We reconsider the theory of the half-filled lowest Landau level using the Chern-Simons formulation and study the grand-canonical potential in the random-phase approximation (RPA). Calculating the unperturbed response functions for current-and charge-density exactly, without any expansion with respect to frequency or wave vector, we find that the integral for the ground-state energy converges rapidly (algebraically) at large wave vectors k, but exhibits a logarithmic divergence at small k. This divergence originates in the k −2 singularity of the Chern-Simons interaction and it is already present in lowest-order perturbation theory. A similar divergence appears in the chemical potential. Beyond the RPA, we identify diagrams for the grand-canonical potential (ladder-type, maximally crossed, or a combination of both) which diverge with powers of the logarithm. We expand our result for the RPA ground-state energy in the strength of the Coulomb interaction. The linear term is finite and its value compares well with numerical simulations of interacting electrons in the lowest Landau level. 71.10.Pm, 73.40.Hm
The r61e of oscillating vertices usually neglected in the Berezinskii diagram technique is analyzed in connection with a random potential, whose spatial correlation is generated by a Markov chain. It appears that the consideration of some of the oscillating vertices is necessary so that the theory can remark the spatial correlation. Correlation mainly leads to an increase of the localization length in comparison with an uncorrelated potential. However, there is a region of the parameter, where the localization length decreases. The Berezinskii diagram technique is used in a modified form.Die Rolle der oszillierenden Vertizes, die ublicherweise in der Berezinskii-Diagrammtechnik vernachlassigt werden, wird im Zusammenhang mit einem zufiilligen Potential, dessen riinmliche Korrelation durch eine Markovkette erzeugt wird, analysiert. Es zeigt sich, daR die Beriicksichtigung einiger oszillierender Vertizes notwendig ist, damit die Theorie die raumliche Korrelation bemerken kann. Korrelation fiihrt meistens zu einer Vergrofierung der Lokalisierungslange gegeniiber dem unkorrelierten Fall. Es gibt jedoch ein Parametergebiet, in dem die Lokalisierungslange verkltinert wird. Die Berezinskii-Diagrammtechnik wird in einer modifizierten Form verwendet.
Functional integral representations are constructed for Fermions with spin 1/2, in which the fields satisfy directly by construction the constraints (e.g., exclusion of double occupancy of a site) appearing in recent models in the theory of high-temperature superconductivity. Thus, the enforcement of the constraints by delta functions in the integration measure is avoided. Perelomov's concept of generalized coherent states is used. However, in constructing such representations, exponential functions of linear combinations of operators (which are difficult to disentangle) are avoided, as is the construction and reduction of the invariant measure. Instead, an ansatz is used for the resolution of the unity operator. This approach aIso provides more freedom in choosing the appropriate fields. Several new and simple representations with only few elementary fields are given. The representation already used by Wiegmann is recovered. In this case and in any other cases the integration measure is explicitly given. In all these representations, the original Fermi operators are substituted by the product of a spin independeiit GraDmann field and a spin dependent bosonic (complex) field in accordance with the physical idea of separation of charge and spin degrees of freedom. It is further shown how a change in the integration measure eliminates also zero occupancy (the case of the Heisenberg antiferromagnet). The absence of an explicit delta function constraint in the functional integral is reflected in a special form of the kinetic part of the action. The considered representations are compared with that of the slave boson method. Funktionalintegral-Darstellungen
On the basis of a previously derived general expression the localization length of an electron moving along two weakly coupled chains is analytically calculated. I n the weak scattering limit an explicit equation is derived for the asymptotic probability density of the five parameters characterizing two matrices e and x which determine the localization length. This equation is solved in the limit of ''strong'' coupling (It11 . ZfAL/(ltlll sink) > 1, although I t l l / l t l l l < 1 ) . The localization length is Zf,$ = 1.7756ZfA2. It is shown that also for arbitrary number of chains and arbitrary dimension of the cross-section the ratio of the strength of disorder and the chain coupling is the essential parameter.Auf der Grundlage eines fruher abgeleiteten allgemeinen dusdrucks wird die Lokalisierungsliinge eines Elektrons, das sich entlang zweier gekoppelter Faden bewegt, berechnet. I m Grenzfall schwacher Streuung wird eine explizite Gleichung fur die asymptotische Wahrscheinlichkeitsdichte der fiinf Parameter abgeleitet, die die Matrizen e und x charakterisieren. Diese Matrizen bestimmen die Lokalisierungslange. Die Gleichung wird im Grenzfall ,,starker" Kopplung
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