# Instabilities in low-dimensional magnetism

In the previous Section, measurements and simulations were discussed of ultrafast magnetization phenomena in three dimensions; here possibilities are considered for using the SwissFEL to investigate the quantum-fluctuating behavior of low-dimensional magnetic systems [13].In many magnetic insulators, magnetic moments interact through an exchange of electrons between neighboring sites. Such exchange interactions are shor t-ranged. If these interactions are isotropic, such systems can be described by the well-known Heisenberg Hamiltonian, which is given by:

where J is the exchange energy, and the summation is over nearest-neighbor spins; If J is positive, the spins S

_{i}and S

_{j}tend to align ferromagnetically. For an ordered magnetic phase, the temperature-dependent change in the saturation magnetization can be calculated [14] in this model as M

_{s}(0) – M

_{s}(T) ∝ N

_{sw}(T), where

is the density of spin-waves excited at the temperature T. For d=3 dimensions, this integral is propor tional to T

^{3/2}, giving the well-known Bloch 3/2-law. For dimensions lower than d=3, the expression for Nsw(T) diverges, implying that fluctuations will prevent the occurrence of long-range magnetic order. This is a fundamental result, which has been rigorously proven by Mermin and Wagner [15], and which means that many types of magnetic systems do not order at any finite temperature.

Some systems are disordered even at zero temperature, where thermal fluctuations are absent, due to the presence of quantum fluctuations in the ground state. This can happen if a static arrangement of magnetic moments is not an eigenstate of the Hamiltonian, causing quantum fluctuations to generate a new type of ground state. These disordered systems form ferromagnetically or antiferromagnetically-coupled spin liquids, and their quantum fluctuations, as described by the intermediate scattering function S(Q,t) (see chapter V), represent a par ticularly rich field of investigation for the SwissFEL.

## Two-dimensional case (d=2)

As an example of the dynamics of a 2d-magnetic structure, consider the case of an infinite in-plane anisotropy (S_{z}=0): the so-called xy-model:

As for the 2d-Heisenberg model, there is no magnetic order at finite temperature in the xy-model. However, it is found that spin correlations cause the formation at low temperature of a disordered array of magnetic vortices, with radii of order R. The cost in exchange energy incurred by the formation of such a vor tex is πJln(R/a), where a is the lattice constant, and the gain in entropy represented by the varying position of the vor tex center is 2k

_{B}ln(R/a). Hence the free energy F = ln(R/a)(πJ-2k

_{B}T) becomes negative below the Kosterlitz-Thouless transition, at the temperature TKT=πJ/2k

_{B}, which separates a non-vor tex from a vor tex phase. At low temperatures, the S=1/2 layered perovskite ferromagnet K

_{2}CuF

_{4}is approximately described by the xy-model, going through a Kosterlitz-Thouless transtion at 5.5 K. A fur ther example of (quasi) 2d-magnetic dynamics is that of vor tex core reversal in thin magnetic nanostructures (see Infobox).

## One-dimensional case (d=1)

Magnetism in one dimension in the zero-temperature limit is par ticularly interesting, because it arises from quantum fluctuations. Consider first the isotropic J>0 Heisenberg model for a one-dimensional chain of N spins S=1/2, with periodic boundary conditions. The (ferromagnetic) ground state can be represented as |Ψ_{0}〉 = |↑↑↑...↑〉. In the Bethe Ansatz, the excited states of the system are built up as a superposition of states with discrete numbers of flipped spins. If we confine ourselves to single-spin (r=1) excitations:

|n 〉= |↑↑↑...↑↓↑...↑〉

(here the nth spin has been reversed), we can write the excited state as

It is then a straightforward exercise to compute from the Schrödinger equation (for convenience, written in terms of the raising and lowering operators S±) the excited- state energy E

_{1}, and one finds, for large N, that excitations exist with arbitrarily small excitation energies E

_{1}– E

_{0}; i.e., the excitation spectrum is gapless. Higher level excitations, involving multiple spin flips r = 2, 3, 4, ..., become increasingly cumbersome to handle, but the gapless spectrum is retained (Figure I.8a shows the analogous result for the 1d-antiferromagnetic spin ½ chain [16]).

## Magnetic vortex core switching

The magnetic vor tex is a very stable, naturally-forming magnetic configuration occurring in thin soft-magnetic nanostructures. Due to shape anisotropy, the magnetic moments in such thin-film elements lie in the film plane. The vor tex configuration is characterized by the circulation of the in-plane magnetic structure around a ver y stable core of only a few tens of nanometers in diameter, of the order of the exchange length. A par ticular feature of this structure is the core of the vor tex, which is perpendicularly magnetized relative to the sample plane. This results in two states: “up” or “down”. Their small size and per fect stability make vor tex cores promising candidates for magnetic data storage. A study by Her tel et al. [27] based on micromagnetic simulations (LLG equation) has shown that, strikingly, the core can dynamically be switched between “up” and “down” within only a few tens of picoseconds by means of an external field. Figure I.i6 below simulates the vortex core switching in a 20 nm thick Permalloy disk of 200 nm diameter after the application of a 60 ps field pulse, with a peak value of 80 mT. Using field pulses as shor t as 5 ps, the authors show that the core reversal unfolds first through the production of a new vor tex with an oppositely oriented core, followed by the annihilation of the original vortex with a transient antivortex structure. To date, no experimental method can achieve the required temporal (a few tens of ps) and spatial (a few tens of nm) resolution to investigate this switching process. The combination of the high-energy THz pump source and circularly-polarized SwissFEL probe pulses will allow such studies.One of the simplest ways for a material to avoid magnetic order and develop macroscopic quantum correlations is through the creation of an energy gap E

_{g}in the excitation spectrum. Since E

_{g}is of the order of the exchange interaction, the gap introduces a time-scale for fluctuations which is typically on the order of femtoseconds. One such phenomenon is the spin Peierls effect. This is related to the better-known charge Peierls metalinsulator transition (see Chapter V). In the spin-Peierls effect, a uniform 1d, S=1/2 spin chain undergoes a spontaneous distor tion, causing dimerization, and hence the appearance of two different exchange couplings J±δJ (see Fig. I.9). For δJ sufficiently large, S=0 singlet pairs are formed on the stronger links, implying a non-magnetic state and a finite energy gap to the excited states (see Fig. I.8b). The Peierls state is stable if the resulting lowering of magnetic energy more than compensates for the elastic energy of the lattice distortion. Note that the distor tion is a distinctive feature which is visible with hard X-ray diffraction. The spin-chain compound CuGeO

_{3}is an inorganic solid which undergoes a spin-Peierels trasition at 14 K.

_{2}BaNiO

_{5}. The Haldane mechanism is also used to describe the dynamic behavior of finite 1d S=1 antiferromagnetic chains, as investigated in Mg-doped Y2BaNiO5 by inelastic neutron scattering [18]. The finite chains are generated by the non-magnetic Mg impurities, and the ends of the chains represent S=1/2 impurities with a strong nano-scale correlation, with the result that the Haldane gap becomes a function of chain length.

## Zero-dimensional case (d=0)

Another example is the creation of magnetic nanodots by sub-monlayer deposition onto a high-index sur face of a metal (see Fig. I.12). If they have a magnetic anisotropy above the superparamagnetic limit, such nanopar ticles may exhibit room-temperature ferro- or antiferromagnetic order, and undergo sub-nanosecond quantum tunnelling between different magnetization directions [21]. Details of this tunnelling, including field-enhancement of the rate, are an attractive topic in ultrafast magnetization dynamics, suitable for study with the SwissFEL.