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arXiv:0810.4584 · Journal of Physics A Mathematical and Theoretical

On thermalization in Kitaev's 2D model

Robert Alicki, M. Fannes, Michał Horodecki

Published Jan 14, 2009• 142 citations
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Abstract

The thermalization process of the 2D Kitaev model is studied within the Markovian weak coupling approximation. It is shown that its largest relaxation time is bounded from above by a constant independent of the system size and proportional to exp(2Δ/ kT ), where Δ is an energy gap over the four-fold degenerate ground state. This means that the 2D Kitaev model is not an example of a memory, neither quantum nor classical.

DOI
10.1088/1751-8113/42/6/065303

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The thermalization process of the 2D Kitaev model is studied within the Markovian weak coupling approximation. It is shown that its largest relaxation time is bounded from above by a constant independent of the system size and proportional to exp(2Δ/ kT ), where Δ is an energy gap over the four-fold degenerate ground state. This means that the 2D Kitaev model is not an example of a memory, neither quantum nor classical.
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Abstract It has been conjectured that the speed of thermalization is universally constrained by a Planckian timescale, τ Pl = ℏ /( k B T ), determined solely by the reduced Planck constant ℏ, Boltzmann’s constant ( k B ) and the temperature ( T ). However, a general and model-independent derivation of this limit has remained elusive. Here we provide a general proof of the emergence of Planckian thermalization based on quantum information geometry and quantum metrology. We formulate thermalization as a process that prepares states close to the corresponding thermal ensemble for a set of distinct Hamiltonians, and show that quantum mechanics imposes a universal lower bound τ ≥ τ Pl /2 on the thermalization time τ at finite temperature. In the low-temperature regime, the spectral gap Δ between the ground and first excited states replaces the temperature as the relevant energy scale and determines the lower bound on the thermalization time, in close connection with the quantum adiabatic theorem. These bounds, rooted in Hamiltonian estimation, establish operational limits on thermalization governed exclusively by fundamental constants and the intrinsic energy scale.
We study ground-state and thermodynamic properties of an S=1S=1S=1 Kitaev model. We first clarify the existence of global parity symmetry in addition to the local symmetry on each plaquette, which enables us to perform the large scale calculations up to 24 sites. It is found that the ground state should be singlet and its energy is estimated as E/N∼−0.65JE/N\sim -0.65JE/N∼−0.65J, where JJJ is the Kitaev exchange coupling. We find that a lowest excited state belongs to the same subspace as the ground state and the gap monotonically decreases with increasing system size, which suggests that the ground state of the S=1S=1S=1 Kitaev model is gapless. By means of the thermal pure quantum states, we clarify finite temperature properties characteristic of the Kitaev models with S≤2S\le 2S≤2.