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Triangular Ising antiferromagnet through a fermionic lens, part 1: free energy, zero-temperature phases and spin-spin correlation

Amir Nourhani, Vincent H. Crespi, Paul E. Lammert

Published Oct 20, 2018• 0 citations
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Abstract

We develop a fermionic formulation of the triangular lattice Ising antiferromagnet (TIAFM) which is both calculationally convenient and intuitively appealing to imaginations steeped in conventional condensed matter physics. It is used to elucidate a variety of aspects of zero-temperature models. Cylindrical systems possess multiple distinguished by the number of circumferential satisfied bonds and by entropy density. On the plane, phases are labelled by densities of satisfied bonds of two different orientations. A local particle (semi)conservation law in the fermionic picture lies behind both these features as well as the classic power-law falloff of the spin-spin correlation function, which is also derived from the fermionic perspective.

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Triangular Ising antiferromagnet through a fermionic lens, part 2: information-theoretic aspects of zero-temperature states on cylinders

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arXiv:1810.08859•Oct 20, 2018•0 citations

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Abstract We present a treatment of the triangular lattice antiferromagnetic Ising model (TAFIM) based on a small number of elementary ideas common to statistical and solid-state physics. The TAFIM is represented as a reduced BCS model in one space, one (imaginary) time dimension. The representation is approximate for nonzero temperature, but allows quick derivation of asymptotically exact thermodynamic functions, and the divergence of the spin–spin correlation length. The fermionic representation is exact at zero temperature. We demonstrate the existence of a two-dimensional continuum of zero-temperature equilibrium macrostates characterized by satisfied bond fractions of the three different orientations, and calculate their entropy densities.

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A nominally two-dimensional spin model wrapped onto a cylinder can profitably be viewed, especially for long cylinders, as a one-dimensional chain. Each site of such a chain is a ring of spins with a complex state space. Traditional correlation functions are inadequate for the study of correlations in such a system and need to be replaced with something like mutual information. Being induced purely by frustration, the disorder of a cylindrical zero-temperature triangular Ising antiferromagnet (TIAFM) and attendant correlations have a chance of evading the consequences of the Perron-Frobenius theorem which describes and constrains correlations in thermally disordered one-dimensional systems. Correlations in such TIAFM systems and the aforementioned evasion are studied here through a fermionic representation. For cylindrical TIAFM models with open boundary conditions, we explain and derive the following characteristics of end-to-end mutual information: period-three oscillation of the decay length, halving of the decay length compared to what Perron-Frobenius predicts on the basis of transfer matrix eigenvalues, and subexponential decay-inverse square in the length-for certain systems.

Physical review. E•Apr 5, 2022•0 citations

Triangular antiferromagnetic Ising model with a transverse field at zero temperature

K. A. Penson, Rémi Jullien, Pierre M. Pfeuty

The ground-state properties of the triangular antiferromagnetic Ising model with Ising constant J in a transverse field h are investigated by a real-space renormalisation group method. For h=0, the known properties (energy=-J, finite entropy, no order) are recovered. For h not=0, the ground state remains highly degenerate, no order exists and the correlation function exhibits power-law behaviour up to (h/J)c approximately=1.41, where a phase transition takes place to a non-degenerate ground state. At h/J=(h/J)c, power-law behaviour is still observed for the correlation function but with a different exponent. Just above this region a gap Delta opens ( Delta approximately ((h/J)-(h/J)c)s with s=0.8) and the coherence length of the correlation function xi diverges as xi varies as ((h/J)-(h/J)c)-v with v=0.8. This corresponds to a three-dimensional classical model where h is replaced by the temperature. In this type of system, where the ground state has a degeneracy proportional to N2/3 it is predicted that a transition will occur at a finite T c to a low-temperature phase without order but with power-law behaviour for the correlation function, as for the classical XY model in two dimensions.

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Robustness of entropy plateaus: A case study of triangular Ising antiferromagnets

Owen Bradley, Chunhan Feng, Richard T. Scalettar +1 more

Residual entropy is a key feature associated with emergence in many-body systems. From a variety of frustrated magnets to the onset of spin-charge separation in Hubbard models and fermion-${Z}_{2}$-flux variables in Kitaev models, the freezing of one set of degrees of freedom and the establishment of local constraints are marked by a plateau in entropy as a function of temperature. Yet, with the exception of the rare-earth pyrochlore family of spin-ice materials, evidence for such plateaus is rarely seen in real materials, raising questions about their robustness. Following recent experimental findings of the absence of such plateaus in the triangular-lattice Ising antiferromagnet (TIAF) ${\mathrm{TmMgGaO}}_{4}$ by Li et al., we explore in detail the existence and rounding of entropy plateaus in TIAF. We use a transfer-matrix method to numerically calculate the properties of the system at different temperatures and magnetic fields, with further neighbor interactions and disorder. We find that temperature windows of entropy plateaus exist only when second-neighbor interactions are no more than a couple of percent of the nearest-neighbor ones, and they are also easily destroyed by disorder in the nearest-neighbor exchange variable, thereby explaining the challenge in observing such effects.

arXiv:1901.08182•Physical review. B./Physical review. B•Aug 20, 2019•7 citations

Zero-temperature properties of quantum spin models on the triangular lattice III: the Heisenberg antiferromagnet

Satoshi Fujiki

The calculation of two- and four-spin correlations of the [Formula: see text] Heisenberg antiferromagnet has been extended to an N = 21 site triangular lattice. The infinite-lattice ground state energy per bond is estimated to be E0/3NJ = −0.3678 ± 0.005 by fitting a quadratic in 1/N to the finite N data. The plaquette chirality order is slightly greater than in the XY antiferromagnet. The two-spin correlation is conjectured to decay as [Formula: see text].

