Frustration-free model for the ν =5 / 2 FQH state Cristian Voinea 1 , Ammar Kirmani, 2 , Songyang Pu 1 , Pouyan Ghaemi 2,3 , Armin Rahmani 4 , Zlatko Papić 1 1 School of Physics and Astronomy, University of Leeds, LS2 9JT, United Kingdom 2 Physics Department, City College of the City University of New York, NY 10031, U.S.A. 3 Physics Program, Graduate Center of City University of New York, NY 10031, U.S.A. 4 Dept. of Physics and Astronomy and Advanced Materials Science and Eng. Center Western Washington University, Bellingham, WA 98225, U.S.A. The Moore-Read state • Fractional Quantum Hall states are famous topological phases of matter, exhibiting anyonic quasiparticles with non-trivial braiding statistics. Re- cently, geometric degrees of freedom have been identified and linked with properties such as the Hall viscosity. • A simple model for the Laughlin state at ν = 1 / 3 on a thin cylinder has been recently studied using quantum simulators [1]. • The Moore-Read (MR) state, representing a FQH state at ν = 5 / 2 , is of particular interest given its non-Abelian quasiparticles and its paired nature. In this work, we aim to construct a simple model that captures some of its physical properties in the thin cylinder limit. The model The Moore-Read state has the following parent Hamiltonian: H MR = − ∑ i<j<k S ijk {∇ 2 i ∇ 4 j } δ 2 ( r i − r j ) δ 2 ( r j − r k ) κ − 1 r i r j r k single particle orbitals Projecting H MR to the lowest Landau level, we construct a fermionic model on the cylin- der using the localised orbitals. The geometric degrees of free- dom emerging from the band mass/interaction anisotropy are also included through the uni- modular metric g ab H F = ∑ i A † i A i + B † i B i + C † i C i A i = 2 e − κ 2 c i c i +1 c i +2 B i = 6 e − 7 κ 2 3 ( c i c i +2 c i +3 + τ c i c i +1 c i +4 ) C i = 6 e − 7 κ 2 3 ( c i +1 c i +2 c i +4 + τ c i c i +3 c i +4 ) Off-diagonal terms: # # B † B ← − → ## # # C † C ← − → ## τ = 2 exp ( − 2 κ 2 (1 − ig 12 ) /g 11 ) is the deformation parameter. In the thin-cylinder limit, the ground state is a classical product state: | ψ 0 ⟩ τ → 0 − − − → | ## . . . ## ⟩ Mapping to the Fredkin spin model We can map the connected component of the Hilbert space containing the ground state to a “Fredkin" spin- 1 / 2 chain [2]: | # ⟩ →|↑⟩ and | # ⟩ →|↓⟩ A ( w ) ↑ ↑ ↑ ↑ ↓ ↓ ↓ ↑ ↓ ↓ The spin ground state can be written as an area-weighted sum of all Dyck words D N In terms of spins, these are con- figurations with S z tot = 0 and ∑ k i =0 S z i ≥ 0 for all k | ψ 0 ⟩ → N − 1 ∑ w ∈D N ( − τ ) A ( w ) / 2 | w ⟩ Every Dyck word can be mapped back to a particular fermionic product state, obtained from squeezing the root state. References [1] A. Kirmani et al: Probing Geometric Excitations of Fractional Quantum Hall States on Quantum Computers , Phys. Rev. Lett. 129 , 056801 (2022) [2] O. Salberger et al: Deformed Fredkin spin chain with extensive entanglement , J. Stat. Mech., 063103 (2017) [3] Z. Liu et al: Geometric quench and nonequilibrium dynamics of fractional quantum Hall states , Phys. Rev. B 98 , 155140 (2018) Capturing the Moore-Read ground state The Fredkin model correctly captures the MR state close to the thin-cylinder limit. Up to L 2 ≈ 7 l B , the overlap between the ground states stays above ∼ 95% The half-chain entanglement entropy is also approximately reproduced up to the same limit (shaded area). One type of partition in the Landau orbitals ( | ) is almost inert in this limit, whereas 2D area law scaling quickly emerges in the other type ( # | # and | # ). Geometric quench E kl B H → H ′ The “graviton" excitation is the k → 0 limit of the mag- netoroton collective mode. A ground state with isotropic metric couples with the gravi- ton when time-evolved by an anisotropic Hamiltonian H ′ ( g ′ ̸ = I ). This leads to oscil- lations in the metric [3], pre- dicted in the 2D limit by lin- earised bimetric theory: Q ( t ) = 2 A sin ( E γ t 2 ) , φ ( t ) = π 2 − E γ t 2 where E γ is the graviton gap, g ab = exp( Q ( ˆ d a ˆ d b − δ a,b )) and ˆ d = (cos( φ/ 2) , sin( φ/ 2)) We find that the Fredkin model obeys bimetric theory close to the 1D limit, which can be explained microscopically using a 2-state system. The model also captures the departure from linearised bimetric theory. Outlook • Can the Fredkin model be used to study the neutral fermion (pairing mode), and possibly the super symmetry between the two collective modes? • Is this a first step in a systematic connection of the root state with the 2D-limit incompressible Moore-Read state? • Can anyons (and their braiding) be realised in the Fredkin model? Scan for contact details & digital version!