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The Berezinskii-Kosterlitz-Thouless (BKT) transition in magnetic systems is an intriguing phenomenon, and estimating the BKT transition temperature is a long-standing problem. In this work, we explore anisotropic classical Heisenberg XY and XXZ models with ferromagnetic exchange on a square lattice and antiferromagnetic exchange on a triangular lattice using an unsupervised machine learning approach called principal component analysis (PCA). The earlier PCA studies of the BKT transition temperature (TBKT) using the vorticities as input fail to give any conclusive results, whereas, in this work, we show that the proper analysis of the first principal component-temperature curve can estimateTBKTwhich is consistent with the existing literature. This analysis works well for the anisotropic classical Heisenberg with a ferromagnetic exchange on a square lattice and for frustrated antiferromagnetic exchange on a triangular lattice. The classical anisotropic Heisenberg antiferromagnetic model on the triangular lattice has two close transitions: theTBKTand Ising-like phase transition for chirality atTc, and it is difficult to separate these transition points. It is also noted that using the PCA method and manipulation of their first principal component not only makes the separation of transition points possible but also determines transition temperature., (© 2024 IOP Publishing Ltd.)

Frustration-driven quantum fluctuation leads to many exotic phases in the ground state (GS) and the study of these quantum phase transitions is one of the most challenging areas of research in condensed matter physics. We study a frustrated HeisenbergJ1-J2model of spin-1/2 chain with nearest exchange interaction J 1 and next nearest exchange interaction J 2 using the principal component analysis (PCA) which is an unsupervised machine learning technique. In this method most probable spin configurations (MPSCs) of GS and first excited state (FES) for differentJ2/J1are used as the input in PCA to construct the covariance matrix. The 'quantified principal component'p1(J2/J1)of the largest eigenvalue of the covariance matrix is calculated and it is shown that the nature and variation ofp1(J2/J1)can accurately predict the phase transitions and degeneracies in the GS. Thep1(J2/J1)calculated from the MPSC of GS only exhibits the signature of degeneracies in the GS, whereas,p1(J2/J1)calculated from the MPSC of FES captures the gapless spin liquid (GSL)-dimer phase transition as well as all the degeneracies of the model system. We show that the jump inp1(J2/J1)of FES atJ2/J1≈0.241, indicates the GSL-dimer phase transition, whereas its kinks give the signature of the GS degeneracies. The scatter plot of the first two principal components of FES shows distinct band formation for different phases. The MPSCs are obtained by using an iterative variational method (IVM) which gives the approximate eigenvalues and eigenvectors., (© 2023 IOP Publishing Ltd.)

Magnetic topological semimetals (TSMs) with broken time-reversal symmetry are very rare and have drawn significant attention in condensed matter physics due to their numerous intriguing topological properties. Among these various magnetic TSMs, Co 2 -based full Heusler compounds are of current interest, since a few of these materials exhibit Weyl and nodal fermions in their topological band structure. In this work, we report a comprehensive study of anomalous Hall effect (AHE) in the ferromagnetic full Heusler compound Co 2 VAl. Recent studies indicate that the intrinsic AHE is closely related to the Berry curvature of the occupied electronic Bloch states. The present study of Co 2 VAl attempts to understand and explore the possibility of topology-induced AHE. The anomalous Hall resistivityρxyAis observed to scale quadratically with the longitudinal resistivity ρ xx . Our experimental results also reveal that the anomalous Hall conductivity (AHC) is ∼85 cm -1 at 2 K with an intrinsic contribution of ∼75.6 S cm -1 , and is nearly insensitive to temperature. The first principle calculations note that the Berry curvature originated from a gapped nodal line and symmetry-protected Weyl nodes near the Fermi level (EF) is the main source of AHE in this compound. Thus, this investigation on Co 2 VAl discloses that it is a ferromagnetic Weyl and nodal-line TSM. The theoretically calculated AHC is in well agreement with the experimentally obtained AHC., (© 2022 IOP Publishing Ltd.)

