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, E. Pérez-Romero

Published Apr 15, 2019• 0 citations

Abstract

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.

Related papers

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

Manuel Calixto, E. Pérez-Romero

Apr 15, 20190 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

arXiv:1904.06932SymmetryApr 24, 20226 citations

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

Manuel Calixto, Alberto Mayorgas, Julio Guerrero

SciPost Physics ProceedingsNov 23, 20230 citations

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

Manuel Calixto, E. Pérez-Romero

arXiv:1312.3148Journal of Physics A Mathematical and TheoreticalFeb 27, 20149 citations

Evolution between quantum Hall and conducting phases: Simple models and some results

Zhihuan Dong, T. Senthil

arXiv:2107.06911Physical review. B./Physical review. BFeb 1, 20226 citations

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

Ying-Hai Wu, Jainendra K. Jain

arXiv:1304.7553Physical Review BJun 24, 201379 citations

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 $\nu=M/f$ 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, E. Pérez-Romero

Published Apr 15, 2019• 0 citations

OpenAlex

Abstract

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.

Related papers

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

Manuel Calixto, E. Pérez-Romero

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 $U(N)/U(M)\times U(N-M)$ .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 $U(N)/U(M)\times U(N-M)$.

Apr 15, 20190 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.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.

arXiv:1904.06932SymmetryApr 24, 20226 citations

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

Manuel Calixto, Alberto Mayorgas, Julio Guerrero

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.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 ProceedingsNov 23, 20230 citations

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.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.

arXiv:1312.3148Journal of Physics A Mathematical and TheoreticalFeb 27, 20149 citations

Evolution between quantum Hall and conducting phases: Simple models and some results

Zhihuan Dong, T. Senthil

Quantum many-particle systems in which the kinetic energy, strong correlations, and band topology are all-important pose an interesting and topical challenge. Here we introduce and study particularly simple models where all of these elements are present. We consider interacting quantum particles in two dimensions in a strong magnetic field such that the Hilbert space is restricted to the lowest Landau level (LLL). This is the familiar quantum Hall regime with rich physics determined by particle filling and statistics. A periodic potential with a unit cell enclosing one flux quantum broadens the LLL into a Chern band with a finite bandwidth. The states obtained in the quantum Hall regime evolve into conducting states in the limit of large bandwidth. We study this evolution in detail for the specific case of bosons at filling factor $\ensuremath{\nu}=1$. In the quantum Hall regime, the ground state at this filling is a gapped quantum hall state (the ``bosonic Pfaffian''), which may be viewed as descending from a (bosonic) composite Fermi liquid. At large bandwidth, the ground state is a bosonic superfluid. We show how both phases and their evolution can be described within a single theoretical framework based on a LLL composite fermion construction. Building on our previous work on the bosonic composite Fermi liquid, we show that the evolution into the superfluid can be usefully described by a noncommutative quantum field theory in a periodic potential.Quantum many-particle systems in which the kinetic energy, strong correlations, and band topology are all-important pose an interesting and topical challenge. Here we introduce and study particularly simple models where all of these elements are present. We consider interacting quantum particles in two dimensions in a strong magnetic field such that the Hilbert space is restricted to the lowest Landau level (LLL). This is the familiar quantum Hall regime with rich physics determined by particle filling and statistics. A periodic potential with a unit cell enclosing one flux quantum broadens the LLL into a Chern band with a finite bandwidth. The states obtained in the quantum Hall regime evolve into conducting states in the limit of large bandwidth. We study this evolution in detail for the specific case of bosons at filling factor $\ensuremath{\nu}=1$. In the quantum Hall regime, the ground state at this filling is a gapped quantum hall state (the ``bosonic Pfaffian''), which may be viewed as descending from a (bosonic) composite Fermi liquid. At large bandwidth, the ground state is a bosonic superfluid. We show how both phases and their evolution can be described within a single theoretical framework based on a LLL composite fermion construction. Building on our previous work on the bosonic composite Fermi liquid, we show that the evolution into the superfluid can be usefully described by a noncommutative quantum field theory in a periodic potential.

arXiv:2107.06911Physical review. B./Physical review. BFeb 1, 20226 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.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:1304.7553Physical Review BJun 24, 201379 citations

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

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.

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)

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.