It is well known that the pair correlation function of the zeros of the Riemann zeta function reproduces the correlation function of the random matrices from the Gaussian unitary ensemble (GUE). Usually, it is said that this is a connection between the zeta function and physics. Nevertheless, most of the uses of random matrices in physics involve the Gaussian orthogonal ensemble (GOE), but not the GUE. What are the applications of the GUE in physics?


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The Gaussian unitary ensemble (GUE), Gaussian orthogonal ensemble (GOE), and Gaussian symplectic ensemble (GSE), are all ensembles of random Hermitian matrices, where the U/O/S indicate the different invariances of the measure over Hermitian matrices.${}^1$

These ensembles are used in physics to study properties of generic or chaotic Hamiltonians. The rough idea (which was Wigner's motivation for writing them down) was that if we have a system of many degrees of freedom with very complicated interactions, certain aspects of the system might behave as if the Hamiltonian were a random matrix. More precisely, the spectral statistics of the Hamiltonian, like the distribution of nearest-neighbor spacings of eigenvalues as well as the eigenvalue correlations, look like those of a random matrix. This has even been proposed as a definition of quantum chaotic system, and a random matrix like distribution of the energies has been observed in many real systems.${}^2$

To your question: the different ensembles describe quantum systems with different symmetries. The GUE is the most general and captures the behavior of systems which break all symmetry, whereas the GOE and GSE correspond to different realizations of time-reversal symmetry.${}^3$ So if you write down some chaotic Hamiltonian without time-reversal symmetry (for instance, a disordered spin system with 1-body and 2-body interactions), the spectral statistics will be that of the GUE. If the system has time-reversal symmetry, the spectral statistics will be that of the GOE or GSE.

  1. i.e. we have $dH=d(UHU^\dagger)$, where $U$ is any unitary/orthogonal/symplectic matrix, depending on the ensemble.
  2. See this review.
  3. There is a precise sense in which there are only three symmetry classes, this is called Dyson's threefold way.

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