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Problems of Thermalization in Closed and Open Quantum Many-Body Systems

濱崎, 立資 東京大学 DOI:10.15083/0002004704

2022.06.22

概要

The main theme of this Thesis is to study problems of thermalization in closed and open quantum many-body systems, which are deeply related to foundations of quantum statistical mechanics. We address how the approach to thermal equilibrium is justified from unitary dynamics of closed quantum systems, and how dynamics and other many- body properties are enriched by external dissipation in open quantum systems. These studies have attracted growing interests due to the recent development of controllable experiments of quantum many-body systems, such as cold atoms and ions.

We first discuss closed quantum systems. Quantum statistical mechanics has become an indispensable framework to understand macroscopic phenomena from the knowledge of quantum mechanics. Nevertheless, why the microcanonical ensemble correctly de- scribes the equilibrium state has not yet been justified. In particular, we ask how and when initially nonequilibrium quantum systems relax to states described by the micro- canonical ensemble only by unitary time evolution. This is the problem of thermalization in closed quantum systems.

This problem was first tacked by von Neumann [J. von Neumann, Zeit. Phys. 57, 30 (1929)]. His seminal ideas have been reformulated and developed over the last decade, motivated by experimental observations of thermalization dynamics in almost closed sys-tems. The first idea is about how a time-evolving quantum pure state |𝜓(𝑡)⟩ = 𝑒 ℏ |𝜓(0)⟩ after a long time can mimic a thermal equilibrium state given by the microcanonical en- semble, 𝜌ˆmic. While the two states are always distinct at the level of density matrices, they can be almost indistinguishable when viewed from physically relevant sets of ob-servables 𝒪ˆ (such as local, few-body, or macroscopic observables). In fact, thermalization requires ⟨𝜓(𝑡)|𝒪ˆ|𝜓(𝑡)⟩ ≃ Tr[𝜌ˆmic𝒪ˆ] for long time 𝑡 (except for rare recurrence times) in the thermodynamic limit.

The second idea is about the sufficient condition for the above thermalization, which is now known as the eigenstate thermalization hypothesis (ETH). The ETH dictates that expectation values of 𝒪𝛼𝛼 := ⟨𝐸𝛼 |𝒪ˆ|𝐸𝛼⟩ with respect to an energy eigenstate |𝐸𝛼⟩ of the Hamiltonian become almost constant over the microcanonical energy shell, i.e., Δ := max|𝐸𝛼⟩,|𝐸𝛽⟩∈microcanonical shell |𝒪𝛼𝛼 − 𝒪𝛽𝛽 | becomes negligibly small in the thermody- namic limit. While the ETH is still a conjecture, it is expected to hold true for generic nonintegrable quantum many-body systems, which exhibit random-matrix-type spectral statistics (e.g., level-spacing distributions) reflecting systems’ complexity. Note that the ETH and statistical mechanics do not hold for several systems, represented by many-body localization (MBL) caused by strong disorder.

The third idea is about why generic systems seem to satisfy the ETH. Von Neumann and his recent followers invoke the typicality argument to explain the ETH [P. Reimann, Phys. Rev. Lett. 115, 010403 (2015)]. This argument consists of two parts. First, we rewrite
𝒪𝛼𝛼 = ∑𝑖 𝑎𝑖 | ⟨𝐸𝛼 |𝑎𝑖⟩ |2, where we have diagonalized the observable as 𝒪ˆ = ∑𝑖 𝑎𝑖 |𝑎𝑖⟩ ⟨𝑎𝑖 |. We also introduce the unitary matrix 𝑈 that represents the basis transformation 𝑈𝛼𝑖 = ⟨𝐸𝛼 |𝑎𝑖⟩. Then, for almost all (i.e., typical) 𝑈’s that are chosen uniformly from the unitary Haar measure, Δ becomes exponentially small with increasing the system size. Second, one assumes that, for physically relevant Hamiltonian and observable, the corresponding 𝑈 (restricted to the microcanonoical energy shell) is typical, unless some special reasons exist, such as the MBL. If this assumption holds true, Δ becomes exponentially small, which leads to the ETH. Note that, while the first part is a mathematical fact, the second part is a conjecture.

