Abstract
As we have seen in Chap. 5, representing determinants as Gaussian integrals over anticommuting alias Grassmann variables makes for great simplifications in computing averages over the underlying matrices.
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Notes
- 1.
A nilpotent quantity has vanishing powers with integer exponents upward of some smallest positive one.
- 2.
Readers not previously familiar with the wedge product of differentials of commuting variables have every right to be momentarily confused. They have the choice of either spending a quiet hour with, e.g., Ref. [7], or simply ignoring the following remark and convincing themselves of the correctness of (6.4.2) in some other way, most simply by expanding the integrand in powers of c.
- 3.
- 4.
Useful generalizations, in particular to higher dimensions, and a watertight proof can be found in Ref. [13].
- 5.
Later we will usually encounter the case of two parameters, K = 2.
- 6.
In our example case μ k∕λ k = (m k + 2∕N)∕(1 − m k) which gives the same condition for massive modes as we have stated in terms of m k.
- 7.
In specifying the saddle (6.6.26), we have immediately restricted ourselves to the energy interval |E∕2| < λ to which the spectrum is confined by the semicircle law.
- 8.
Different such choices are possible and lead to different but of course equivalent appearances of \(\hat {e}_\pm \).
- 9.
This ordering of matrices in AR⊗BF is often called ‘AR notation’. Unless noted explicitly otherwise we shall stick to that ordering.
- 10.
The equivalence of M F with the two-sphere and of M B with the two-hyperboloid will be shown in Sect. 6.8.5.
- 11.
This subsection is lifted from Ref. [17], word by word.
- 12.
Recall from Sect. 5.13 that ordinary (over commuting variables, that is) real Gaussian integrals lead to inverse square roots of determinants while such integrals over anti-commuting variables give Pfaffians (which also square to determinants). It is straightforward to generalize these two cases to ‘real’ Gaussian super-integrals where one accordingly arrives at square roots of superdeterminants. See also Sect. 5.14.4 for an application of Pfaffians .
- 13.
The block Z D is called diffusion mode and Z C the Cooperon mode, names invented in the disordered systems community [3]. In the CXE sigma model these two modes are coupled as they expressed as blocks in a larger supermatrix. If one breaks time-reversal symmetry continuously the diffusion mode remains massless but the Cooperon mode acquires a mass. If that mass is sufficiently large (greater than 1∕N) one may neglect the Cooperon mode and what is left is the standard CUE result (i.e. time-reversal is fully broken).
- 14.
- 15.
Our unscrupulous change of the order of integrations as well as the naive use of the saddle-point approximation for the N-fold energy integral later admittedly give a certain heuristic character to this section.
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Haake, F., Gnutzmann, S., Kuś, M. (2018). Supersymmetry and Sigma Model for Random Matrices. In: Quantum Signatures of Chaos. Springer Series in Synergetics. Springer, Cham. https://doi.org/10.1007/978-3-319-97580-1_6
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