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arXiv:cond-mat/0604649 · Journal of Physics A Mathematical and General

Linear independence of localized magnon states

Heinz-Jürgen Schmidt, Johannes Richter, Roderich Moessner

Published Aug 9, 2006• 50 citations
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Abstract

At the magnetic saturation field, certain frustrated lattices have a class of states known as "localized multi-magnon states" as exact ground states. The number of these states scales exponentially with the number $N$ of spins and hence they have a finite entropy also in the thermodynamic limit $N\to \infty$ provided they are sufficiently linearly independent. In this article we present rigorous results concerning the linear dependence or independence of localized magnon states and investigate special examples. For large classes of spin lattices including what we called the orthogonal type and the isolated type as well as the kagomé, the checkerboard and the star lattice we have proven linear independence of all localized multi-magnon states. On the other hand the pyrochlore lattice provides an example of a spin lattice having localized multi-magnon states with considerable linear dependence.

DOI
10.1088/0305-4470/39/34/006

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At the magnetic saturation field, certain frustrated lattices have a class of states known as "localized multi-magnon states" as exact ground states. The number of these states scales exponentially with the number NNN of spins and hence they have a finite entropy also in the thermodynamic limit N→∞N\to \inftyN→∞ provided they are sufficiently linearly independent. In this article we present rigorous results concerning the linear dependence or independence of localized magnon states and investigate special examples. For large classes of spin lattices including what we called the orthogonal type and the isolated type as well as the kagomé, the checkerboard and the star lattice we have proven linear independence of all localized multi-magnon states. On the other hand the pyrochlore lattice provides an example of a spin lattice having localized multi-magnon states with considerable linear dependence.
Abstract. At the magnetic saturation field, certain frustrated lattices have a class of states known as “localized multi-magnon states ” as exact ground states. The number of these states scales exponentially with the number N of spins and hence they have a finite entropy also in the thermodynamic limit N → ∞ provided they are sufficiently linearly independent. In this article we present rigorous results concerning the linear dependence or independence of localized magnon states and investigate special examples. For large classes of spin lattices including what we called the orthogonal type and the isolated type as well as the kagomé, the checkerboard and the star lattice we have proven linear independence of all localized multi-magnon states. On the other hand the pyrochlore lattice provides an example of a spin lattice having localized multi-magnon states with considerable linear dependence. ‡ To whom correspondence should be addressed (hschmidt@uos.de) Localized magnon states 2 1. Introduction and
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.
For a class of highly frustrated antiferromagnetic quantum spin lattices the ground state exhibits a huge degeneracy in high magnetic fields due to the existence of localized magnon states. For some of these spin lattices (in particular, the 1D dimer-plaquette, sawtooth and kagom\'elike chains as well as the 2D kagom\'e lattice) we calculate rigorously the ground-state entropy at the saturation field. We find that the ground-state entropy per site remains finite at saturation. This residual ground-state entropy produces a maximum in the field dependence of the isothermal entropy at low temperatures. By numerical calculation of the field dependence of the low-temperature entropy for the sawtooth chain we find that the enhancement of isothermal entropy is robust against small deviations in exchange constants. Moreover, the effect is most pronounced in the extreme quantum case of spin 12\frac{1}{2}21​.
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 clarify the existence of several magnetization plateaus for the kagome S=12S=\frac{1}{2}S=21​ antiferromagnetic Heisenberg model in a magnetic field. Using approximate or exact localized magnon eigenstates, we are able to describe in a similar manner the plateau states that occur for magnetization per site m=13m=\frac{1}{3}m=31​, 59\frac{5}{9}95​, and 79\frac{7}{9}97​ of the saturation value. These results are confirmed using large-scale exact diagonalization on lattices up to 63 sites.
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.