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Theoretical and Mathematical Physics

Local Hamiltonians for integrable quantum models on a lattice

V. Tarasov, Leon Armenovich Takhtadzhyan, L. D. Faddeev

Published Nov 1, 1983• 164 citations
Publisher
DOI
10.1007/bf01018648

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The freedom to change Fock basis seems to ensure a minimum amount of locality in lattice theories in the following sense: If $\lbrace (\hat a_i^\dagger\,,\,\hat a_i)\rbrace$ for $i=1,\dots,n$ is a lattice of creation and annihilation operators and if a given Hamiltonian $\hat H$ induces highly non-local dynamics on that lattice, then it will usually be possible to change to a new set of operators $\lbrace (\hat b_i^\dagger\,,\,\hat b_i)\rbrace$ in terms of which the dynamics appear less non-local. We demonstrate this by turning a highly non-local random matrix model into a local, 1D lattice theory where particles can propagate in localized wave packets. More generally, we show that any Hamiltonian can be made to look like such a theory, with the lattice dispersion relation and the non-integrability of the theory depending on the spectrum of $\hat H$. We argue that our results are a step towards quantum mereology for fields.

quant-pharXiv:2606.10004•Jun 10, 2026•0 citations
Introduction Lattice variants of integrable models, both classical and quantum, are formulated in the present chapter. The nonlinear Schrödinger (NS) equation and the sine-Gordon (SG) model are considered. QISM makes it possible to put continuous models of field theory on a lattice while preserving the property of integrability. In addition, the explicit form of the R-matrix is kept unchanged; this means that the structure of the action-angle variables for the classical models is unchanged. For quantum models, the analogue is the preservation of the S -matrix (see the end of section VII.7). The critical exponents, which characterize the power-law decay of correlators for large distances, are also preserved. For relativistic models of quantum field theory, lattice models may be used to rigorously solve the problem of ultraviolet divergences. The construction of local Hamiltonians for lattice models in quantum field theory is given much attention in this chapter. It is of interest to note that the L-operators of lattice models depend on some additional parameter Δ (which is absent in the R-matrix). This is the lattice spacing. However, the L-operator can be continued in Δ to the whole complex plane. Based on this fact, the most general L-operator may be constructed which is intertwined by a given R-matrix. This solves the problem of enumerating all the integrable models connected with a given R-matrix.
The freedom to change Fock basis seems to ensure a minimum amount of locality in lattice theories in the following sense: If {(a^i† , a^i)}\lbrace (\hat a_i^\dagger\,,\,\hat a_i)\rbrace{(a^i†​,a^i​)} for i=1,…,ni=1,\dots,ni=1,…,n is a lattice of creation and annihilation operators and if a given Hamiltonian H^\hat HH^ induces highly non-local dynamics on that lattice, then it will usually be possible to change to a new set of operators {(b^i† , b^i)}\lbrace (\hat b_i^\dagger\,,\,\hat b_i)\rbrace{(b^i†​,b^i​)} in terms of which the dynamics appear less non-local. We demonstrate this by turning a highly non-local random matrix model into a local, 1D lattice theory where particles can propagate in localized wave packets. More generally, we show that any Hamiltonian can be made to look like such a theory, with the lattice dispersion relation and the non-integrability of the theory depending on the spectrum of H^\hat HH^. We argue that our results are a step towards quantum mereology for fields.