Decimation 1 1

Consider only paths involving a,b,c,d. What is the probability p1 that nodes A and B of new lattice are connected? A 1 B 2 a g b d C D 2 2

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Onset of conductivity (R. Rosman-B. Shapiro Phys. Rev Onset of conductivity (R. Rosman-B. Shapiro Phys. Rev. B16, 5117 (1977)) A 1 B 2 a g b d C D 5 5

Evaluation of conductance s1 of renormalized bond AB 2 a g b d C D 6 6

Evaluation of critical exponent t 1 B 2 a g b d C D 7 7

Renormalization group for the 1d Ising model 2 3 4 Decimation Eliminating even sites: we must sum over even sites. Sum over spin 2: Sum over all even sites: 8 8

Decimation Eliminating even sites: we must sum over even sites AND set the result back in the original form (as far as we can). 9 9

Renormalization group for the 2d Ising model Fixed point: no ferromagnetism, no phase transition, no long-range order! Renormalization group for the 2d Ising model (see Grosso-Pastori,McComb)

Eliminate site 0

New couplings are needed to represent all possibilities: Indeed we can fix the uknowns by solving the 16 equations that arise from the possible spin orientations and that we can summarize in three cases: 12 12

no ferromagnetism, no phase transition, no long-range order! Iterating, This is unlike the original H and interations bring interactions of increasing complexity. Approximations needed. no ferromagnetism, no phase transition, no long-range order! 13 13

H.J. Maris-L.P.Kadanoff,am.J.Phys.46, 652 (1978): 14 14

Peierls transition in 1d chains regular 1d chain a a a Consider a 1d chain of identical equally spaced atoms at hal filling. Peierls in the thirties has shown that in the Born-Oppenheimer approximation the system is unstable and the ground state shows alternating long and short bond lengths. Peierls distorted 1d chain a-u a+u The lattice parameter doubles, the BZ becomes the half and a gap opens.This is called a charge density wave (CDW) state, which can be seen in X-Ray diffraction. All the occupied states go down in energy..

Polyacetylene with its anternating singe and double bond is the most common example of the Peierls distortion. Often 1d systems are poor models of 3d crystals because there are qualitative differences. See e.g. Sander van Smaalen, Acta Crystallographica (2005)

Jordan-Wigner string In 1d one can make Fermions out of Bosons Consider a boson field is defined on a chain, with annihilation operators ci How can we use them to define anticomuting fermions? In fact, we can in 1d. The fermions are: 18 18 18 18

There is also a continuous version So, we can verify that in the following situation, a b There is also a continuous version

Jordan-Wigner string with Pauli matrices tx ty tz Let us associate Pauli matrices to sites of a chain. Matrices of different sites commute. Then introduce the matrices They anticommute and are Fermion annihilation and creation operators Indeed the Jordan-Wigner factors can be removed when they occur twice, so on site 20 20

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The Onsager solution of the 2d Ising model The partition function is: Notation: The geometry of a torus is adopted, identifying column N+1 with column 1 and row N+1 with row 1, that is, imposing pbc

First step: write Z in terms of the eigenvalues of a Hermitean matrix-the transfer matrix V. This will be chosen with the structure of a one-dimensional array. 23 23

Row-by-row description Indeed, in the l.h.s. we are summing on all the configurations taking them site by site, in the r.h.s. we do the same thing row by row. 24 24

For each lattice configuration Z has a contribution

Example:N=3 For each lattice configuration Z has a contribution