltonian 8/18

18 Globally driven systems Lattice shaking shaking mirrors or periodically detuning lasers of optical lattice shaking of the lattice (without tilt = 0) modulated tilting V (t) = V 0 x cos(ωt) in co-moving frame time averaged hopping acquires a phase (if certain time-reflection symmetries are broken) J x J x J 0 ( V0 ω ) e iφ J. Struck et al., Phys. Rev. Lett. 108, (2012) 9/18

19 Globally driven systems Simulation of (classical) magnetism Lattice shaking induced staggered flux through the triangular plaquettes CSF D 0.05 D J. Struck et al., Nature Physics 9, (2013) 0 SF t 2 Phases in a bosonic zig-zag ladder with strong interactions 10/ 18

20 The experiments in Munich and Boston Realizing the Hofstadter-Model with cold atoms Local optical potential by two far-detuned running-wave beams realizes spatial dependent phase of the shaking Φ = q r/2 V 0 cos 2 (ωt/2+q r/2) tilt given by magnetic field gradient, acceleration, gravity,... with J x = J x J 1 ( 2 V 0 ω sin(φ Φ x+1,y )/2 ) and J y = J y J 0 (...) H eff = J x e iφ a a x+1,y J y a a +1 +h.c. 11/18

21 The experiments in Munich and Boston Cyclotron dynamics Dynamics in four site plaquette in a superlattice X = N right N left and Y = N up N down Inverse flux and dynamics for -component Homogeneity shown by shift of one lattice site 12/18

22 Hofstadter model with graphene In the meantime in solid state physics C.R. Dean et al., Nature 497 (2013) coupling between graphene and hexagonal boron nitride results in a periodic Moiré pattern measurements of Quantum Hall conductivity shows anomalous behavoir related to fractionalized band structure 13/18

23 Ladders Ladder with magnetic flux *T. Atala et al., arxiv: (2014) Direct observation of edge states? Go to a simpler model! Non-interacting BH-model on a ladder with fluxes Φ. What happens with interactions? 14/18

24 Outlook Going further Observation of edge states, measure topological invariants,... Study effect of interactions (only in certain limits theoretical predictions possible) Related systems non-abelian magnetic fields... Spin orbit-coupling... Dynamical lattice gauge theories: condensed matter, high energy physics (QCD simulator?) In the following: Anyons and density depended magnetic (gauge) fields 15/18

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26 Density dependent magnetic fields Density dependent Peierls phases Anyonic Hubbard model (Keilmann et al., Nature Commun. 2, 361 (2011)) H = J x b xb x+1 e iφnx +h.c. AB-model with modulated interactions (Greschner, Sun, Poletti, Santos, arxiv: ) H eff = J 2 a 2x ei Φna 2x b 2x+1 x +b 2x+1 ei Φna 2x+2 a2x+2 +h.c. Density depended drift in momentum space of ground state and and statistically induced phase transitions! Outlook / Work in Progress: Density dependent magnetic fields in ladders, 2D,... 17/18

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