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Quantum to classical mapping of the two-dimensional toric code in an external field

Sydney R. Timmerman, Zvonimir Z. Bandic, Roger G. Melko

SciPost Phys. Lect. Notes 57 (2022) · published 20 July 2022


Kitaev's toric code Hamiltonian in dimension D=2 has been extensively studied for its topological properties, including its quantum error correction capabilities. While the Hamiltonian is quantum, it lies within the class of models that admits a D+1 dimensional classical representation. In these notes, we provide details of a Suzuki-Trotter expansion of the partition function of the toric code Hamiltonian in the presence of an external magnetic field. By coupling additional degrees of freedom in the form of a matter field that can subsequently be gauged away, we explicitly derive a classical Hamiltonian on a cubic lattice which takes the form of a non-isotropic 3D Ising gauge theory.

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