arXiv subjects

M. A. H. Ahsan

Publications and source records attributed to M. A. H. Ahsan.

At least 19 recordsLinked to original sources

arXiv2026

Interaction-rotation driven localization-delocalization crossover in Fock space: An exact diagonalization study on trapped Bose gas

We study the localization-delocalization crossover and quantum entanglement with varying numbers of interacting bosons for the nonrotating and the rotating cases. The many-body ground state within subspaces of given total angular momentum is obtained via exact diagonalization and analyzed using the inverse participation ratio (IPR), the information (Shannon) entropy and the von Neumann entanglement entropy. In the non-rotating case, a crossover from localization to delocalization is observed with increasing interaction, characterized by decreasing value of IPR with corresponding increase in information entropy and von Neumann entanglement entropy, arising from interaction-induced depletion of Bose-Einstein condensate. When subjected to rotation, the system is driven further towards delocalization due to rotation-induced depletion of Bose-Einstein condensate. The crossover from localization to delocalization is indicated by distribution of the eigenstate weights over an increasing number of basis states in the many-body Hilbert space. The consistent behavior of IPR, information entropy and von Neumann entanglement entropy with respect to interaction and rotation demonstrates that these quantities provide a unified characterization of the localization-delocalization crossover. Results on computed von Neumann entropy show that localized states exhibit weak entanglement while delocalized states are characterized by strong entanglement within the quantum many-body system.

cond-mat.quant-gas↗

arXiv2026

Persistent nonlinear Hall effect driven by parallel field across a topological phase transition and intraband sign-reversing integer quantum Hall effect

We investigate the linear and nonlinear Hall effects in a two-dimensional Rashba spin-orbit coupled nodal ring electronic system. We consider both the cases of linear Hall effect, the Berry curvature induced anomalous Hall and the perpendicular magnetic field-induced integer quantum Hall effect. We observe that the system exhibits a topological gap along the boundary of the nodal ring that strongly depends on the radius of the nodal ring and the strength of the Rashba spin-orbit interaction, resulting in the quantum anomalous Hall effect. Subsequently, we include a perpendicular uniform magnetic field and obtain the exact Landau levels that exhibit sign reversal in the slope with magnetic field around the ring boundary. This causes a sign reversal in the quantum Hall conductivity in the same band by tuning the magnetic field. Most importantly, we also show that when the magnetic field is strictly parallel to the system, it can induce an anisotropy to the Berry curvature that leads to the emergence of the nonlinear Hall effect. Additionally, tuning the parallel field can close and reopen the topological gap, resulting in the reverse topological phase transition from the topological Chern insulator to a trivial insulator. The nonlinear Hall effect remains persistent in both phases, but its dependence on the chemical potential exhibits distinct signatures. Noticeably, the nonlinear Hall response exhibits a sign-changing peak structure in Chern insulating phase whereas it displays a single-sign response in trivial insulating phase. These distinct signatures suggest that nonlinear Hall can serve as a probe to study such a topological phase transition.

cond-mat.mes-hall↗

arXiv2026

Spectral form factor and power spectrum for trapped interacting rotating bosons: Crossover from integrability to quantum chaos

The emergence of quantum chaos in a system of trapped interacting bosons with externally impressed rotation is studied through spectral form factor (SFF) and power spectrum using exact diagonalization. Two distinct interaction regimes are considered: the moderate, when the interaction energy is small compared to the trap energy and the strong, when the interaction energy is comparable to the trap energy. In the moderate interaction regime, the SFF for the non-rotating case exhibits a dip-plateau structure with absence of linear ramp, indicating integrable behavior, while for the single-vortex state the SFF exhibits a discernible linear ramp consistent with pseudo-integrable behavior. In the strong interaction regime, the non-rotating case exhibits emergence of a linear ramp with small time span in SFF, indicating that the system has moved further towards chaotic regime but continues to be pseudo-integrable. For the single-vortex and the multi-vortex states in strong interaction regime, the span of the linear ramp in SFF increases progressively with rotation, indicating the system has moved into strong chaotic regime consistent with Gaussian orthogonal ensemble. The power spectrum results with exponent lying in the interval $1 \lesssim α\lesssim 2$ are consistent with the findings of SFF. An understanding of the observed crossover from integrable to quantum chaos is presented in terms of the macroscopic occupation of a single-particle quantum state---the Bose-Einstein condensation---and its depletion driven by interaction and rotation.

