582-2008-5 - Physics 582, Fall Semester 2008 Professor...

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Unformatted text preview: Physics 582, Fall Semester 2008 Professor Eduardo Fradkin Problem Set No. 5: Due Date: November 9, 2008, 9:00 pm 1 Femions in one dimension In this problem we will consider an application of the Dirac theory to a problem in condensed matter physics: polyacetylene. Polyacetylene is a long polymer chain of the type ( CH ) n . The motion of the conduction electrons in polyacety- lene can be described by the following model due to Su, Schrieffer and Heeger. In this model, one considers a linear chain of carbon atoms ( C ) with classical equilibrium positions at the regularly spaced sites { x ( n ) | x ( n ) = n a } (where a is the lattice constant). The carbon atoms share their -orbital electrons, one per carbon atom. These electrons are allowed to hop from site to site. This hopping process is modulated by the lattice vibrations. Since the mass M of the atoms is much larger than the mass of the electrons (or, what is the same, the tunneling hopping (kinetic) energy t of the electrons is much larger than the kinetic energy of the atoms), we can give an approximate description by treat- ing the atoms classically while treating the electrons as quantum mechanical objects. The Hamiltonian , for a lattice with N (even) sites, is H = N 2 summationdisplay n = N 2 +1 summationdisplay = , [ t ( x ( n ) x ( n + 1))] bracketleftbig c ( n ) c ( n + 1) + h . c . bracketrightbig + N 2 summationdisplay n = N 2 +1 bracketleftbigg P 2 n 2 M + D 2 ( x ( n ) x ( n + 1)) 2 bracketrightbigg (1) where c ( n ) and c ( n ) are fermion operators which create and destroy a - electron with spin at the n th site of the chain, the x ( n )s are the coordi- nates of the carbon atoms (measured from their classical equilibrium positions), P ( n ) are their momenta, M is the carbon mass, D is the elastic constant, t is the electron hopping matrix element (for the undistorted lattice) and is the electron-phonon coupling constant. Polyacetylene has one electron per carbon atom and, hence, it is a half-filled system and there are N electrons in a chain with N atoms. The study of this problem is greatly simplified by considering a continuum version of the model. If the coupling constant is not too large, the only physical processes which are important are those which mix nearly degenerate states, i.e., the only electronic states that will matter are those within a narrow 1 band of width 2 E c centered at the Fermi energy E F = 0. In this limit, the single particle dispersion law becomes E ( p ) v F ( p p F ). These states are right moving electrons ( with p p F ) and left moving electrons ( with p F )....
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582-2008-5 - Physics 582, Fall Semester 2008 Professor...

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