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It is an important challenge to understanding how isolated quantum systems reach thermal equilibrium from the microscopic dynamics, which is linked to foundations of statistical mechanics. Such a problem is also related to the question of how to characterize chaos in quantum systems.
As a sufficient condition for relaxation to thermal equilibrium, the eigenstate thermalization hypothesis (ETH) has attracted much attention. The ETH is a hypothesis which states that the expectation values of physical quantities are approximately equal for the energy eigenstates belonging to the same energy shell. It has been confirmed numerically for many non-integrable systems and few-body physical quantities.
エネルギーと物理量の固有状態間の変換を表すユニタリー行列の集合を考えると、典型的なものに対し「物理量の固有状態期待値の差がミクロカノニカルシェル内で指数的に小さい」という条件を満たします。これはETHの十分条件です。
一方我々は、few-bodyのハミルトニアンとランダムなfew-bodyの物理量に対して、対応するユニタリー行列がほとんどの場合非典型的に振る舞うことを示しました。
一方我々は、few-bodyのハミルトニアンとランダムなfew-bodyの物理量に対して、対応するユニタリー行列がほとんどの場合非典型的に振る舞うことを示しました。
It was argued by von Neumann that ETH can be justified if we assume that the energy eigenstates and the eigenstates of physical quantities are randomly oriented in the sense of unitary Haar measure (the typicality argument).
However, we have rigorously shown that, when the energy shell is not exponentially small as in the case of microcanonical shells, the naive typicality argument is not applicable in most cases for setups with few-body (or local interaction) nature that we are usually interested in. This means the typicality argument by von Neumann cannot provide justification of ETH.
Phys. Rev. Lett. 120, 080603 (2018). [arXiv:1708.04772]
So, is the ETH really universally valid in locally interacting systems? To answer this question, we considered an ensemble of Hamiltonians with randomized local interactions. We then showed numerically that the ETH holds for most of the samples in the ensemble. This is the first evidence that ETH is universally valid in realistic systems.
Phys. Rev. Lett. 126, 120602 (2021). [arXiv:2005.06379]
We further extended this result to long-range interacting systems with power decay. We obtained strong evidence that the ETH holds where the power index a is greater than 0.6, while there is no sign of the ETH where a is less than 0.5, up to 20 spins. This means that the symmetry that exists for a=0 (which leads to the breakdown of the ETH for a=0) approximately remains and prohibits the ETH up to relatively large sizes, which are experimentally relevant.
Among many quantum many-body systems, the transverse field Ising model is one of the most fundamental models. However, in higher dimensions, the model is non-integrable and its treatment is difficult. In particular, property of thermalization has remained elusive.
We have analytically shown that thermalization is prohibited in the high-dimensional transverse Ising model in the weak-field limit because of the recently discovered mechanism called the Hilbert Space Fragmentation (HSF). The HSF denotes a non-trivial decomposition of the Hilbert space (which cannot be explained by ordinary locally conserved quantities) due to dynamical constraints, which prevents the ETH and thermalization. We also point out that, while most of the previous works on the HSF were based on more than one conservation law, the HSF in our work is caused by a single conservation law of the domain wall number.
Phys. Rev. Lett. 129, 090602 (2022). [arXiv:2111.05586]
Moreover, this fact can be applied to quantum technology.
Controlling entanglement and coherence of a multiple qubit system is a crucial task in today’s quantum technology. One important example is the quantum sensing, where states with appropriate quantum coherence increases the sensitivity (called the Heisenberg limit) beyond the reach of classical schemes. However, when one actually tries to perform sensing using a quantum many-body system, the problem of interaction arises. That is, while proper use of interactions can produce coherence and entanglement, interactions generally thermalize and dephase the system over time.
We have shown that in high-dimensional quantum many-body spin systems with strong proximity Ising interactions, coherence is preserved by the HSF, which allows stable quantum sensing of transverse fields. Specifically, we analytically show that the emergent HSF caused by strong Ising interactions enables us to design a stable state where part of the spins is effectively decoupled from the rest of the system. Using the decoupled spins as a probe to measure a transverse field, we demonstrate that the Heisenberg limited sensitivity is achieved without suffering from thermalization.
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