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Simulation MethodUsing the Metropolis algorithm, we first equilibrate the lattice at a given temperature T without itinerant electrons. When equilibrium is reached, we randomly add N_{0} polarized itinerant spins into the lattice. Each itinerant electron interacts with lattice spins in a sphere of radius Di centered at its position, and with other itinerant electrons in a sphere of radius D_{2}. We next equilibrate the itinerant spins using the following updating. We calculate the energy E_{0}i_{d} of an itinerant electron taking into account all interactions described above. Then we perform a trial move of length i taken in an arbitrary direction with random modulus in the interval [/?ь ft_{2}] where Ri = 0 and R_{2} = a (NN distance), a being the lattice constant. Note that the move is rejected if the electron falls in a sphere of radius r_{0} centered at a lattice spin or at another itinerant electron. This excluded space emulates the Pauli exclusion. We calculate the new energy E_{new} and use the Metropolis algorithm to accept or reject the electron displacement. We choose another itinerant electron and begin again this procedure. When all itinerant electrons are considered, we say that we have made a MC sweeping, or one MC step/spin. We have to repeat a large number of MC steps/spin to reach a stationary transport regime. We then perform the averaging to determine physical properties such as magnetic resistivity, electron velocity, energy, etc., as functions of temperature. We define the dimensionless spin resistivity p as
where n_{e} is the number of itinerant electron spins crossing a unit slice perpendicular to the x direction per unit of time. An example with real units is shown in Section 17.7. In order to have sufficient statistical averages on microscopic states of both the lattice spins and the itinerant spins, we use what we call "multistep averaging procedure”: after averaging the resistivity over N steps for "each” lattice spin configuration, we thermalize again the lattice with N_{2} steps in order to take another disconnected lattice configuration. Then we take back the averaging of the resistivity for Ni steps for the new lattice configuration. We repeat this cycle for N_{3} times, usually several hundreds of thousands times. The total MC steps for averaging is about 4 x 10^{5} steps per spin in our simulations. This procedure reduces strongly thermal fluctuations observed in our previous work [4]. Of course, the larger N_{2} and N_{3} are, the better the statistics become. The question is what is the correct value of Ni for averaging with one lattice spin configuration at a given T7 This question is important because this is related to the relaxation time zl of the lattice spins compared to that of the itinerant spins, г/. The two extreme cases are (i) tl — zi, one should take Ni = 1, namely the lattice spin configuration should change with each move of itinerant spins (ii) z_{L} » z_{t}, in this case, itinerant spins can travel in the same lattice configuration for many times during the averaging. In order to choose a right value of N_{x}, we consider the following temperature dependence of z_{L} in nonfrustrated spin systems. The relaxation time is expressed in this case as [154, 267]
where A is a constant, v the correlation critical exponent, and z the dynamic exponent which depend on the spin model and space dimension. For 3D Ising model, i> = 0.638 and z = 2.02. From this expression, we see that as T tends to 7c, zl diverges. In the critical region around T_{c} the system encounters thus the so called "critical slowing down": the spin relaxation is extremely long due to the divergence of the spinspin correlation. When we take into account the temperature dependence of z_{L}, the shape of the resistivity is modified strongly at T_{c} where zl is very long, and in the paramagnetic phase where the relaxation time is very short due to rapid thermal fluctuations. On the other hand, at low T, z_{L} does not modify p because in the ordered phase the spin landscape from one microscopic state to another does not change significantly to affect the motion of the itinerant spin (see the discussion in Ref. [220]). In simulations, we consider a film with a thickness of N_{z} cubic cells in the z direction. Each of the xy planes contains N_{x} x N_{y }cells. The periodic boundary conditions are used on the xy planes to ensure that the itinerant electrons that leave the system at the second end are to be reinserted at the first end. For the z direction, we use the mirror reflection at the two surfaces. These boundary conditions conserve thus the average density of itinerant electrons. Dynamics of itinerant electrons is created by an electric field applied along the x axis. Spin Resistivity in Ferromagnets and AntiferromagnetsIn ferromagnets, experimental data mentioned above show a peak at T_{c}. The peak is related to the critical slowing down where the relaxation time diverges. Direct MC simulations in the case of Ising spin give a pronounced peak at T_{c} as shown in Fig. 17.1 in agreement with experiments. Note that p increases at low T. The reasons for this are multiple: It can stem from the freezing or crystallization of itinerant spins at low T or just from the smallness of the number of conduction electrons in such a lowГ region. The shape of p depends on many factors: lattice structure, various interactions encountered by itinerant spins, electron concentration, relaxation time, spin model, magneticfield amplitude, etc. For example, a decrease in the interaction between itinerant spins K_{0} will reduce the increase of p as T > 0, an applied magnetic field will decrease the peak height, the larger carrier concentration will reduce p in particular at T_{c}. All of these have been discussed in Ref. [218]. We note a strong effect of the temperature dependence of tl on p for T > Tc. This is very important because r_{L} depends intrinsically on the material via v and z. For a quantitative comparison with experiments for a given material, it is necessaiy to take into account the specific parameters of that material. This is what we do in Section 17.7. In antiferromagnets much less is known because there have been very few theoretical investigations