Porter-Thomas fluctuations in complex quantum systems

The Gaussian orthogonal ensemble (GOE) of random matrices has been widely employed to describe diverse phenomena in strongly coupled quantum systems. In particular, it has often been invoked to explain the fluctuations in decay rates that follow the $\chi$-squared distribution for one degree of freedom, as originally proposed by Brink and by Porter and Thomas. However, we find that the coupling to the decay channels can change the effective number of degrees of freedom from one to two. Our conclusions are based on a configuration-interaction Hamiltonian originally constructed to test the validity of transition-state theory, also known as the Rice-Ramsperger-Kassel-Marcus theory in chemistry. The internal Hamiltonian consists of two sets of GOE reservoirs connected by an internal channel. We find that the effective number of degrees of freedom depends on the control parameter $\rho \mathrm{\Gamma }$, where $\rho$ is the level density in the first reservoir and $\mathrm{\Gamma }$ is the level decay width. The distribution for two degrees of freedom is a well-known property of the Gaussian unitary ensemble (GUE); our model demonstrates that the GUE fluctuations can be present under much milder conditions. Our treatment of the model permits an analytic derivation for $\rho \mathrm{\Gamma }\gtrsim 1$.

Figure 1

Distribution of numerically sampled decay probabilities $P_b$ (black circles) compared with the PT distribution (dashed line) and $\chi$-squared distribution for two degrees of freedom (dashed line). The dimensions of the two GOE spaces are $N_g=100$ and their Hamiltonian parameters $v_g,v_k,v_{k^{\prime }},{\mathrm{\Gamma }}_g$ are set to 0.1. The hopping matrix elements in the channel spaces are taken as $t_i=1$. The mean values and the root-mean-square (rms) deviations for the numerical sampling are calculated for 50 histogrammed runs, each of which is constructed for 500 samples.

Figure 2

The distribution of the transmission probability for the second reservoir, $P_b$, for several values of ${\mathrm{\Gamma }}_a$ and $t_2=-{\left(10{\mathrm{\Gamma }}_a\right)}^{1/2}$ as explained in the text. The dots with error bars were calculated with 50 histogrammed samplings as in Fig. 1. The dashed and the solid curves denote the PT distribution and the $\chi$-squared distribution for two degrees of freedom, respectively.

Figure 3

Fitted values to the number of degrees of freedom $\nu$ as a function of the control parameter ${\mathrm{\Gamma }}_a{\rho }_{0a}$. The Hamiltonians are defined in the same way as in Fig. 2. The dashed line shows an empirical fit, $\nu \left(y\right)=(1+8.28y^2)/(1+3.81y^2)$ with $y={\rho }_{0a}{\mathrm{\Gamma }}_a$. Note that the $\nu$ exceeds 2 in the asymptotic region ${\rho }_{0a}{\mathrm{\Gamma }}_a\gg 1$. This may be a finite-size effect, but we have not examined this possibility.