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arXiv:0910.3662 · Physical Review B

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

Masaaki Nakamura, Emil J. Bergholtz, Juha Suorsa

Published Apr 5, 2010• 11 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.81.165102

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We study a strong coupling expansion of the \ensuremathν=1/3\ensuremath{\nu}=1/3\ensuremathν=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 \ensuremathν=1/3\ensuremath{\nu}=1/3\ensuremathν=1/3 fractional quantum Hall state is adiabatically connected to both Haldane and large-DDD 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 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=1m=1m=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=1S=1S=1 chains with ppp 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 ppp is increased which is similar to S=pS=pS=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 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 consider spin-polarized electrons in a single Landau level on a torus. The quantum Hall problem is mapped onto a one-dimensional lattice model with lattice constant $2\ensuremath{\pi}/{L}_{1}$, where L1{L}_{1}L1​ is a circumference of the torus (in units of the magnetic length). In the Tao-Thouless limit ${L}_{1}\ensuremath{\rightarrow}0$, the interacting many-electron problem is exactly diagonalized at any rational filling factor $\ensuremath{\nu}=p/q\ensuremath{\le}1$. For odd qqq, the ground state has the same qualitative properties as a bulk $({L}_{1}\ensuremath{\rightarrow}\ensuremath{\infty})$ quantum Hall hierarchy state and the lowest-energy quasiparticle excitations have the same fractional charges as in the bulk. These states are the ${L}_{1}\ensuremath{\rightarrow}0$ limits of the Laughlin and Jain wave functions for filling fractions where these exist. We argue that the exact solutions generically, for odd qqq, are continuously connected to the two-dimensional bulk quantum Hall hierarchy states---i.e., that there is no phase transition as ${L}_{1}\ensuremath{\rightarrow}\ensuremath{\infty}$ for filling factors where such states can be observed. For even-denominator fractions, a phase transition occurs as L1{L}_{1}L1​ increases. For $\ensuremath{\nu}=1/2$ this leads to the system being mapped onto a Luttinger liquid of neutral particles at small but finite L1{L}_{1}L1​; this then develops continuously into the composite fermion wave function that is believed to describe the bulk $\ensuremath{\nu}=1/2$ system. The analysis generalizes to non-Abelian quantum Hall states.
The authors consider here the behavior of quantum Hall edges away from the Luttinger liquid fixed point that describes the low energy, large system limit. Using the links between quantum Hall wave functions and conformal field theories, they construct effective Hamiltonians and constrain their forms by considering bulk symmetries. In examining the effect of bulk interactions on this edge, they find remarkable simplifications to these effective theories that allow a very accurate description of the low-energy physics of quantum Hall edges relatively far away from the Luttinger liquid fixed point, and that apply to small systems and higher energies.