Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1406.5891 · Physical Review B

Absence of a gap in the Gaffnian state

Th. Jolicœur, T. Mizusaki, P. Lecheminant

Published Aug 11, 2014• 15 citations
arXivHTMLPDF

Abstract

We study the Gaffnian trial wave function proposed to describe fractional quantum Hall correlations at Bose filling factor $\ensuremath{\nu}=2/3$ and Fermi filling $\ensuremath{\nu}=2/5$. A family of Hamiltonians interpolating between a hard-core interaction for which the physics is known and a projector whose ground state is the Gaffnian is studied in detail. We give evidence for the absence of a gap by using large-scale exact diagonalizations in the spherical geometry. This is in agreement with recent arguments based on the fact that this wave function is constructed from a nonunitary conformal field theory. By using the cylinder geometry, we discuss in detail the nature of the underlying minimal model and we show the appearance of heterotic conformal towers in the edge energy spectrum where left and right movers are generated by distinct primary operators.

DOI
10.1103/physrevb.90.075116

Related papers

Gaffnian and Haffnian: Physical relevance of nonunitary conformal field theory for the incompressible fractional quantum Hall effect

Bo Yang

We motivate a close look on the usefulness of the Gaffnian and Haffnian quasihole manifold (null spaces of the respective model Hamiltonians) for well-known gapped fractional quantum Hall phases. The conformal invariance of these subspaces is derived explicitly from microscopic many-body states. The resultant conformal field theory (CFT) description leads to an intriguing emergent primary field with $h=2,\phantom{\rule{0.28em}{0ex}}c=0$, and we argue the quasihole manifolds are quantum mechanically well defined and well behaved. Focusing on the incompressible phases at $\ensuremath{\nu}=\frac{1}{3}$ and $\frac{2}{5}$, we show the low-lying excitations of the Laughlin phase are quantum fluids of Gaffnian and Haffnian quasiholes, and give a microscopic argument showing that the Haffnian model Hamiltonian is gapless against Laughlin quasielectrons. We discuss the thermal Hall conductance and shot-noise measurements at $\ensuremath{\nu}=\frac{2}{5}$, and argue that the experimental observations can be understood from the dynamics within the Gaffnian quasihole manifold. A number of detailed predictions on these experimental measurements are proposed, and we discuss their relationships to the conventional CFT arguments and the composite fermion descriptions.

arXiv:2010.13799•Physical review. B./Physical review. B•Mar 2, 2021•14 citations

Quantum Hall wave functions based on S3 conformal field theories

Steven H. Simon, Edward H. Rezayi, Nicolas Regnault

We construct a family of quantum Hall Hamiltonians whose ground states, at least for small system sizes, give correlators of the ${S}_{3}$ conformal field theories. The ground states are considered as trial wave functions for quantum Hall effect of bosons at filling fraction $\ensuremath{\nu}=3/4$ interacting either via delta function interaction or delta function plus dipole interaction. While the ${S}_{3}$ theories can be either unitary or nonunitary, we find high overlaps with exact diagonalizations only for the nonunitary case, suggesting that these wave functions may correspond to critical points, possibly analogous to the previously studied Gaffnian wave function. These wave functions give an explicit example which cannot be fully characterized by their thin-torus limit or by their pattern of zeros.

arXiv:0908.0947•Physical Review B•Mar 2, 2010•19 citations

S3 Quantum Hall Wavefunctions

Steven H. Simon, Edward H. Rezayi, Nicolas Regnault

We construct a family of quantum Hall Hamiltonians whose ground states, at least for small system sizes, give correlators of the S3 conformal field theories. The ground states are considered as trial wavefunctions for quantum Hall effect of bosons at filling fraction nu=3/4 interacting either via delta function interaction or delta function plus dipole interaction. While the S3 theories can be either unitary or nonunitary, we find high overlaps with exact diagonalizations only for the nonunitary case, suggesting that these wavefunctions may correspond to critical points, possibly analogous to the previously studied Gaffnian wavefunction. These wavefunctions give an explicit example which cannot be fully characterized by their thin-torus limit or by their pattern of zeros.

