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Composite fermions, rectangular Young tableaux and Gelfand-Tsetlin basis in multicomponent fractional quantum Hall systems

Manuel Calixto, E. Pérez-Romero

Published Apr 15, 2019• 0 citations
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Abstract

We put forward a group-theoretical approach to describe the Hilbert space of $M$ fermions, with $N$ components (spin, layer, valley, sub-lattice, etc), per Landau site in the lowest Landau level, at fractional filling factor $\nu=M/\lambda$. The Hilbert space turns out to be the carrier space of a unitary irreducible representation of $U(N)$ described by Gelfand-Tsetlin patterns corresponding to rectangular Young tableaux of $M$ rows and $\lambda$ columns. Here $\lambda$ represents the number of magnetic flux quanta vortex lines penetrating the sample, which must be odd for antisymmetry requirements. The construction can be considered an extension of Haldane's $U(2)$ picture for spherical geometry to a more involved $U(N)$ description for Grassmannian geometry. This construction relies on the boson realization of isospin $U(N)$ operators as bilinear products of creation and annihilation operators of magnetic flux quanta, with an embedding of the Hilbert space into Fock space. Coherent state excitations above a spontaneously chosen ground state are labeled by points $Z$ on the Grassmannian coset $U(N)/U(M)\times U(N-M)$.

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Quantization on Grassmann phase spaces $U(N)/U(M)\times U(N-M)$ and rectangular $M\times f$ Young tableaux with applications to $U(N)$ quantum Hall ferromagnets at filling factor $\nu=M/f$

Manuel Calixto, Julio Guerrero, Alberto Mayorgas +1 more

We develop a group-theoretical approach to describe $N$-component composite bosons as planar electrons attached to an odd number $f$ of Chern-Simons flux quanta. This picture arises when writing the Coulomb exchange interaction as a quantum Hall ferromagnet in terms of collective $U(N)$-spin operators. A spontaneously chosen ground state of $M$ electrons per Landau site breaks the symmetry from $U(N)$ to the stability subgroup $U(M)\times U(N-M)$, so that coherent state excitations are labeled by points on the Grassmannian coset $U(N)/U(M)\times U(N-M)$. The quantization of this Grassmann phase space corresponds to the carrier Hilbert space of unitary irreducible representations of $U(N)$ described by rectangular Young tableaux of $M$ rows and $f$ columns. We construct an embedding of the Hilbert space into Fock space by using a Schwinger realization of collective $U(N)$-spin operators as bilinear products of composite boson operators. We also build a system of Grassmann coherent states and discuss the classical limit of $U(N)$ quantum Hall ferromagnets in terms of nonlinear sigma models on Grasmannians.

•Apr 15, 2019•0 citations

Hilbert Space Structure of the Low Energy Sector of U(N) Quantum Hall Ferromagnets and Their Classical Limit

Manuel Calixto, Alberto Mayorgas, Julio Guerrero

Using the Lieb–Mattis ordering theorem of electronic energy levels, we identify the Hilbert space of the low energy sector of U(N) quantum Hall/Heisenberg ferromagnets at filling factor M for L Landau/lattice sites with the carrier space of irreducible representations of U(N) described by rectangular Young tableaux of M rows and L columns, and associated with Grassmannian phase spaces U(N)/U(M)×U(N−M). We embed this N-component fermion mixture in Fock space through a Schwinger–Jordan (boson and fermion) representation of U(N)-spin operators. We provide different realizations of basis vectors using Young diagrams, Gelfand–Tsetlin patterns and Fock states (for an electron/flux occupation number in the fermionic/bosonic representation). U(N)-spin operator matrix elements in the Gelfand–Tsetlin basis are explicitly given. Coherent state excitations above the ground state are computed and labeled by complex (N−M)×M matrix points Z on the Grassmannian phase space. They adopt the form of a U(N) displaced/rotated highest-weight vector, or a multinomial Bose–Einstein condensate in the flux occupation number representation. Replacing U(N)-spin operators by their expectation values in a Grassmannian coherent state allows for a semi-classical treatment of the low energy (long wavelength) U(N)-spin-wave coherent excitations (skyrmions) of U(N) quantum Hall ferromagnets in terms of Grasmannian nonlinear sigma models.

