Chapter 6. The Ising model on a square lattice 6.1 The basic Ising model The Ising model is a minimal model to capture the spontaneous emergence of macroscopic ferromagnetism from the interaction of individual microscopic magnetic dipole moments. Consider a two-dimensional system of spin 1=2 magnetic dipoles arrayed on a square L L lattice. At each lattice point .i; j /, the dipole can assume only one of two spin states, i;j 1. The dipoles interact with an externally applied constant magnetic field B, and the energyD˙ of a dipole interacting with the external field, Uext, is given by U .i;j / Hi;j ext D where H is a non-dimensionalized version of B. The energy of each spin (dipole) is lower when it is aligned with the external magnetic field. The dipoles also interact with their nearest neighbors. Due to quantum effects the energy of two spins in a ferromagnetic material is lower when they are aligned with each other1. As a model of this, it is assumed in the Ising model that the interaction energy for the dipole at .i; j / is given by Uint.i;j / Ji;j .i 1;j i 1;j i;j 1 i;j 1/ D C C C C C where J is a non-dimensionalized coupling strength. Summing over all sites, we get the total energy for a configuration given by E L L J L L U./ H i;j i;j .i 1;j i 1;j i;j 1 i;j 1/: E D 2 C C C C C i 1 j 1 i 1 j 1 XD XD XD XD The second sum is divided by two to account for each interaction being counted twice. For this exercise, assume the domain is periodic, so for example, 1;j has left neighbor L;j , and i;L has top neighbor i;1, etc. An observable of interest in this case is the magnetization of the system, i.e., the expected value of the alignment with the external field 1 L L M.T / M./p.; T / ; where M./ i;j h i D E E E D L2 i 1 j 1 XE XD XD 1The dipole interaction between classical magnetic dipoles does not lead to ferromagnetic ordering. ©kath2020esam448notes 76 The Ising model on a square lattice and p.; T / is the Boltzmann probability density function at temperature T that gives the likelihoodE of any particular energy state, 1 U./=kB T p.; T / e E : E D Z Here the partition function Z is given by U./=kB T Z e E D L L I 1;1 E2fXg L L this normalizes the Boltzmann probabilities. This sum is over 2 terms, a number which becomes very large very quickly as L is increased. Since ferromagnetism (and any phase transition) is a macroscopic phenomenon that arises, strictly speaking, only in the infinite-size limit L , the numerical computation of Z is typically forbiddingly expensive. ! 1 6.2 Metropolis simulation of the Ising model on a square lattice Computing statistical quantities for the Ising model directly for a given temperature T would require summing over 2L2 configurations, an enormous value for even modest values of L. One therefore uses Markov Chain Monte Carlo integration to compute M.T / , U.T / , etc. The Metropolis algorithm is used to generate a sample of configurationsh with distributioni h i de- termined by p.; T / and the sum becomes 1 K M.T / M. .k// h i K k 1 XD where .k/ are the generated sample configurations. The Metropolis algorithm is constructed using transition probability 1 L L 2 ; if i 1 j 1 i;j0 i;j 2; L D D j j D q. 0 / 8 j D P P <ˆ0 otherwise ; ˆ i.e. only transitions in which a: single, randomly selected spin is flipped are considered. In other words, this transition probability means that one should randomly select one of the L2 sites and flip the spin at that site. The construction of the acceptance probability then follows the discussion in the previous sections. Note, however, that the acceptance probability can be ©kath2020esam448notes 6.3 A very brief overview of thermodynamics and statistical mechanics 77 computed efficiently by computing only the change in energy caused by the proposed spin flip at site .i; j /, .old/ .new/ .old/. This change in energy is given by i;j ! i;j D i;j U U .new/ U .old/ Á i;j i;j D Á Á .new/ .old/ .old/ .old/ .old/ .old/ i;j i;j H J.i 1;j i 1;j i;j 1 i;j 1/ D C C C C C .old/ Á.old n / .old/ .old/ .old/ .old/ o 2Hi;j 2Ji;j .i 1;j i 1;j i;j 1 i;j 1/: D C C C C C C One then accepts this proposed change of spin with probability U=kB T min e ; 1 I h i otherwise one rejects the flip of that spin and the previous configuration is duplicated. One then repeats the steps to generate more configurations. These configurations, of course, are sampled from the Boltzmann distribution at this particular temperature, and thus can be used to compute statistics of interest. Thus, the algorithm is (using ˇ 1=kB T ) D 1. Pick a random spin .i; j / and propose the candidate spin flip .i;j / .i;j / ! 