We propose a detailed analysis of datasets generated from simulations of two-dimensional quantum spin systems using the quantum Ising model at absolute zero temperature. Our focus is on examining how fundamental physical properties, energy, magnetization, and entanglement entropy, evolve under varying external transverse magnetic fields and system sizes. From the Quantum Toolbox in Python (QuTiP), we simulate systems with 4, 8, and 16 spins arranged in square lattices, generating extensive datasets with 5000 samples per magnetic field value. The Hamiltonian operator incorporates quantum mechanical effects such as superposition and tunneling, challenging classical interpretations of spin states. We compute extended Pauli operators and construct the Hamiltonian to include spin-spin interactions and transverse field terms. Our analysis reveals that as the system size increases, fluctuations in energy and entanglement entropy become more evident, indicating lifted sensitivity to external perturbations and suggesting the onset of quantum phase transitions. Spin-spin correlation functions demonstrate that interactions are predominantly local, but larger systems exhibit more complex and fluctuating correlations. These findings provide valuable insights into the behavior of quantum spin systems and lay the groundwork for future machine learning applications aimed at predicting physical quantities and identifying phase transitions from a quantum perspective.
We propose a detailed analysis of datasets generated from simulations of two-dimensional quantum spin systems using the quantum Ising model at absolute zero temperature. Our focus is on examining how fundamental physical properties, energy, magnetization, and entanglement entropy, evolve under varying external transverse magnetic fields and system sizes. From the Quantum Toolbox in Python (QuTiP), we simulate systems with 4, 8, and 16 spins arranged in square lattices, generating extensive datasets with 5000 samples per magnetic field value. The Hamiltonian operator incorporates quantum mechanical effects such as superposition and tunneling, challenging classical interpretations of spin states. We compute extended Pauli operators and construct the Hamiltonian to include spin-spin interactions and transverse field terms. Our analysis reveals that as the system size increases, fluctuations in energy and entanglement entropy become more evident, indicating lifted sensitivity to external perturbations and suggesting the onset of quantum phase transitions. Spin-spin correlation functions demonstrate that interactions are predominantly local, but larger systems exhibit more complex and fluctuating correlations. These findings provide valuable insights into the behavior of quantum spin systems and lay the groundwork for future machine learning applications aimed at predicting physical quantities and identifying phase transitions from a quantum perspective.
For a system at absolute zero temperature, all thermal fluctuations are frozen out, while quantum fluctuations prevail.These microscopic quantum fluctuations can induce a quantum phase transition (QPT) in the ground state of a many-body system when the relative strength of two competing energy terms is varied across a critical value.This paper presents a detailed and quantitative study of the magnetic properties of Ising two-spin system at zero temperature in the presence of longitudinal, transverse and mixed fields, and also at finite temperatures.At the critical longitudinal magnetic field, the system is found to undergo a QPT to a ferromagnetic state.For this system, an infinitesimal departure from zero transverse field is found to trigger off QPT and also remove the degeneracies that persisted even in the presence of longitudinal field.These quantum fluctuations, leading to QPT are more evident in the mixed field than the separate effect of longitudinal and transverse fields.The observed quantum effects at zero temperature for this system are found to be completely wiped out at finite temperatures due to thermal fluctuations.A temperature increase favours antiferromagnetic alignment of spins, while a decrease in temperature favours ferromagnetic alignment.
The quantum phase transitions provide a paradigm for studying collective quantum phenomena that are a result of competing non-commuting interactions. This paper will study the ground state properties and quantum critical dynamics of the one-dimensional transverse field Ising model through a combined perspective that includes exact diagonalisation, variational quantum eigensolver (VQE) simulations, and simulations on realistic physical quantum devices. We focus on a lattice of four spins, where we calculate the ground-state energies, magnetic order parameters and correlation functions at uniformly applied conditions, which is repeated by all systems. Precise diagonalisation provides both a benchmark, which is symmetry-conserving, and a depth-two, physics inspired variational approximation, which provides simulations accessible to hardware. The circuits that have been optimised identically are then placed on the IQM Garnet quantum processor, using a resource-efficient batched protocol. We find that the ground-state energies of shallow variational circuits are reliably captured by the circuit over the entire parameter space; the magnetic arrangement parameters and observables sensitive to correlation signal significantly more noise. The error analysis of quantitative analysis reveals a strong broadening of critical crossover on hardware, which is consistent with the noise attenuation of long-range correlations. These findings highlight the current capabilities as well as the fundamental limitations of noisy intermediate-scale quantum systems in modelling quantum critical phenomena as a benchmark to future enhancements in obtaining quantum hardware and quantum algorithms development.
