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Faculty of Physics, Lomonosov Moscow State University Advanced Quantum Field Theory: Mo dern Applications in HEP, Astro & Cond-Mat Instructor: O. Kharlanov

Handout (Fall 2015 term) ­ PRELIMINARY VERSION
1. Relying upon the well-known theorem from quantum mechanics on simultaneous diagonal^ izability of commuting Hermitian operators, show that the translation operator Td ed· 3 has a complete orthonormalized set of eigenfunctions in L2 (R ) that are also eigenfunctions ^ p2 ^ of the Hamiltonian H = 2m + V (x), V (x + d) = V (x). 2. Find the Brillouin zone for a 2D honeycomb lattice (lattice parameter d) and transform it into a hexagon. 3. Find the ground state energy of a helium atom using the Hartree­Fock­Roothaan equations based on (a) 1s orbitals, (b) orbitals with n = 1, . . . , nmax for some reasonable nmax ; (c) using the post-Hartree­Fock configuration interaction scheme. Compare the energies and the RMS charge radii electrons, in units of e).
1 2

(r )d3 x thus obtained ((x) is the total charge density of two

4. Find the density profile n(r) for an atom with the nucleus charge -Z e using the Thomas­ Fermi equation. Note: resort to spherically-symmetric solutions; find the form of the r and the r 0 asymptotics; estimate the unknown parameters in the asymptotic expressions numerically. 5. Find the leading coefficient g0 in the exchange-correlation energy for the dilute homogeneous electron gas of density n,
xc

= g0 n

1/3

+ O(n

1 /2

),

n 0.

6. Show that for a lattice with the primitive cell containing only one lattice site, the electron Green's function in the Tight-binding model is G


(x, y , t) -i FS| Tc (x, t)c (y , 0) |FS ^ ^
+

=

, -

e-it d 2

kBZ

eik(x-y Nsites

)

( k - F ) (F - -k ) + - k + i0 - -k - i0

,

where Nsites is the total number of the lattice sites (= the number of points in the Brillouin zone), k is the spectrum of 1-particle energies, and F is the Fermi energy. Note: prove and then use the identity
+

(t)e

-it

=
-

e-it id . 2 - + i0

7. Consider a strained graphene with the Hamiltonian ^ H=-
xA j =1,2,3

(t + tj )^ (x + j )a (x) + h.c., b ^

(1)

where 1,2,3 connect a site x of sublattice A with its nearest neighbours of sublattice B and |tj | t (in general, tj C). In the leading order in |tj |/t, find the shifted Fermi points (and justify that the Fermi surface is still degenerate, consisting of only two points ). Introducing the wavefunction (x) containing the low-energy degrees of freedom, find the effective wave equation for it. 1


8. Consider a graphene sheet with a small Kekul´ distortion of the lattice, which corresponds e to Q k+ - k- , tj tj (x) = ei(Q·x+k+ ·j ) + ei(-Q·x+k- ·j ) 0, in the above Hamiltonian (1). Find the modification of the effective Dirac equation resulting from such a perturbation. 9. Show that the rotation about the z axis normal to the graphene plane results in the transformation of the effective wavefunction ^ (x, y ) = ei
3

/2

^ (x cos - y sin , x sin + y cos ),

^ where is the rotation angle. The definition of the spinor field (x, y ) is as follows: its Fourier image with the momentum p contains the a/b operators corresponding to the momentum states k± + p near the two Fermi points k± of the rotated graphene. 10. Find the one-particle eigenstates and their energies for the -band electrons in graphene in the external magnetic field B = B ez orthogonal to the graphene plane (neglecting the Zeeman interaction µB B ). Use the potential A(x) = eB xey . ¯ 11. In the = 1 gauge, integrate out the photon field Aµ in the partition function Z [ , ] ¯ for the electrons in graphene ( (x), (x) are the sources conjugate to the spinor fields ¯ (x), (x), respectively). As a result, find the effective action for the interacting graphene and resort to its nonrelativistic limit ¯ Seff [ , ] = - ¯ d3 x (x) i( 0 0 + vF · e2 4 dtd2 xd2 y ) (x)

¯ ¯ (x) 0 (x) (y ) 0 (y ) . |x - y |

2