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Coherent states on the GrassmannianU(4)/U(2)2: oscillator realization and bilayer fractional quantum Hall systems

Manuel Calixto, E. Pérez-Romero

Bilayer quantum Hall (BLQH) systems, which underlie a U (4) symmetry, display unique quantum coherence effects. We study coherent states (CS) on the complex Grassmannian , orthonormal basis, U (4) generators and their matrix elements in the reproducing kernel Hilbert space of analytic square-integrable holomorphic functions on , which carries a unitary irreducible representation of U (4) with index . A many-body representation of the previous construction is introduced through an oscillator realization of the U (4) Lie algebra generators in terms of eight boson operators. This particle picture allows us to make a physical interpretation of our abstract mathematical construction in the BLQH jargon. In particular, the index λ is related to the number of flux quanta bound to a bi-fermion in the composite fermion picture of Jain for fractions of the filling factor ν = 2. The simpler, and better known, case of spin- s CS on the Riemann–Bloch sphere is also treated in parallel, of which Grassmannian -CS can be regarded as a generalized (matrix) version.

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Hilbert space structure and classical limit of the low energy sector of U(N) quantum Hall ferromagnets

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Using the Lieb–Mattis ordering theorem of electronic energy levels, we identify and construct the Hilbert space of the low energy sector of U(N) quantum Hall/Heisenberg ferromagnets at filling factor M for L Landau/lattice sites. The carrier Hilbert space of irreducible representations of U(N) is described by rectangular Young tableaux of M rows and L columns, and associated with Grassmannian phase spaces U(N)/U(M)×U(N-M). Replacing U(N)-spin operators by their expectation values in a Grassmannian coherent state allows for a semi-classical treatment of the low energy U(N)-spin-wave coherent excitations (skyrmions) of U(N) quantum Hall ferromagnets in terms of Grasmannian nonlinear sigma models.

SciPost Physics Proceedings•Nov 23, 2023•0 citations

Composite Fermions in the Hilbert Space of the Lowest Electronic Landau Level

Jainendra K. Jain, Rajiv K. Kamilla

Single particle basis functions for composite fermions are obtained from which many-composite fermion states confined to the lowest electronic Landau level can be constructed in the standard manner, i.e. by building Slater determinants. This representation enables a Monte Carlo study of systems containing a large number of composite fermions, yielding new quantitative and qualitative information. The ground state energy and the gaps to charged and neutral excitations are computed for a number of fractional quantum Hall effect (FQHE) states, earlier off-limits to a quantitative investigation. The ground state energies are estimated to be accurate to ~0.1% and the gaps at the level of a few percent. It is also shown that at Landau level fillings smaller than or equal to 1/9 the FQHE is unstable to a spontaneous creation of excitons of composite fermions. In addition, this approach provides new conceptual insight into the structure of the composite fermion wave functions, resolving in the affirmative the question of whether it is possible to motivate the composite fermion theory entirely within the lowest Landau level, without appealing to higher Landau levels.

arXiv:cond-mat/9704031•International Journal of Modern Physics B•Sep 10, 1997•232 citations

Quantum Hall effect of two-component bosons at fractional and integral fillings

Ying-Hai Wu, Jainendra K. Jain

We investigate the feasibility of many candidate quantum Hall states for two-component bosons in the lowest Landau level. We identify interactions for which spin-singlet incompressible states occur at filling factors $\ensuremath{\nu}=2/3$, $4/5$, and $4/3$, and partially spin-polarized states at filling factors $3/4$ and $3/2$, where ``spin'' serves as a generic label for the two components. We study ground states, excitations, edge states, and entanglement spectrum for systems with up to 16 bosons and construct explicit trial wave functions to clarify the underlying physics. The composite fermion theory very accurately describes the ground states as well as excitations at $\ensuremath{\nu}=2/3$, $4/5$, and $3/4$, although it is less satisfactory for the $\ensuremath{\nu}=3/2$ state. For $\ensuremath{\nu}=4/3$ a ``non-Abelian spin-singlet'' state, which is the exact ground state of a three-body contact interaction, has been proposed to occur even for a two-body contact interaction; our trial wave functions are very accurate for the excitations of the three-body interaction, but they do not describe the excitations of the two-body interaction very well. Instead, we find that the $\ensuremath{\nu}=4/3$ state is more likely to be a spin-singlet state of reverse-flux-attached composite fermions at filling ${\ensuremath{\nu}}^{*}=4$. We also consider incompressible states at integral filling factors $\ensuremath{\nu}=1$ and 2. The incompressible state at $\ensuremath{\nu}=1$ is shown to be well described by the parton-based Jain spin-singlet wave function, and the incompressible state at $\ensuremath{\nu}=2$ as the spin-singlet state of reverse-flux-attached composite fermions at ${\ensuremath{\nu}}^{*}=2$, which provides an example of the bosonic integer topological phase.

