Topological states of matter with cold gases and Rydberg atoms Hans Peter Büchler Institut für theoretische Physik III, Universität Stuttgart, Germany Collaboration Norman Yao, Mikhail Lukin, Sebastian Huber Research group: David Peter, Nicolai Lang, Sebastian Weber Adam Bühler, Przemek Bienias Krzysztof Jachymski, Jan Kumlin Strongly interacting Rydberg slow light polaritons SFB TRR21: Tailored quantum matter Outline Topological band structures with dipolar interactions - realized with polar molecules and Rydberg atoms - Chern number C=2 Majorana Edge modes - exact solvable system with fixed particle number - double wire system Topological band structure singel particle band structure independent on statistics of particles - topological quantum numbers - edge states Requirements for topological states/phase Fermions Bosons integer filling+ weak interactions topological insulators flat bands + strong interactions fractional topological insulators flat bands + strong interactions bosonic fractional topological insulators motivation: difficulties Topological band structure Homogeneous magnetic fields - integer quantum Hall effect - Hofstadter butterfly Time-reversal symmetry breaking - lattice shaking - artificial gauge fields - synthetic dimension (Esslinger, Bloch, Spielman, Fallani,… - Haldane model Here: product state of lowest band Topological band structure characterized by a Chern number 1 C= 2⇡ Z F (✓x , ✓y ) = Im h@✓y |@✓x i ✓y 2⇡ d✓x 0 h@✓x |@✓y i Z 2⇡ d✓y F (✓x , ✓y ) 0 ✓x Dipolar interactions Dipole-dipole interaction di - anisotropic interaction nij dipole operator Vdd = di · dj 3 (di · nij ) (di · nij ) |ri rj |3 - coupling between orbital degree of freedom and internal degree of freedom dj Observation in cold gases - Einstein de Haas effect (Y. Kawaguchi et al, PRL 2006) “spin-orbit” coupling - demagnetization cooling (M. Fattori et al, Nature Phys 2006) - pattern formation in spinor condensates (D. Stamper-Kurn, M. Ueda, RMP 2013). Dipolar interactions di Exploit this spin orbit coupling for the generation of topological band structures nij dj title Requirements: Systems: - particles in a 2D lattice - polar molecules - Rydberg atoms - internal “Spin” structure - NV centers - atoms with large magnetic dipole moments - strong dipole-dipole interaction 1 Rydberg atoms in an optical lattice Setup - one atom per lattice site with quenched tunneling - static external electric field and magnetic field - select three internal states | ii |+ii |0ii : two excited states : ground state Mapping onto two hard-core bosons: - bosonic creation operators for excitations |+ii = b†i,+ |0i | ii = b†i, |0i Rydberg atoms in an optical lattice Setup - one atom per lattice site with quenched tunneling - static external electric field and magnetic field | ii |+ii - select three internal states | ii |+ii |0ii : ground state : two excited states Mapping onto two hard-core bosons: - bosonic creation operators for excitations |+ii = b†i,+ |0i | ii = b†i, |0i |0ii Dipolar exchange interactions Hamiltonian - dipolar interaction restricted to the three internal levels Hij = 1 |ri rj |3 d0i d0j d0 ⌘ dz 1 + + di dj + di d+ j 2 dipole interaction d+ ⌘ dx + idy 3 + + di dj e 2 i2 d ⌘ dx ij + di dj ei2 spin-orbit coupling exchange interaction di nij ij dj hopping of excitation idy ij Dipolar exchange interactions Hamiltonian - dipolar interaction restricted to the three internal levels Hij = 1 |ri rj |3 d0i d0j d0 ⌘ dz 1 + + di dj + di d+ j 2 dipole interaction d+ ⌘ dx + idy 3 + + di dj e 2 i2 d ⌘ dx ij + di dj ei2 spin-orbit coupling exchange interaction di nij ij dj hopping with spin flip idy ij Single excitation Hamiltonian Hamiltonian i - single particle hoping with spin orbit interaction H= X i6=j a |ri 3 rj |3 † i ✓ t e + i2 Time-reversal symmetry breaking - different hopping for excitations t 6= t+ - energy shift on levels H= X i † i ✓ µ 0 0 µ ◆ i ij we i2 t ij ◆ i = ✓ b+,i b ,i ◆ Single excitation Hamiltonian X 1 ik·ri +i2 ✏ˆk = e |ri |3 Hamiltonian i6=0 - single particle hoping with spin orbit interaction H= X † k k = X k † k ⇥ n0k ✓ + t ✏k + µ wˆ ✏⇤k + nk ⇤ wˆ ✏ t ✏k + µ X 1 ik·ri ✏k = e |ri |3 k i6=0 - characterized by a Chern number C= dk nk · v0 ✓ @nk @nk ^ @kx @ky ◆ k Chern number k as a winding number Topological character of band Z i ◆ 2Z Topological band structure band structure With time reversal symmetry - square lattice - two bands with quadratic band touching points Linear dispersion: - consequence of slow dipolar decay of hopping 12 quadratic band touching point Dirac points in rectangular lattice see also Syzranov et al Nat. Comm. 2014 Topological band structure Without time reversal symmetry band structure - square lattice topological ph - gap openings Topological bands - Chern number C= Z dk nk · v0 ✓ @nk @nk ^ @kx @ky ◆ 2Z 12 Topological band structure Without time reversal symmetry - optimal experimental parameters - Chern number C=2 band Edge states Finite system in y-direction - bulk edge correspondence (Hatsugai PRL 1993) - exponential localization in presence of long-range hopping C= 2 implies two edge states edge states Stability under disorder Disorder Chern number in the disordered system - missing molecules in the lattice - stabilized by long-range hopping edge states: disorder 18 26 Flat topological bands Honeycomb lattice - much flatter bands accessible - very rich topological structure C 2 {0, ±1, ±2, ±3, ±4} honeycomb lattice: flat bands - even richer for Kagame lattice honeycomb lattice: topological phase diagram 0 µs 0.1 µs 0.2 µs Energy (MHz) 20 10 0 Finite systems Propagation of edge excitation -10 Minimal system for observation of edge states -20 K localized at edge 3 µs - exponentially 0.4K µs 0.5 µs 0.0 µs M 0.2 µs M 0.4 µsKK - uni-directional motion 0.0 0.3 0.60.6 0.0 0.3 Density (1 / volume BZ) Density (1 / of volume of BZ) 0.6 µs K 0.8 µs Band structure DOS along path DOS Band structure M Bulk densityBand structure Density of states S 0. 1.0 µs excitation ofOverlap states edge Overlap Overlapwith with with excitation (A) with(B) excitation (B) Location Location 20 0 0 µs -10 0.2 µs 6 µs 0.4 µs -20 0.8 µs 1.0 Kµs 0 10 Propagation of edge excitation 0 -10 -20 M0.0 0 µs K 0 0.6 µs 0 f revolutions 10 Energy (MHz) Energy (MHz) 20 5 4 3 0. 0 0 µs -10 0.4 µs 0 0 -10 0.8 µs Finite systems -20for observation 0-20 Minimal system µs of 1.0edge µs states 0 K 00 M K Direct observation of excitation 0 of edge excit atPropagation edge - round structure with a single edge 0.0Bulk 0.3 0.6 Edge Bulk 0.0 0.0 0.3 0.6 0.3 0.6 Density (1 / volume Density (1 / volume of(1BZ) Density / volume of BZ) of BZ) Edge Spectroscopy DOS Unwrapped DOS Unwrapped band structure band structure Band -DOS round structure withstructure a single edge 0.0 µs 0.6 µs Overlap with excitation (C) Location Overlap with excitation (C) Location Overlap with excitation (B) Location 20 Energy (MHz) 0 201.0 0 100.8 0 00.6 0-100.4 0it-200.2 0 0.0 Density of 10 1.2 µs 0 -10 1.8 µs 5 4 3 2 1 0 0 Linear fit Linear fit -20 50 0 100 1.0 µs Number of revolutions Energy ( 0.6 µs 00 50 50 100 0100 0 Density of states in round sample states with 0.000 0.0025 0.050 Bulk 0.000 Bulk Edge Bulk Edge 0.00.0025 0.3 0.050 0.6 edge Density (1 / volume of BZ) 50 2.4 µs 100 3.0 µs Edge Propagation of edge excit Outlook on topological bands Dipolar interaction provides natural spin orbit coupling - topological band structures with Chern number C=2 Robust to disorder - topological nature is very robust to missing particles in the lattice Towards bosonic fractional Chern insulators - strongly interacting system - is flatness high enough for topological phases - candidate expected at 2/3 filling edge states: disorder Outline Topological band structures with dipolar interactions - realized with polar molecules and Rydberg atoms - Chern number C=2 Majorana Edge modes - exact solvable system with fixed particle number - double wire system Kitaev’s Majorana chain Kitaev’s Majorana chain - fermions on a 1D lattice with supefluid pairing term H= L X1 h wa†i ai+1 ai ai+1 + h.c. i=1 µ L X a†i ai i Trivial phase: Dominant chemical potential i=1 - exact solution by introducing Majorana fermions c2i 1 + ic2i ai = 2 Topological phase: Dominant hopping & pairing Topological