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arXiv:2008.10614 · Condensed Matter Physics

Loop-gas description of the localized-magnon states on the kagome lattice with open boundary conditions

A. Honecker, Richter, Schnack, Wietek

Published Dec 1, 2020• 6 citations

Abstract

The high-field regime of the spin-s XXZ antiferromagnet on the kagome lattice gives rise to macroscopically degenerate ground states thanks to a completely flat lowest single-magnon band.The corresponding excitations can be localized on loops in real space and have been coined "localized magnons".Thus, the description of the many-body ground states amounts to characterizing the allowed classical loop configurations and eliminating the quantum mechanical linear relations between them.Here, we investigate this loop-gas description on finite kagome lattices with open boundary conditions and compare the results with exact diagonalization for the spin-1/2 XY model on the same lattice.We find that the loop gas provides an exact account of the degenerate ground-state manifold while a hard-hexagon description misses contributions from nested loop configurations.The densest packing of the loops corresponds to a magnon crystal that according to the zero-temperature magnetization curve is a stable ground state of the spin-1/2 XY model in a window of magnetic fields of about 4% of the saturation field just below this saturation field.We also present numerical results for the specific heat obtained by the related methods of thermal pure quantum (TPQ) states and the finite-temperature Lanczos method (FTLM).For a field in the stability range of the magnon crystal, one finds a low-temperature maximum of the specific heat that corresponds to a finite-temperature phase transition into the magnon crystal at low temperatures.

DOI: 10.5488/cmp.23.43712

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The high-field regime of the spin-s XXZ antiferromagnet on the kagome lattice gives rise to macroscopically degenerate ground states thanks to a completely flat lowest single-magnon band.The corresponding excitations can be localized on loops in real space and have been coined "localized magnons".Thus, the description of the many-body ground states amounts to characterizing the allowed classical loop configurations and eliminating the quantum mechanical linear relations between them.Here, we investigate this loop-gas description on finite kagome lattices with open boundary conditions and compare the results with exact diagonalization for the spin-1/2 XY model on the same lattice.We find that the loop gas provides an exact account of the degenerate ground-state manifold while a hard-hexagon description misses contributions from nested loop configurations.The densest packing of the loops corresponds to a magnon crystal that according to the zero-temperature magnetization curve is a stable ground state of the spin-1/2 XY model in a window of magnetic fields of about 4% of the saturation field just below this saturation field.We also present numerical results for the specific heat obtained by the related methods of thermal pure quantum (TPQ) states and the finite-temperature Lanczos method (FTLM).For a field in the stability range of the magnon crystal, one finds a low-temperature maximum of the specific heat that corresponds to a finite-temperature phase transition into the magnon crystal at low temperatures.

We investigate the spin-1 kagome-lattice Heisenberg antiferromagnet using large-scale Lanczos diagonalization and the finite-temperature Lanczos method. The zero-temperature magnetization process exhibits plateaus at m = 0 , 1 / 3 , 7 / 9 , and 8 / 9 , where m is the normalized magnetization. The m = 0 plateau is identified as a trimer valence-bond-crystal state, while the high-field plateaus at m = 7 / 9 and 8 / 9 are identified as magnon crystals. In particular, the m = 8 / 9 plateau corresponds to the exact localized-magnon crystal state. A smoothed zero-temperature magnetization curve constructed using the Gaussian-kernel smoothing method indicates magnetization jumps at the lower-field edge of the m = 1 / 3 plateau and at the upper-field edges of the m = 7 / 9 and 8 / 9 plateaus. At finite temperatures, the specific heat exhibits a double-peak structure with peaks around T / J ≃ 0.1 and T / J ≃ 1.1 , and the low-temperature peak may be related to trimer valence-bond-crystal ordering. The finite-temperature magnetization curves show that the m = 1 / 3 plateau remains visible at low temperatures, whereas the high-field plateaus are rapidly smeared out by thermal effects. These results provide benchmark data for thermodynamic and high-field magnetization measurements in candidate spin-1 kagome-lattice materials.

Low-energy degrees of freedom of a spin-

1∕2

kagom\'e antiferromagnet in the vicinity of the saturation field are mapped to a hard-hexagon model on a triangular lattice. The latter model is exactly solvable. The presented mapping allows us to obtain the quantitative description of the magnetothermodynamics of a quantum kagom\'e antiferromagnet up to exponentially small corrections as well as predict the critical behavior for the transition into a magnon crystal state. Analogous mapping is presented for the sawtooth chain, which is mapped onto a model of classical hard dimers on a chain.

Plateaus can be observed in the zero-temperature magnetization curve of quantum spin systems at rational values of the magnetization. In one dimension, the appearance of a plateau is controlled by a quantization condition for the magnetization which involves the length of the local spin and the volume of a translational unit cell of the ground state. We discuss examples of geometrically frustrated quantum spin systems with large (in general unbounded) periodicities of spontaneous breaking of translational symmetry in the ground state. In two dimensions, we discuss the square, triangular and Kagomé lattices using exact diagonalization (ED) for up to N = 40 sites. For the spin-1/2 XXZ model on the triangular lattice we study the nature and stability region of a plateau at one third of the saturation magnetization. The Kagomé lattice gives rise to particularly rich behaviour with several plateaus in the magnetization curve and a jump due to local magnon excitations just below saturation.

We consider a spin-[Formula: see text] kagome-like chain with competing ferro- and antiferromagnetic anisotropic exchange interactions. The ground state phase diagram of this model consists of the ferromagnetic and ferrimagnetic phases. We study the ground state and the low-temperature properties on the phase boundary between these phases. The ground state on this phase boundary is macroscopically degenerate and consists of localized magnon states. We calculate the ground state degeneracy and corresponding residual entropy. The spontaneous magnetization has a jump on the phase boundary confirming the first-order type of the phase transition. In the limit of a strong anisotropy, the spectrum of the low-energy excitations has multi-scale structure governing the peculiar features of the specific heat behavior.

We consider the S = 1 / 2 antiferromagnetic Heisenberg model on a frustrated kagome-lattice bilayer with strong nearest-neighbor interlayer coupling and examine its low-temperature magnetothermodynamics using a mapping onto a rhombi gas on the kagome lattice. Besides, we use finite-size numerics to illustrate the validity of the classical lattice-gas description. Among our findings there are (i) the absence of an order-disorder phase transition and (ii) the sensitivity of the specific heat at low temperatures to the shape of the system just below the saturation magnetic field even in the thermodynamic limit.

For a class of frustrated spin lattices including the Kagomé lattice we construct exact eigenstates consisting of several independent, localized one-magnon states and argue that they are ground states for high magnetic fields. If the maximal number of local magnons scales with the number of spins in the system, which is the case for the Kagomé lattice, the effect persists in the thermodynamic limit and gives rise to a macroscopic jump in the zero-temperature magnetization curve just below the saturation field. The effect decreases with increasing spin quantum number and vanishes in the classical limit. Thus it is a true macroscopic quantum effect.