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International Journal For Multidisciplinary Research

Phase Transitions in Quantum Spin System With Long-Range Interactions: A Nonlinear Variational Inequality Approach

Amarjeet

Published Sep 17, 2025• 0 citations
Publisher

Abstract

This Paper presents a novel mathematical framework for modeling phase transitions in quantum spin system.
DOI
10.36948/ijfmr.2025.v07i05.55954

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Phase Transitions in Quantum Spin Systems with Long-Range Interactions: A Nonlinear Variational Inequality Approach

Amarjeet

This paper presents a novel mathematical framework for modeling phase transitions in quantum spin systems with long-range interactions using nonlinear variational inequalities (NVIs). Traditional methods such as mean-field theory and Landau-Ginzburg models often fall short in capturing spatial inhomogeneities, critical phenomena, and constraints inherent in realistic spin systems. Our approach addresses these limitations by reformulating the energy-based governing equations as an NVI problem defined over a suitable convex set of physically admissible spin configurations. We incorporate both Ising and Heisenberg-type quantum lattice models with power-law decaying interactions to accurately represent nonlocal effects. This research provides rigorous mathematical preliminaries, including operator-theoretic foundations in Hilbert and Banach spaces, and prove existence and uniqueness results under monotonicity and coercivity assumptions. Analytical insights include bifurcation analysis, critical point theory, and regularity of solutions, offering a robust theoretical foundation for understanding phase transitions. A finite element–Galerkin discretization is developed to simulate the variational system numerically. Simulation results demonstrate the emergence of ordered phases, spin textures, and domain formation, consistent with known physical behavior. Comparisons with classical models reveal the advantages of the NVI framework in capturing phase boundaries and metastable states with higher fidelity. The proposed framework also opens pathways for extending the analysis to quantum entanglement, topological order, and non-equilibrium phenomena. This research thus contributes both to the mathematical theory of variational inequalities and to the physical modeling of quantum spin systems, providing tools for deeper exploration of complex condensed matter phenomena.

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This paper presents a novel mathematical framework for modeling phase transitions in quantum spin systems with long-range interactions using nonlinear variational inequalities (NVIs). Traditional methods such as mean-field theory and Landau-Ginzburg models often fall short in capturing spatial inhomogeneities, critical phenomena, and constraints inherent in realistic spin systems. Our approach addresses these limitations by reformulating the energy-based governing equations as an NVI problem defined over a suitable convex set of physically admissible spin configurations. We incorporate both Ising and Heisenberg-type quantum lattice models with power-law decaying interactions to accurately represent nonlocal effects. This research provides rigorous mathematical preliminaries, including operator-theoretic foundations in Hilbert and Banach spaces, and prove existence and uniqueness results under monotonicity and coercivity assumptions. Analytical insights include bifurcation analysis, critical point theory, and regularity of solutions, offering a robust theoretical foundation for understanding phase transitions. A finite element–Galerkin discretization is developed to simulate the variational system numerically. Simulation results demonstrate the emergence of ordered phases, spin textures, and domain formation, consistent with known physical behavior. Comparisons with classical models reveal the advantages of the NVI framework in capturing phase boundaries and metastable states with higher fidelity. The proposed framework also opens pathways for extending the analysis to quantum entanglement, topological order, and non-equilibrium phenomena. This research thus contributes both to the mathematical theory of variational inequalities and to the physical modeling of quantum spin systems, providing tools for deeper exploration of complex condensed matter phenomena.
We investigate the zero-temperature and finite-temperature phase transitions of quantum Ising and quantum rotor models. We here assume a long-range (falling off as ${1/r}^{d+\ensuremath{\sigma}},$ where r is the distance between two spins/rotors in units of lattice spacing) ferromagnetic interaction among the spins or rotors. We find that the long-range behavior of the interaction drastically modifies the universal critical behavior of the system. The corresponding upper critical dimension and the hyperscaling relation and exponents associated with the quantum transition are modified and, as expected, they attain values of short-range system when $\ensuremath{\sigma}=2.$ The dynamical exponent varies continuously as the parameter \ensuremath{\sigma} and is unity for $\ensuremath{\sigma}=2.$ The one-dimensional long-range quantum Ising system shows a phase transition at T=0T=0T=0 for all values of \ensuremath{\sigma}. The most interesting observation is that the phase diagram for $\ensuremath{\sigma}=d=1$ shows a line of Kosterlitz-Thouless transition at finite temperature even though the T=0T=0T=0 transition is a simple order-disorder transition. These finite temperature transitions are studied near the phase boundary using renormalisation group equations and a region with diverging susceptibility is located. We have also studied one-dimensional quantum rotor model which exhibits a rich and interesting transition behavior depending upon the parameter \ensuremath{\sigma}. We explore the phase diagram extending the short-range quantum nonlinear \ensuremath{\sigma} model renormalisation group equations to the present case.
We investigate a nonequilibrium phase transition in a dissipative and coherent quantum spin system using the quantum Langevin equation and mean-field theory. Recently, the quantum contact process (QCP) was theoretically investigated using the Rydberg antiblockade effect, in particular, when the Rydberg atoms were excited in s states so that their interactions were regarded as being between the nearest neighbors. However, when the atoms are excited to d states, the dipole-dipole interactions become effective, and long-range interactions must be considered. Here we consider a quantum spin model with a long-range QCP, where the branching and coagulation processes are allowed not only for the nearest-neighbor pairs but also for long-distance pairs, coherently and incoherently. Using the semiclassical approach, we show that the mean-field phase diagram of our long-range model is similar to that of the nearest-neighbor QCP, where the continuous (discontinuous) transition is found in the weak (strong) quantum regime. However, at the tricritical point, we find a new universality class, which was neither that of the QCP at the tricritical point nor that of the classical directed percolation model with long-range interactions. Implementation of the long-range QCP using interacting cold gases is discussed.