which have been carried out. Haas [138] has shown that while in ferromagnets the resistivity p shows a sharp peak at the magnetic transition of the lattice spins, in antiferromagnets there is no such a peak. We found that the peak exists in antiferromagnets but it is less pronounced as seen in Fig. 17.1. The alternate change of sign of the spinspin correlation with distance may have something to do with the absence of a Figure 17.1 BCC ferromagnetic and antiferromagnetic films: Resistivity R with temperaturedependent relaxation for ferro (black circles) and antiferromagnet (white circles) in arbitrary unit versus temperature T, in zero magnetic field, with electric field e = 1, Io = 2, Ко = 0.5, A = 1. sharp peak. We have tested, for example, the effect of the cutoff distance D [219]: When D_{x} increases, it will include successively upspin shells and downspin shells in the sphere of radius D. As a consequence, the difference between the numbers of up and down spins in the sphere oscillates with varying D, making an oscillatory behavior of p at small D, unlike in ferromagnets. It is interesting to note that in the presence of an itinerant spin, the ferromagnet and its antiferromagnet counterpart are no more invariant by the local Mattis transformation {Jij *■ —Jij, S_{y} *■ —S_{;}). Note that we can calculate the spin resistivity using Boltzmann's equation combined with Monte Carlo simulations. The method consists of two steps:
Spin Resistivity in Frustrated SystemsSimple Cubic J1 − J2 ModelWe consider the simple cubic lattice shown in Fig. 17.2. The Hamiltonian is given by where S, is the Ising spin at the lattice site /, ^ is made over the NN spin pairs with interaction J_{b} while Xlp.m) *^{s} performed over the NNN pairs with interaction J_{2}■ We are interested in the frustrated regime. Therefore, hereafter we suppose that ] i= — J {J > 0, antiferromagnetic interaction, and J_{2} = —t]J where r/ is a positive parameter. The ground state (GS) of this system is easy to obtain either by minimizing the energy, or by comparing the energies of different spin configurations, or just a numerical minimizing by a steepest descent method [248]. We obtain the antiferromagnetic configuration shown by the upper figure of Fig. 17.3 for J_{2} < 0.251711, or the configuration shown in the lower figure for /21 > 0.251711. Note that this latter configuration is 3fold degenerate by choosing the parallel NN spins onx,y or z axis. With the permutation of black and white spins, the total degeneracy is thus 6. The phase transition in the case of the Heisenberg model in the frustrated region [/_{2} > 0.251711) has been found to be of first order [274]. The system is very unstable due to its large degeneracy. We find that the case of the Ising spin shows an even stronger first order transition [151]. It is interesting to note that the resistivity Figure 17.2 Simple cubic lattice with nearest and nextnearest neighbor interactions, у 1 and J_{2}, indicated. Figure 17.3 Simple cubic lattice. Upspins: white circles, downspins: black circles. Top: Ground state when /_{2} < 0.251/d Bottom: Ground state when IM > 0.25IM. Figure 17.4 Left: Sublattice magnetization M versus T, Right: Energy versus T, for }_{2} = 0.261/il, N_{x} = N_{y} = 20, N_{z} = 6. of itinerant spins in systems with a firstorder transition undergoes a discontinuity at T_{c} just as the system energy and the order parameter. We show p in Fig. 17.7 for several cutoff distance D. One observes here that p can jump or fall at the transition depending on the interaction range D_{x}. The resistivity discontinuity has been confirmed in another system with firstorder transition, the frustrated FCC antiferromagnet [219]. This seems to be a general rule. We show first the result of the lattice alone, namely without itinerant spins. The lattice in the frustrated region, i.e., I/2/./1I > 0.25, shows a strong firstorder transition as seen in Fig. 17.4: The sublattice magnetization and the energy per spin as functions of T for /2=  0.26711 for the lattice size N_{x} = N_{y} = 20, N_{z} = 6 Figure 17.5 Energy histogram taken at the transition temperature T_{c} for J_{2} = —0.261Уj: black circles are for N_{x} = N_{y} = 20, N_{z} = 6, T_{c} = 1.320, void circles for N_{x} = N_{y} = 30, N_{z} = 6, Tc = 1.320 and black triangles for N_{x} = N_{y} = 20, N_{z} = 10, T_{c} = 1.305. Other parameters are I_{0} = K_{0} = 0.5, Di = 0.8a, D_{2} = a, D = 1, e = 1. show a discontinuity at the transition temperature. To check further the firstorder nature of the transition, we have calculated the energy histogram at the transition temperature T_{c}. This is shown in Fig. 17.5. The doublepeak structure indicates the coexistence of the ordered and disordered phases at T_{c}. The distance between two peaks represents the latent heat. Now we consider the lattice with the presence of itinerant spins. As far as the interaction between itinerant spins is attractive, we need a chemical potential to avoid the collapse of the system. The strength of the chemical potential D depends on K_{0}■ We show in Fig. 17.6 the collapse phase diagram which allows to choose for a given K_{0}, an appropriate value of D. We show now the main result on the spin resistivity versus T for y21 = 0.26/i for several values of D. Other parameters are the same as in Fig. 17.4. As said in Section 17.2.2, within the physical constraints, the variation of most of the parameters does not change qualitatively the physical effects observed in simulations, except for the parameter D. Due to the AF ordering, increasing D means that we include successively neighboring down and up spins Figure 17.6 Phase diagram in the plane (Ко, D). The collapse region is in black, for /_{2} = 0.26y!. Other parameters are D_{:} = D_{2} = a, I_{0} = 0.5, f = 1. surrounding a given itinerant spin. As a consequence, the energy of the itinerant spin oscillates with varying D_{1;} giving rise to the change of behavior of p: p can make a down fall or an upward jump at T_{c} depending on the value of Di as shown in Fig. 17.7. Note the discontinuity of p at T_{c}. This behavior has been observed and analyzed in terms of the averaged magnetization in the sphere of radius Di in the frustrated FCC antiferromagnet [219]. 
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