We discuss the relationship between fractional quantum Hall (FQH) states at filling factor $\ensuremath{\nu}=p/(2p+1)$ and quantum spin chains. This series corresponds to the Jain series $\ensuremath{\nu}=p/(2mp+1)$ with $m=1$ where the composite fermion picture is realized. We show that the FQH states with toroidal boundary conditions beyond the thin-torus limit can be mapped to effective quantum spin $S=1$ chains with $p$ spins in each unit cell. We calculate energy gaps and the correlation functions for both the FQH systems and the corresponding effective spin chains, using exact diagonalization and the infinite time-evolving block decimation (iTEBD) algorithm. We confirm that the mass gaps of these effective spin chains are decreased as $p$ is increased which is similar to $S=p$ integer Heisenberg chains. These results shed new light on a link between the hierarchy of FQH states and the Haldane conjecture for quantum spin chains.We discuss the relationship between fractional quantum Hall (FQH) states at filling factor $\ensuremath{\nu}=p/(2p+1)$ and quantum spin chains. This series corresponds to the Jain series $\ensuremath{\nu}=p/(2mp+1)$ with $m=1$ where the composite fermion picture is realized. We show that the FQH states with toroidal boundary conditions beyond the thin-torus limit can be mapped to effective quantum spin $S=1$ chains with $p$ spins in each unit cell. We calculate energy gaps and the correlation functions for both the FQH systems and the corresponding effective spin chains, using exact diagonalization and the infinite time-evolving block decimation (iTEBD) algorithm. We confirm that the mass gaps of these effective spin chains are decreased as $p$ is increased which is similar to $S=p$ integer Heisenberg chains. These results shed new light on a link between the hierarchy of FQH states and the Haldane conjecture for quantum spin chains.

arXiv:1205.4850Physical Review BOct 2, 201215 citations

Beyond the Tao-Thouless limit of the fractional quantum Hall effect: spin chains and Fermi surface deformation

Masaaki Nakamura, Zheng-Yuan Wang, Emil J. Bergholtz

We discuss the relationship between the fractional quantum Hall effect in the vicinity of the thin-torus, a.k.a. Tao-Thouless (TT), limit and quantum spin chains. We argue that the energetics of fractional quantum Hall states in Jain sequence at filling fraction ν = p /(2 p + 1) (and ν = 1 − p /(2 p + 1)) in the lowest Landau level is captured by S = 1 spin chains with p spins in the unit cell. These spin chains naturally arise at sub-leading order in e −2π 2 / L 2 1 which serves as an expansion parameter away from the TT limit ( L 1 → 0). We also corroborate earlier results on the smooth Fermi surface deformation of the gapless state at ν = 1/2, interpolating between a state described by a critical S = 1/2 chain and the bulk.We discuss the relationship between the fractional quantum Hall effect in the vicinity of the thin-torus, a.k.a. Tao-Thouless (TT), limit and quantum spin chains. We argue that the energetics of fractional quantum Hall states in Jain sequence at filling fraction ν = p /(2 p + 1) (and ν = 1 − p /(2 p + 1)) in the lowest Landau level is captured by S = 1 spin chains with p spins in the unit cell. These spin chains naturally arise at sub-leading order in e −2π 2 / L 2 1 which serves as an expansion parameter away from the TT limit ( L 1 → 0). We also corroborate earlier results on the smooth Fermi surface deformation of the gapless state at ν = 1/2, interpolating between a state described by a critical S = 1/2 chain and the bulk.

arXiv:1105.2105Journal of Physics Conference SeriesJul 20, 20119 citations

Ground State for Fractional Quantum Hall Effect

Ahmed Jellal

International Journal of Theoretical PhysicsFeb 1, 20000 citations

Link between the hierarchy of fractional quantum Hall states and Haldane’s conjecture for quantum spin chains

Masaaki Nakamura, Emil J. Bergholtz, Juha Suorsa

We study a strong coupling expansion of the $\ensuremath{\nu}=1/3$ fractional quantum Hall state away from the Tao-Thouless limit and show that the leading quantum fluctuations lead to an effective spin-1 Hamiltonian that lacks parity symmetry. By analyzing the energetics, discrete symmetries of low-lying excitations, and string order parameters, we demonstrate that the $\ensuremath{\nu}=1/3$ fractional quantum Hall state is adiabatically connected to both Haldane and large- $D$ phases and is characterized by a string order parameter which is dual to the ordinary one. This result indicates a close relation between (a generalized form of) the Haldane conjecture for spin chains and the fractional quantum Hall effect.We study a strong coupling expansion of the $\ensuremath{\nu}=1/3$ fractional quantum Hall state away from the Tao-Thouless limit and show that the leading quantum fluctuations lead to an effective spin-1 Hamiltonian that lacks parity symmetry. By analyzing the energetics, discrete symmetries of low-lying excitations, and string order parameters, we demonstrate that the $\ensuremath{\nu}=1/3$ fractional quantum Hall state is adiabatically connected to both Haldane and large-$D$ phases and is characterized by a string order parameter which is dual to the ordinary one. This result indicates a close relation between (a generalized form of) the Haldane conjecture for spin chains and the fractional quantum Hall effect.

arXiv:0910.3662Physical Review BApr 5, 201011 citations

m=1

where the composite fermion picture is realized. We show that the FQH states with toroidal boundary conditions beyond the thin-torus limit can be mapped to effective quantum spin

S=1

chains with

p

spins in each unit cell. We calculate energy gaps and the correlation functions for both the FQH systems and the corresponding effective spin chains, using exact diagonalization and the infinite time-evolving block decimation (iTEBD) algorithm. We confirm that the mass gaps of these effective spin chains are decreased as

p

is increased which is similar to

S=p

integer Heisenberg chains. These results shed new light on a link between the hierarchy of FQH states and the Haldane conjecture for quantum spin chains.

D

phases and is characterized by a string order parameter which is dual to the ordinary one. This result indicates a close relation between (a generalized form of) the Haldane conjecture for spin chains and the fractional quantum Hall effect.