We study a frustrated two-leg spin ladder with alternate isotropic Heisenberg and Ising rung exchange interactions, whereas, interactions along legs and diagonals are Ising-type. All the interactions in the ladder are anti-ferromagnetic in nature and induce frustration in the system. This model shows four interesting quantum phases: (i) stripe rung ferromagnetic (SRFM), (ii) stripe rung ferromagnetic with edge singlet (SRFM-E), (iii) anisotropic antiferromagnetic (AAFM), and (iv) stripe leg ferromagnetic (SLFM) phase. We construct a quantum phase diagram for this model and show that in stripe rung ferromagnet (SRFM), the same type of sublattice spins (either isotropic S -type or discrete anisotropic σ -type spins) are aligned in the same direction. Whereas, in anisotropic antiferromagnetic phase, both S and σ -type of spins are anti-ferromagnetically aligned with each other, two nearest S spins along the rung form an anisotropic singlet bond whereas two nearest σ spins form an Ising bond. In large Heisenberg rung exchange interaction limit, spins on each leg are ferromagnetically aligned, but spins on different legs are anti-ferromagnetically aligned. The thermodynamic quantities like specific heat C v ( T ), magnetic susceptibility χ ( T ) and thermal entropy S ( T ) are also calculated using the transfer matrix method for various phases. The magnetic gap in the SRFM and the SLFM can be noticed from χ ( T ) and C v ( T ) curves., (© 2021 IOP Publishing Ltd.)

A Heisenberg spin- s chain with alternating ferromagnetic (FM) (-J1F<0) and antiferromagnetic (J1A>0) nearest-neighbor (NN) interactions, exhibits the dimer and spin-2 s Haldane phases in the limitsJ1F/J1A→0andJ1F/J1A→∞respectively. These two phases are understood to be topologically equivalent. Induction of the frustration through the next NN FM interaction (-J2F<0) produces a very rich quantum phase diagram. With frustration, the whole phase diagram is divided into a FM and a nonmagnetic (NM) phase. For s = 1/2, the full NM phase is seen to be of Haldane-dimer type, but for s > 1/2, a spiral phase comes between the FM and the Haldane-dimer phases. The study of a suitably defined string-order parameter and spin-gap at the phase boundary indicates that the Haldane-dimer and spiral phases have different topological characters. We also find that, along theJ2F=12J1Fline in the NM phase, an NN dimer state is the exact groundstate, providedJ1A>JC=κJ1Fwhere κ ⩽ s + h for applied magnetic field h . Without magnetic field, the position of J C is on the FM-NM phase boundary when s = 1/2, but for s > 1/2, the location of J C is on the phase separation line between the Haldane-dimer and spiral phases., (© 2020 IOP Publishing Ltd.)

The spin-1/2 chain with isotropic exchange J1, J2 > 0 between first and second neighbors is frustrated for either sign of J1 and has a singlet ground state (GS) for J1/J2 ⩾ -4. Its rich quantum phase diagram supports gapless, gapped, commensurate (C), incommensurate (IC) and other phases. Critical points J1/J2 are evaluated using exact diagonalization and density matrix renormalization group calculations. The wave vector qG of spin correlations is related to GS degeneracy and obtained as the peak of the spin structure factor S(q). Variable qG indicates IC phases in two J1/J2 intervals, [-4, - 1.24] and [0.44, 2], and a C-IC point at J1/J2 = 2. The decoupled C phase in [-1.24, 0.44] has constant qG = π/2, nondegenerate GS, and a lowest triplet state with broken spin density on sublattices of odd and even numbered sites. The lowest triplet and singlet excitations, E m and Eσ, are degenerate in finite systems at specific frustration J1/J2. Level crossing extrapolates in the thermodynamic limit to the same critical points as qG. The S(q) peak diverges at qG = π in the gapless phase with J1/J2 > 4.148 and quasi-long-range order (QLRO(π)). S(q) diverges at ±π/2 in the decoupled phase with QLRO(π/2), but is finite in gapped phases with finite-range correlations. Numerical results and field theory agree at small J2/J1 but disagree for the decoupled phase with weak exchange J1 between sublattices. Two related models are summarized: one has an exact gapless decoupled phase with QLRO(π/2) and no IC phases; the other has a single IC phase without a decoupled phase in between.

The spin-1/2 chain with isotropic Heisenberg exchange J1, J2  >  0 between first and second neighbors is frustrated for either sign of J1. Its quantum phase diagram has critical points at fixed J1/J2 between gapless phases with nondegenerate ground state (GS) and quasi-long-range order (QLRO) and gapped phases with doubly degenerate GS and spin correlation functions of finite range. In finite chains, exact diagonalization (ED) estimates critical points as level crossing of excited states. GS spin correlations enter in the spin structure factor S(q) that diverges at wave vector qm in QLRO(q(m)) phases with periodicity 2π/q(m) but remains finite in gapped phases. S(q(m)) is evaluated using ED and density matrix renormalization group (DMRG) calculations. Level crossing and the magnitude of S(q(m)) are independent and complementary probes of quantum phases, based respectively on excited and ground states. Both indicate a gapless QLRO(π/2) phase between  -1.2  <  J1/|J2|  <  0.45. Numerical results and field theory agree well for quantum critical points at small frustration J2 but disagree in the sector of weak exchange J1 between Heisenberg antiferromagnetic chains on sublattices of odd and even-numbered sites.