arXiv:1304.7553•Physical Review B•Jun 24, 2013•79 citations
We put forward a group-theoretical approach to describe the Hilbert space of MMM fermions, with NNN components (spin, layer, valley, sub-lattice, etc), per Landau site in the lowest Landau level, at fractional filling factor ν=M/λ\nu=M/\lambdaν=M/λ. The Hilbert space turns out to be the carrier space of a unitary irreducible representation of U(N)U(N)U(N) described by Gelfand-Tsetlin patterns corresponding to rectangular Young tableaux of MMM rows and λ\lambdaλ columns. Here λ\lambdaλ represents the number of magnetic flux quanta vortex lines penetrating the sample, which must be odd for antisymmetry requirements. The construction can be considered an extension of Haldane's U(2)U(2)U(2) picture for spherical geometry to a more involved U(N)U(N)U(N) description for Grassmannian geometry. This construction relies on the boson realization of isospin U(N)U(N)U(N) operators as bilinear products of creation and annihilation operators of magnetic flux quanta, with an embedding of the Hilbert space into Fock space. Coherent state excitations above a spontaneously chosen ground state are labeled by points ZZZ on the Grassmannian coset U(N)/U(M)×U(N−M)U(N)/U(M)\times U(N-M)U(N)/U(M)×U(N−M).
Quantization on Grassmann phase spaces U(N)/U(M)×U(N−M)U(N)/U(M)\times U(N-M)U(N)/U(M)×U(N−M) and rectangular M×fM\times fM×f Young tableaux with applications to U(N)U(N)U(N) quantum Hall ferromagnets at filling factor ν=M/f\nu=M/fν=M/f
We develop a group-theoretical approach to describe NNN-component composite bosons as planar electrons attached to an odd number fff of Chern-Simons flux quanta. This picture arises when writing the Coulomb exchange interaction as a quantum Hall ferromagnet in terms of collective U(N)U(N)U(N)-spin operators. A spontaneously chosen ground state of MMM electrons per Landau site breaks the symmetry from U(N)U(N)U(N) to the stability subgroup U(M)×U(N−M)U(M)\times U(N-M)U(M)×U(N−M), so that coherent state excitations are labeled by points on the Grassmannian coset U(N)/U(M)×U(N−M)U(N)/U(M)\times U(N-M)U(N)/U(M)×U(N−M). The quantization of this Grassmann phase space corresponds to the carrier Hilbert space of unitary irreducible representations of U(N)U(N)U(N) described by rectangular Young tableaux of MMM rows and fff columns. We construct an embedding of the Hilbert space into Fock space by using a Schwinger realization of collective U(N)U(N)U(N)-spin operators as bilinear products of composite boson operators. We also build a system of Grassmann coherent states and discuss the classical limit of U(N)U(N)U(N) quantum Hall ferromagnets in terms of nonlinear sigma models on Grasmannians.
Using the Lieb–Mattis ordering theorem of electronic energy levels, we identify the Hilbert space of the low energy sector of U(N) quantum Hall/Heisenberg ferromagnets at filling factor M for L Landau/lattice sites with the carrier space of irreducible representations of U(N) described by rectangular Young tableaux of M rows and L columns, and associated with Grassmannian phase spaces U(N)/U(M)×U(N−M). We embed this N-component fermion mixture in Fock space through a Schwinger–Jordan (boson and fermion) representation of U(N)-spin operators. We provide different realizations of basis vectors using Young diagrams, Gelfand–Tsetlin patterns and Fock states (for an electron/flux occupation number in the fermionic/bosonic representation). U(N)-spin operator matrix elements in the Gelfand–Tsetlin basis are explicitly given. Coherent state excitations above the ground state are computed and labeled by complex (N−M)×M matrix points Z on the Grassmannian phase space. They adopt the form of a U(N) displaced/rotated highest-weight vector, or a multinomial Bose–Einstein condensate in the flux occupation number representation. Replacing U(N)-spin operators by their expectation values in a Grassmannian coherent state allows for a semi-classical treatment of the low energy (long wavelength) U(N)-spin-wave coherent excitations (skyrmions) of U(N) quantum Hall ferromagnets in terms of Grasmannian nonlinear sigma models.