•Jan 1, 2010•0 citations

Spin-singlet Gaffnian wave function for fractional quantum Hall systems

Simon C. Davenport, Eddy Ardonne, Nicolas Regnault +1 more

We characterize in detail a wave function conceivable in fractional quantum Hall systems where a spin or equivalent degree of freedom is present. This wave function combines the properties of two previously proposed quantum Hall wave functions, namely the non-Abelian spin-singlet state and the nonunitary Gaffnian wave function. This is a spin-singlet generalization of the spin-polarized Gaffnian, which we call the ``spin-singlet Gaffnian'' (SSG). In this paper we present evidence demonstrating that the SSG corresponds to the ground state of a certain local Hamiltonian, which we explicitly construct, and, further, we provide a relatively simple analytic expression for the unique ground-state wave functions, which we define as the zero energy eigenstates of that local Hamiltonian. In addition, we have determined a certain nonunitary, rational conformal field theory which provides an underlying description of the SSG and we thus conclude that the SSG is ungapped in the thermodynamic limit. In order to verify our construction, we implement two recently proposed techniques for the analysis of fractional quantum Hall trial states: The ``spin dressed squeezing algorithm,'' and the ``generalized Pauli principle.''

arXiv:1210.8143•Physical Review B•Jan 22, 2013•7 citations

Bubble phase at ν= 13 for spinless hollow-core interaction

Grégoire Misguich, Th. Jolicœur, T. Mizusaki

We investigate fractional quantum Hall states for model interactions restricted to a repulsive hard core. When the hard core excludes relative angular momentum $m=1$ between spinless electrons the ground state at Landau level filling factor $\ensuremath{\nu}=\frac{1}{3}$ is known to be exactly given by the Laughlin wave function. When we exclude relative angular momentum 3 only, W\'ojs, Quinn, and Yi have suggested the appearance of a liquid state with non-Laughlin correlations. We study this special hard-core interaction at filling factor $\frac{1}{3}$ on the sphere, torus, and cylinder geometry. An analysis of the charged and neutral gaps on the sphere geometry points to a gapless state. On the torus geometry the projected static structure factor has a two-peak feature pointing to one-dimensional density ordering. To clarify the nature of the ground state we perform extended density matrix renormalization group studies on the cylinder geometry for up to 30 particles. The pair correlation function allows us to conclude that the ground state is a two-particle bubble phase.

arXiv:1703.07095•Physical review. B./Physical review. B•Dec 4, 2020•4 citations

Modified Coulomb gas construction of quantum Hall states from nonunitary conformal field theories

Milica V. Milovanović, Th. Jolicœur, Ivana Vidanović

Some fractional quantum Hall states observed in experiments may be described by first-quantized wavefunctions with special clustering properties such as the Moore-Read Pfaffian for filling factor $\ensuremath{\nu}=5/2$. This wavefunction has been constructed by constructing correlation functions of a two-dimensional conformal field theory (CFT) involving a free boson and a Majorana fermion. By considering other CFTs many other clustered states have been proposed as candidate fractional quantum Hall states under appropriate circumstances. It is believed that the underlying CFT should be unitary if one wants to describe an incompressible, i.e., gapped liquid state. We show that by changing the way one derives the wavefunction from its parent CFT it is possible to obtain an incompressible candidate state when starting from a nonunitary parent. The construction mimics a global change in parameters in the phase space of the electron system. We explicit our construction in the case of the so-called Gaffnian state (a state for filling factor 2/5) and also for the Haldane-Rezayi state (a spin-singlet state at filling 1/2). We note that there are obstructions along this path in the case of the permanent spin-singlet state of Read and Rezayi which can be characterized as a robust gapless state.