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.
Using the trace minimization algorithm, we carried out an exact calculation of entanglement in a 19-site two-dimensional transverse Ising model. This model consists of a set of localized spin-21 particles in a two-dimensional triangular lattice coupled through exchange interaction J and subject to an external magnetic field of strength h. We demonstrate, for such a class of two-dimensional magnetic systems, that entanglement can be controlled and tuned by varying the parameter $\ensuremath{\lambda}=h/J$ in the Hamiltonian and by introducing impurities into the systems. Examining the derivative of the concurrence as a function of $\ensuremath{\lambda}$ shows that the system exhibits a quantum phase transition at about ${\ensuremath{\lambda}}_{c}=3.01$, a transition induced by quantum fluctuations at the absolute zero of temperature.
Quantum spin networks represent configurations of spins arranged on a topological lattice, where the spin interactions are governed by the system's Hamiltonian. These networks are critical for understanding magnetic materials, as the arrangement of spins and the type of interaction between neighboring spins determine the macroscopic behavior of the system. The behavior of these systems is further influenced by the presence of external magnetic fields. In this paper, we first investigate the various phases of the two-dimensional Ising lattice with periodic boundary conditions under the influence of a uniform external magnetic field. The exploration of these phases is performed using the Metropolis Monte Carlo (MP-MN) algorithm, a well-established statistical method for simulating spin systems. Subsequently, we explore the potential of deep learning, specifically convolutional neural networks (CNN), in identifying and predicting these phases of spin lattices. The CNN's ability to classify different phases of the two-dimensional Ising model in the presence of a homogeneous magnetic field at a constant temperature is examined. The study aims to demonstrate how machine learning models, particularly CNNs, can effectively detect phase transitions and predict the system's behavior, which traditionally requires extensive computational methods. Finally, the performance of the CNN algorithm is evaluated by assessing its accuracy in predicting different phases of the Ising model.
The quantum Ising model is perhaps the simplest possible model of a quantum magnetic material. Despite its simplicity, its versatility and wide range of applications, from quantum computation, to combinatorial optimization, to biophysics, make it one of the most important models of modern physics. In this thesis, we develop a general framework for studying quantum Ising systems with an arbitrary single ion Hamiltonian, with emphasis on the effects of quantum fluctuations, and the quantum phase transition between paramagnetic and ferromagnetic states that occurs when a magnetic field is applied transverse to the easy axis of the system. The magnetic insulating crystal LiHoF₄ is a physical realization of the quantum Ising model, with the additional features that the dominant coupling between spins is the long range dipolar interaction, and each electronic spin is strongly coupled to a nuclear degree of freedom. These nuclear degrees of freedom constitute a spin bath environment acting on the system. In this thesis, we present an effective low temperature Hamiltonian for LiHoF₄ that incorporates both these features, and we analyze the effects of the nuclear spin bath on the system. We find the lowest energy crystal field excitation in the system is gapped at the quantum critical point by the presence of the nuclear spins, with spectral weight being transferred down to a lower energy electronuclear mode that fully softens to zero at the quantum critical point. Furthermore, we present a toy model, the spin half spin half model, that illustrates the effects of an anisotropic hyperfine interaction on a quantum Ising system. We find the critical transverse field is increased when the longitudinal hyperfine coupling is dominant, as well as an enhancement of both the longitudinal electronic susceptibility and an applied longitudinal field. In addition, we present a field theoretic formalism for incorporating the effects of fluctuations beyond the random phase approximation in general quantum Ising systems. We find that any regular on site interaction, such as a nuclear spin bath, does not fundamentally alter the critical properties of a quantum Ising system. This formalism is used to calculate corrections to the magnetization of LiHoF₄.