2. Compute the energy difference U 3. If U < 0, accept the step. ˇU 4. If U > 0, accept the step with probability e < 1. 5. Repeat. If a spin flip is rejected the previous configuration is duplicated. Iterating this procedure even- tually generates configurations sampled from the Boltzmann distribution at temperature T . Note that when the temperature is lower it is less likely that a spin flip will be accepted if it would raise the energy; thus, the system will be more confined to energy minima and less likely to climb over energy barriers. This is just an overview of the method, of course; there are a number of other practical details associated with a good implementation of the method that we will not discuss in detail here. 6.3 A very brief overview of thermodynamics and statistical mechanics To understand the key elements of the Ising model we need to have a few basic concepts from statistical mechanics and as they relate to thermodynamics. To use a system for which we have more intuition, consider a gas comprised of many molecules. The molecules are bounc- ing off each other all the time. At any given moment the system can be characterized by its ©kath2020esam448notes 78 The Ising model on a square lattice microstate, i.e. by the positions ri .t/ and velocities vi .t/ of all of its molecules. All macro- scopic measurements involve a very large number of molecules and take a long time compared to the time between collisions of the molecules. Macroscopic measurements could therefore be considered as time averages over the evolving microstates. However, since computing the temporal average requires knowledge of the dynamics of each individual molecule, it is better to consider an ensemble consisting of a very large number of systems, each having different initial conditions, and to assume that the time average can be replaced by an average across this ensemble at a given time. 1 2 In closed systems the total energy E.rj ; vj / j 2 mvj U.r1; r2;::: rN /, composed of the kinetic and the potential energy of the molecules,Á is conserved.C P Ergodic hypothesis: Since the motion of the molecules is extremely complicated, we assume that they explore the complete energy surface E.rj ; vj / E0 and that they do so uniformly. D How does the notion of a temperature arise? For that we need to consider two separated systems S1;2 with energies E1;2. If they are brought in thermal contact with each other, how will the total energy E E1 E2 be distributed between the two systems? A state with a given value of the macroscopicD C parameters, e.g. the energy, can be realized by a large number ˝ of different microstates. For the two combined systems we have ˝.E/ ˝1.E1/ ˝2.E2/ with E E1 E2: D D C Postulate: The final equilibrium state that is reached when two systems are brought into thermal contact is the state that maximizes the number of microstates compatible with the macrostate (i.e., here, total energy E). To find how E1 and E2 distribute, we use Lagrange multipliers to extremize ˝1.E1/ ˝2.E2/ .E1 E2/: C We find @˝1 @˝2 ˝2 0 and ˝1 0 ; @E1 D @E2 D or 1 @˝ 1 @˝ 1 2 ; ˝1 @E1 D ˝2 @E2 or @ @ ln ˝1.E1/ ln ˝2.E2/: @E1 D @E2 The final equilibrium is characterized by the equality of the quantity @ ln ˝=@E in both sys- tems, suggesting that this quantity has a special meaning. Thermodynamics relates this quan- tity to the temperature, @ ln ˝.E/ 1 @E D kB T ©kath2020esam448notes 6.3 A very brief overview of thermodynamics and statistical mechanics 79 and defines S kB ln ˝ D as the entropy. The entropy ‘counts’ the number of microstates compatible with the macro- scopic variables. To describe a system S at a fixed temperature rather than at a fixed total energy we couple it to a very large second system SR, which acts as a heat reservoir. We need then the probability for that system to have an energy in the interval ŒE; E dE C ˝R.Etot E/ ˝.E/ dE p.E/ dE D 1 ˝R.Etot E/ ˝.E/ dE 0 where R SR.Etot E/=kB ˝R.Etot E/ e : D We are looking for a description of system S without explicit reference to SR.