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Journal of Applied Physics

Jordan block structure of two-dimensional quantum neural cellular network

Bi Qiao, Harry E. Ruda

Published Jul 1, 1999• 2 citations
Publisher

Abstract

We present a Jordan type spectral decomposition for the Hamiltonian operator of a two-dimensional (2D) Ising model driven by a local external field. The results reveal that the external field induces a dynamic contribution to the Jordan block structure, while the interaction between lattice spins contributes to the energy spectrum, tending to attract the system to a minimum energy state. Both interactions contributing to the principle vectors of the Hamiltonian operator reflect their correlation. Since we adopt a first-principle approach the results are useful to general 2D Ising spin systems subject to time-independent external fields irrespective of whether the system is in equilibrium.

DOI
10.1063/1.370777

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We present a Jordan type spectral decomposition for the Hamiltonian operator of a two-dimensional (2D) Ising model driven by a local external field. The results reveal that the external field induces a dynamic contribution to the Jordan block structure, while the interaction between lattice spins contributes to the energy spectrum, tending to attract the system to a minimum energy state. Both interactions contributing to the principle vectors of the Hamiltonian operator reflect their correlation. Since we adopt a first-principle approach the results are useful to general 2D Ising spin systems subject to time-independent external fields irrespective of whether the system is in equilibrium.
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Quantum spin networks represent systems in which quantum spins are arranged on a topological lattice, where the nature of spin interactions and the influence of external magnetic fields can lead to complex collective behaviors. The Hamiltonian governing such systems determines the energy landscape and phase transitions under various thermodynamic conditions. In this paper, we focus on the two-dimensional Ising spin model with periodic boundary conditions subjected to a uniform external magnetic field. Initially, we employ the Metropolis Monte Carlo (MP-MN) algorithm to simulate the system and identify its phase transitions at different magnetic field strengths. The emergence of ordered and disordered phases in response to thermal fluctuations and magnetic interactions is systematically analyzed. Subsequently, we explore the application of convolutional neural networks (CNNs), a powerful class of deep learning models, to detect and classify the phases of the Ising model based on spin configurations generated at a fixed temperature. The CNN is trained using labeled data representing different magnetic field values, and its performance in phase prediction is quantitatively evaluated. The results demonstrate that CNNs can successfully learn complex spin patterns and provide accurate classification of spin phases, highlighting their potential for analyzing phase transitions in statistical physics models.
We will discuss on the quantum dynamics on the following three cases. (1) First, we will study structure of adiabatic energy levels for a triangle magnetic cluster with the Dzyaloshinsky-Moriya (DM) interaction. Generally, in single molecular magnets, the magnetic energy levels are discrete and we can find various interesting magnetic behavior when we sweep the external magnetic field. For the dynamics, the adiabatic energy level structure, in particular, avoided level structures at the energy level crossings, plays an important role. The mechanism to open the gap is attributed to some interactions which do not commute with the magnetization. We will discuss dependence of the dynamical magnetization process under the field cycling on the angle between the DM interaction vector and the external magnetic fields.[1,2] We also discuss a hidden conserved quantity due to the chiral symmetry.[3] The dynamical DM effect will be also discussed. (2) We will study the quantum dynamics of one-dimensional transverse Ising model which exhibits a quantum phase transition at zero temperature. There, we find a metastable state and a kind of spinodal decomposition phenomena in the sense of the quantum dynamics. We also refer to the cases of very fast sweep where some size independent collective motions are observed.[4] (3) Finally, we will also study an itinerant electron system in which the Mott insulator-Nagaoka ferromagnetism transition. There, we study the adiabatic change from the Mott insulator to the Nagaoka ferromagnetic state under removal of an electron from the lattice. We also study the magnetization process as a function of the magnetic field.
We propose a new approach to extract the important degrees of freedom in quantum dynamics induced by an external stimulus. We calculate the coefficient matrix numerically, where the i−li-li−l element of the matrix is the coefficient of the lth basis state at the ith discretized time in the solution of the time-dependent Schrödinger equation induced by the external stimulus. By performing a randomized singular value decomposition of the coefficient matrix, a practically exact solution is obtained using a linear combination of the important modes, where the number of modes is much smaller than the dimensions of the Hilbert space in many cases. We apply this method to analysis of the light absorption spectrum in two-dimensional (2D) Mott insulators using an effective model of the 2D Hubbard model in the strong interaction case. From the dynamics induced by an ultrashort weak light pulse, we find that the practically exact light absorption spectrum can be reproduced by as few as 1000 energy eigenstates in the 1.7×1071.7 \times 10^{7}1.7×107-dimension Hilbert space of a 26-site cluster. These one-photon active energy eigenstates are classified into free holon and doublon (H-D) and localized H-D states. In the free H-D states, the main effect of the spin degrees of freedom on the transfer of a holon (H) and a doublon (D) is the phase shift, and the H and the D move freely. In the localized H-D states, an H and a D are localized with relative distances of 5\sqrt{5}5​ or 13\sqrt{13}13​. The antiferromagnetic (AF) spin orders in the localized H-D states are much stronger than those in the free H-D states, and the charge localization is of magnetic origin. There are sharp peaks caused by excitations to the localized H-D states below the broad band caused by excitations to the free H-D states in the light absorption spectrum.
Spectral functions are central to link experimental probes to theoretical models in condensed matter physics. However, performing exact numerical calculations for interacting quantum matter has remained a key challenge especially beyond one spatial dimension. In this work, we develop a versatile approach using neural quantum states to obtain spectral properties based on simulations of the dynamics of excitations initially localized in real or momentum space. We apply this approach to compute the dynamical structure factor in the vicinity of quantum critical points (QCPs) of different two-dimensional quantum Ising models, including one that describes the complex density wave orders of Rydberg atom arrays. When combined with deep network architectures we find that our method reliably describes dynamical structure factors of arrays with up to 24×24 spins, including the diverging timescales at critical points. Our approach is broadly applicable to interacting quantum lattice models in two dimensions and consequently opens up a route to compute spectral properties of correlated quantum matter in yet inaccessible regimes.
Real-time dynamics of quantum observables provide direct access to excitation spectra and correlation functions in quantum many-body systems, but currently available quantum devices are limited to short evolution times due to decoherence. We propose a neural ordinary differential equation (Neural ODE) framework with a physics-driven neural network to reconstruct long-time operator dynamics from short-time quantum data. By expanding observables in the Pauli basis and exploiting locality and symmetry constraints, the operator evolution is reduced to a tractable set of coefficients whose dynamics are learned from the measurement data. Applied to the Ising spin chain with mixed fields, the operator learning method accurately extrapolates its long-time correlation function and resolves its excitation spectrum. We validate the framework by collecting short-time data from IBM's superconducting quantum processor and successfully reconstruct the Fourier spectrum of an N=8N=8N=8 Ising spin chain with 5 times enhanced precision. Our results demonstrate a scalable and data-efficient strategy for extracting dynamical and spectral information from practical quantum hardware.