state - robust ground state degeneracy - non-local order parameter - localized edge states ⇒ Zero-energy edge mode ai = c2i 1 + ic2i 2 Why should we care Topological invariant edge states - ground state degeneracy is robust to local perturbations robust quantum memory? Non-abelian anyons - localized edge modes obey non-abelian braiding statistics topological quantum computation Topological Phase: (Alicea et al Nat. Phys. 2011) Anyonic Statistics: Topological invariant protects - Novel state of matter ground state degeneracy Localized edge modes obey non-abelian anyonic statistics & ↓ - Toy model of a topological phase Inherently robust way of storing & manipulating quantum information Beyond mean-field H= L X1 h wa†i ai+1 ai ai+1 + h.c. i=1 i - mean-field coupling - violates particle conservation exist a particle conserving theory with Majorana modes in one-dimension? Beyond mean-field Previous work in this context: → M. Cheng and H.-H. Tu (2011). Physical Review B, 84(9), 094503. Majorana edge states in interacting two-chain ladders of fermions. → J. D. Sau et al. (2011). Physical Review B, 84(14), 144509. Number conserving theory for topologically protected degeneracy in one-dimensional fermions. → L. Fidkowski et al. (2011). Physical Review B, 84(19), 195436. Majorana zero modes in one-dimensional quantum wires without long-ranged superconducting order. Bosonization Bosonization → J. Ruhman et al. (2014). arXiv:1412.3444 Topological States in a One-Dimensional Fermi Gas with Attractive Interactions. → C. V. Kraus et al. (2013). Physical Review Letters, 111(17), 173004. Majorana Edge States in Atomic Wires Coupled by Pair Hopping. Numerical (DMRG) Numerical → G. Ortiz et al. (2014). arXiv:1407.3793 Many-body characterization of topological superconductivity: The Richardson-Gaudin-Kitaev chain. Long-Range Long-range Here: Sort-range theory ⇒ Here: Short-range interactinginteracting Theory exact Exact ground stateground state Majorana edge modes Majorana Modes on edges N. Lang and H. P. Büchler, Phys. Rev. B 92, 041118 (2015). F.Iemini,et al., Phys. Rev. Lett. 115, 156402 (2015). Microscopic model Hamiltonian - double wire system H = Ha + Hb + Hab - intra-chain contribution Ha = X Aai (1 + Aai ) i - inter-chain contribution Hab = X Bi (1 + Bi ) i Symmetries - total number of particles N - time reversal symmetry T P - sub-chain parity Microscopic model Inter-chain Hamiltonian Ha = X Aai (1 + Aai ) i Aai = a†i ai+1 + a†i+1 ai Intrachain Interactions: - positive Hamiltonian - zero-energy state is ground state Single-particle hopping & NN density-density inter X |ni = {ni } | . . . , ni , ni+1 , . . .i equal weight superposition of all possible distribution of n fermions Inter-chain Hamiltonian (expanded) Expanded form: Hia = ai a†i+1 + ai+1 a†i + nai 1 nai+1 + nai+1 (1 nai ) Microscopic model Intra-chain Hamiltonian Hab = X Bi (1 + Bi ) i Bi = a†i a†i+1 bi bi+1 + b†i b†i+1 ai ai+1 pair-hopping between chains Interchain Interaction: - positive Hamiltonian & Pair-density-densityX Pair-hopping interactions - zero-energy state is ground state - fixed total number of particles | i= n |ni|N ni equal weight superposition of all possible distribution of N fermions between the two wires Intra-chain Hamiltonian (expanded) i Hab = a†i a†i+1 bi bi+1 +b†i b†i+1 ai ai+1 +nai nai+1 1 nbi 1 nbi+1 + nbi nbi+1 (1 nai ) 1 nai+1 Ground state degeneracy GS = Equal-weight superposition Do the math ... total particle number & subc Two-open chains Degeneracy for open boundar Even total number of particles Two GS for each particle - two-fold ground state degeneracy | | even i = X n2even X |ni|N Do the math ... even odd even odd Degeneracy for open boundary conditions GS = Equal-weight superposition Two GS for with eachfixed particle number s ni Note: total particle number & subchain parity even not work odd for periodic Does even Do the math ... odd with GS = Equal-weight superposition with fixed ni Note: total particle number & subchain parity superposition with fixed Does not work for periodic boundary total particle number & subchain parity i =math ... |ni|N Do oddthe GS = Equal-weight