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arXiv:2007.07201 · Physical review. B./Physical review. B

Hall viscosity of the composite-fermion Fermi seas for fermions and bosons

Songyang Pu

Published Oct 2, 2020• 8 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.102.165101

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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 \ensuremathν=1/m\ensuremath{\nu}=1/m\ensuremathν=1/m, where mmm is even for fermions and odd for bosons. A well-defined value for the Hall viscosity is not obtained by viewing the 1/m1/m1/m composite-fermion Fermi seas as the n\ensuremath→\ensuremath∞n\ensuremath{\rightarrow}\ensuremath{\infty}n\ensuremath→\ensuremath∞ limit of the Jain \ensuremathν=n/(nm\ifmmode±\else\textpm\fi1)\ensuremath{\nu}=n/(nm\ifmmode\pm\else\textpm\fi{}1)\ensuremathν=n/(nm\ifmmode±\else\textpm\fi1) states, whose Hall viscosities (\ifmmode±\else\textpm\fin+m)\ensuremathℏ\ensuremathρ/4(\ifmmode\pm\else\textpm\fi{}n+m)\ensuremath{\hbar}\ensuremath{\rho}/4(\ifmmode±\else\textpm\fin+m)\ensuremathℏ\ensuremathρ/4 (\ensuremathρ\ensuremath{\rho}\ensuremathρ is the two-dimensional density) approach \ifmmode±\else\textpm\fi\ensuremath∞\ifmmode\pm\else\textpm\fi{}\ensuremath{\infty}\ifmmode±\else\textpm\fi\ensuremath∞ in the limit n\ensuremath→\ensuremath∞n\ensuremath{\rightarrow}\ensuremath{\infty}n\ensuremath→\ensuremath∞. 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 \ensuremathτ\ensuremath{\tau}\ensuremathτ space. I find that the \ensuremathν=1/2\ensuremath{\nu}=1/2\ensuremathν=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 1/21/21/2 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 ppp-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 5/25/25/2 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 sss-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.
In a two-dimensional electron gas under a strong magnetic field, correlations generate emergent excitations distinct from electrons. It has been predicted that "composite fermions"--bound states of an electron with two magnetic flux quanta--can experience zero net magnetic field and form a Fermi sea. Using infinite-cylinder density matrix renormalization group numerical simulations, we verify the existence of this exotic Fermi sea, but find that the phase exhibits particle-hole symmetry. This is self-consistent only if composite fermions are massless Dirac particles, similar to the surface of a topological insulator. Exploiting this analogy, we observe the suppression of 2k(F) backscattering, a characteristic of Dirac particles. Thus, the phenomenology of Dirac fermions is also relevant to two-dimensional electron gases in the quantum Hall regime.
Motivated by the issue of particle-hole symmetry for the composite fermion Fermi sea at the half filled Landau level, Dam T. Son has made an intriguing proposal [Phys. Rev. X {\bf 5}, 031027 (2015)] that composite fermions are Dirac particles. We ask what features of the Dirac-composite fermion theory and its various consequences may be reconciled with the well established microscopic theory of the fractional quantum Hall effect and the 1/2 state, which is based on {\em non-relativistic} composite fermions. Starting from the microscopic theory, we derive the assertion of Son that the particle-hole transformation of electrons at filling factor ν=1/2\nu=1/2ν=1/2 corresponds to an effective time reversal transformation (i.e. {k⃗j}\{\vec{k}_j\}{kj​}→\rightarrow→{−k⃗j}\{-\vec{k}_j\}{−kj​}) for composite fermions, and discuss how this connects to the absence of 2kF2k_{\rm F}2kF​ backscattering in the presence of a particle-hole symmetric disorder. By considering bare holes in various composite-fermion Λ\LambdaΛ levels (analogs of electronic Landau levels) we determine the Λ\LambdaΛ level spacing and find it to be very nearly independent of the Λ\LambdaΛ level index, consistent with a parabolic dispersion for the underlying composite fermions. Finally, we address the compatibility of the Chern-Simons theory with the lowest Landau level constraint, and find that the wave functions of the mean-field Chern-Simons theory, as well as a class of topologically similar wave functions, are surprisingly accurate when projected into the lowest Landau level. These considerations lead us to introduce a normal form for the unprojected wave functions of the n/(2pn−1)n/(2pn-1)n/(2pn−1) states that correctly capture the topological properties even without lowest Landau level projection.
We construct explicit lowest-Landau-level wave functions for the composite-fermion Fermi sea and its low-energy excitations following a recently developed approach [Pu, Wu, and Jain, Phys. Rev. B 96, 195302 (2017)] and demonstrate them to be very accurate representations of the Coulomb eigenstates. We further ask how the Berry phase associated with a closed loop around the Fermi circle, predicted to be $\ensuremath{\pi}$ in a Dirac composite fermion theory satisfying particle-hole symmetry [D. T. Son, Phys. Rev. X 5, 031027 (2015)], is affected by Landau-level mixing. For this purpose, we consider a simple model wherein we determine the variational ground state as a function of Landau-level mixing within the space spanned by two basis functions: the lowest-Landau-level projected and the unprojected composite-fermion Fermi sea wave functions. We evaluate Berry phase for a path around the Fermi circle within this model following a recent prescription, and find that it rotates rapidly as a function of Landau-level mixing. We also consider the effect of a particle-hole symmetry-breaking three-body interaction on the Berry phase while confining the Hilbert space to the lowest Landau level. Our study deepens the connection between the $\ensuremath{\pi}$ Berry phase and the exact particle-hole symmetry in the lowest Landau level.
We derive the definition of the Berry phase for the adiabatic transport of a composite fermion (CF) in a half-filled composite Fermi liquid (CFL).It is found to be different from that adopted in previous investigations by Geraedts et al.For the standard CFL wave function, we analytically show that the Berry curvature is uniformly distributed in the momentum space.For the Jain-Kamilla wave function, we numerically show that its Berry curvature has a continuous distribution inside the Fermi sea and vanishes outside.We conclude that the CF with respect to both the microscopic wave functions is not a massless Dirac particle.