By Philip W. Anderson

The name of the publication could be deceptive. recognition, this e-book is for complex readers in Condensed subject physics. truly, the e-book is usually consisted of a few reliable papers chosen through by way of Anderson. A newbie can learn this after he get to grasp the "basic notions" from easy books.

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1(b)). Since the energy denominator is zero we may have an instability of the perturbation theory. This is an antiferromagnetic instability since the momentum transfer is π. Conversely, instability in the forward-scattering channel is a symptom of ferromagnetism. In dimensions higher than one, the situation is more complex due to the intricacies of the Fermi surface. For instance, in the case of a half-filled square lattice the Fermi surface is a square (see Fig. 4). A scattering process involving the (nesting) wave vectors shown in Fig.

103) Both results can be derived quite easily by expanding and ¯ in a basis of eigenstates of M, see Faddeev (1976) or Negele and Orland (1988). 3 Path integrals and mean-field theory We now turn to the mean-field theory for the Hubbard model in path-integral form. The advantage of this description is that we will be able to extract an effective-field theory for the low-lying modes in the Néel state: spin waves. The Lagrangian density for the Hubbard model in two dimensions, in real time and at zero temperature is, from Eqs.

1(b)). This state has an energy U above that of the degenerate ground states. The matrix element (squared) is t 2 . There is also a multiplicity factor of 2 since this process can occur in two different ways. Hence we expect that the relevant parameter of the effective Hamiltonian should be 2t 2 /U . Also, the final state has to be either the same one as the initial state or it can differ at most by a spin exchange (see Fig. 1(c)). The natural candidate for the effective Hamiltonian is the quantum Heisenberg antiferromagnet.