1 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

2 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.

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

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6 Definition: direct product of matrices Multi-index matrices Example: this is a special multi-index matrix. With many indices multidimensional block structure.

77 Example: consider a chain with N sites. Starting from the Pauli matrices X,Y,Z one defines by direct product: letting 1 denote the 2X2 identity, mth place Property: This represents X at site m.

8 Factorization of V Indeed,

Example:N=3

10 Block matrices inside V1

11 V2 matrix in terms of Pauli and related matrices Pauli matrices X,Y,Z mth place Remark: For eack site k we have a matrix, and V is a direct product of all these matrices. It would be fine if we could have a sum, instead! How can we take logarithms?

12  IDEA:all in exponential form!

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14 Simplifying (e.g. by Mathematica)

15 Canonical transformation (the method by Schulz, Mattis and Lieb, Rev. Mod. Physics (1964)) makes the problem much easier. Start by a  /2 ‘spin’ rotation around y axis

16 Symmetrized form of V We can redefine which has the same trace as the previous form since TrAB=TrBA but is preferable for its symmetry. In particular it will turn out that this new form is Hermitean.

17 Fermionization of the Transfer Matrix V mth place We already introduced those Pauli matrices on every site: However we must write everything in terms of shift operators.

18 Pauli matrices: write everything by shift operators Operators defined on different sites obviously commute, and Moreover,

19 Fermionization of the Transfer Matrix V Thus we shall work with a chain of sites with Fermions living there upon.

20 Next, we must rewrite in terms of Fermions, using Fermionization of the Transfer Matrix V

21 But wait! Annihilation on m brings a – sign!

We succeded in converting V to a second-quantization operator with the structure of an exponential. At the exponent there is something similar to a periodic tight-binding chain with nn hoppings + interaction terms. This fictitious system will contain an unspecified number of interacting particles and we must seek the largest eigenvalue. We shall be able to solve the problem by using the periodicity and the fact that the interactions are of the pairing type. 1d periodic Fermi system. Therefore,

23 Fourier, of course Jean Baptiste Joseph Fourier

24 and restricting to positive q Similarly We need V 2 too All toghether

25 Thus, skipping some complications arising from periodic/antiperiodic boundary conditions and odd/even numbers of particles (since all such distinctions lose importance in the thermodynamic limit) we write:

26 Different q are decoupled and the problem reduces to diagonalizing That is to diagonalizing simultaneously Since simultanous eigenstates of those operators are eigenstates of V q.