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We present the first successful application of the matrix product state (MPS)representing a thermal quantum pure state (TPQ) in equilibrium in two spatialdimensions over almost the entire temperature range. We use the Kitaevhoneycomb model as a prominent example hosting a quantum spin liquid (QSL)ground state to target the two specific-heat peaks previously solved nearlyexactly using the free Majorana fermionic description. Starting from thehigh-temperature random state, our TPQ-MPS framework on a cylinder preciselyreproduces these peaks, showing that the quantum many-body description based onspins can still capture the emergent itinerant Majorana fermions in a ${\mathbbZ}_2$ gauge field. The truncation process efficiently discards the high-energystates, eventually reaching the long-range entangled topological stateapproaching the exact ground state for a given finite size cluster. Anadvantage of TPQ-MPS over exact diagonalization or purification-based methodsis its lowered numerical cost coming from a reduced effective Hilbert spaceeven at finite temperature.

Thermal pure matrix product state in two dimensions: tracking thermal equilibrium from paramagnet down to the Kitaev honeycomb spin liquid state