Tags: Fixed Point Method

Intro

We want to solve the Bethe Equations,

\(\xi_i^L = (-1)^{N-1} \prod_{j=1}^{N} \frac{1 + \xi_i \xi_j - \Delta \xi_j}{1 + \xi_i \xi_j - \Delta \xi_i}, \quad i= 1, \dots, N\),

via a fixed point method, i.e. a sequnce \(\xi(k), k \in \mathbb{Z}_{\geq 1}\) that converges to the solution. See this post.

The question is how to define the sequence. We have attempted two definition; see previous post. Both case, it seems that we run into issues when we applyt the Fixed Point Theorem. In both cases, we defined the sequence via a function \(F: \mathbb{C}^N \rightarrow \mathbb{C}^N\) so that \(F: \xi(k) \mapsto \xi(k+1)\). We couldn’t apply the Fixed Point Theorem becasue the function was not continuous. In the first case, the function involved \(L^{th}\) roots of unit and, in the second, the function has a pole singularity along an exceptional divsor. One may resolve these sigularities or show that the points of the sequence avoids the singularities. For instance, in the second case, one may resolve the pole singularity by extending the domain and range of the function to \((\mathbb{CP}^1)^{N}\). The set of points where the function is singular is called the exceptional divisor. We may control and show that the sequence is uniformly bounded away from the exceptional divisor if we have a convenient/efficient/clear description of the exceptional divisor. Thus, we consider yet a third defintion of the sequence below.

New sequence

We consider another way to define the sequence:

\[\begin{align} &(-1)^{N-1} \xi_i^L \prod_{j=1}^{N}(\xi_j^{-1} + \xi_i - \Delta \xi_i\xi_j^{-1})\\ &= \prod_{j=1}^{N} ( \xi_i - (\Delta- (\xi_j^{-1}) )\\ &= \sum_{m=0}^{N} (-1)^{N-m} e_{N-m}(x_1, x_2, \dots, x_n) \xi_i^m\\ &= (-1)^{N} e_{N}(x_1, \dots, x_N) +\sum_{m= 1}^{N} (-1)^{N-m} e_{N-m}(x_1, x_2, \dots, x_n) \xi_i^m\\ &= (-1)^{N} (\Delta - \xi_i^{-1})\prod_{j \neq i} x_j +\sum_{m= 1}^{N} (-1)^{N-m} e_{N-m}(x_1, x_2, \dots, x_n) \xi_i^m \end{align}\]

where \(x_j= \Delta - \xi_j^{-1}\) and \(e_{m}\) is the elemtary symmetric polynomial. Thus, we may consider the following sequence

\[\begin{align} &\xi_i(k+1)^{-1}\\ &= \frac{\Delta + (-1)^N \left(\xi_i(k)^L \prod_{j=1}^{N}(\xi_j(k)^{-1} + \xi_i(k) - \Delta \xi_i(k)\xi_j(k)^{-1}) + (-1)^N \sum_{m= 1}^{N} (-1)^{N-m} e_{N-m}(x_1(k), \dots, x_N(k)) \xi_i(k)^m\right)}{\prod_{j \neq i}x_j}. \end{align}\]

Let us denote the right-side of the equation above as \(F_i(\xi(k))\) and set \(F=(F_1, \dots, F_N)\). Then, the sequence is given as follows

\[\xi(k+1) = F(\xi(k)).\]

Then, to use the Fixed Point Theorem, we need to show that \(F\) is continuous (in the appropate region) and that the map is a contraction (or, sufficiently, that the norm of the Jacobian of \(F\) is less than one).

The exeptional divisor in this case is \(\{\xi_j \neq \Delta^{-1}: j=1, \dots, N\}\), where the function has a pole singularity. This divisor appears to be easier to control for theoretical bounds. There doesn’t appear to be a way to define the function of the sequence without any sort of singularity (… otherwise the function would be constant by the Louiville Theorem in complex analysis.)

Next Steps

  1. Implement this in Python.
  2. Develop theoretical bounds for the anisotropic parameter \(\Delta \in \mathbb{R}\) based on the number of up-spins \(N\) and the length of the ring \(L\).