[PDF] Feldtheorie der Kondensierten Materie RG und Quantenkritikalität

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Karlsruher Institut f

ur Technologie Institut fur Theorie der Kondensierten Materie Feldtheorie der Kondensierten Materie, RG und Quantenkritikalit at SS 2013

Prof. Dr. J. SchmalianBlatt 01

Dr. P. OrthBesprechung 23.04.20131. Ising4-eld theory(10 + 20 + 10 + 10 + 10 + 10 = 70 Punkte) In this exercise we derive the scalar4-eld theory starting from the microscopic Ising

Hamiltonian

H=12 N X i;j=1J ijSiSj;(1) wherei;jrun over lattice sites (e.g.of ad-dimensional cubic lattice), the Ising variables take valuesSi=1,Jij>0 denotes a ferromagnetic coupling between spins. (a) Prove the following identity for multi-dimensional integrals over real variables Z 1 1dx

1dxn(2)n=2e12

P i;jxiAijxj+P ixiJi= [detA]1=2e12 P ijJi(A1)ijJj; whereAis a real symmetric positive denite matrix. (b) Use this identity to make the exponent of the partition sum for the Ising model linear in the spin variablesSi(Hubbard-Stratonovich transformation). The partition sum reads Z=X fSigexp(H): The price to pay is to introduce a (Hubbard-Stratonovich) real variablexiat each lattice site. (c) It is now possible to exactly perform the summation over Ising variablesP fSig, which leads to a spatially local (potential) term inxj. Then perform the variable transformationi=1p2 P jK1 ijxjwhereKij=Jijandxjare the elds introduced in part (b). (d) Now expand the exponentHe[j] inZRD[j]exp(He[j]) up to quartic order injand then go to momentum space. Here,D[j] denotes the integrati- on measure. Explicitly calculate the coupling functionJ(k) in momentum space (assuming translational invarianceJij=Jr=rirj)

J(k) =X

rJ reik(rirj) assuming ferromagnetic nearest-neighbor coupling on ad-dimensional cubic lattice. (e) ExpandJ(k) up to second order ink2, and identify the temperatureTcat which the (T-dependent) constant in front of thek-independent part of the quadratic term vanishes and changes sign. Then expand the coecient in front of the quadratick2term as well as the quartic term to lowest order inTTcto bring the expression into the form H e[k] =12 Z ddk(2)dka0(TTc)+bk2k+u4 Z ddk1(2)dd dk2(2)dd dk3(2)dk1k2k3k1k2k3:

Identify the constantsTc,a0,bandu.

(f) Transform back to real space to obtain the real-space form ofHe[x]. How is this expression related to Landau theory ?

2. Field theory for the three-state Potts model(10 + 10 + 10 = 30 Punkte)

An extension of the Ising model is the Potts model, which assumes that there areq states at each lattice site and the energy of two sites depends only on whether they are in the same state or not. The Hamiltonian reads H=J2 X hi;ji qqi;qj1 withqi2 f1;:::;qgandhi;jidenotes the sum over nearest-neighbors. We want to derive the mean-eld theory for theq= 3 state Potts model using the same prescription as in exercise 1. (a) Show that one can represent the energy of this system byE=P i;jJijsisjwithsi being a two-dimensional normalized vector at sitei. Give an explicit representation of these vectorssi. (b) Following the same prescription as in exercise 1 derive the eective eld theory up to quartic order in the Hubbard-Stratonovich elds starting from the microscopic

HamiltonianH=P

i;jJijsisjwithsibeing one of the three possible vectors at sitei. (c) Check to each order in the elds that the terms inHe[x] obey the symmetries of the system.quotesdbs_dbs26.pdfusesText_32

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