Finite-temperature tensor-network methods for quantum spin systems and their application to the extended Kitaev model

Tensor network methods have become powerful and versatile tools for investigating strongly correlated quantum many-body systems. Their applicability now extends far beyond ground-state calculations and also applies to finite-temperature properties and real-time dynamics. In this talk, we introduce tensor-network approaches for representing finite-temperature states, focusing on tensor-network constructions of thermal density matrices. We represent the density matrix as a single-layer tensor network, such as a matrix product operator (MPO) or tensor product operator (TPO), and optimize it at a target temperature. Although this representation does not strictly guarantee the positivity of the density operator, it nevertheless provides a controlled and efficient framework for exploring thermodynamic quantities and temperature-driven phenomena in practical computations.

We apply this approach to the extended Kitaev model on the two-dimensional honeycomb lattice, where additional off-diagonal interactions are added to the exactly solvable Kitaev Hamiltonian that hosts a quantum spin-liquid ground state. We demonstrate that the thermal Hall conductivity under a magnetic field can be computed by employing open boundary conditions. We also discuss the possibility of a finite-temperature phase transition associated with the breaking of lattice rotational symmetry.