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Physical Review Letters

Off-diagonal long-range order, oblique confinement, and the fractional quantum Hall effect

S. M. Girvin, Allan H. MacDonald

Published Mar 23, 1987• 536 citations
Publisher

Abstract

We demonstrate the existence of a novel type of off-diagonal long-range order in the fractional-quantum-Hall-effect (FQHE) ground state. This is revealed for the case of fractional filling factor \ensuremath{\nu}=1/m by application of Wilczek's ``anyon'' gauge transformation to attach m quantized flux tubes to each particle. The binding of the zeros of the wave function to the particles in the FQHE is a (2+1)-dimensional analog of oblique confinement in which a condensation occurs, not of ordinary particles, but rather of composite objects consisting of particles and gauge flux tubes.

DOI
10.1103/physrevlett.58.1252

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A natural generalization of the Heisenberg uncertainty principle inequality holding for non-Hermitian operators is presented and applied to the fractional quantum Hall effect (FQHE). This inequality was used in a previous paper to prove the absence of long-range order in the ground state of several one-dimensional (1D) systems with continuous-group symmetries. In this paper, we use it to rule out the occurrence of Bose-Einstein condensation in the bosonic representation of the FQHE wave function proposed by Girvin and MacDonald. We show that the absence of off-diagonal long-range order in this two-dimensional (2D) problem is directly connected with the ${\mathit{q}}^{2}$ behavior of the static structure function S(q) at small momenta.

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Boson-fermion mapping and off-diagonal long-range order in fractional quantum Hall effect

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A new nonsingular mapping between bosons and fermions is introduced in connection with the fractional quantum Hall effect. The mapping connects a fermion system of filling ${\ensuremath{\nu}}_{\mathit{f}}$ to a boson system of filling ${\ensuremath{\nu}}_{\mathit{b}}$ through the relation ${\ensuremath{\nu}}_{\mathit{f}}^{\mathrm{\ensuremath{-}}1}$=${\ensuremath{\nu}}_{\mathit{b}}^{\mathrm{\ensuremath{-}}1}$+1. The ground state of the boson system is shown to map accurately to that of the fermion system if and only if they are both incompressible quantum Hall states. The accuracy of the mapping is shown to be related to the validity of the mean-field theory. The relevance of this mapping to off-diagonal long-range order is discussed by numerically studying the binding of wave-function zeros to particle positions.

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A recently developed hierarchy theory of the fractional quantum Hall effect (FQHE) is reviewed in this article. We use a many quasiparticle (QP) wave function to obtain the effective partition function of a FQHE system in the path-integral representation, which has been shown to be formally equivalent to an anyon system confined to the lowest Landau level. However the two systems are different in the dimension of the Hilbert space, which implies that the standard hierarchy theory is invalid beyond second hierarchical states. The only way to overcome the difficulty is to place anyons in higher Landau levels. By making a mean-field approximation to the statistical gauge field, the QPs are converted to bosons and a constraint boson system is derived. We find a new type of condensate of the bosonic QPs, which results in a new type of incompressible states of the electrons. These states which involve only two hierarchies should have higher stability over others. That explains why some sequences of filling fractions occur prominently in experiment. Finally, we show that there is a one-to-one correspondence between the new hierarchy theory and the composite fermion theory proposed by Jain.

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How low can the fractional quantum Hall effect go

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While it is generally believed that a crystal state is favored at filling factors $\nu< 1/6$ in the lowest Landau level, experiments show, at elevated temperatures, minima in the longitudinal resistance that are associated with fractional quantum Hall effect (FQHE) at the standard sequences $\nu=n/(6n\pm 1)$ and $\nu=n/(8n\pm 1)$. To address this paradox, we investigate the nature of the state at filling factors $\nu=1/7$, 1/9, 1/11 and 2/13 by several methods and find that the competition between the crystal and liquid states is more subtle than previously thought. Exact diagonalization study in the spherical geometry shows behavior consistent with an FQHE liquid for system sizes available. Density matrix renormalization group study in the cylindrical geometry shows that the nature of the state depends on the particle number, with crystal favored for certain particle numbers but not all; these studies do not allow us to make conclusive statements regarding the thermodynamic nature of the state. A variational study shows that the composite-fermion crystal and the renormalized FQHE liquid are very close in energy. Based on these studies, it is not possible to rule out, definitively, an FQHE liquid with strong short-range crystalline correlations at these filling factors.

•Feb 21, 2020•0 citations

Quantum Hall Wave Functions on the Torus

Mikael Fremling

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.

