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
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
Magnetic fields and lattice systems The Hofstadter Butterfly 1 3 2 1 Flux φ 0.5 Ε k 0 1 2 0-4 -2 0 2 4 Energy 3 0 2Π k q 2 p = 1 and q = 2 5/18
Magnetic fields and lattice systems The Hofstadter Butterfly 1 3 2 1 Flux φ 0.5 Ε k 0 1 2 0-4 -2 0 2 4 Energy 3 0 2Π k q 3 p = 1 and q = 3 5/18
Magnetic fields and lattice systems The Hofstadter Butterfly 1 3 2 1 Flux φ 0.5 Ε k 0 1 2 0-4 -2 0 2 4 Energy 3 0 2Π k q 4 p = 1 and q = 4 5/18
Magnetic fields and lattice systems The Hofstadter Butterfly 1 3 2 1 Flux φ 0.5 Ε k 0 1 2 0-4 -2 0 2 4 Energy 3 0 2Π k q 8 p = 2 and q = 5 5/18
Magnetic fields and lattice systems The Hofstadter Butterfly Ε k 3 2 1 0 1 2 3 0 2Π 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
Effective models Engineering Magnetic fields with cold atoms D. Jaksch and P. Zoller, New J. Phys. 5 56 (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
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
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
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
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
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, 225304 (2012) 9/18
Globally driven systems Simulation of (classical) magnetism Lattice shaking induced staggered flux through the triangular plaquettes - - + + 0.15 0.1 CSF D 0.05 D J. Struck et al., Nature Physics 9, 738 743 (2013) 0 SF 0 0.4 0.8 1.2 t 2 Phases in a bosonic zig-zag ladder with strong interactions 10/ 18
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
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
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
Ladders Ladder with magnetic flux *T. Atala et al., arxiv:1402.0819 (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
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
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:1311.3150) 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