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arXiv:2311.04665 · Physica A Statistical Mechanics and its Applications

A complex network analysis on the eigenvalue spectra of random spin systems

Qiaomu Xue, Wen-Jia Rao

Published Feb 1, 2024• 1 citation
arXivHTMLPDF
DOI
10.1016/j.physa.2024.129572

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We present a statistical analysis of spectra of transfer matrices of classical lattice spin models; this continues the work on the eight-vertex model of the preceding paper [H. Meyer, J.-C. Angl`es d'Auriac, and J.-M. Maillard, Phys. Rev. E 55, 5261 (1997)]. We show that the statistical properties of these spectra can serve as a criterion of integrability. It also provides an operational numerical method to locate integrable varieties. In particular, we distinguish the notions of integrability and criticality, considering the two examples of the three-dimensional Ising critical point and the two-dimensional three-state Potts critical point. For complex spectra, which appear frequently in the context of transfer matrices, we show that the notion of independence of eigenvalues for integrable models still holds.
We present an alternative procedure for solving the eigenvalue problem of replicated transfer matrices describing disordered spin systems with (random) 1D nearest neighbour bonds and/or random fields, possibly in combination with (random) long range bonds. Our method is based on transforming the original eigenvalue problem for a 2 n × 2 n matrix (where n → 0) into an eigenvalue problem for integral operators. We first develop our formalism for the Ising chain with random bonds and fields, where we recover known results. We then apply our methods to models of spins which interact simultaneously via a one-dimensional ring and via more complex long-range connectivity structures, e.g., (1 + ∞ )-dimensional neural networks and 'small-world' magnets. Numerical simulations confirm our predictions satisfactorily.
Dynamic mode decomposition (DMD) is an equation-free, data-driven method for the prediction and control of complex dynamical systems. A DMD method for data-driven quantum control was proposed recently and numerically demonstrated in a single spin system where time series of a complete orthonormal set of Hamiltonian is available [1]. In quantum spin networks, it is generally difficult to access all spins but only a small set of spins is practically accessible. In this paper, we formulate a Hankel-DMD method for open quantum systems and extend the applicability of the DMD framework to quantum spin networks with limited access. We demonstrate that Hankel DMD can precisely evaluate the eigenvalues and decompose the dynamics into the respective oscillatory eigenmodes from the observed data. In particular, it can reveal the decoherence-free dynamics in spin networks possessing eigenvalues on the imaginary axis.