LowD magnetism is not related to 2D lattices (on surfaces) or 2D electron gases (in quantum wells) or quantum dots.

## Magnets in restricted dimensions

Exchange interactions which lead to magnetic coupling are much stronger in one or two spatial directions than in the remaining ones.

Real materials are 3D but behave effectively as low-dimensional systems if the dominant exchange interactions are intra-chain (1D) or intra-planar (2D).

## Ising model

Interaction favors alignment of neighboring magnetic moments along a preferred direction, and thus lowers the energy of the system, making a negative contribution -J.

**What about thermal fluctuations? kT ****~**** J
**

**What about quantum fluctuations?**

- Ising considered linear chain of magnetic moments with the interaction of the nearest neighbors. He showed that the spontaneous magnetization can not be explained within the framework of a one-dimensional model.
- Extended to 3D case – erroneously!
*Heisenberg W.*// Zeitschrift f. Physik. 1928. Bd.49. S.619-636.:**Ising showed that even sufficiently large force between any two adjacent atoms in the chain does not explain the appearance of ferromagnetism**.*Peierls R.*// On the Ising model of ferromagnetism, Proc. Cambridge Phil. Soc. 1936. V.32. P.477-481 coined the term “Ising model”. In 1D case ferromagnetism exists, but T_{c }= 0.*Kramers H.A., Wannier G.H.*// Phys. Rev. 1941. V.60. P.252-262.*Onsager L.*// Phys. Rev. 1944. V.65. P.117-149 found the solution for the 2D model (T_{c}= 2.27 J).

**The interest in low-dimensional magnets **developed into a field of its own because these materials provide:

- a unique possibility to study ground and excited states of quantum models;
- possible new phases of matter;
- interplay of quantum fluctuations and thermal fluctuations.

## Ising, XY and Heisenberg models

**LowD magnetism can be traced back 90 years ago:**

- In 1925 Ernst Ising followed a suggestion of his academic teacher Lenz and investigated the 1D version of the model which is now well known under his name in an effort to provide a microscopic justification for Weiss’ molecular field theory of cooperative behavior in magnets;
- In 1931 Hans Bethe wrote his famous paper entitled ’Zur Theorie der Metalle. I.Eigenwerte und Eigenfunktionen der linearen Atomkette’ describing the ’Bethe ansatz’ method to find the exact quantum mechanical ground state of the antiferromagnetic Heisenberg model for the 1D case;
- Both papers were actually not to the complete satisfaction of their authors: The 1D Ising model failed to show any spontaneous order whereas Bethe did not live up to the expectation expressed in the last sentence of his text: ’in a subsequent publication the method is to be extended to cover 3D lattices’.

**For the first 40 years LowD magnetism was an exclusively theoretical field.**

- Theorists were attracted by the chance to find interesting exact results without having to deal with the hopelessly complicated case of models in 3D.
- They succeeded in extending the solution of Ising’s (classical) model to 2D (which, as Onsager showed,
*did*exhibit spontaneous order). - Succeeded in calculating excitation energies, correlation functions and thermal properties for the quantum mechanical 1D Heisenberg.
- An important characteristic of low-dimensional magnets is the absence of long range order in models with a continuous symmetry at any finite temperature as stated in the theorem of Mermin and Wagner, and sometimes even the absence of long range order in the ground state (Coleman).

Question:** Does a lower dimension (e.g., 2D instead of 3D), i.e. ‘less neighbouring spins’ change the ordering behavior?**

Answer:** Yes.**

**Fundamental statement**

‘At any non-zero temperature, a one- or two-dimensional **isotropic** spin-S Heisenberg model with **finite-range** exchange interaction can be neither ferromagnetic nor antiferromagnetic.’

*Mermin, n.D. & Wagner, H. (1966), ‘Absence of Ferromagnetism or Antiferromagnetism in One- or Two- Dimensional Isotropic Heisenberg Models’*

## Experimental 1D and 2D magnetism

- It was only around 1970 when it became clear that the one- and two-dimensional might also be
**relevant for real materials.** **Magnets in restricted dimensions**have a natural realization: Real bulk crystals with**exchange interactions much stronger**in one or two spatial directions than in the remaining ones.- Most studies of lowD magnetism concentrate on Сu or Ni compounds
**which realize spin-½ or 1 correspondingly.**

World-wide progress in **material design** has provided a **multitude of materials** in which selected degrees of freedom like spins, orbitals, or charges are dispersing effectively only along one or two dimensions like in the **spin-1/2 Heisenberg antiferromagnet chains** (HAFC).

## Quantum spin magnetism in organic radical crystals

## Milestones in Low D Magnetism

**1925/31**Ernst Ising, Hans Bethe (Heisenberg chain)**1944**Lars Onsager: 2D Ising modell**1966**Mermin-Wagner theorem: strong temperature fluctuations**1983**Haldane conjecture: strong quantum fluctuations**1986**High Tc superconductivity based on 2D AF’s**since 1990**quantum phase diagrams / magnetization plateaus / order from disorder / BEC of quantum magnets / quantum solitons**21**: quantum communications^{th}century

## Recommended literature

- H.-J. Mikeska and A.K. Kolezhuk, One-Dimensional Magnetism, Lect. Notes Phys.
**645**, 1–83 (2004). - Christopher P. Landee , Mark M. Turnbull. Review: A gentle introduction to magnetism: units, fields, theory, and experiment.
*Journal of Coordination Chemistry*2014;**67**: 375.