cond-mat.quant-gas↗

arXiv2026

Finite-size effects and interaction-driven crossovers in quarter-filled attractive Hubbard model: Exact diagonalization, DMRG and machine-learning analysis

We investigate the quarter-filled attractive Hubbard model on finite-width cylindrical lattices using exact diagonalization (ED), density-matrix renormalization group (DMRG) and unsupervised machine-learning-based techniques. Analysis of the ground-state energetics, local observables and correlation functions reveals a continuous interaction-driven crossover from weakly correlated fermions to a regime dominated by tightly bound singlet pairs. This crossover originates from the competition between kinetic-energy-driven fermionic itinerancy and interaction-driven onsite pair formation and exhibits behavior consistent with the BCS--BEC crossover in the thermodynamic limit. Hole-binding-energy calculations provide direct energetic evidence for pair formation: the two-hole binding energy remains negative throughout the attractive regime whereas three-hole binding emerges only at sufficiently strong attraction and exhibits pronounced finite-size dependence. To obtain an unbiased characterization of the correlation landscape, we apply principal component analysis (PCA) and uniform manifold approximation and projection (UMAP) to the real-space correlation matrices. PCA reveals a systematic redistribution of correlation variance whereas UMAP identifies a clear separation between weak- and strong-pairing regimes. Both machine-learning-based approaches independently identify the same crossover region inferred from conventional observables while providing an order-parameter-independent characterization of the underlying reorganization of many-body correlations. Finite-size scaling analyses of the pairing structure factor and the leading PCA variance ratio demonstrate that these signatures remain robust with increasing system size.

cond-mat.str-el↗

arXiv2026

Emergence of correlation-driven altermagnetism in Hubbard model on geometrically frustrated lattice-clusters

We investigate the emergence of correlation-driven altermagnetism in simple and extended Hubbard model on geometrically frustrated lattice-clusters using exact diagonalization. By systematically tuning the degree of geometric frustration across different cluster geometries -- including the fully frustrated 3x3 torus, the partially frustrated 2x3 cylinder and the unfrustrated 2x4 cylindrical lattice -- we isolate the necessary conditions for compensated anisotropic spin order. At half-filling on 3x3 lattice, the average altermagnetic spin-response <Δ_spin> remains small despite robust local moment formation, indicating that localized moments alone are insufficient to break directional symmetry. In contrast, the introduction of a single mobile charge carrier (hole or electron) induces a finite <Δ_spin> that increases monotonically with on-site interaction U . This altermagnetic phase is characterized by a d-wave-like alternating sign structure in real-space correlations, a broad momentum-space distribution in the altermagnetic structure factor S_alm(q) and a distinct particle-hole asymmetry with significantly enhanced response for one-electron-doped system. We demonstrate that these correlations remain zero on the unfrustrated 2x4 lattice, establishing geometric frustration as a fundamental prerequisite. The inclusion of nearest-neighbor Coulomb repulsion V weakens the altermagnetic correlations in 3x3 lattice through enhanced electronic localization but facilitates the onset of altermagnetic order in 2x3 lattice beyond a critical threshold. Finally, we show that the anisotropy parameter A is non-zero only for degenerate ground states, revealing that macroscopic symmetry breaking on finite clusters requires a confluence of geometric frustration and ground-state degeneracy.

cond-mat.str-el↗

arXiv2026

Principal Component Analysis of Competing Correlations in Quarter-Filled Hubbard Models

We present an unsupervised learning analysis of correlation hierarchies in the quarter-filled simple and extended Hubbard models by applying principal component analysis (PCA) to exact-diagonalization (ED) data on 3x4 and 4x4 cylindrical clusters. While the non-interacting limit (U=0) provides a finite-size reference, increasing on-site repulsion U induces localization and reorganizes the low-energy spectrum. For the extended model, we examine moderate (U=4) and strong (U=10) coupling regimes, where conventional structure factors reveal familiar crossovers among charge, spin and local-pairing correlations. PCA of the corresponding correlation matrices captures these crossovers directly from the data, without assuming predefined order parameters by identifying charge-dominated, spin-dominated and pairing-dominated regimes through variance condensation into leading components. This establishes PCA as a transparent, model-agnostic framework for uncovering the hierarchy and competition of correlation channels in finite Hubbard clusters, providing a bridge between exact diagonalization and modern machine-learning diagnostics in strongly correlated systems.

