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arXiv:1609.03818 · Journal of Statistical Physics

Rigidity of the Laughlin Liquid

Élliott H. Lieb, Nicolas Rougerie, Jakob Yngvason

Published Jun 21, 2018• 15 citations
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Abstract

We consider general N-particle wave functions that have the form of a product of the Laughlin state with filling factor $$1/\ell $$ and an analytic function of the N variables. This is the most general form of a wave function that can arise through a perturbation of the Laughlin state by external potentials or impurities, while staying in the lowest Landau level and maintaining the strong correlations of the original state. We show that the perturbation can only shift or lower the 1-particle density but nowhere increase it above a maximum value. Consequences of this bound for the response of the Laughlin state to external fields are discussed.

DOI
10.1007/s10955-018-2082-1

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We consider general N-particle wave functions that have the form of a product of the Laughlin state with filling factor 1/ℓ1/\ell 1/ℓ and an analytic function of the N variables. This is the most general form of a wave function that can arise through a perturbation of the Laughlin state by external potentials or impurities, while staying in the lowest Landau level and maintaining the strong correlations of the original state. We show that the perturbation can only shift or lower the 1-particle density but nowhere increase it above a maximum value. Consequences of this bound for the response of the Laughlin state to external fields are discussed.
The Laughlin state is an ansatz for the ground state of a system of 2D quantum particles submitted to a strong magnetic field and strong interactions. The two effects conspire to generate strong and very specific correlations between the particles. I present a mathematical approach to the rigidity these correlations display in their response to perturbations. This is an important ingredient in the theory of the fractional quantum Hall effect. The main message is that potentials generated by impurities and residual interactions can be taken into account by generating uncorrelated quasi-holes on top of Laughlin's wave-function. An appendix contains a conjecture (not due to me) that should be regarded as a major open mathematical problem of the field, relating to the spectral gap of a certain zero-range interaction.
The most robust fractional quantum Hall states occur in the lowest Landau level at filling factors, 1/3 and 1/5. Such states are very well described by Laughlin's wave function. In this work, we have succeeded in calculating exactly the one-particle density function of the Laughlin states for some finite systems of particles in a disk geometry. The exact results we provide are not only important for the Laughlin states, but also for the general field of numerical calculations because they can serve as benchmarks to test the accuracy of various approaches, numerical schemes and computational methods used in studies of strongly correlated electronic systems.
The Laughlin state is an ansatz for the ground state of a system of 2D quantum particles submitted to a strong magnetic field and strong interactions. The two effects conspire to generate strong and very specific correlations between the particles. I present a mathematical approach to the rigidity these correlations display in their response to perturbations. This is an important ingredient in the theory of the fractional quantum Hall effect. The main message is that potentials generated by impurities and residual interactions can be taken into account by generating uncorrelated quasi-holes on top of Laughlin’s wave-function. An appendix contains a conjecture (not due to me) that should be regarded as a major open mathematical problem of the field, relating to the spectral gap of a certain zero-range interaction. Expository text based on joint works with Elliott H. Lieb, Alessandro Olgiati, Sylvia Serfaty and Jakob Yngvason.
Motivated by the structure of the quasiparticle wavefunction in the composite fermion (CF) theory for fractional quantum Hall filling factor [Formula: see text] (m odd), I consider a suitable quasiparticle operator in differential form, as a modified form of Laughlin's quasiparticle operator, that reproduces quasiparticle wave function as predicted in the CF theory, without a priori assumption of the presence of CFs. I further consider the conjugate of this operator as quasihole operator for obtaining a novel quasihole wave function for 1/m state. Each of these wave functions is interpreted as expelling an electron into a different Hilbert subspace from the original Hilbert space of the Laughlin condensate while still maintaining its correlation (although changed) with the electrons in the condensate such that the expelled electron behaves as a CF with respect to the electrons in the condensate. With this interpretation, I show that the ground state wavefunctions for general states at filling fractions [Formula: see text], respectively, can be constructed as coherent superposition of n coupled Laughlin condensates and their 'conjugates', formed at different Hilbert subspaces. The corresponding wave functions, specially surprising for [Formula: see text] sequence of states, are identical with those proposed in the theory of noninteracting CFs. The states which were considered as fractional quantum Hall effect of interacting CFs, can also be treated in the same footing as for the prominent sequences of states describing as the coupled condensates among which one is a non-Laughlin condensate in a different Hilbert subspace. Further, I predict that the half filling of the lowest Landau level is a quantum critical point for phase transition between two topologically distinct phases each corresponding to a family of states: one consists of large number of coupled Laughlin condensates of filling factor 1/3 and the other corresponds to large number of coupled conjugate Laughlin condensates of filling factor 1, which may be distinguished, respectively, by the absence and presence of upstream edge modes.
We investigate lattice effects on wave functions that are lattice analogs of bosonic and fermionic Laughlin wave functions with number of particles per flux $\ensuremath{\nu}=1/q$ in the Landau levels. These wave functions are defined analytically on lattices with $\ensuremath{\mu}$ particles per lattice site, where $\ensuremath{\mu}$ may be different than $\ensuremath{\nu}$. We give numerical evidence that these states have the same topological properties as the corresponding continuum Laughlin states for different values of qqq and for different fillings $\ensuremath{\mu}$. These states define, in particular, particle-hole symmetric lattice fractional quantum Hall states when the lattice is half filled. On the square lattice it is observed that for $q\ensuremath{\le}4$ this particle-hole symmetric state displays the topological properties of the continuum Laughlin state at filling fraction $\ensuremath{\nu}=1/q$, while for larger qqq there is a transition towards long-range ordered antiferromagnets. This effect does not persist if the lattice is deformed from a square to a triangular lattice, or on the kagome lattice, in which case the topological properties of the state are recovered. We then show that changing the number of particles while keeping the expression of these wave functions identical gives rise to edge states that have the same correlations in the bulk as the reference lattice Laughlin states but a different density at the edge. We derive an exact parent Hamiltonian for which all these edge states are ground states with different number of particles. In addition this Hamiltonian admits the reference lattice Laughlin state as its unique ground state of filling factor 1/q1/q1/q. Parent Hamiltonians are also derived for the lattice Laughlin states at other fillings of the lattice, when $\ensuremath{\mu}\ensuremath{\le}1/q$ or $\ensuremath{\mu}\ensuremath{\ge}1\ensuremath{-}1/q$ and when q=4q=4q=4 also at half filling.
We propose a set of spin system wave functions that is very similar to lattice versions of the Laughlin states. The wave functions are conformal blocks of conformal field theories and for a filling factor of ν = 1/2 we provide a parent Hamiltonian, which is valid for any even number of spins and is at the same time a 2D generalization of the Haldane-Shastry model. We also demonstrate that the Kalmeyer-Laughlin state is reproduced as a particular case of this model. Finally, we discuss various properties of the spin states and point out several analogies to known results for the Laughlin states.