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arXiv:2603.07200

Thermal Properties of Gauge-Invariant Graphene in Noncommutative Phase-Space

Ilyas Haouam

math-phquant-phSubmitted Mar 7, 2026• Announced Mar 10, 2026• Revised Jul 28, 2026 (v2)
v1 · Mar 7, 2026v2 · Jul 28, 2026
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Abstract

We study graphene in an external magnetic field within a noncommutative (NC) framework. A gauge-invariant NC Hamiltonian is derived, and the system is analyzed using the ladder-operator formalism, yielding deformed Landau levels and eigenstates. A mapping to the anti-Jaynes-Cummings model is established, providing a bridge to quantum-optical interpretation. The thermal properties of gauge-invariant NC graphene are then investigated via the partition function, constructed using Euler and zeta functions. Analytical expressions for the partition function, free energy, internal energy, entropy, and specific heat are obtained and numerically evaluated. The results show that the NC phase-space deformation modifies the effective Landau-level spacing and consequently alters the thermal behavior of the graphene system. In particular, the deformation suppresses the thermal accessibility of excited states and produces measurable deviations from the commutative case while preserving the correct low- and high-temperature asymptotic limits.

Journal reference
Physica A: Statistical Mechanics and its Applications, 699, 2026, 15 pp
DOI
10.1016/j.physa.2026.131868
Comments
15 pages, 5 figures