n2odd Degeneracy for open boundary conditions: with Odd total number of particles Two GS for each particle number sector: Two- closed chains even with odd even odd Degeneracy for open boundary conditions: even odd odd even - only one zero energy state for GS for each particle number sector: Two Degeneracy open boundary conditions: total even number offor particles Note: even oddfor periodic boundary even Two GS for each particleDoes number sector: not work conditions odd in all sectors! even even Note: odd Note: odd even odd even odd odd even odd even Does not work for periodic boundary conditions in all sectors! Do the math ... Do the math ... GS = Equal-weight w GS = Equal-weight superposition withsuperposition fixed total&particle number total particle number subchain parity with & subc Wire networks Degeneracy for open boundary conditions: Degeneracy for open boundary Do the math ...number sector: Two GS for each particle n Two GS for each particle Networks of wires GS = Equal-weight superposition with fixed with even even even odd odd odd total particle number & subchain parity Degeneracy for open boundary conditions: even odd even even odd Two GSodd for each particle number sector: even Degeneracy even oddconditions: for open boundary withconditions Note: Degeneracy for open boundary - exact ground states for arbitraryNote: networks Does not work fornumber periodic b Two GS for each particle Two for GS periodic for each particle number sector: even Does not work boundary conditions in all sectors! odd odd se - degeneracy consistent with majorana modes at edges 2E/2 1 number of edges even even odd odd Note: Degeneracy boundary conditions: even even for openodd odd even odd Does notfor work for particle periodic boundary conditions in all Two GS each number sector: Note: even Note: odd even Does not work for periodic boundary in all sect Does not work forconditions periodic boundary c even odd odd Note: Does not work for periodic boundary conditions in all Ground state properties Density-density correlations i - independent on ground state 0 hni nj i = ⇢2 i 6= j i Superfluid correlations ha†i a†i+1 aj aj+1 i = ⇢(1 j ⇢) - long-range superfluid p-wave pairing i+l Greens Function: - exponential decay j+l Vanishes exponentially in the bulk & revival at t Green’s function ha†i aj i j existence of edge modes - revival at the edge We conclude: Ground state properties Stability of ground state degeneracy of edge states - stable under all local perturbations - splitting decays exponentially Stability of ground state degeneracy for open wires - Protected by either time-reversal symmetry or subchain parity a†i bi + b†i ai : stable under time reversal hopping ia†i bi : finite overlap between two ground states ib†i ai Excitation spectrum Low-energy excitations - Goldstone mode due to broken U(1) symmetry - exact wave function for single phase kink excitation |k, i = X j h eikj ( 1) na j + ( 1) - quadratic excitation spectrum ✏k = 4 sin2 k/2 ⇠ k 2 nbj System is in a critical state i - vanishing compressibility | i - Goldstone mode with quadratic dispersion Non-abelian Braiding statistics Idea: Setup for braiding of two edge states Braid edge-modes on subchains by adiabat - wire network with two edges - restriction to the low energy diabatic deformation of Hamiltonian sector "Bath" - very weak coupling terms: adiabatic switching between them Majorana Low-energy 8 relevant states sector: Negative total parity states Weak couplings - characterized by → 8parity ground subchain Subchain-pari Non-abelian Braiding statistics Idea: Braid edge-modes on subchains by adiabatic deformatio "Bath" Negative t → 8 groun Adiabatic switching of coupling - transformation of the ground state according to the non-abelian statistic of Majorana modes Low-energy Majorana Weak couplings Subchain-parities: Conclusion Topological band structure with dipolar exchange interactions - spin-orbit coupling natural present in dipolar system - existence of topological band structures Majorana Edge modes - exact solvable system with fixed particle number - analytical demonstration of Majorana edge modes