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

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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 ) = na 0 } (where a 0 is the lattice constant). The carbon atoms share their π -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 σ = , [ t α ( x ( n ) x ( n + 1))] bracketleftbig c σ ( n ) c σ ( 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 σ ( n ) and c σ ( n ) are fermion operators which create and destroy a π - electron with spin σ 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 α 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 α 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

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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 ). Here v F is the Fermi velocity. These considerations motivate the following way of writing the electron operators c σ ( n ) = e ip F n R σ ( n ) + e ip F n L σ ( n ) (2) Likewise, since the only processes in which phonons mix electrons near ± p F have momentum q 0 (forward scattering) or q 2 p F (backward scattering), it is also natural to split the phonon fields into two terms x ( n ) = δ ( n ) + e 2 ip F n Δ + ( n ) + e 2 ip F n Δ ( n ) (3) where p F = π 2 a 0 .
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582-2008-5 - Physics 582 Fall Semester 2008 Professor...

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