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arXiv:2608.04395

Zero-Clustering Geometry in Realistic Fractional Quantum Hall Wave Functions

Xin Wan, Ziang Wang, Zi-Xiang Hu, Zhao Liu

Published Aug 5, 2026• 0 citations
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Abstract

The clustering pattern of zeros in the ground state of a fractional quantum Hall system is a defining feature of its topological properties. We analyze the geometrical fluctuations of the zeros around individual electrons and propose to use the displacement ratio of the zeros to visualize and measure the distance of a realistic state to a model wave function. The distribution of the zero displacement ratio behaves like an order parameter in the transition from a Laughlin phase to a topologically trivial one. The statistical comparison between quantum Hall states belonging to different Jain sequences leads to a composite fermion fluid description of the $ν= 1/5$ ground state with long-range Coulomb interaction that agrees almost perfectly for as few as $3$-$5$ electrons, overcoming the long-standing difficulties of accommodating the competing liquid and crystal orders at short distances.

DOI
10.48550/arxiv.2608.04395

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The clustering pattern of zeros in the ground state of a fractional quantum Hall system is a defining feature of its topological properties. We analyze the geometrical fluctuations of the zeros around individual electrons and propose to use the displacement ratio of the zeros to visualize and measure the distance of a realistic state to a model wave function. The distribution of the zero displacement ratio behaves like an order parameter in the transition from a Laughlin phase to a topologically trivial one. The statistical comparison between quantum Hall states belonging to different Jain sequences leads to a composite fermion fluid description of the ν=1/5ν= 1/5ν=1/5 ground state with long-range Coulomb interaction that agrees almost perfectly for as few as 333-555 electrons, overcoming the long-standing difficulties of accommodating the competing liquid and crystal orders at short distances.
The topological morphology--order of zeros at the positions of electrons with respect to a specific electron--of Laughlin state at filling fractions 1/m1/m1/m (mmm odd) is homogeneous as every electron feels zeros of order mmm at the positions of other electrons. Although fairly accurate ground state wave functions for most of the other quantum Hall states in the lowest Landau level are quite well-known, it had been an open problem in expressing the ground state wave functions in terms of flux-attachment to particles, {\em a la}, this morphology of Laughlin state. With a very general consideration of flux-particle relations only, in spherical geometry, we here report a novel method for determining morphologies of these states. Based on these, we construct almost exact ground state wave-functions for the Coulomb interaction. Although the form of interaction may change the ground state wave-function, the same morphology constructs the latter irrespective of the nature of the interaction between electrons.
We comprehensively show the topological structure--order of zeros felt by an electron at the positions of other electrons --for the enigmatic filling factor 5/2 at the "Pfaffian-shift" in a spherical geometry. A set of linearly independent antisymmetric functions that are constructed from the possible graphs preserving this topological structure provides complete basis for determining an accurate ground state wave function for any interaction, in particular the Coulomb interaction. One of the graphs describes Z2Z_2Z2​ para-fermion equivalently the Pfaffian wave function, but some of the other graphs also contribute to form the exact ground state for the Coulomb interaction. We further show that our wave function for the Coulomb interaction supports clustering up to half of the composite bosons, signifying a strong-coupling regime.
The fractional quantum Hall effect (FQHE), now entering it's fourth decade, continues to draw attention from the condensed matter community. New experiments in recent years are raising hopes that it will be possible to observe quasi-particles with non-abelian anyonic statistics. These particles could form the building blocks of a quantum computer. The quantum Hall states have topologically protected energy gaps to the low-lying set of excitations. This topological order is not a locally measurable quantity but rather a non-local object, and it is one of the keys to it's stability. From an early stage understanding of the FQHE has been facilitate by constructing trial wave functions. The topological classification of these wave functions have given further insight to the nature of the FQHE. An early, and successful, wave function construction for filling fractions ν=p/(2p+1) was that of composite fermions on planar and spherical geometries. Recently, new developments using conformal field theory have made it possible to also construct the full Haldane-Halperin hierarchy wave functions on planar and spherical geometries. In this thesis we extend this construction to a toroidal geometry, i.e. a flat surface with periodic boundary conditions. One of the defining features of topological states of matter in two dimensions is that the ground state is not unique on surfaces with non trivial topology, such as a torus. The archetypical example is the fractional quantum Hall effect, where a state at filling fraction ν=p/q, has at least a q-fold degeneracy on a torus. This has been shown explicitly for a few cases, such as the Laughlin states and the the Moore-Read states, by explicit construction of candidate electron wave functions with good overlap with numerically found states. In this thesis, we construct explicit torus wave functions for a large class of experimentally important quantum liquids, namely the chiral hierarchy states in the lowest Landau level. These states, which includes the prominently observed positive Jain sequence at filling fractions ν=p/(2p+1), are characterized by having boundary modes with only one chirality. Our construction relies heavily on previous work that expressed the hierarchy wave functions on a plane or a sphere in terms of correlation functions in a conformal field theory. This construction can be taken over to the torus when care is taken to ensure correct behaviour under the modular transformations that leave the geometry of the torus unchanged. Our construction solves the long standing problem of engineering torus wave functions for multi-component many-body states. Since the resulting expressions are rather complicated, we have carefully compared the simplest example, that of ν=2/5, with numerically found wave functions. We have found an extremely good overlap for arbitrary values of the modular parameter τ, that describes the geometry of the torus. Having explicit torus wave functions allows us to use the methods developed by Read and Read \\& Rezayi to numerically compute the quantum Hall viscosity. Hall viscosity is conjectured to be a topologically protected macroscopic transport coefficient characterizing the quantum Hall state. It is related to the shift of the same QH-fluid when it is put on a sphere. The good agreement with the theoretical prediction for the 2/5 state strongly suggests that our wave functions encodes all relevant topologically information. We also consider the Hall viscosity in the limit of a very thin torus. There we find that the viscosity changes as we approach the thin torus limit. Because of this we study the Laughlin state in that limit and see how the change in viscosity arises from a change in the Hamiltonian hopping elements. Finally we conclude that there are both qualitative and quantitative difference between the thin and the square torus. Thus, one has to be careful when interpreting results in the thin torus limit.
We present a detailed study of the ground-state entanglement in disordered fractional quantum Hall liquids. We consider electrons at various filling fractions fff in the lowest Landau level, with Coulomb interactions. At f=1/3,1/5f=1/3,1/5f=1/3,1/5, and 2/52/52/5 where an incompressible ground-state manifold exists at zero disorder, we observe a pronounced minimum in the derivative of entanglement entropy with respect to disorder. At each filling, the position of this minimum is stable against increasing system size, but its magnitude grows monotonically and appears to diverge in the thermodynamic limit. We consider this behavior of the entropy derivative as a compelling signal of the expected disorder-driven phase transition from a topological fractional quantum Hall phase to an insulating phase. On the contrary, at f=1/2f=1/2f=1/2 where a compressible composite fermion sea is present at zero disorder, the entropy derivative exhibits much greater, almost chaotic, finite-size effects, without a clear phase transition signal for system sizes within our exact diagonalization limit. However, the dependence of entanglement entropy with system size changes with increasing disorder, consistent with the expectation of a phase transition from a composite fermion sea to an insulator. Finally, we consider f=1/7f=1/7f=1/7 where compressible Wigner crystals are quite competitive at zero disorder, and analyze the level statistics of entanglement spectrum at f=1/3f=1/3f=1/3.