cond-mat.str-el↗

arXiv2026

Spin-resolved Mott crossover and entanglement in the half-filled Hubbard model

We investigate the interaction-driven reorganization of spin and charge correlations in finite Hubbard clusters using exact diagonalization. Focusing on half-filled and lightly doped square lattices, we analyze spin-resolved charge-gaps, local observables, two-point correlation functions, entanglement measures, principal component analysis (PCA) of correlation matrices and quantum-geometry-based distance metrics. At half-filling, we observe the emergence of a robust Mott gap whose spin-dependent component is controlled by the effective exchange energy scale J~4t^2/U at strong coupling, confirming that residual spin dynamics govern the separation between the lowest-spin and the next higher-spin charge excitation channels. Distinct cluster geometries and boundary conditions reveal how spin-singlet versus finite-spin ground-state influence charge and spin responses. Upon one-hole doping, the spin-resolved charge-gaps collapses, indicating restored compressibility and metallic behavior. PCA and quantum-geometry-based analysis provide complementary data-driven and wavefunction-based perspectives on correlation-driven Mott crossover phenomena. Our results demonstrate that finite-size Hubbard clusters exhibit clear signatures of the Mott physics, spin-charge interplay and emergent exchange energy scales, offering a unified microscopic picture of interaction-induced electronic reorganization.

cond-mat.str-el↗

arXiv2026

Exact diagonalization study of energy level statistics in harmonically confined interacting bosons

We present an exact diagonalization study of the spectral properties of bosons harmonically confined in a quasi-2D plane and interacting via repulsive Gaussian potential. We consider the lowest $100$ energy levels for systems of $N=12, 16$ and $20$ bosons in two distinct regimes: (a) when the interaction energy is small compared to the trap energy (moderate interaction) and (b) when the interaction energy is comparable to the trap energy (strong interaction), for the non-rotating ($L_{z}=0$) as well as the rotating single-vortex state ($L_{z}=N$). For higher angular momenta, $L_{z}=2N$ and $L_{z}=3N$, only the strong interaction regime is considered. While the nearest-neighbor spacing distribution (NNSD) $P(s)$ and the ratios of consecutive level spacings distribution $P(r)$ are used to study the short-range correlations, the Dyson-Mehta $Δ_3$ statistic and the level number variance $Σ^2(L)$ are used to examine the long-range correlations. In the moderate interaction regime, the non-rotating system exhibits Poisson distribution, a characteristic of the regular energy spectra. In the strong interaction regime, the non-rotating system exhibits chaotic behavior signified by GOE distribution. Furthermore, in the rotating case for the single-vortex state ($L_{z} = N$) in the moderate interaction regime, the system exhibits signatures of weak chaos with some degree of regularity in the energy-level spectra. However, in the strong interaction regime for the rotating case with $L_{z} = N$, $2N$ and $3N$, the system exhibits strong chaotic behavior. The rotation is found to contribute to enhancement of chaotic behavior in the system for both the moderate and the strong interaction regimes. Our results of NNSD analysis are supported by the analysis of the ratios of consecutive level spacings distribution $P(r)$, which does not involve unfolding.

cond-mat.quant-gas↗

arXiv2025

Two rotating particles interacting via two-body Gaussian potential harmonically confined in two spatial dimensions

We study two spinless bosons interacting via two-body Gaussian potential subjected to an externally impressed rotation about an axis confined in a harmonic trap in two-spatial dimensions. We obtain a transcendental equation for the relative angular momentum $|m|$ state with various values of the two-body interaction range $σ$ and the two-body interaction strength $g_{2}$ to study the resulting energy spectrum and analyze the role of Hilbert space dimensions $\widetilde{N}$. We compare results for both attractive and repulsive interaction for $δ$-function potential and Gaussian potential for various values of interaction range. We study the effects of interaction parameters and relative angular momentum on the ground state energy and its various components, namely, kinetic energy, trap potential and interaction potential. For a given $|m|$ and non-interacting case, we observe that the ground state energy becomes independent of interaction range. However, for a given relative angular momentum and interaction strength $g_{2}>0$, there is an increase in ground state energy with an increase in interaction range. Below the interaction strength $g_{2}V(r)\leq -1$, ground state energy diverges to physically unacceptable negative-infinity for $|m|=0$ state. Further, for $|m|=1$, the ground state energy becomes independent of the interaction strength. For a $|m|$, we present a comparative study between the Gaussian interaction potential and the $δ$-function potential. Further, we observe that for a given $g_{2}$ and $|m|$, for $δ$-function potential {\it i.e.} $σ\to 0$, to achieve the convergence of ground state energy, we require a considerably large critical Hilbert space. Whereas, in the case of Gaussian interaction potential with $σ\to 1$, the ground state energy converges for a considerably small critical Hilbert space.

