University of Applied Sciences Bielefeld  Department of Engineering and Mathematics Prof. Dr. rer. nat. Christian SchrĂ¶der

Stochastic Spin Dynamics Methods (Tutorial)Currently there exists a number of computational approaches, where the socalled MonteCarloapproach is the most prominent one. Whereas this approach will be described elsewhere, here I want to introduce a technique which is called "Stochastic Spin Dynamics". 1. Conservative spin dynamicsFirst let's answer the question what happens if we put a classical spin or equivalently a classical magnetic moment (red vector) into an external magnetic field (brown vector). If these two vectors are noncollinear we will have a resulting precession of the moment around the external field direction, according the equations of motion. As long as we solve this equations without any heat bath coupling we always end up with a more or less complicated interaction of precessional dynamics with different frequencies. Look for example at this case of a ring consisting of 4 spins coupled ferromagnetically by assuming the Heisenberg exchange. Without any heat bath interaction you will see a deterministic dynamics which preserves the total spin (indicated by a blue arrow). Although these precessional dynamics shows a complicated trajectory for each spin there are quantities which only precess around the total spin. These quantities are the sums of the diagonally displaced spins (green and brown arrows). The only differences are the choices of initial conditions and hence the total spin of the system. By watching the movies you recognize the conservation of the total spin in space and time. 2. Dissipative spin dynamicsIn order to simulate heat bath interactions we first couple dissipative terms to the equations of motion. With the result that the system can loose its intrinsic energy. Clearly, the total spin isn't a conserved quantity any longer. Now, the yellow arrow shows the influence of the LandauLifshitzdamping. As expected by construction of this term, the direction of the resulting 'force' is generally perpendicular to that of the spin and its precessional motion. Because this force depends on the angle between the local field and the spin, the motion will drive the system towards a collinear state or in other words towards the ground state with T=0. 3. Heat bath spin dynamicsIn order to simulate a heat bath we have to include a competing force to the dissipation described above. For doing this, the principles of statistical physics will help us when looking at these useful definition of temperature. We can realize this requirements by adding fluctuational terms (green arrow) to our equations of motion. If we do this in the 'correct' way we will end up with a set of stochastic differential equations, which can by solved numerically when mapped onto a RungeKuttascheme developed by Milstein et al. 4. Thermal equilibrium spin dynamicsNow, let's see what happens if we apply this method to the very simple problem of two interacting spins. We again use the Heisenberg model and assume a ferromagnetic coupling. Suppose we want to study this system at very low temperature. First we want to see if we can produce the nearly ground state configuration which means in this case a collinear alignment of both spins or of each spin with its local field, respectively. Therefore we start with an antiparallel alignment and watch the dynamics while the system develops in space and time. One can clearly see the dissipation of energy due to the LandauLifshitzdamping and the remaining fluctuations of the angle between the spins or it's deviation from exact parallel alignment due to the (small but finite) temperature. We can compare this simulation to a similar situation of two antiferromagnetically coupled spins, starting from a parallel alignment. To complete our picture of dynamical behavior of simple spin
systems we simulate an arrangement of three spins, coupled
antiferromagnetically Compared to the two spin cases we observe a complete different behavior, the spins move around in a seemingly chaotic manner. But isn't there really any order?? Well, we have to look at the correct quantity which can be used to characterize the system. In this case we have a frustrating situation. Each spin wants to align antiparallel to each other. But the geometric constraints doesn't allow for it! As a result, the system has to find a compromise, which means in this case an 120 degree angle between every pair of spins. But this arrangement can be realized by (infinite) many configurations which are separated by small energy barriers that can be overcome by small fluctuations. So, if we look at the time evolution of the angle between two spins we recognize that it remains more or less constant and fluctuates around 120 degrees according to the adjusted temperature. Up to now we realize that this method is well suited for the search of (nearly) ground state configurations. To conclude, let's have a look at a ring of 20 spins which are coupled antiferromagnetically. 5. An approach to nonequilibrium spin dynamicsAlthough the above method is developed only for equilibrium studies it is generally accepted that the structure of the equations of motion remains unchanged also in the case of 'slight' nonequilibrium situations, at least where we can guarantee that the characteristic frequencies of the perturbation are much smaller than the fluctuations which simulate the heat bath interaction. To illustrate a typical situation we can think of a perturbation caused by an external (spatial) homogeneous magnetic field which varies sinusoidal in time. The following movie shows a ring consisting of 100 spins which are coupled ferromagnetically. We apply very low temperatures and start the simulation from a near ground state configuration of aligned spins. The external magnetic field is sketched by a blue vector in the center of the ring. We see that the spins follow the external field not instantaneous but with a short delay. This kind of behavior is generally called a hysteresis, which means that the response of the system or equivalently the susceptibility has a frequency dependence according to the frequency of the applied field. If we change our exchange interaction from ferromagnetic to antiferromagnetic we observe a complete different behavior of the system. The spins do not follow the external field any longer, but arrange themselves nearly perpendicular to the applied field with a small canting depending on the strength of the magnetic field. As a result we only will measure a small response or susceptibility compared to the ferromagnetic case. 
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