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Symmetry

Absence of Spontaneous Spin Symmetry Breaking in 1D and 2D Quantum Ferromagnetic Systems with Bilinear and Biquadratic Exchange Interactions

Roberto Zivieri

Published Dec 11, 2020• 6 citations

Abstract

Some measurements have shown that the second-order exchange interaction is non-negligible in ferromagnetic compounds whose microscopic interactions are described by means of half-odd integer quantum spins. In these spin systems the ground state is either ferromagnetic or antiferromagnetic when the bilinear exchange interaction is dominant. Instead, in ferromagnetic systems characterized by bilinear and biquadratic exchange interactions of comparable magnitude, the energy minimum occurs when spins are in a canting ground-state. To this aim, a one-dimensional (1D) quantum spin chain and a two-dimensional (2D) lattice of quantum spins subjected to periodic boundary conditions are modeled via the generalized quantum Heisenberg Hamiltonian containing, in addition to the isotropic and short-range bilinear exchange interaction of the Heisenberg type, a second-order interaction, the isotropic and short-range biquadratic exchange interaction between nearest-neighbors quantum spins. For these 1D and 2D quantum systems a generalization of the Mermin–Wagner–Hohenberg theorem (also known as Mermin–Wagner–Berezinksii or Coleman theorem) is given. It is demonstrated, by means of quantum statistical arguments, based on Bogoliubov’s inequality, that, at any finite temperature, (1) there is absence of long-range order and that (2) the law governing the vanishing of the order parameter is the same as in the bilinear case for both 1D and 2D quantum ferromagnetic systems. The physical implications of the absence of a spontaneous spin symmetry breaking in 1D spin chains and 2D spin lattices modeled via a generalized quantum Heisenberg Hamiltonian are discussed.

DOI: 10.3390/sym12122061

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Some measurements have shown that the second-order exchange interaction is non-negligible in ferromagnetic compounds whose microscopic interactions are described by means of half-odd integer quantum spins. In these spin systems the ground state is either ferromagnetic or antiferromagnetic when the bilinear exchange interaction is dominant. Instead, in ferromagnetic systems characterized by bilinear and biquadratic exchange interactions of comparable magnitude, the energy minimum occurs when spins are in a canting ground-state. To this aim, a one-dimensional (1D) quantum spin chain and a two-dimensional (2D) lattice of quantum spins subjected to periodic boundary conditions are modeled via the generalized quantum Heisenberg Hamiltonian containing, in addition to the isotropic and short-range bilinear exchange interaction of the Heisenberg type, a second-order interaction, the isotropic and short-range biquadratic exchange interaction between nearest-neighbors quantum spins. For these 1D and 2D quantum systems a generalization of the Mermin–Wagner–Hohenberg theorem (also known as Mermin–Wagner–Berezinksii or Coleman theorem) is given. It is demonstrated, by means of quantum statistical arguments, based on Bogoliubov’s inequality, that, at any finite temperature, (1) there is absence of long-range order and that (2) the law governing the vanishing of the order parameter is the same as in the bilinear case for both 1D and 2D quantum ferromagnetic systems. The physical implications of the absence of a spontaneous spin symmetry breaking in 1D spin chains and 2D spin lattices modeled via a generalized quantum Heisenberg Hamiltonian are discussed.

Abstract The Mermin-Wagner theorem states that spontaneous continuous symmetry breaking is prohibited in systems with short-range interactions at spatial dimension D ≤ 2. For long-range interactions with a power-law form (1/ r α ), the theorem further forbids ferromagnetic or antiferromagnetic order at finite temperature when α ≥ 2 D . However, the situation for α ∈ (2, 4) at D = 2 is not covered by the theorem. To address this, we conduct large-scale quantum Monte Carlo simulations and field theoretical analysis. Our findings show spontaneous breaking of S U (2) symmetry in the ferromagnetic Heisenberg model with 1/ r α -form long-range interactions at D = 2. We determine critical exponents through finite-size analysis for α < 3 (above the upper critical dimension with Gaussian fixed point) and 3 ≤ α < 4 (below the upper critical dimension with non-Gaussian fixed point). These results reveal new critical behaviors in 2D long-range Heisenberg models, encouraging further experimental studies of quantum materials with long-range interactions beyond the Mermin-Wagner theorem’s scope.

