We consider quantum Hamiltonians of the form $H={H}_{0}\ensuremath{-}U{\ensuremath{\sum}}_{j}cos({C}_{j})$, where
H0 is a quadratic function of position and momentum variables ${{x}_{1},{p}_{1},{x}_{2},{p}_{2},\ensuremath{\cdots}}$ and the
Cj's are linear in these variables. We allow
H0 and
Cj to be completely general with only two restrictions: we require that (1) the
Cj's are linearly independent and (2)
[Cj,Ck] is an integer multiple of $2\ensuremath{\pi}i$ for all
j,k so that the different cosine terms commute with one another. Our main result is a recipe for solving these Hamiltonians and obtaining their exact low-energy spectrum in the limit $U\ensuremath{\rightarrow}\ensuremath{\infty}$. This recipe involves constructing creation and annihilation operators and is similar in spirit to the procedure for diagonalizing quadratic Hamiltonians. In addition to our exact solution in the infinite
U limit, we also discuss how to analyze these systems when
U is large but finite. Our results are relevant to a number of different physical systems, but one of the most natural applications is to understanding the effects of electron scattering on quantum Hall edge modes. To demonstrate this application, we use our formalism to solve a toy model for a fractional quantum spin Hall edge with different types of impurities.
We obtain an exact solution for a generalized three-chain Ising tube (TCGIT) of length
L with toroidal boundary conditions and the most general
C3-invariant Hamiltonian on an elementary prism, containing 20 independent coupling constants, including an external magnetic field. Using an
8×8 transfer matrix, we derive the exact partition function of the finite system and obtain the free energy, internal energy, specific heat, magnetization, magnetic susceptibility, and entropy in the thermodynamic limit
L→∞. In the general case,
λmax is determined by a quartic equation, whereas in the principal special case with even-spin interactions (PSC) the spectrum simplifies substantially: the characteristic polynomial factorizes, and
λmax is given by the root of a quadratic equation. For mirror-symmetric subfamilies, we derive explicit formulas for the pair correlation functions and express the magnetization in terms of the components of the eigenvector associated with
λmax; in the even-spin case with
h=0, the magnetization vanishes. Important special cases include the width-three planar model with nearest-neighbor, next-nearest-neighbor, and plaquette interactions, including the entropy limit
S(T→0+)=(ln2)/3 for
k≥0 and
S(T→0+)=0 for
k<0, as well as the width-three planar triangular model with distinct nearest-neighbor couplings, three-spin interactions involving neighboring triangles, and an external field.