The Hall viscosity has been proposed as a topological property of incompressible fractional quantum Hall states and can be evaluated as Berry curvature. This paper reports on the Hall viscosities of composite-fermion Fermi seas at $\ensuremath{\nu}=1/m$, where $m$ is even for fermions and odd for bosons. A well-defined value for the Hall viscosity is not obtained by viewing the $1/m$ composite-fermion Fermi seas as the $n\ensuremath{\rightarrow}\ensuremath{\infty}$ limit of the Jain $\ensuremath{\nu}=n/(nm\ifmmode\pm\else\textpm\fi{}1)$ states, whose Hall viscosities $(\ifmmode\pm\else\textpm\fi{}n+m)\ensuremath{\hbar}\ensuremath{\rho}/4$ ($\ensuremath{\rho}$ is the two-dimensional density) approach $\ifmmode\pm\else\textpm\fi{}\ensuremath{\infty}$ in the limit $n\ensuremath{\rightarrow}\ensuremath{\infty}$. A direct calculation shows that the Hall viscosities of the composite-fermion Fermi sea states are finite and also relatively stable with system size variation, although they are not topologically quantized in the entire $\ensuremath{\tau}$ space. I find that the $\ensuremath{\nu}=1/2$ composite-fermion Fermi sea wave function for a square torus yields a Hall viscosity that is expected from particle-hole symmetry and is also consistent with the orbital spin of $1/2$ for Dirac composite fermions. I compare my numerical results with some theoretical conjectures.The Hall viscosity has been proposed as a topological property of incompressible fractional quantum Hall states and can be evaluated as Berry curvature. This paper reports on the Hall viscosities of composite-fermion Fermi seas at $\ensuremath{\nu}=1/m$, where $m$ is even for fermions and odd for bosons. A well-defined value for the Hall viscosity is not obtained by viewing the $1/m$ composite-fermion Fermi seas as the $n\ensuremath{\rightarrow}\ensuremath{\infty}$ limit of the Jain $\ensuremath{\nu}=n/(nm\ifmmode\pm\else\textpm\fi{}1)$ states, whose Hall viscosities $(\ifmmode\pm\else\textpm\fi{}n+m)\ensuremath{\hbar}\ensuremath{\rho}/4$ ($\ensuremath{\rho}$ is the two-dimensional density) approach $\ifmmode\pm\else\textpm\fi{}\ensuremath{\infty}$ in the limit $n\ensuremath{\rightarrow}\ensuremath{\infty}$. A direct calculation shows that the Hall viscosities of the composite-fermion Fermi sea states are finite and also relatively stable with system size variation, although they are not topologically quantized in the entire $\ensuremath{\tau}$ space. I find that the $\ensuremath{\nu}=1/2$ composite-fermion Fermi sea wave function for a square torus yields a Hall viscosity that is expected from particle-hole symmetry and is also consistent with the orbital spin of $1/2$ for Dirac composite fermions. I compare my numerical results with some theoretical conjectures.

arXiv:2007.07201Physical review. B./Physical review. BOct 2, 20208 citations

Bardeen-Cooper-Schrieffer pairing of composite fermions

Anirban Sharma, Songyang Pu, Jainendra K. Jain

Topological pairing of composite fermions has led to remarkable ideas, such as excitations obeying non-Abelian braid statistics and topological quantum computation. We construct a $p$ -wave paired Bardeen-Cooper-Schrieffer (BCS) wave function for composite fermions in the torus geometry, which is a convenient geometry for formulating momentum space pairing as well as for revealing the underlying composite-fermion Fermi sea. Following the standard BCS approach, we minimize the Coulomb interaction energy at half filling in the lowest and the second Landau levels, which correspond to filling factors $\ensuremath{\nu}=1/2$ and $\ensuremath{\nu}=5/2$ in GaAs quantum wells, by optimizing two variational parameters that are analogous to the gap and the Debye cutoff energy of the BCS theory. Our results show no evidence for pairing at $\ensuremath{\nu}=1/2$ but a clear evidence for pairing at $\ensuremath{\nu}=5/2$. To a good approximation, the highest overlap between the exact Coulomb ground state at $\ensuremath{\nu}=5/2$ and the BCS state is obtained for parameters that minimize the energy of the latter, thereby providing support for the physics of composite-fermion pairing as the mechanism for the $5/2$ fractional quantum Hall effect. We discuss the issue of modular covariance of the composite-fermion BCS wave function, and calculate its Hall viscosity and pair correlation function. By similar methods, we look for but do not find an instability to $s$ -wave pairing for a spin-singlet composite-fermion Fermi sea at half-filled lowest Landau level in a system where the Zeeman splitting has been set to zero.Topological pairing of composite fermions has led to remarkable ideas, such as excitations obeying non-Abelian braid statistics and topological quantum computation. We construct a $p$-wave paired Bardeen-Cooper-Schrieffer (BCS) wave function for composite fermions in the torus geometry, which is a convenient geometry for formulating momentum space pairing as well as for revealing the underlying composite-fermion Fermi sea. Following the standard BCS approach, we minimize the Coulomb interaction energy at half filling in the lowest and the second Landau levels, which correspond to filling factors $\ensuremath{\nu}=1/2$ and $\ensuremath{\nu}=5/2$ in GaAs quantum wells, by optimizing two variational parameters that are analogous to the gap and the Debye cutoff energy of the BCS theory. Our results show no evidence for pairing at $\ensuremath{\nu}=1/2$ but a clear evidence for pairing at $\ensuremath{\nu}=5/2$. To a good approximation, the highest overlap between the exact Coulomb ground state at $\ensuremath{\nu}=5/2$ and the BCS state is obtained for parameters that minimize the energy of the latter, thereby providing support for the physics of composite-fermion pairing as the mechanism for the $5/2$ fractional quantum Hall effect. We discuss the issue of modular covariance of the composite-fermion BCS wave function, and calculate its Hall viscosity and pair correlation function. By similar methods, we look for but do not find an instability to $s$-wave pairing for a spin-singlet composite-fermion Fermi sea at half-filled lowest Landau level in a system where the Zeeman splitting has been set to zero.

