Entanglement spectrum statistics of a time reversal invariant spin chain system: insights from random matrix theory

The entanglement spectrum statistics (ESS) of a disordered and generalised time reversal invariant XXZ model is inspected in the bipartite framework using exact finite-N results from the fixed trace Wishart–Laguerre (FTWL) ensemble of random matrices. Despite significant interest in entanglement spectrum of various spin models, exact finite-N RMT results had hitherto remained unutilized in the study of ESS of short and moderate sized spin chains. In this particular model, disorder has been introduced via the addition of a random z-field (transverse field), defect field at a particular site, or both. Next-nearest neighbour interactions are also put in place, to witness the effect of competing interactions on the ESS. One noteworthy feature of this model is that even in presence of the z-field, the results adhere to the orthogonal ensemble (OE) of random matrices, as a consequence of the inherent non-conventional time reversal symmetry associated with a $$\pi /2$$ π / 2 rotation about the x-axis. Additionally we examine the eigenvector statistics using the distribution of eigenvector components which as expected, follow that of the trace normalised real Ginibre matrices. The empirical results show significantly good agreement with exact RMT results of the $$\beta = 1$$ β = 1 FTWL ensemble when the system parameters are properly adjusted. In particular for fine tuning of system parameters, the smallest and largest Schmidt eigenvalue distributions, which are sensitive measures among the ESS, are in good agreement to analytical results. In some previous work its has been shown that for Hamiltonian systems, the entanglement spectrum exhibits level statistics matching RMT predictions and are governed by the same random matrix ensemble as the energy spectrum. This the authors have concluded to be an evidence in favour of a strong version of the eigenvalue thermalization hypothesis. We opine through the results of this work, that for short and moderate sized spin chains which have been bi-partitioned and therefore possesses even smaller subsystems, the information provided by just the level spacings or their ratios is insufficient to conclude whether the system is in a thermalizing or many-body localized phase. In this scenario one needs to examine other statistical measures of the Schmidt eigenvalues using exact finite-N RMT results or those from corresponding random matrix model simulations which are better suited to detect quantum chaotic behaviour for the composite as well as reduced density matrices of these systems. Graphical abstract