# Is it possible to efficiently update the ground state of an Ising lattice after a local change in the fields?

The Hamiltonian of an Ising model can be written as:

$$H(\mathbf s) = \sum_{i<j}J_{ij}s_i s_j + \sum_i h_i s_i$$

where $s_i \in \{0,1\}$ are the spins on each site. The ground state is the spin configuration that minimizes $H(\mathbf s)$. Computing the ground state is computationally hard.

But suppose you have computed the ground-state $\mathbf s^*$ for a given set of fields $J_{ij},h_i$, and that you make a local update affecting only one field, say $J_{rs} \rightarrow J_{rs}^\prime$, while $J_{ij}$ is not affected for $r\ne i,j\ne s$. From the information of the ground-state of the original fields, is it possible to compute efficiently the ground-state of the perturbed fields?

• Just to be completely clear (maybe you could edit the question) can you say if you mean the random field/random coupling Ising model in which the coupling constants $J_{ij}$ and/or fields $h_i$ are chosen from some probability distribution? The ground states of such models are well-known to be highly sensitive to very small global changes $\delta J$ etc, but I don't know the literature well enough to offer a definitive answer for a localized change. Hopefully a real expert will see your question. But I'd be surprised if it were possible to do as you ask. – user197851 Aug 5 '18 at 16:11
• As you said, computing the ground state is in general computationally hard. Now, note that if there was an efficient way of determining the new ground state after changing one of the parameters, then this would yield an efficient way of computing a general ground state: start with all $h_i$ and $J_{ij}$ nonnegative. In this case, computing the ground state is trivial; then modify one of the parameters at a time, applying your efficient algorithm at each step. – Yvan Velenik Aug 5 '18 at 18:53
• @LonelyProf No random. $J_{ij},h_i$ are fixed numbers. – becko Aug 5 '18 at 20:35
• @YvanVelenik Of course. You can add an answer if you like and I'll accept it. Thanks. – becko Aug 5 '18 at 20:36

Now, note that if there was an efficient way of determining the new ground state after changing one of the parameters, then this would yield an efficient way of computing a general ground state. One would just start with all $h_i$ and $J_{ij}$ nonnegative, in which case the computation of the ground state is trivial. One could then modify one of the parameters at a time, applying your efficient algorithm at each step.