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:1312.3148 · Journal of Physics A Mathematical and Theoretical

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

Manuel Calixto, E. Pérez-Romero

Published Feb 27, 2014• 9 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1751-8113/47/11/115302

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

•Apr 15, 2019•0 citations

Husimi function and phase-space analysis of bilayer quantum Hall systems at ν = 2/ λ

Manuel Calixto, Carlos Peón-Nieto

Abstract We propose localization measures in phase space of the ground state of bilayer quantum Hall systems at fractional filling factors , to characterize the three quantum phases (shortly denoted by spin, canted and ppin) for arbitrary -isospin λ . We use a coherent state (Bargmann) representation of quantum states, as holomorphic functions in the 8-dimensional Grassmannian phase-space (a higher-dimensional generalization of the Haldane’s 2-dimensional sphere ). We quantify the localization (inverse volume) of the ground state wave function in phase-space throughout the phase diagram (i.e. as a function of Zeeman, tunneling, layer distance, etc, control parameters) with the Husimi function second moment, a kind of inverse participation ratio that behaves as an order parameter. Then we visualize the different ground state structure in phase space of the three quantum phases, the canted phase displaying a much higher delocalization (a Schrödinger cat structure) than the spin and ppin phases, where the ground state is highly coherent. We find a good agreement between analytic (variational) and numeric diagonalization results.

arXiv:1706.04793•Journal of Statistical Mechanics Theory and Experiment•May 1, 2018•7 citations

Some properties of Grassmannian $U(4)/U{(2)}^{2}$ coherent states and an entropic conjecture

Manuel Calixto, Emilio Pérez-Romero

We analyze mathematical and physical properties of a previously introduced (Calixto and Pérez-Romero 2014 J. Phys. A: Math. Theor. 47 023027 ) family of U (4) coherent states (CS). They constitute a matrix version of standard spin U (2) CS when we add an extra (pseudospin) dichotomous degree of freedom: layer, sublattice, two-well, nucleon, etc. Applications to bilayer quantum Hall systems at fractions of filling factor are discussed, where Haldane’s sphere picture is generalized to a Grassmannian picture. We also extend Wehrl’s definition of entropy from Glauber to Grassmannian CS and state a conjecture on the entropy lower bound.

arXiv:1506.01541•Journal of Physics A Mathematical and Theoretical•Nov 18, 2015•5 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.

arXiv:1904.06932•Symmetry•Apr 24, 2022•6 citations

Interlayer coherence and entanglement in bilayer quantum Hall states at filling factorν= 2/λ

Manuel Calixto, E Pérez-Romero

We study coherence and entanglement properties of the state space of a composite bi-fermion (two electrons pierced by λ magnetic flux lines) at one Landau site of a bilayer quantum Hall system. In particular, interlayer imbalance and entanglement (and its fluctuations) are analyzed for a set of U(4) coherent (quasiclassical) states generalizing the standard pseudospin U(2) coherent states for the spin-frozen case. The interplay between spin and pseudospin degrees of freedom opens new possibilities with regard to the spin-frozen case. Actually, spin degrees of freedom make interlayer entanglement more effective and robust under perturbations than in the spin-frozen situation, mainly for a large number of flux quanta λ. Interlayer entanglement of an equilibrium thermal state and its dependence with temperature and bias voltage is also studied for a pseudo-Zeeman interaction.

arXiv:1403.2443•Journal of Physics Condensed Matter•Oct 29, 2014•6 citations

Interlayer coherence and entanglement in bilayer quantum Hall states at $\nu=2/\lambda$

Manuel Calixto, E. Pérez-Romero

We study coherence and entanglement properties of the state space of a composite bi-fermion (two electrons pierced by $\lambda$ magnetic flux lines) at one Landau site of a bilayer quantum Hall system. In particular, interlayer imbalance and entanglement (and its fluctuations) are analyzed for a set of $U(4)$ coherent (\emph{quasiclassical}) states generalizing the standard pseudospin $U(2)$ coherent states for the spin-frozen case. The interplay between spin and pseudospin degrees of freedom opens new possibilities with regard to the spin-frozen case. Actually, spin degrees of freedom make interlayer entanglement more effective and robust under perturbations than in the spin-frozen situation, mainly for a large number of flux quanta $\lambda$.

•Mar 11, 2014•0 citations
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.
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).
Abstract We propose localization measures in phase space of the ground state of bilayer quantum Hall systems at fractional filling factors , to characterize the three quantum phases (shortly denoted by spin, canted and ppin) for arbitrary -isospin λ . We use a coherent state (Bargmann) representation of quantum states, as holomorphic functions in the 8-dimensional Grassmannian phase-space (a higher-dimensional generalization of the Haldane’s 2-dimensional sphere ). We quantify the localization (inverse volume) of the ground state wave function in phase-space throughout the phase diagram (i.e. as a function of Zeeman, tunneling, layer distance, etc, control parameters) with the Husimi function second moment, a kind of inverse participation ratio that behaves as an order parameter. Then we visualize the different ground state structure in phase space of the three quantum phases, the canted phase displaying a much higher delocalization (a Schrödinger cat structure) than the spin and ppin phases, where the ground state is highly coherent. We find a good agreement between analytic (variational) and numeric diagonalization results.
Some properties of Grassmannian U(4)/U(2)2U(4)/U{(2)}^{2}U(4)/U(2)2 coherent states and an entropic conjecture
We analyze mathematical and physical properties of a previously introduced (Calixto and Pérez-Romero 2014 J. Phys. A: Math. Theor. 47 023027 ) family of U (4) coherent states (CS). They constitute a matrix version of standard spin U (2) CS when we add an extra (pseudospin) dichotomous degree of freedom: layer, sublattice, two-well, nucleon, etc. Applications to bilayer quantum Hall systems at fractions of filling factor are discussed, where Haldane’s sphere picture is generalized to a Grassmannian picture. We also extend Wehrl’s definition of entropy from Glauber to Grassmannian CS and state a conjecture on the entropy lower bound.
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.
We study coherence and entanglement properties of the state space of a composite bi-fermion (two electrons pierced by λ magnetic flux lines) at one Landau site of a bilayer quantum Hall system. In particular, interlayer imbalance and entanglement (and its fluctuations) are analyzed for a set of U(4) coherent (quasiclassical) states generalizing the standard pseudospin U(2) coherent states for the spin-frozen case. The interplay between spin and pseudospin degrees of freedom opens new possibilities with regard to the spin-frozen case. Actually, spin degrees of freedom make interlayer entanglement more effective and robust under perturbations than in the spin-frozen situation, mainly for a large number of flux quanta λ. Interlayer entanglement of an equilibrium thermal state and its dependence with temperature and bias voltage is also studied for a pseudo-Zeeman interaction.
Interlayer coherence and entanglement in bilayer quantum Hall states at ν=2/λ\nu=2/\lambdaν=2/λ
We study coherence and entanglement properties of the state space of a composite bi-fermion (two electrons pierced by λ\lambdaλ magnetic flux lines) at one Landau site of a bilayer quantum Hall system. In particular, interlayer imbalance and entanglement (and its fluctuations) are analyzed for a set of U(4)U(4)U(4) coherent (\emph{quasiclassical}) states generalizing the standard pseudospin U(2)U(2)U(2) coherent states for the spin-frozen case. The interplay between spin and pseudospin degrees of freedom opens new possibilities with regard to the spin-frozen case. Actually, spin degrees of freedom make interlayer entanglement more effective and robust under perturbations than in the spin-frozen situation, mainly for a large number of flux quanta λ\lambdaλ.