Wigner surmise for Hermitian and non-Hermitian chiral random matrices

We use the idea of a Wigner surmise to compute approximate distributions of the first eigenvalue in chiral random matrix theory, for both real and complex eigenvalues. Testing against known results for zero and maximal non-Hermiticity in the microscopic large-$N$ limit, we find an excellent agreement valid for a small number of exact zero eigenvalues. Compact expressions are derived for real eigenvalues in the orthogonal and symplectic classes and at intermediate non-Hermiticity for the unitary and symplectic classes. Such individual Dirac eigenvalue distributions are a useful tool in lattice gauge theory, and we illustrate this by showing that our results can describe data from two-color quantum chromodynamics simulations with chemical potential in the symplectic class.

Figure 1

(Color online) (a) Surmise $p_{WS}^{\left(\beta \right)}\left(s\right)$ (red) vs exact result [20] (dashed blue) from $\beta =1,2,4$ (bottom to top). (b) $p_{Gin}^{\left(2\right)}\left(r\right)$ for $N=2,3,4,20$ in red to blue (bottom to top).

Figure 2

(Color online) $p_{\nu }^{\left(\beta \right)}\left(s\right)$ with $\nu =0,1,2$ in dashed blue to green (left to right) for (a) $\beta =2$, (b) $\beta =4$, and (c) $\beta =1$ vs $N=2$ surmise is in red. Our result Eq. (11) is compared to a numerical simulation at $N=20$ (black dots).

Figure 3

(Color online) Integrated first eigenvalue $P_{\nu }^{\left(\beta \right)}\left(r\right)$ at $\mu =1$ for $\nu =0,1,2$ in blue to green dashes (left to right) vs $N=2$ (red): (a) $\beta =2$ and (b) $\beta =4$.

Figure 4

(Color online) Integrated first eigenvalue $P_{\nu }^{\left(4\right)}\left(r\right)$ (red) vs lattice data [17] (blue histograms) with volume $V=4^4$, gauge coupling 1.3, ${\mu }_{\text{Lat}}=0.2$ and mass 20 in lattice units, using a very large number ${10}^5$ configurations.

Figure 5

(Color online) Top: contour plots for Lattice data as in Fig. 4 but with (a) ${\mu }_{\text{Lat}}=0.1$ vs (b) surmise Eq. (18). Bottom: a single peak cut in (c) $x$ and (d) $y$ directions.