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.
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.
Quantization on Grassmann phase spaces U(N)/U(M)×U(N−M) and rectangular M×f Young tableaux with applications to U(N) quantum Hall ferromagnets at filling factor ν=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)×U(N−M), so that coherent state excitations are labeled by points on the Grassmannian coset 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) 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 ν=M/λ. 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 λ columns. Here λ 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)×U(N−M).
The Lieb-Mattis theorem orders the lowest-energy states of total spin s of a system of P interacting fermions. We generalize these predictions to fermionic mixtures of P particles with more than N=2 spinor components or species in the thermodynamic limit P→∞. The lowest-energy state inside each permutation symmetry sector h, arising in the P-fold tensor product decomposition, is well approximated by a U(N) coherent (quasiclassical variational) state, especially in the limit P→∞. In particular, the ground state of the system belongs the most symmetric (dominant Young tableau h_{0}) configuration. We exemplify our construction with the N=3 level Lipkin-Meshkov-Glick model, with a previous motivation on pairing correlations and U(N)-invariant quantum Hall ferromagnets. In the limit P→∞, each lowest-energy state within each permutation symmetry sector h undergoes a quantum phase transition for a critical value λ_{c}(h) of the exchange coupling constant λ, depending on h. This generalizes standard quantum phase transitions and their phase diagrams corresponding to the ground state belonging to the most symmetric sector h_{0}.
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.
Quantum Hall systems have a one-body energy spectrum consisting of dispersion-less Landau levels. Electron-electron interactions thus dominate in partially filled Landau levels, which exhibit a myriad of strongly correlated phases such as quantum hall ferromagnets and fractional quantum Hall phases. We study two examples of these phenomena. In the first project, we explore the ground state of a system with an interface between two semi-infinite regions with fillings ν= 4 and ν= 3 respectively. The width of the interface can be controlled by varying the background potential, which provides an additional tuning parameter. For a certain range of interaction strengths, the ν= 4 bulk is unpolarized whereas the ν= 3 bulk is fully polarized. In the parameter space spanned by the interaction strength and width of the interface, we find two phases at the interface. Phase A has spin as a good quantum number, and the long-wavelength spin edge excitations are gapped. In phase B, spin rotation symmetry is spontaneously broken at the mean-field level. Using symmetry arguments we find the effective theory near the interface of phase B. This effective theory is known to have gapless long-wavelength spin excitations. In the second project, we study the ground state of a tunnel-decoupled double-layer graphene system when both layers are undoped. We find a simple Hamiltonian in the continuum limit from symmetries of the system. Using the Hartree-Fock approximation we find a state with inter-layer coherence with broken layer U(1) symmetry. This phase becomes magnetized in presence of a non-zero Zeeman field. A first-order phase transition can be driven from the ferromagnetic phase to the magnetized inter-layer coherent phase by increasing the Zeeman field. We predict the number of gapless modes in the bulk.