arXiv:0902.1719•Physical Review B•Oct 23, 2009•15 citations
We study the Gaffnian trial wave function proposed to describe fractional quantum Hall correlations at Bose filling factor \ensuremathν=2/3\ensuremath{\nu}=2/3\ensuremathν=2/3 and Fermi filling \ensuremathν=2/5\ensuremath{\nu}=2/5\ensuremathν=2/5. A family of Hamiltonians interpolating between a hard-core interaction for which the physics is known and a projector whose ground state is the Gaffnian is studied in detail. We give evidence for the absence of a gap by using large-scale exact diagonalizations in the spherical geometry. This is in agreement with recent arguments based on the fact that this wave function is constructed from a nonunitary conformal field theory. By using the cylinder geometry, we discuss in detail the nature of the underlying minimal model and we show the appearance of heterotic conformal towers in the edge energy spectrum where left and right movers are generated by distinct primary operators.
We motivate a close look on the usefulness of the Gaffnian and Haffnian quasihole manifold (null spaces of the respective model Hamiltonians) for well-known gapped fractional quantum Hall phases. The conformal invariance of these subspaces is derived explicitly from microscopic many-body states. The resultant conformal field theory (CFT) description leads to an intriguing emergent primary field with h=2,c=0h=2,\phantom{\rule{0.28em}{0ex}}c=0h=2,c=0, and we argue the quasihole manifolds are quantum mechanically well defined and well behaved. Focusing on the incompressible phases at $\ensuremath{\nu}=\frac{1}{3}$ and 25\frac{2}{5}52​, we show the low-lying excitations of the Laughlin phase are quantum fluids of Gaffnian and Haffnian quasiholes, and give a microscopic argument showing that the Haffnian model Hamiltonian is gapless against Laughlin quasielectrons. We discuss the thermal Hall conductance and shot-noise measurements at $\ensuremath{\nu}=\frac{2}{5}$, and argue that the experimental observations can be understood from the dynamics within the Gaffnian quasihole manifold. A number of detailed predictions on these experimental measurements are proposed, and we discuss their relationships to the conventional CFT arguments and the composite fermion descriptions.
We construct a family of quantum Hall Hamiltonians whose ground states, at least for small system sizes, give correlators of the S3{S}_{3}S3​ conformal field theories. The ground states are considered as trial wave functions for quantum Hall effect of bosons at filling fraction $\ensuremath{\nu}=3/4$ interacting either via delta function interaction or delta function plus dipole interaction. While the S3{S}_{3}S3​ theories can be either unitary or nonunitary, we find high overlaps with exact diagonalizations only for the nonunitary case, suggesting that these wave functions may correspond to critical points, possibly analogous to the previously studied Gaffnian wave function. These wave functions give an explicit example which cannot be fully characterized by their thin-torus limit or by their pattern of zeros.
We construct a family of quantum Hall Hamiltonians whose ground states, at least for small system sizes, give correlators of the S3 conformal field theories. The ground states are considered as trial wavefunctions for quantum Hall effect of bosons at filling fraction nu=3/4 interacting either via delta function interaction or delta function plus dipole interaction. While the S3 theories can be either unitary or nonunitary, we find high overlaps with exact diagonalizations only for the nonunitary case, suggesting that these wavefunctions may correspond to critical points, possibly analogous to the previously studied Gaffnian wavefunction. These wavefunctions give an explicit example which cannot be fully characterized by their thin-torus limit or by their pattern of zeros.
We characterize in detail a wave function conceivable in fractional quantum Hall systems where a spin or equivalent degree of freedom is present. This wave function combines the properties of two previously proposed quantum Hall wave functions, namely the non-Abelian spin-singlet state and the nonunitary Gaffnian wave function. This is a spin-singlet generalization of the spin-polarized Gaffnian, which we call the ``spin-singlet Gaffnian'' (SSG). In this paper we present evidence demonstrating that the SSG corresponds to the ground state of a certain local Hamiltonian, which we explicitly construct, and, further, we provide a relatively simple analytic expression for the unique ground-state wave functions, which we define as the zero energy eigenstates of that local Hamiltonian. In addition, we have determined a certain nonunitary, rational conformal field theory which provides an underlying description of the SSG and we thus conclude that the SSG is ungapped in the thermodynamic limit. In order to verify our construction, we implement two recently proposed techniques for the analysis of fractional quantum Hall trial states: The ``spin dressed squeezing algorithm,'' and the ``generalized Pauli principle.''
We investigate fractional quantum Hall states for model interactions restricted to a repulsive hard core. When the hard core excludes relative angular momentum m=1m=1m=1 between spinless electrons the ground state at Landau level filling factor $\ensuremath{\nu}=\frac{1}{3}$ is known to be exactly given by the Laughlin wave function. When we exclude relative angular momentum 3 only, W\'ojs, Quinn, and Yi have suggested the appearance of a liquid state with non-Laughlin correlations. We study this special hard-core interaction at filling factor 13\frac{1}{3}31​ on the sphere, torus, and cylinder geometry. An analysis of the charged and neutral gaps on the sphere geometry points to a gapless state. On the torus geometry the projected static structure factor has a two-peak feature pointing to one-dimensional density ordering. To clarify the nature of the ground state we perform extended density matrix renormalization group studies on the cylinder geometry for up to 30 particles. The pair correlation function allows us to conclude that the ground state is a two-particle bubble phase.
Some fractional quantum Hall states observed in experiments may be described by first-quantized wavefunctions with special clustering properties such as the Moore-Read Pfaffian for filling factor $\ensuremath{\nu}=5/2$. This wavefunction has been constructed by constructing correlation functions of a two-dimensional conformal field theory (CFT) involving a free boson and a Majorana fermion. By considering other CFTs many other clustered states have been proposed as candidate fractional quantum Hall states under appropriate circumstances. It is believed that the underlying CFT should be unitary if one wants to describe an incompressible, i.e., gapped liquid state. We show that by changing the way one derives the wavefunction from its parent CFT it is possible to obtain an incompressible candidate state when starting from a nonunitary parent. The construction mimics a global change in parameters in the phase space of the electron system. We explicit our construction in the case of the so-called Gaffnian state (a state for filling factor 2/5) and also for the Haldane-Rezayi state (a spin-singlet state at filling 1/2). We note that there are obstructions along this path in the case of the permanent spin-singlet state of Read and Rezayi which can be characterized as a robust gapless state.