Noncommutative geometry governs the physics of quantum Hall (QH) effects. We introduce the Weyl ordering of the second quantized density operator to explore the dynamics of electrons in the lowest Landau level. We analyze QH systems made of N -component electrons at the integer filling factor ν = k ≤ N . The basic algebra is the SU(N)-extended W∞. A specific feature is that noncommutative geometry leads to a spontaneous development of SU(N) quantum coherence by generating the exchange Coulomb interaction. The effective Hamiltonian is the Grassmannian G N,k sigma model, and the dynamical field is the Grassmannian G N,k field, describing k(N − k) complex Goldstone modes and one kind of topological solitons (Grassmannian solitons).

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The Abelian Higgs model with a fermionic current nonconserved due to an anomaly is considered in 1 + 1 dimensions. The one-loop expression for the rate of the fermionic-number nonconservation at high temperatures is obtained analytically for arbitrary values of the Lagrangian parameters.The non-Abelian nature of the standard electroweak theory remains a subject of intense interest. The existence of the 9 vacuum in this SU(2) x U ( 1 ) gauge theory leads to the nonconservation of leptonic and baryonic numbers which is, however, negligibly small at zero temperature. Nevertheless, as was pointed out in Ref. 2 in matter at high temperatures which took place in the early Universe anomalous nonconservation of fermionic numbers is not suppressed. The relevant considerations are usually performed in the Ao=O gauge. There is a static energy barrier (SEB) between the classical vacua with different values of the Chern-Simons n~m b e r .~ At high temperatures the system has enough energy to pass through the SEB (Ref. 4) via classical thermodynamical fluctuations. At temperatures smaller than the height E, of the SEB the probability l-of the transitions over the barrier is small and may be evaluated in the semiclassical approximation l-=Aexp( -E,/T). Here the preexponential factor A is important.' Exact analytical evaluation of the ~r e e x~o n e n t i a l factor A in 3+ 1 dimensions is a serious So the semiclassical calculations in various tov models are v a l~a b l e .~-'~ In Ref. 10 the r q version of , -the Abelian Higgs model in 1 + 1 dimensions was shown to reproduce many essential features of the real case. It was solved analytically in the limit g '/h >> 1 (where g is gauge and h is scalar self-coupling constants) for integer values of the ratio g / m . In this paper we give an analytical solution for arbitrary values of the coupling constants g and h.The theory under consideration is defined by the Lagrangian of the form where @, W, and A, are scalar, spinor, and vector gauge fields, respectively. The particle spectrum contains vector and Higgs bosons with masses rn; "g 'c ' and rnh =2hc 2.The gauge-invariant fermionic current J, =-1Jy,ty is not conserved due to an anomaly:Nonconservation of the fermionic number is associated with the fluctuations of gauge fields which in the AoPO gauge change the value of the Chern-Simons number. The theory has a 9-vacuum structure. The classical vacua with different values of the Chern-Simons number are separated by SEB, the minimum height of which E, is nonzero. A statistical system built in the vicinity of one such vacua is slightly unstable with respect to penetration through the SEB. The decay rate of such a state coincides with the rate of anomalous fermionic-number nonconservation in hot plasma. l 2 In the one-loop approximation it is related to the imaginary part of thk free energy y:5,12 the coefficient K is to be defined later. The relation (3) is useful because there is a regular representation for the free energy in terms of the Matsubara functional integral. The functio...

At the filling factor ν=2, the bilayer quantum Hall system has three phases, the spin-ferromagnet phase, the spin singlet phase and the canted antiferromagnet (CAF) phase, depending on the relative strength between the Zeeman energy and interlayer tunneling energy. We present a systematic method to derive the effective Hamiltonian for the Goldstone modes in these three phases. We then investigate the dispersion relations and the coherence lengths of the Goldstone modes. To explore a possible emergence of the interlayer phase coherence, we analyze the dispersion relations in the zero tunneling energy limit. We find one gapless mode with the linear dispersion relation in the CAF phase. PACS. 73.21.-b Collective excitations in nanoscale systems -73.43.Nq Phase transitions quantum Hall effects

We present a microscopic theory of the Hall current in the bilayer quantum Hall system on the basis of noncommutative geometry. By analyzing the Heisenberg equation of motion and the continuity equation of charge, we demonstrate the emergence of the phase current in a system where the interlayer phase coherence develops spontaneously. The phase current arranges itself to minimize the total energy of the system, as induces certain anomalous behaviors in the Hall current in the counterflow geometry and also in the drag experiment. They explain the recent experimental data for anomalous Hall resistances due to Kellogg et al. [M.