Canadian Journal of Physics•May 1, 1987•48 citations
We develop a fermionic formulation of the triangular lattice Ising antiferromagnet (TIAFM) which is both calculationally convenient and intuitively appealing to imaginations steeped in conventional condensed matter physics. It is used to elucidate a variety of aspects of zero-temperature models. Cylindrical systems possess multiple distinguished by the number of circumferential satisfied bonds and by entropy density. On the plane, phases are labelled by densities of satisfied bonds of two different orientations. A local particle (semi)conservation law in the fermionic picture lies behind both these features as well as the classic power-law falloff of the spin-spin correlation function, which is also derived from the fermionic perspective.
A classical lattice spin model wrapped on a cylinder is profitably viewed as a chain of rings of spins. From that perspective, mutual information between ring configurations plays much the same role as spin-spin correlation functions in simpler settings. We study zero-temperature states of triangular lattice Ising antiferromagnet (TIAFM) systems from this point of view using a fermionic representation presented in a companion paper (Part 1). On infinite cylinders, ring-to-ring mutual information falls off asymptotically at a rate which decreases smoothly with cylinder circumference, but the end-to-end mutual information for finite cylinders depends strongly on the residue class modulo 3 of the circumference as well as on whether spin periodicity or antiperiodicity is imposed in the circumferential direction. In some cases, the falloff is only as the inverse square of the cylinder length. These features, puzzling within the original spin formulation, are easily understood and calculated within the fermionic formulation.
Abstract We present a treatment of the triangular lattice antiferromagnetic Ising model (TAFIM) based on a small number of elementary ideas common to statistical and solid-state physics. The TAFIM is represented as a reduced BCS model in one space, one (imaginary) time dimension. The representation is approximate for nonzero temperature, but allows quick derivation of asymptotically exact thermodynamic functions, and the divergence of the spin–spin correlation length. The fermionic representation is exact at zero temperature. We demonstrate the existence of a two-dimensional continuum of zero-temperature equilibrium macrostates characterized by satisfied bond fractions of the three different orientations, and calculate their entropy densities.
A nominally two-dimensional spin model wrapped onto a cylinder can profitably be viewed, especially for long cylinders, as a one-dimensional chain. Each site of such a chain is a ring of spins with a complex state space. Traditional correlation functions are inadequate for the study of correlations in such a system and need to be replaced with something like mutual information. Being induced purely by frustration, the disorder of a cylindrical zero-temperature triangular Ising antiferromagnet (TIAFM) and attendant correlations have a chance of evading the consequences of the Perron-Frobenius theorem which describes and constrains correlations in thermally disordered one-dimensional systems. Correlations in such TIAFM systems and the aforementioned evasion are studied here through a fermionic representation. For cylindrical TIAFM models with open boundary conditions, we explain and derive the following characteristics of end-to-end mutual information: period-three oscillation of the decay length, halving of the decay length compared to what Perron-Frobenius predicts on the basis of transfer matrix eigenvalues, and subexponential decay-inverse square in the length-for certain systems.
The ground-state properties of the triangular antiferromagnetic Ising model with Ising constant J in a transverse field h are investigated by a real-space renormalisation group method. For h=0, the known properties (energy=-J, finite entropy, no order) are recovered. For h not=0, the ground state remains highly degenerate, no order exists and the correlation function exhibits power-law behaviour up to (h/J)c approximately=1.41, where a phase transition takes place to a non-degenerate ground state. At h/J=(h/J)c, power-law behaviour is still observed for the correlation function but with a different exponent. Just above this region a gap Delta opens ( Delta approximately ((h/J)-(h/J)c)s with s=0.8) and the coherence length of the correlation function xi diverges as xi varies as ((h/J)-(h/J)c)-v with v=0.8. This corresponds to a three-dimensional classical model where h is replaced by the temperature. In this type of system, where the ground state has a degeneracy proportional to N2/3 it is predicted that a transition will occur at a finite T c to a low-temperature phase without order but with power-law behaviour for the correlation function, as for the classical XY model in two dimensions.
Residual entropy is a key feature associated with emergence in many-body systems. From a variety of frustrated magnets to the onset of spin-charge separation in Hubbard models and fermion-Z2{Z}_{2}Z2​-flux variables in Kitaev models, the freezing of one set of degrees of freedom and the establishment of local constraints are marked by a plateau in entropy as a function of temperature. Yet, with the exception of the rare-earth pyrochlore family of spin-ice materials, evidence for such plateaus is rarely seen in real materials, raising questions about their robustness. Following recent experimental findings of the absence of such plateaus in the triangular-lattice Ising antiferromagnet (TIAF) TmMgGaO4{\mathrm{TmMgGaO}}_{4}TmMgGaO4​ by Li et al., we explore in detail the existence and rounding of entropy plateaus in TIAF. We use a transfer-matrix method to numerically calculate the properties of the system at different temperatures and magnetic fields, with further neighbor interactions and disorder. We find that temperature windows of entropy plateaus exist only when second-neighbor interactions are no more than a couple of percent of the nearest-neighbor ones, and they are also easily destroyed by disorder in the nearest-neighbor exchange variable, thereby explaining the challenge in observing such effects.