The Mermin-Wagner theorem is strengthened so as to rule out magnetic long-range order at

T>0

in one- or two-dimensional Heisenberg and XY systems with long-range interactions decreasing as ${R}^{\ensuremath{-}\ensuremath{\alpha}}$ with a sufficiently large exponent $\ensuremath{\alpha}$. For oscillatory interactions, ferromagnetic long-range order at

T>0

is ruled out if $\ensuremath{\alpha}\ensuremath{\ge}1(D\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1)$ or $\ensuremath{\alpha}>5/2(D\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2)$. For systems with monotonically decreasing interactions, ferro- or antiferromagnetic long-range order at

T>0

is ruled out if $\ensuremath{\alpha}\ensuremath{\ge}2D$.

Spontaneous symmetry breaking of 2D isotropic Heisenberg magnet is restricted by Mermin-Wagner theorem at any finite temperature in presence of short-range exchange interaction.Kosterlitz and Thouless using XY spin model showed that how an order state could developed in 2D spin system in presence of short range isotropic interaction.Very recent discovery of several van der waals magnet revised and redefined our understanding on 2D Heisenberg magnet and its ground state properties.After a rigorous and careful study of several 2D magnetic material we have realized from both experimentally and numerically that the finite size of a 2D system has great impact on the ground state symmetry breaking.Because of that finite size effect more often an anisotropic residual magnetic moment is generated and trigger the spontaneous symmetry breaking at finite temperature(T) and even only presence of short-range interaction we observed the phase transition of that Heisenberg spin system.In this present work we have shown the basic role of finite size,anisotropy during the symmetry breaking of 2D Heisenberg XY magnet.Here we have simulated Kosterlitz-Thouless transition using classical Monte-carlo simulation and study the effect of anisotropy during the phase transition.We presented the behaviour of different thermodynamic properties of 2D XY spin model system during the Kosterlitz-Thouless(KT) transition.The generic characteristic of KT transition which make it distinct from other critical phenomena is that the peak of heat capacity is not diverging with increase of system size rather peak is decreasing with the increasing of system size near at transition temperature.Here we are observing that behaviour in our present simulation and that specific behaviour help us for classifying the present transition as Kosterlitz-Thouless(KT) transition.

Quantum corrections of the biquadratic interaction in the 1D spin-1/2 frustrated ferromagnetic Heisenberg model are studied. The biquadratic interaction for spin-1/2 chains is eliminated and transformed to the quadratic interaction. Doing a numerical experiment, new insight as to how the classical phases get modified on the inclusion of quantum fluctuations is provided. Observed results suggest the existence of an intermediate region in the ground state phase diagram of the frustrated ferromagnetic spin-1/2 chains with combination of dimer and chiral orders. In addition, from the quantum entanglement view point, differences between quantum phases are also obtained. The nearest neighbor spins never be entangled in the frustrated ferromagnetic chains but are entangled up to the Majumdar-Ghosh point in the frustrated antiferromagnetic chains. On the other hand, the next nearest neighbor spins in the mentioned intermediate region are entangled.

The Hohenberg-Mermin-Wagner theorem states that there is no spontaneous breaking of continuous internal symmetries in spatial dimensions d≤2 at finite temperature. At zero temperature, the quantum-to-classical mapping further implies the absence of such symmetry breaking in one dimension, which is also known as Coleman's theorem in the context of relativistic quantum field theories. One route to violate this "folklore" is requiring an order parameter to commute with a Hamiltonian, as in the classic example of the Heisenberg ferromagnet and its variations. However, a systematic understanding has been lacking. In this Letter, we propose a family of one-dimensional models that display spontaneous breaking of a U(1) symmetry at zero temperature, where the order parameter does not commute with the Hamiltonian. While our models can be deformed continuously within the same phase, there exist symmetry-preserving perturbations that render the observed symmetry breaking fragile. We argue that a more general condition for this behavior is that the Hamiltonian is frustration-free.

We study quantum phase transitions in Heisenberg antiferromagnetic chains with a staggered power-law decaying long-range interactions. Employing the density-matrix renormalization group (DMRG) algorithm and the fidelity susceptibility as the criticality measure, we establish more accurate values of quantum critical points than the results obtained from the spin-wave approximation, quantum Monte Carlo, and DMRG in literatures. The deviation is especially evident for strong long-range interactions. We extend isotropic long-range interactions to the anisotropic cases and find that kaleidoscope of quantum phases emerge from the interplay of anisotropy of the long-range exchange interaction and symmetry breaking. We demonstrate nonfrustrating long-range interactions induce the true long-range order in Heisenberg antiferromagnetic chains with a continuous symmetry breaking, lifting the restrictions imposed by the Mermin-Wagner theorem.