cond-mat.quant-gas↗

arXiv2025

Binding of holes and competing spin-charge order in simple and extended Hubbard model on cylindrical lattice: An exact diagonalization study

We investigate the binding of holes and the emergence of competing spin-charge order in the simple and extended Hubbard model using exact diagonalization on the 3x4 cylindrical lattice. For the simple Hubbard model (V=0), we find weakly bound hole pairing mediated by magnetic correlations at intermediate repulsive U, without any evidence of phase separation. Introducing nearest-neighbor interaction V reveals a rich phase diagram: attractive V drives multi-hole clustering and phase separation with localized magnetic quenching, while repulsive V stabilizes charge-density-wave (CDW) order that coexists with bound hole pairs within a modulated magnetic background. At strong coupling (U=10), the competition sharpens, with attractive V overcoming on-site repulsion to form magnetically quenched clusters and repulsive V producing robust CDW order that constrains pairing. Real-space analysis of spin and charge correlations provides microscopic evidence of distinct binding mechanisms -- phase separation versus correlation-mediated pairing -- depending on the sign and strength of intersite interaction V . Our results establish a comprehensive picture of how nonlocal Coulomb interactions reshape the landscape of hole-binding and collective order in correlated electron systems.

cond-mat.str-el↗

arXiv2025

Derivation of the deformed Heisenberg algebra from discrete spacetime

Although the deformation of the Heisenberg algebra by a minimal length has become a central tool in quantum gravity phenomenology, it has never been rigorously obtained and is often derived using heuristic reasoning. In this study, we move beyond the heuristic derivation of the deformed Heisenberg algebra and explicitly derive it using a model of discrete spacetime, which is motivated by quantum gravity. Initially, we investigate the effects of the leading order Planckian lattice corrections and demonstrate that they precisely match those suggested by the heuristic arguments commonly used in quantum gravity phenomenology. Furthermore, we rigorously obtain deformations from the higher-order Planckian lattice corrections. In contrast to the leading-order corrections, these higher-order corrections are model dependent. We select a specific model that breaks the rotational symmetry, as the importance of such rotational symmetry breaking lies in the relationship between CMB anisotropies and quantum gravitational effects. Based on the mathematical similarity of the Planckian lattice used here with the graphene lattice, we propose that graphene can serve as an analogue system for the study of quantum gravity. Finally, we examine the deformation of the covariant form of the Heisenberg algebra using a four-dimensional Euclidean lattice.

gr-qc↗

arXiv2023

Novel phases in rotating Bose-condensed gas: vortices and quantum correlation

We present the exact diagonalization study of rotating Bose-condensed gas interacting via finite-range Gaussian potential confined in a quasi-2D harmonic trap. The system of many-body Hamiltonian matrix is diagonalized in given subspaces of quantized total angular momentum to obtain the lowest-energy eigenstate employing the beyond lowest-Landau-level approximation. In the co-rotating frame, the quantum mechanical stability of angular momentum states is discussed for the existence of phase transition between the stable states of interacting system. Thereby analyzing the von Neumann entanglement entropy and degree of condensation provide the information about quantum phase correlation in the many-body states. Calculating the conditional probability distribution, we further probe the internal structure of quantum mechanically stable and unstable states. Much emphasis is put on finding the spatial correlation of bosonic atoms in the rotating system for the formation and entry of singly quantized vortices, and then organizing into canonical polygons with and without a central vortex at the trap center. Results are summarized in the form of a movie depicting the vortex patterns having discrete p-fold rotational symmetry with $p = 2,3,4,5,6$.

cond-mat.quant-gas↗

arXiv2020

Physical Properties of Nano-crystalline Sm$_2$CoMnO$_6$: structure, magnetism, spin-phonon coupling and dielectric study

