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arXiv:cond-mat/0204309 · Physical review. B, Condensed matter

Green’s function of the half-filled Landau level Chern-Simons theory in the temporal gauge

J. Dietel

Published Nov 27, 2002• 0 citations
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Abstract

We study the Green's function of the $\ensuremath{\nu}=1/2$ Chern-Simons system in the temporal (Weyl) gauge. We derive the Chern-Simons path integral in the temporal gauge. In order to do this, we gauge transform the path integral in the Coulomb gauge which represents the partition function of the correct normal ordered Chern-Simons Hamiltonian. We calculate the self-energy of this path integral in the random-phase approximation (RPA) for temperature $T=0.$ This self-energy does not have the divergence with the logarithm of the area, which is known to imply the vanishing of the exact Green's function in the Coulomb gauge for an infinite area. By Chern-Simons retransforming the path integral representing the Green's function in the temporal gauge we calculate explicitly the exact Green's function by neglecting the interaction between the electrons, getting a finite value. Furthermore, we give arguments that the Green's function of the interacting system is also finite. The nonvanishing of the Green's function for infinite area is due to a dynamical creation of the phase factors linking the created and annihilated particles with the particles in the ground state. The absence of these phase factors is the reason for the vanishing of the Green's function in the Coulomb gauge.

DOI
10.1103/physrevb.66.195323

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We study the Green's function of the \ensuremathν=1/2\ensuremath{\nu}=1/2\ensuremathν=1/2 Chern-Simons system in the temporal (Weyl) gauge. We derive the Chern-Simons path integral in the temporal gauge. In order to do this, we gauge transform the path integral in the Coulomb gauge which represents the partition function of the correct normal ordered Chern-Simons Hamiltonian. We calculate the self-energy of this path integral in the random-phase approximation (RPA) for temperature T=0.T=0.T=0. This self-energy does not have the divergence with the logarithm of the area, which is known to imply the vanishing of the exact Green's function in the Coulomb gauge for an infinite area. By Chern-Simons retransforming the path integral representing the Green's function in the temporal gauge we calculate explicitly the exact Green's function by neglecting the interaction between the electrons, getting a finite value. Furthermore, we give arguments that the Green's function of the interacting system is also finite. The nonvanishing of the Green's function for infinite area is due to a dynamical creation of the phase factors linking the created and annihilated particles with the particles in the ground state. The absence of these phase factors is the reason for the vanishing of the Green's function in the Coulomb gauge.
The author considers the quantum mechanics coupled to the gauge field which has the Chern-Simons term as the action. The generalized Hamiltonian formalism of the system is constructed and canonical and path-integral quantizations are performed. In solving the Gauss law constraint, the gauge field is represented by a multivalued function and the charge density operator of the particle. The gauge potential is singular and also induces a singular magnetic field through the magnetic flux is finite. By a phase transformation, the singular gauge field is eliminated. Then the transformed wavefunction absorbing the gauge field induces a novel phase factor under 2 pi -rotation in space. Thus the author shows that the transformed wavefunction describes exotic spin state. The extension of these analyses to the relativistic case is also presented.
We have previously studied the phase structure at finite temperatures of the Chern-Simons (CS) gauge theory coupled with fermions by using lattice gauge theory. In this paper, we formulate the “chargeon-fluxon ” representation of electrons and use it to reinvestigate the phenomenon of particle-flux separation (PFS) of electrons in the halffilled Landau level. We start with a lattice system of fermions interacting with a CS gauge field, and introduce two slave operators named chargeon and fluxon that carry the CS charge and flux, respectively. The original fermion, the composite fermion of Jain, is a composite of a chargeon and a fluxon. We further rewrite the model by introducing an auxiliary link field, the phase of which behaves as a gauge field gluing chargeons and fluxons. Then we study a confinement-deconfinement transition of that gauge field by using the theory of separation phenomena as in the previous paper. The residual four-fermi interactions play an important role to determine the critical temperature TPFS, below which the PFS takes place. The new representation
We consider electrons in two dimensions in a strong magnetic field at half filling of the lowest Landau level using the Chern-Simons approach. Starting from a lattice Hamiltonian for the electrons, we derive a path integral (PI) formulation for the composite fermions (CF) which respects the order of the operators. We use a time lattice with intermediate times in order to have a PI expressed in density fluctuations. This formulation reveals that there is no infrared (IR) singularity in the grand-canonical potential in lowest order perturbation theory.
The Chern-Simons bosonization with $\mathrm{U}(1)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(2)$ gauge field is applied to the two-dimensional $t\ensuremath{-}J$ model in the limit $t\ensuremath{\gg}J,$ to study the normal-state properties of underdoped cuprate superconductors. We prove the existence of an upper bound on the partition function for holons in a spinon background, and we find the optimal spinon configuration saturating the upper bound on average---a coexisting flux phase and s+ids+ids+id-like resonating-valence-bond state. After neglecting the feedback of holon fluctuations on the U(1) field B and spinon fluctuations on the SU(2) field VVV, the holon field is a fermion and the spinon field is a hard-core boson. Within this approximation we show that the B field produces a \ensuremath{\pi} flux phase for the holons, converting them into Dirac-like fermions, while the VVV field, taking into account the feedback of holons produces a gap for the spinons vanishing in the zero-doping limit. The nonlinear-\ensuremath{\sigma} model with a mass term describes the crossover from the short-ranged antiferromagnetic (AF) state in doped samples to long-range AF order in reference compounds. Moreover, we derive a low-energy effective action in terms of spinons, holons and a self-generated U(1) gauge field. Neglecting the gauge fluctuations, the holons are described by the Fermi-liquid theory with a Fermi surface consisting of four ``half-pockets'' centered at (\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2,\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2) and one reproduces the results for the electron spectral function obtained in the mean-field approximation, in agreement with the photoemission data on underdoped cuprates. The gauge fluctuations are not confining due to coupling to holons, but nevertheless yield an attractive interaction between spinons and holons leading to a bound state with electron quantum numbers. The renormalization effects due to gauge fluctuations give rise to non-Fermi-liquid behavior for the composite electron, in certain temperature range showing the linear in T resistivity. This formalism provides a new interpretation of the spin gap in the underdoped superconductors (mainly due to the short-ranged AF order) and predicts that the minimal gap for the physical electron is proportional to the square root of the doping concentration. Therefore the gap does not vanish in any direction. All these predictions can be checked explicitly in experiment.
We calculated the exact self energy and the exact Green's function for the Bethe lattice in the hopping model using the Dyson's equation and from the fourth order of perturbation theory. The self energy is found from processes where an electron hops to its neighbors and includes terms where the particle hops twice to a next nearest neighbor (NNN) and then hops back. carried out by counting all non-self intersecting paths, leading to an implicit equation for the local Green function. By integrating out branches of the Bethe lattice, the same equation is obtained from a path integral approach for the partition function. In this study, the electron Green's function will be used to calculate the exact Green's function and self energy for the Bethe lattice in the hopping model. Though it is not one of the goals of our study here, it is pertinent to point out that the self energy can be used to obtain the vertex function which contains all the information on the kinetic and interaction behaviour of the particles in the system. Interestingly the vertex function can be decoupled into the charge vertex function and spin vertex function, thereby enabling the studying of either non-spin ordering systems or spin ordering system independently.