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We study graphene in an external magnetic field within a noncommutative (NC) framework. A gauge-invariant NC Hamiltonian is derived, and the system is analyzed using the ladder-operator formalism, yielding deformed Landau levels and eigenstates. A mapping to the anti-Jaynes-Cummings model is established, providing a bridge to quantum-optical interpretation. The thermal properties of gauge-invariant NC graphene are then investigated via the partition function, constructed using Euler and zeta functions. Analytical expressions for the partition function, free energy, internal energy, entropy, and specific heat are obtained and numerically evaluated. The results show that the NC phase-space deformation modifies the effective Landau-level spacing and consequently alters the thermal behavior of the graphene system. In particular, the deformation suppresses the thermal accessibility of excited states and produces measurable deviations from the commutative case while preserving the correct low- and high-temperature asymptotic limits.
In this paper, we study the two-dimensional Schrodinger oscillator within a dynamical noncommutative (DNC) space. By leveraging perturbation theory, we derive the energy eigenvalues and eigenvectors and systematically analyze the effects of both dynamical and non-dynamical noncommutative settings. First-order corrections to the eigensystem are obtained, revealing that the energy shift explicitly depends on the DNC parameter tau. Furthermore, we explore the thermal properties of the system by employing the partition function. Numerical results are presented to provide a comprehensive analysis of the system behavior under the considered effects. Notably, in the DNC framework, the commutation relations and the deformation parameter are position-dependent. Using the two-dimensional Bopp-shift, we effectively map the noncommutative problem to its commutative counterpart.
This work investigates a quantum system described by a Hamiltonian operator in a two dimensional noncommutative space. The system consists of an electron subjected to a perpendicular magnetic field B\mathbf{B}B, coupled to a harmonic potential and an external electric field E\mathbf{E}E, within the context of non-extensive statistical thermodynamics. The noncommutative geometry introduces a fundamental minimal length that modifies the phase space structure. The thermodynamics of this quantum system is developed within the framework of Tsallis statistics through the derivation of qqq-generalized versions of the partition function, magnetization, and magnetic susceptibility, following the application of a generalized Hilhorst transformation adapted to non-commutative geometry. The combined effects of the non-extensivity parameter qqq and the noncommutativity parameter θθθ are analyzed by considering the limit q→1q \rightarrow 1q→1, revealing new thermodynamic regimes and anomalous electromagnetic properties specific to quantum systems in non-commutative geometry.
The Weyl-Wigner formalism for evaluating the intrinsic information of Dirac bispinors as correlated qubits (localized) in a magnetic field is investigated in the extension to statistical ensembles. The confining external field quantizes the quantum correlation measures implied by the spin-parity qubit structure of the Dirac equation in 3+1 dimensions, which simplifies the computation of the entanglement quantifier for mixed states in relativistic Landau levels. This allows for the evaluation of quantum and classical correlations in terms of entropy measures for Dirac structures that are eventually mixed. Our results are twofold. First, a family of mixed Gaussian states is obtained in phase space, and its intrinsic correlation structure is computed in closed form. Second, the partition function for the low-dimensional Dirac equation in a magnetic field is derived through complex integration techniques. It describes the low-temperature regime in terms of analytically continued Zeta functions and the high temperature limit as a polynomial on the temperature variable. The connection with lower dimensional systems is further elicited by mapping the spin-parity qubits to valley-sublattice bispinors of the low-energy effective Hamiltonian of graphene.
Landau-like quantization of the Anandan system in a special electromagnetic field is studied. Unlike the cases of the Aharonov-Casher (AC) system and the He-McKellar-Wilkens (HMW) system, the torques of the system on the magnetic dipole and the electric dipole do not vanish. By constructing Heisenberg algebra, the Landau analog levels and eigenstates on commutative space, noncommutative (NC) space, and NC phase space are obtained, respectively. By using the coherent state method, some statistical properties of such free-atom gas are studied and the expressions of some thermodynamic quantities related to revolution direction are obtained. Two particular cases of temperature are discussed and the more simple expressions of the free energy on the three spaces are obtained. We give the relation between the value of $\ensuremath{\sigma}$ and revolution direction clearly and find Landau like levels of the Anandan system are invariant and the levels between the AC system and the HMW system are interchanged each other under Maxwell dual transformations on the three spaces. The two sets of eigenstates labeled by $\ensuremath{\sigma}$ can be related by a supersymmetry transformation on commutative space, but the phenomenon do not occur on NC situation. We emphasize that some results relevant to Anandan interaction are suitable for the cases of AC interaction and HMW interaction under special conditions.
The Weyl-Wigner formalism for evaluating the intrinsic information of Dirac bispinors as correlated qubits (localized) in a magnetic field is investigated in the extension to statistical ensembles. The confining external field quantizes the quantum correlation measures implied by the spin-parity qubit structure of the Dirac equation in 3+13+13+1 dimensions, which simplifies the computation of the entanglement quantifier for mixed states in relativistic Landau levels. This allows for the evaluation of quantum and classical correlations in terms of entropy measures for Dirac structures that are eventually mixed. Our results are twofold. First, a family of mixed Gaussian states is obtained in phase space, and its intrinsic correlation structure is computed in closed form. Second, the partition function for the low-dimensional Dirac equation in a magnetic field is derived through complex integration techniques. It describes the low-temperature regime in terms of analytically continued zeta functions and the high-temperature limit as a polynomial on the temperature variable. The connection with lower dimensional systems is further elicited by mapping the spin-parity qubits to valley-sublattice bispinors of the low-energy effective Hamiltonian of graphene.
In this paper, we obtained the three-dimensional Pauli equation for a spin-1/2 particle in the presence of an electromagnetic field in a noncommutative phase-space as well as the corresponding deformed continuity equation, where the cases of a constant and non-constant magnetic fields are considered. Due to the absence of the current magnetization term in the deformed continuity equation as expected, we had to extract it from the noncommutative Pauli equation itself without modifying the continuity equation. It is shown that the non-constant magnetic field lifts the order of the noncommutativity parameter in both the Pauli equation and the corresponding continuity equation. However, we successfully examined the effect of the noncommutativity on the current density and the magnetization current. By using a classical treatment, we derived the semi-classical noncommutative partition function of the three-dimensional Pauli system of the one-particle and N-particle systems. Then, we employed it for calculating the corresponding Helmholtz free energy followed by the magnetization and the magnetic susceptibility of electrons in both commutative and noncommutative phase-spaces. Knowing that with both the three-dimensional Bopp-Shift transformation and the Moyal-Weyl product, we introduced the phase-space noncommutativity in the problems in question.