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Register a new userCoherent states on the Grassmannian $U(4)/U(2)^2$: Oscillator realization and bilayer fractional quantum Hall systems
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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 $mathbb G_2^4=U(4)/U(2)^2$, orthonormal basis, $U(4)$ generators and their matrix elements in the reproducing kernel Hilbert space $mathcal H_lambda(mathbb G_2^4)$ of analytic square-integrable holomorphic functions on $mathbb G_2^4$, which carries a unitary irreducible representation of $U(4)$ with index $lambdainmathbb N$. 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 for a physical interpretation of our abstract mathematical construction in the BLQH jargon. In particular, the index $lambda$ 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 $ u=2$. The simpler, and better known, case of spin-$s$ CS on the Riemann-Bloch sphere $mathbb{S}^2=U(2)/U(1)^2$ is also treated in parallel, of which Grassmannian $mathbb G_2^4$-CS can be regarded as a generalized (matrix) version.
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We analyze mathematical and physical properties of a previously introduced [J. Phys. A47, 115302 (2014)] 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 $ u=2$ are discussed, where Haldanes sphere picture is generalized to a Grassmannian picture. We also extend Wehrls definition of entropy from Glauber to Grassmannian CS and state a conjecture on the entropy lower bound.
We revise the unireps. of $U(2,2)$ describing conformal particles with continuous mass spectrum from a many-body perspective, which shows massive conformal particles as compounds of two correlated massless particles. The statistics of the compound (boson/fermion) depends on the helicity $h$ of the massless components (integer/half-integer). Coherent states (CS) of particle-hole pairs (excitons) are also explicitly constructed as the exponential action of exciton (non-canonical) creation operators on the ground state of unpaired particles. These CS are labeled by points $Z$ ($2times 2$ complex matrices) on the Cartan-Bergman domain $mathbb D_4=U(2,2)/U(2)^2$, and constitute a generalized (matrix) version of Perelomov $U(1,1)$ coherent states labeled by points $z$ on the unit disk $mathbb D_1=U(1,1)/U(1)^2$. Firstly we follow a geometric approach to the construction of CS, orthonormal basis, $U(2,2)$ generators and their matrix elements and symbols in the reproducing kernel Hilbert space $mathcal H_lambda(mathbb D_4)$ of analytic square-integrable holomorphic functions on $mathbb D_4$, which carries a unitary irreducible representation of $U(2,2)$ with index $lambdainmathbb N$ (the conformal or scale dimension). Then we introduce a many-body representation of the previous construction through an oscillator realization of the $U(2,2)$ Lie algebra generators in terms of eight boson operators with constraints. This particle picture allows us for a physical interpretation of our abstract mathematical construction in the many-body jargon. In particular, the index $lambda$ is related to the number $2(lambda-2)$ of unpaired quanta and to the helicity $h=(lambda-2)/2$ of each massless particle forming the massive compound.
We have measured the Hall-plateau width and the activation energy of the bilayer quantum Hall (BLQH) states at the Landau-level filling factor $ u=1$ and 2 by tilting the sample and simultaneously changing the electron density in each quantum well. The phase transition between the commensurate and incommensurate states are confirmed at $ u =1$ and discovered at $ u =2$. In particular, three different $ u =2$ BLQH states are identified; the compound state, the coherent commensurate state, and the coherent incommensurate state.
We investigate the finite frequency noise of a quantum point contact at filling factor { u} = 5/2 using a weakly coupled resonant LC circuit as a detector. We show how one could spectroscopically address the fractional charged excitations inspecting separately their charge and scaling dimensions. We thus compare the behaviour of the Pfaffian and the anti-Pfaffian non-Abelian edge states models in order to give possible experimental signatures to identify the appropriate model for this fractional quantum Hall states. Finally we investigate how the temperature of the LC resonant circuit can be used in order to enhance the sensibility of the measurement scheme.
We study the nature of the u=5/2 quantum Hall state in wide quantum wells under the mixing of electronic subbands and Landau levels. We introduce a general method to analyze the Moore-Read Pfaffian state and its particle-hole conjugate, the anti-Pfaffian, under periodic boundary conditions in a quartered Brillouin zone scheme containing both even and odd numbers of electrons. We examine the rotational quantum numbers on the torus, and show spontaneous breaking of the particle-hole symmetry can be observed in finite-size systems. In the presence of electronic-subband and Landau-level mixing the particle-hole symmetry is broken in such a way that the anti-Pfaffian is unambiguously favored, and becomes more robust in the vicinity of a transition to the compressible phase, in agreement with recent experiments.
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