Bilayer quantum Hall (BLQH) systems, which underlie a U (4) symmetry, display unique quantum coherence effects. We study coherent states (CS) on the complex Grassmannian , orthonormal basis, U (4) generators and their matrix elements in the reproducing kernel Hilbert space of analytic square-integrable holomorphic functions on , which carries a unitary irreducible representation of U (4) with index . A many-body representation of the previous construction is introduced through an oscillator realization of the U (4) Lie algebra generators in terms of eight boson operators. This particle picture allows us to make a physical interpretation of our abstract mathematical construction in the BLQH jargon. In particular, the index λ is related to the number of flux quanta bound to a bi-fermion in the composite fermion picture of Jain for fractions of the filling factor ν = 2. The simpler, and better known, case of spin- s CS on the Riemann–Bloch sphere is also treated in parallel, of which Grassmannian -CS can be regarded as a generalized (matrix) version.
Using the Lieb–Mattis ordering theorem of electronic energy levels, we identify and construct the Hilbert space of the low energy sector of U(N) quantum Hall/Heisenberg ferromagnets at filling factor M for L Landau/lattice sites. The carrier Hilbert space of irreducible representations of U(N) is described by rectangular Young tableaux of M rows and L columns, and associated with Grassmannian phase spaces U(N)/U(M)×U(N-M). Replacing U(N)-spin operators by their expectation values in a Grassmannian coherent state allows for a semi-classical treatment of the low energy U(N)-spin-wave coherent excitations (skyrmions) of U(N) quantum Hall ferromagnets in terms of Grasmannian nonlinear sigma models.
Single particle basis functions for composite fermions are obtained from which many-composite fermion states confined to the lowest electronic Landau level can be constructed in the standard manner, i.e. by building Slater determinants. This representation enables a Monte Carlo study of systems containing a large number of composite fermions, yielding new quantitative and qualitative information. The ground state energy and the gaps to charged and neutral excitations are computed for a number of fractional quantum Hall effect (FQHE) states, earlier off-limits to a quantitative investigation. The ground state energies are estimated to be accurate to ~0.1% and the gaps at the level of a few percent. It is also shown that at Landau level fillings smaller than or equal to 1/9 the FQHE is unstable to a spontaneous creation of excitons of composite fermions. In addition, this approach provides new conceptual insight into the structure of the composite fermion wave functions, resolving in the affirmative the question of whether it is possible to motivate the composite fermion theory entirely within the lowest Landau level, without appealing to higher Landau levels.
We investigate the feasibility of many candidate quantum Hall states for two-component bosons in the lowest Landau level. We identify interactions for which spin-singlet incompressible states occur at filling factors $\ensuremath{\nu}=2/3$, 4/54/54/5, and 4/34/34/3, and partially spin-polarized states at filling factors 3/43/43/4 and 3/23/23/2, where ``spin'' serves as a generic label for the two components. We study ground states, excitations, edge states, and entanglement spectrum for systems with up to 16 bosons and construct explicit trial wave functions to clarify the underlying physics. The composite fermion theory very accurately describes the ground states as well as excitations at $\ensuremath{\nu}=2/3$, 4/54/54/5, and 3/43/43/4, although it is less satisfactory for the $\ensuremath{\nu}=3/2$ state. For $\ensuremath{\nu}=4/3$ a ``non-Abelian spin-singlet'' state, which is the exact ground state of a three-body contact interaction, has been proposed to occur even for a two-body contact interaction; our trial wave functions are very accurate for the excitations of the three-body interaction, but they do not describe the excitations of the two-body interaction very well. Instead, we find that the $\ensuremath{\nu}=4/3$ state is more likely to be a spin-singlet state of reverse-flux-attached composite fermions at filling ${\ensuremath{\nu}}^{*}=4$. We also consider incompressible states at integral filling factors $\ensuremath{\nu}=1$ and 2. The incompressible state at $\ensuremath{\nu}=1$ is shown to be well described by the parton-based Jain spin-singlet wave function, and the incompressible state at $\ensuremath{\nu}=2$ as the spin-singlet state of reverse-flux-attached composite fermions at ${\ensuremath{\nu}}^{*}=2$, which provides an example of the bosonic integer topological phase.