•Jan 1, 2015•1 citation

Off-Diagonal Long-Range Order in Fractional Quantum-Hall-Effect States

Edward H. Rezayi, F. D. M. Haldane

We present evidence of off-diagonal long-range order in the incompressible fractional quantum Hall states at $\ensuremath{\nu}=\frac{1}{3} \mathrm{and} \frac{2}{5}$ Landau-level fillings. We construct correlation functions that exhibit this property only for incompressible quantum-Hall-effect states. In a compressible state no off-diagonal long-range order is observed.

Physical Review Letters•Oct 24, 1988•57 citations
We demonstrate the existence of a novel type of off-diagonal long-range order in the fractional-quantum-Hall-effect (FQHE) ground state. This is revealed for the case of fractional filling factor \ensuremath{\nu}=1/m by application of Wilczek's ``anyon'' gauge transformation to attach m quantized flux tubes to each particle. The binding of the zeros of the wave function to the particles in the FQHE is a (2+1)-dimensional analog of oblique confinement in which a condensation occurs, not of ordinary particles, but rather of composite objects consisting of particles and gauge flux tubes.
A natural generalization of the Heisenberg uncertainty principle inequality holding for non-Hermitian operators is presented and applied to the fractional quantum Hall effect (FQHE). This inequality was used in a previous paper to prove the absence of long-range order in the ground state of several one-dimensional (1D) systems with continuous-group symmetries. In this paper, we use it to rule out the occurrence of Bose-Einstein condensation in the bosonic representation of the FQHE wave function proposed by Girvin and MacDonald. We show that the absence of off-diagonal long-range order in this two-dimensional (2D) problem is directly connected with the q2{\mathit{q}}^{2}q2 behavior of the static structure function S(q) at small momenta.
A new nonsingular mapping between bosons and fermions is introduced in connection with the fractional quantum Hall effect. The mapping connects a fermion system of filling ${\ensuremath{\nu}}_{\mathit{f}}$ to a boson system of filling ${\ensuremath{\nu}}_{\mathit{b}}$ through the relation ${\ensuremath{\nu}}_{\mathit{f}}^{\mathrm{\ensuremath{-}}1}$=${\ensuremath{\nu}}_{\mathit{b}}^{\mathrm{\ensuremath{-}}1}$+1. The ground state of the boson system is shown to map accurately to that of the fermion system if and only if they are both incompressible quantum Hall states. The accuracy of the mapping is shown to be related to the validity of the mean-field theory. The relevance of this mapping to off-diagonal long-range order is discussed by numerically studying the binding of wave-function zeros to particle positions.
A recently developed hierarchy theory of the fractional quantum Hall effect (FQHE) is reviewed in this article. We use a many quasiparticle (QP) wave function to obtain the effective partition function of a FQHE system in the path-integral representation, which has been shown to be formally equivalent to an anyon system confined to the lowest Landau level. However the two systems are different in the dimension of the Hilbert space, which implies that the standard hierarchy theory is invalid beyond second hierarchical states. The only way to overcome the difficulty is to place anyons in higher Landau levels. By making a mean-field approximation to the statistical gauge field, the QPs are converted to bosons and a constraint boson system is derived. We find a new type of condensate of the bosonic QPs, which results in a new type of incompressible states of the electrons. These states which involve only two hierarchies should have higher stability over others. That explains why some sequences of filling fractions occur prominently in experiment. Finally, we show that there is a one-to-one correspondence between the new hierarchy theory and the composite fermion theory proposed by Jain.
While it is generally believed that a crystal state is favored at filling factors ν<1/6\nu< 1/6ν<1/6 in the lowest Landau level, experiments show, at elevated temperatures, minima in the longitudinal resistance that are associated with fractional quantum Hall effect (FQHE) at the standard sequences ν=n/(6n±1)\nu=n/(6n\pm 1)ν=n/(6n±1) and ν=n/(8n±1)\nu=n/(8n\pm 1)ν=n/(8n±1). To address this paradox, we investigate the nature of the state at filling factors ν=1/7\nu=1/7ν=1/7, 1/9, 1/11 and 2/13 by several methods and find that the competition between the crystal and liquid states is more subtle than previously thought. Exact diagonalization study in the spherical geometry shows behavior consistent with an FQHE liquid for system sizes available. Density matrix renormalization group study in the cylindrical geometry shows that the nature of the state depends on the particle number, with crystal favored for certain particle numbers but not all; these studies do not allow us to make conclusive statements regarding the thermodynamic nature of the state. A variational study shows that the composite-fermion crystal and the renormalized FQHE liquid are very close in energy. Based on these studies, it is not possible to rule out, definitively, an FQHE liquid with strong short-range crystalline correlations at these filling factors.
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.