arXiv:2107.07628Physical review. B./Physical review. BNov 10, 202116 citations

Modular Transformations of Fermionic Fractional Quantum Hall Phases

Alan Tran, Parsa Bonderson, Meng Cheng

Bulletin of the American Physical SocietyMar 13, 20171 citation

Incompressible composite fermion liquids

Arkadiusz Wójs, John J. Quinn

Physica E Low-dimensional Systems and NanostructuresDec 4, 20071 citation

Hall viscosity and putative quantum Hall states without positive-definite K-matrix

Emanuele Di Salvo, Dirk Schuricht, Johannes K. Slingerland +1 more

We investigate putative quantum Hall effect states, labeled by their K-matrix equal to (1 1 3), by defining them on the torus and computing their Hall viscosity. Such states have been introduced on the sphere as a phase distinct from Pfaffian and anti-Pfaffian ones. This was done in order to explain certain results on thermal Hall conductivity in favor of particle-hole symmetric Pfaffian topological order in presence of Landau level mixing. The requirements of boundary conditions, modular invariance and ground state degeneracy are enough to uniquely fix the form of the proposed wave functions. We generalize a method to enforce them which we call monodromy matching and check our results on wave functions and Hall viscosity against realizations on the torus of Laughlin and hierarchical states. We highlight the issues in the realization of these states, which turn out to exhibit the formation of clusters. We show that the effect of anti-symmetrization on the system is not enough to prevent clustering; we compute the Hall viscosity for the Halperin version of these states and the fully anti-symmetrized one and we find them being dependent on the geometry and the particle number.We investigate putative quantum Hall effect states, labeled by their K-matrix equal to (1 1 3), by defining them on the torus and computing their Hall viscosity. Such states have been introduced on the sphere as a phase distinct from Pfaffian and anti-Pfaffian ones. This was done in order to explain certain results on thermal Hall conductivity in favor of particle-hole symmetric Pfaffian topological order in presence of Landau level mixing. The requirements of boundary conditions, modular invariance and ground state degeneracy are enough to uniquely fix the form of the proposed wave functions. We generalize a method to enforce them which we call monodromy matching and check our results on wave functions and Hall viscosity against realizations on the torus of Laughlin and hierarchical states. We highlight the issues in the realization of these states, which turn out to exhibit the formation of clusters. We show that the effect of anti-symmetrization on the system is not enough to prevent clustering; we compute the Hall viscosity for the Halperin version of these states and the fully anti-symmetrized one and we find them being dependent on the geometry and the particle number.

arXiv:2603.21794Mar 23, 20260 citations

m

is even for fermions and odd for bosons. A well-defined value for the Hall viscosity is not obtained by viewing the

1/m

composite-fermion Fermi seas as the $n\ensuremath{\rightarrow}\ensuremath{\infty}$ limit of the Jain $\ensuremath{\nu}=n/(nm\ifmmode\pm\else\textpm\fi{}1)$ states, whose Hall viscosities $(\ifmmode\pm\else\textpm\fi{}n+m)\ensuremath{\hbar}\ensuremath{\rho}/4$ ($\ensuremath{\rho}$ is the two-dimensional density) approach $\ifmmode\pm\else\textpm\fi{}\ensuremath{\infty}$ in the limit $n\ensuremath{\rightarrow}\ensuremath{\infty}$. A direct calculation shows that the Hall viscosities of the composite-fermion Fermi sea states are finite and also relatively stable with system size variation, although they are not topologically quantized in the entire $\ensuremath{\tau}$ space. I find that the $\ensuremath{\nu}=1/2$ composite-fermion Fermi sea wave function for a square torus yields a Hall viscosity that is expected from particle-hole symmetry and is also consistent with the orbital spin of

1/2

for Dirac composite fermions. I compare my numerical results with some theoretical conjectures.

Topological pairing of composite fermions has led to remarkable ideas, such as excitations obeying non-Abelian braid statistics and topological quantum computation. We construct a

p

-wave paired Bardeen-Cooper-Schrieffer (BCS) wave function for composite fermions in the torus geometry, which is a convenient geometry for formulating momentum space pairing as well as for revealing the underlying composite-fermion Fermi sea. Following the standard BCS approach, we minimize the Coulomb interaction energy at half filling in the lowest and the second Landau levels, which correspond to filling factors $\ensuremath{\nu}=1/2$ and $\ensuremath{\nu}=5/2$ in GaAs quantum wells, by optimizing two variational parameters that are analogous to the gap and the Debye cutoff energy of the BCS theory. Our results show no evidence for pairing at $\ensuremath{\nu}=1/2$ but a clear evidence for pairing at $\ensuremath{\nu}=5/2$. To a good approximation, the highest overlap between the exact Coulomb ground state at $\ensuremath{\nu}=5/2$ and the BCS state is obtained for parameters that minimize the energy of the latter, thereby providing support for the physics of composite-fermion pairing as the mechanism for the

5/2

fractional quantum Hall effect. We discuss the issue of modular covariance of the composite-fermion BCS wave function, and calculate its Hall viscosity and pair correlation function. By similar methods, we look for but do not find an instability to

s

-wave pairing for a spin-singlet composite-fermion Fermi sea at half-filled lowest Landau level in a system where the Zeeman splitting has been set to zero.