Transfer matrix for 1D chains Until recently I believed that the transfer matrix method such as used in solving the 1D Ising model could be used to solve the thermodynamics of any system that is:


*

*1D

*Translationally invariant

*Has only nearest-neighbor interactions (or any fixed finite range), and

*Has finite local dimension.


Besides being used for Ising spin-1/2, Heisenberg, and Ising spin-1 models, papers like this one use it for chains with local dimension 4. (Since it has next-nearest-neighbor interactions, it actually becomes local dimension 16.) In particular, the ground state energy is the lowest eigenvalue of the transfer matrix.
But then, there is Gottesman, Irani 2009 which seemed to create a very hard problem on a system that has all the above properties. Bausch et al. extended the work, reducing the local dimension to about 40. Given that finding the ground state energy of these Hamiltonians is QMAEXP-Complete, there certainly aren't solvable with a simple transfer matrix -- but why not?
My two guesses is that there's some additional condition (bosonic vs. fermionic operators, perhaps?) that I'm missing, or that somehow the finite system size of those 1D chains ends up contributing finite size effects that end up being more relevant than expected.
 A: Answering my own question, feeling silly now. The key additional requirement is that local interactions can be broken up into commuting terms. For some general nearest-neighbor interaction $J_{ij}$ acting on sites $i$ and $j$, the partition function reads
$$Z = \exp(-\beta H) = \exp(\sum -\beta J_{i,i+1})\quad \neq\quad \exp(-\beta J_{1,2})\exp(-\beta J_{2,3}) \dots = T_{12}T_{23} \dots$$
On the left we have the partition function, and on the right are the transfer matrices $\exp(-\beta J)$ that we want. But we do not in general have equality in the middle, unless all $J$'s commute. (Although you could do an expansion in terms of e.g. the BCH formula.) Of the models I gave as examples,


*

*The Ising model can only be solved with transfer matrices if it the interactions are aligned with the field, i.e. $\sum J S^z_{i} S^z_{i+1} + h S^z_i$, the classical Ising model. The transverse field Ising model with $h S^x_i$ cannot be solved with transfer matrices, and is solved through other methods. The same is true for spin-1 Ising models.

*The Heisenberg model cannot (as far as I can tell, upon more careful reading) be solved with transfer matrices. It also requires Jordan-Wigner or Bethe ansatz solutions.

*The other paper I linked, with the Hubbard models and local dimension 16, makes a narrow-bandwidth approximation. This allows them to drop the hopping terms and have only number terms, which all commute.


Lots of other 1D models cannot be written in this commuting form of course, which prevents them from being solved this way. Having a commuting form can be understood as the model being "inherently classical", in which case the partition function becomes a counting problem on a path graph.
Personally, I think of this in relation to fixed-parameter tractable algorithms for counting solutions to constraint problems on graphs of fixed path-width, e.g. Courcelle's theorem.