Structural, magnetic, and dielectric properties of Sm$_2$CoMnO$_6$ have been studied. X-ray diffraction and Rietveld analysis show that the sample crystallizes in the monoclinic structure with \textit{P2$_1$/n} space group. Magnetic study shows that the sample undergoes a paramagnetic to ferromagnetic phase transition around $T_c$ $\sim$148 K and a low-temperature ordering of Sm$^{3+}$ moments. Temperature-dependent Raman study shows spin-phonon coupling present in this material. Dielectric response of Sm$_2$CoMnO$_6$ shows strong frequency dependence with large dispersion and large dielectric constant. Deviated from ideal Debye's model is found in impedance spectroscopy. Further, the activation energy obtained from tangent loss and complex impedance suggests that the relaxation process is due to electron hopping. AC conductivity has been studied to understand the conduction mechanism. The detailed analysis of AC conductivity shows that quantum mechanical tunneling of charges is involved in the conduction mechanism. The exponent obtained from AC conductivity measurements confirms the non-Debye's nature of Sm$_2$CoMnO$_6$.

cond-mat.str-el↗

arXiv2020

Ground state properties of trapped boson system with finite-range Gaussian repulsion: Exact diagonalization study

We use exact diagonalization to study an interacting system of $N$ spinless bosons with finite-range Gaussian repulsion, confined in a quasi-two-dimensional harmonic trap with and without an introduced rotation. The diagonalization of the Hamiltonian matrix using Davidson algorithm in subspaces of quantized total angular momentum $L_{z}$ is carried out to obtain the $N$-body lowest eigenenergy and eigenstate. To bring out the effect of quantum (Bose) statistics and consequent phase stiffness (rigidity) of the variationally obtained many-body wavefunction on various physical quantities, our study spans from few-body ($N=2$) to many-body ($N=16$) systems. Further, to examine the finite-range effect of the repulsive Gaussian potential on many-body ground state properties of the Bose-condensate, we obtain the lowest eigenstate, the critical angular velocity of single vortex state and the quantum correlation (measured) in terms of von Neumann entanglement entropy and degree of condensation. It is found that for small values of the range (measured by the parameter $σ$) of Gaussian potential, the ground state energy increases for few-boson ($2\le N\le 8$) systems but decreases for many-boson ($N>8$) systems. On the other hand for relatively large values of the range of Gaussian potential, the ground state energy exhibits a monotonic decrease, regardless of the number of bosons $N$. For a given $N$, there is found an optimal value of the range of Gaussian potential for which the first vortex (with $L_{z}=N$) nucleates at a lower value of the rotational angular velocity $Ω_{\bf c1}$ compared to the zero-range ($δ$-function) potential. Further, we observe that the inter-particle interaction and the introduced rotation are competing effects with latter being dominant over the former.

cond-mat.quant-gas↗

arXiv2020

Structure, magnetic and dielectric properties in nano-crystalline Yb$_2$CoMnO$_6$

Structural, magnetic and dielectric properties have been studied for Yb$_2$CoMnO$_6$. Nano-crystalline sample of Yb$_2$CoMnO$_6$ synthesized by sol-gel method and structural analysis shows that the sample crystallizes in monoclinic crystal structure with \textit{P2$_1$/n} phase group. To understand the charge state of Co, Mn and Yb we have performed the XPS study. Magnetic study shows that the sample undergoes a paramagnetic to ferromagnetic phase transition around $T_c$ $\sim$56 K and an additional magnetic ordering at a lower temperature around 14 K due to ordering of Yb$^{3+}$ magnetic ions. Temperature dependent Raman study reveals that spin-phonon interaction is present in this material. Further, we have studied the dielectric properties of this material. We observed that the material shows a relaxation behavior that obeys the thermally activated relaxation mechanism. Impedance spectroscopy reveals that the material shows non-Debye's behavior. AC conductivity study is performed to understand the conduction mechanics which involve the quantum mechanical tunneling phenomenon.

