10.1. Ising model - Notes#

10.1.1. Introduction#

10.1.2. Time evolution#

10.1.2.1. Metropolis-Hastings#

Metropolis-Hastings algorithm is designed to generate a collection of states according to a desired distribution \(p(s)\), using a Markov process with a unique stationary distribution \(\pi(s) = p(s)\).

Algorithm

  • Initialize state \(s_0\) with energy \(E_0\)

  • while not terminal condition:

    • flip the spin of a randomly chosen site, to get the possible state \(s_\widetilde{n}\)

    • calculate the change in energy of the state \(\Delta \widetilde{E}_n := \widetilde{E}_n - E_{n-1}\)

    • update state:

      • if \(\Delta \widetilde{E}_n < 0\) accept the change, \(s_n = s_{\widetilde{n}}\)

      • if \(\Delta \widetilde{E}_n > 0\) accept the change with probability \(\exp\left( - \frac{\Delta \widetilde{E}_n}{T} \right)\)

Starting from the detailed balance (sufficient but not necessary condition for the existence of a (unique?) stationary state), \(P(s'|s) p(s) = P(s|s') p(s')\),

\[\frac{P(s'|s)}{P(s|s')} = \frac{p(s')}{p(s)} \ ,\]

each transition is splitted in two sub-steps: proposal and acceptance-rejection steps. Transition is written as \(P(s'|s) = g(s'|s) A(s',s)\) being \(g(s'|s)\) the proposal conditional probability, and \(A(s',s)\) the acceptance probability. Acceptance probability is choosen in a way to satisfy the conditions of time-reversal and ergodicity. Metropolis choice reads

\[A(s', s) = \min \left( 1, \frac{p(s')}{p(s)} \frac{g(s|s')}{g(s'|s)} \right) \ .\]

todo check if Metropolis choice fulfills the necessary conditions.

Example: Boltmann distribution with uniform proposal. Let a system have Boltzmann stationary distribution, \(p(s) \propto \exp\left( - \beta E(s) \right)\), and uniform proposal conditional probability \(g(s'|s) = \frac{1}{N}\), with \(N\) the number of accessible states from the state \(s\).

Example 10.1 (Ising model on a 2D lattice with 1 flip per update)

As an example, for an Ising model on a 2D lattice with dimension \(m \times n\) with proposal update with \(1\) flip \(N = mn\).

Example 10.2 (Ising model on a 2D lattice with \(k\) flips per update)

todo Is a probability built with independent successive flips ok? Target states at each flip have the same probability, s.t. the probability of reaching steps \(s_k\) after \(k\) flips starting from \(s_0\) is not uniform.

Let \(g(s|s') = g(s'|s)\). The acceptance probability of the transition between different states \(s' \ne s\) reads

\[A(s',s) = \min \left( 1, e^{ -\beta ( E(s') - E(s) ) } \right) \ ,\]

and the corresponding element of transition matrix is

\[P(s'|s) = \frac{1}{N} \min \left( 1, e^{ -\beta ( E(s') - E(s) ) } \right) \ .\]

The diagonal elements of the probability transition matrix - representing no state transition - are evaluated as the probability that makes \(\sum_{s'} P(s'|s) = P(s|s) + \sum_{s' \in T(s)} P(s'|s) = 1\), i.e.

\[\begin{split}P(s|s) = \frac{1}{N} \sum_{\begin{aligned} s' & \in T(s) \\ E(s') & > E(s) \end{aligned}} \left( 1 - e^{-\beta (E(s') - E(s))} \right) \ .\end{split}\]
\(P(s|s)\)
\[\begin{split}\begin{aligned} P(s|s) & = 1 - \sum_{s' \in T(s)} P(s'|s) = \\ & = 1 - \sum_{s' \in T(s)} \frac{1}{N} \min \left( 1, e^{-\beta (E(s') - E(s))} \right) = \\ & = \frac{1}{N} \sum_{s' \in T(s)} \left[ 1 - \min \left( 1, e^{-\beta (E(s') - E(s))} \right) \right] = \\ & = \frac{1}{N} \sum_{s' \in T(s)} \max \left( 0, 1 - e^{-\beta (E(s') - E(s))} \right) = \\ & = \frac{1}{N} \sum_{\begin{aligned} s' & \in T(s) \\ E(s') & > E(s) \end{aligned}} \left( 1 - e^{-\beta (E(s') - E(s))} \right) \ . \end{aligned}\end{split}\]

It’s easy to prove that this probability functions are related by the detailed balance equations,

\[\frac{P(s'|s)}{P(s|s')} = \frac{\min \left( 1, e^{ -\beta ( E(s') - E(s) ) } \right) }{\min \left( 1, e^{ -\beta ( E(s) - E(s') ) } \right) } = e^{ -\beta ( E(s') - E(s) ) } = \frac{e^{-\beta E(s')}}{e^{-\beta E(s)}} = \frac{p(s')}{p(s)} \ .\]