^{1}, J. Richter

^{2,a)}, A. Honecker

^{3}and H.-J. Schmidt

^{4}

### Abstract

The purpose of the present paper is twofold. On the one hand, we review some recent studies on the low-temperature strong-field thermodynamic properties of frustrated quantum spin antiferromagnets which admit the so-called localized-magnon eigenstates. On the other hand, we provide some complementary new results. We focus on the linear independence of the localized-magnon states, the estimation of their degeneracy with the help of auxiliary classical lattice-gas models, and the analysis of the contribution of these states to thermodynamics.

O. D. and J. R. acknowledge the kind hospitality of the MPIPKS-Dresden in the end of 2006. O. D. is indebted to Magdeburg University and to Wroclaw University for their hospitality in the autumn of 2006. A part of the numerical calculations was performed using Jörg Schulenburg’s *spinpack*. We would like to thank M. E. Zhitomirsky for useful discussions and comments on this manuscript.

I. INTRODUCTION

II. LOCALIZED-MAGNON STATES

III. LINEAR INDEPENDENCE AND COMPLETENESS

IV. LOW-TEMPERATURE STRONG-FIELD THERMODYNAMICS. LATTICE-GAS DESCRIPTION

A. Hard-monomer universality class

B. One-dimensional hard-dimer universality class

C. Two-dimensional lattice gases, hard-square universality class

D. Region of validity

V. CONCLUSIONS

### Key Topics

- Magnons
- 56.0
- Ground states
- 32.0
- Diamond
- 23.0
- Antiferromagnetism
- 18.0
- Classical spin models
- 16.0

## Figures

Three examples of one-dimensional spin lattices admitting localized magnons: the diamond chain (hard-monomer universality class) (a), the frustrated two-leg ladder (one-dimensional hard-dimer universality class) (b), the kagomé-like chain, originally introduced in Ref. 29 (one-dimensional hard-dimer universality class) (c).

Three examples of one-dimensional spin lattices admitting localized magnons: the diamond chain (hard-monomer universality class) (a), the frustrated two-leg ladder (one-dimensional hard-dimer universality class) (b), the kagomé-like chain, originally introduced in Ref. 29 (one-dimensional hard-dimer universality class) (c).

Three examples of two-dimensional spin lattices admitting localized magnons: the kagomé lattice (hard-hexagon universality class) (a), the checkerboard lattice (large-hard-square universality class) (b), the frustrated bilayer lattice (hard-square universality class) (c). We also show auxiliary lattice-gas models (hard hexagons on a triangular lattice, large hard squares on a square lattice, and hard squares on a square lattice) which describe low-energy degrees of freedom of the spin models in strong magnetic fields.

Three examples of two-dimensional spin lattices admitting localized magnons: the kagomé lattice (hard-hexagon universality class) (a), the checkerboard lattice (large-hard-square universality class) (b), the frustrated bilayer lattice (hard-square universality class) (c). We also show auxiliary lattice-gas models (hard hexagons on a triangular lattice, large hard squares on a square lattice, and hard squares on a square lattice) which describe low-energy degrees of freedom of the spin models in strong magnetic fields.

The checkerboard lattice. A localized magnon which occupies a larger area than the smallest possible one (a). A three-magnon state which is not a large-hard-square state because of the nested “defect” state at the lower left side of the lattice; one magnon is localized on each of the three loops (b).

The checkerboard lattice. A localized magnon which occupies a larger area than the smallest possible one (a). A three-magnon state which is not a large-hard-square state because of the nested “defect” state at the lower left side of the lattice; one magnon is localized on each of the three loops (b).

, , and versus at low temperatures; , and versus around the saturation field. From left to right: diamond chain with , , frustrated two-leg ladder with , , kagomé-like chain with . We set the field range and the temperature range . The exact diagonalization (ED) data (symbols) refer to finite systems of sizes (diamond chain and frustrated ladder) and (kagomé-like chain). The analytical predictions for hard monomers (HM) and one-dimensional hard dimers (HD) are shown by lines.

, , and versus at low temperatures; , and versus around the saturation field. From left to right: diamond chain with , , frustrated two-leg ladder with , , kagomé-like chain with . We set the field range and the temperature range . The exact diagonalization (ED) data (symbols) refer to finite systems of sizes (diamond chain and frustrated ladder) and (kagomé-like chain). The analytical predictions for hard monomers (HM) and one-dimensional hard dimers (HD) are shown by lines.

versus around the saturation field for the frustrated bilayer lattice with , . The exact diagonalization data (filled symbols) refer to a finite spin system of sites. The analytical results (unfilled symbols) refer to a finite hard-square system of sites. The Monte Carlo simulation data (lines) are obtained for the hard-square system on finite lattices with up to .

versus around the saturation field for the frustrated bilayer lattice with , . The exact diagonalization data (filled symbols) refer to a finite spin system of sites. The analytical results (unfilled symbols) refer to a finite hard-square system of sites. The Monte Carlo simulation data (lines) are obtained for the hard-square system on finite lattices with up to .

## Tables

The degeneracies and the energy gaps for various finite frustrated bilayer lattices: exact diagonalization data for finite systems with versus hard-square predictions. is the number of sites in the spin lattice, DGS is the degeneracy of the ground state as follows from exact diagonalization for a given , #HSS is the number of configurations with hard squares, and is the energy gap between the ground state and the first excited state in units of . The dots for indicate omitted sectors with .

The degeneracies and the energy gaps for various finite frustrated bilayer lattices: exact diagonalization data for finite systems with versus hard-square predictions. is the number of sites in the spin lattice, DGS is the degeneracy of the ground state as follows from exact diagonalization for a given , #HSS is the number of configurations with hard squares, and is the energy gap between the ground state and the first excited state in units of . The dots for indicate omitted sectors with .

The degeneracies and the energy gaps for the kagomé lattice: exact diagonalization data for finite systems versus hard-hexagon predictions. is the number of sites in the spin lattice, DGS is the degeneracy of the ground state as follows from exact diagonalization for a given , #HHS is the number of configurations with hard hexagons, and is the energy gap between the ground state and the first excited state in units of . Dots indicate some sectors with which have been omitted for larger values of . For the two-magnon sector one has and .

The degeneracies and the energy gaps for the kagomé lattice: exact diagonalization data for finite systems versus hard-hexagon predictions. is the number of sites in the spin lattice, DGS is the degeneracy of the ground state as follows from exact diagonalization for a given , #HHS is the number of configurations with hard hexagons, and is the energy gap between the ground state and the first excited state in units of . Dots indicate some sectors with which have been omitted for larger values of . For the two-magnon sector one has and .

The degeneracies and the energy gaps for various finite checkerboard lattices: exact diagonalization data for finite systems versus large-hard-square predictions. is the number of sites in the spin lattice, DGS is the degeneracy of the ground state as follows from exact diagonalization for a given , #LHSS is the number of configurations with large hard squares, is the energy gap between the ground state and the first excited state in units of . Dots indicate some sectors with which have been omitted for larger values of . For the two-magnon sector one has and .

The degeneracies and the energy gaps for various finite checkerboard lattices: exact diagonalization data for finite systems versus large-hard-square predictions. is the number of sites in the spin lattice, DGS is the degeneracy of the ground state as follows from exact diagonalization for a given , #LHSS is the number of configurations with large hard squares, is the energy gap between the ground state and the first excited state in units of . Dots indicate some sectors with which have been omitted for larger values of . For the two-magnon sector one has and .

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