cond-mat.str-el↗

arXiv2019

Physical properties in nano-crystalline Ho$_2$CoMnO$_6$

3$d$ based double perovskite materials have received much attention in recent years due to their exotic magnetic structure and magneto-electric coupling. In this work we have prepared and studied the nano-crystalline sample of Ho$_2$CoMnO$_6$. Structural, magnetic, Raman and dielectric properties have been studied in detail. The structural analysis shows that the sample crystallize in monoclanic crystal structure with $\textit{P2$_1$/n}$ phase group. The X-ray photoelectron spectroscopy have been employed to confirms the charge state of cations presents in the material. Magnetic study shows that the sample undergoes a paramagnetic to ferromagnetic phase transition around $T_c$ $\sim$85 K. The isothermal magnetization measurements shows hysteresis curve hence confirm ferromagnetic behavior at low temperature. Temperature dependent Raman study reveals that there is spin phonon coupling in the sample marked by deviation in phonon mode from anharmonic behavior. Dielectric response of Ho$_2$CoMnO$_6$ shows the large dispersion and large dielectric constant. Impedance spectroscopy and electrical modulus study reveal that system shows deviation from ideal Debye model. AC conductivity have been studied as a function of both temperature and frequency. We found that the conduction mechanism is obeyed by Jonscher's model. The exponent factor $n$ is suggest that the material deviates from ideal Debye model.

cond-mat.str-el↗

arXiv2018

Critical behavior and magnetocaloric effect in ferromagnetic nano-crystalline Pr$_2$CoMnO$_6$

To understand the nature of magnetic phase transition in nano-crystalline Pr$_2$CoMnO$_6$, in present study we have investigated the critical behavior and magnetocaloric effect. To estimate the critical exponents, various methods have been adopted like; the modified Arrott plots (MAP), the Kouvel-Fisher method (KF) and the critical isotherm analysis. This material shows a second order type a paramagnetic (PM) to ferromagnetic (FM) phase transition around 160 K. The critical exponents obtained from modified Arrott plots are $β$ = 0.531, $γ$ = 0.935 and $T_c$ = 160 K. Kouvel-Fisher method gives the exponents as $β$ = 0.533$\pm$0.001 with $T_c$ = 160.72$\pm$0.03 K and $γ$ = 0.932$\pm$0.003 with $T_c$ = 160.15$\pm$0.05 K. The third exponent $δ$ = 2.763$\pm$0.005 obtained from critical isothermal is in agreement with Wisdom scaling rule. The estimated critical exponents do not exactly match with any established universality class, however, the deduced exponent values suggest that spin interaction in the present material is close to mean-field model which suggests the existence of long-range ferromagnetic order in nano-crystalline Pr$_2$CoMnO$_6$. The reliability of critical exponents is checked by universal scaling hypothesis on magnetic data across T$_c$. We have calculated the magnetic entropy change from magnetic data and found maximum value of -$Δ$S = 2.05 (J kg$^{-1}$ K$^{-1}$) for 50 kOe at 180 K. Moreover, field dependent change in magnetic entropy obeys scaling and also indicates that the magnetic interaction is close to mean-field type.

cond-mat.str-el↗

arXiv2018

Magnetic behavior, Griffiths phase and magneto-transport study in 3$d$ based nano-crystalline double perovskite Pr$_2$CoMnO$_6$

Double perovskite (DP) oxide material receive extensive research interest due to exciting physical properties with potential technological application. 3$d$ based DP oxides are promising for exciting physics like magnetodielectric, ferroelectric, Griffith phase etc., specially Co/Mn DPs are gaining much research interest. In this paper we present the study of magnetic phase and transport properties in nano-crystalline Pr$_2$CoMnO$_6$ a 3$d$ based double perovskite compound. This material shows a paramagnetic (PM) to ferromagnetic (FM) phase transition below 173 K marked by a rapid increase in magnetic moment due to spin ordering. We found divergence in inverse magnetic susceptibility ($χ$$^{-1}$) from Curie weiss behavior around 206 K which indicates the evolution of Griffiths phase before actual PM-FM transition. We found that the Griffiths phase suppressed with increasing applied magnetic filed. For the understanding of charge transport in this material we have measured temperature dependent electrical resistivity. Pr$_2$CoMnO$_6$ is a strong insulator where resistivity increase abruptly below magnetic phase transition. To understand the effect of magnetic field on transport behavior we have also measured the magnetoresistance (MR) at different temperatures. Sample shows the negative MR with maximum value $\sim$22 $\%$ under applied magnetic field of 50 kOe at 125 K. MR follows quadratic field dependency above $T_c$ however below $T_c$ the MR shows deviation from this field dependency at low field.

cond-mat.str-el↗