As a first original result in this Thesis, we revisit the typicality argument and show that it does not hold true, i.e., 𝑈 is atypical, for physically relevant setups of few-body (or local) Hamiltonians and observables (see Chapter 3). To be precise, we show that Δ does not decrease exponentially with increasing the system size for few-body (local) Hamiltonians and most few-body (local) observables, when the width of the energy shell decreases at most polynomially. This work asserts that a different scenario that does not rely on the typicality argument should be required to justify the ETH.

From Chapter 4, we discuss open quantum systems, where the system exhibits non- unitary dynamics due to the coupling to external environments or ancillas. State-of-the- art technologies now enable us to manipulate such dissipation in experiments. In other words, we are at the stage of understanding nonequilibrium phenomena of open quantum many-body systems, which can be richer than those of closed systems.

One of the interesting setups for open quantum systems is continuously or repeat- edly measured quantum systems. Here, we keep measuring the stochastic outcomes due to dissipation (such as particle loss), from which we obtain a set of quantum trajecto- ries depending on the outcomes. Each of the trajectories consists of (i) non-Hermitian time-evolution and (ii) stochastic quantum jumps that suddenly change the state. If the quantum jumps are all traced out, the averaged dynamics is described by the Lindblad master equation in a certain limit. In turn, we can postselect some trajectories and dis- cuss their dynamics: for example, trajectories with null quantum jumps obey purely non-Hermitian dynamics.

As a second original result in this Thesis, we study how the non-Hermiticity in open systems affects for the physics of the MBL, which has been an important con- cept for thermalization in closed systems (see Chapter 5). We first consider an interacting model with disorder potential and asymmetric hopping that causes non-Hermiticity while keeping time-reversal symmetry (TRS). While non-interacting disordered particles with asymmetric-hopping have been well known as one of the prototypical models with non- Hermiticity [N. Hatano and D. R. Nelson, Phys. Rev. Lett. 77, 570 (1996)], the interaction considered here forces us to study many-body properties of the model. Consequently, we find a novel real-complex phase transition of many-body energy eigenvalues associated with the TRS: almost all eigenvalues become complex (real) below (above) the critical disorder. The real-complex transition alters the stability of thermalization dynamics in this open system. We also argue that the real-complex transition occurs upon the MBL extended to non-Hermitian systems. We demonstrate that the non-Hermitian MBL can be characterized by the level-spacing distributions on the complex plane, which changes from the non-Hermitian random-matrix (Ginibre) statistics to the Poisson statistics with increasing disorder. The non-Hermitian MBL can also be characterized by the eigenstates that obey the area law of entanglement, as in the Hermitian MBL. We also consider the non-Hermitian interacting disordered model with gain and loss, which breaks the TRS. Due to the absence of the TRS, this model does not exhibit the real-complex transition while the non-Hermitian MBL still persists.

As mentioned above, certain properties of many-body systems, such as delocalization and nonintegrability, are characterized by the level-spacing distributions of random ma- trices. This is a demonstration of the universality: some spectral statistics are described by those of the Gaussian random matrices irrespective of the detail of the matrix elements. In Hermitian matrices, such universal statistics are classified by the TRS: three distinct universal level-spacing distributions appear according to three different symmetry classes A (with no TRS), AI (with TRS whose square equals to +1), and AII (with TRS whose square equals to −1) introduced by Dyson.

On the other hand, the situation is different for non-Hermitian random matrices with TRS. Indeed, matrices in non-Hermitian classes A, AI, and AII introduced by Ginibre lead to only a single universality class in stark contrast with the Hermitian case.

As a third original result in this Thesis, we solve this problem by considering different symmetry classes called classes AI† and AII† involving transposition symmetry, which is distinct from TRS (see Chapter 6). For example, real matrices (𝐻 = 𝐻∗) in class A are distinct from the symmetric matrices (𝐻 = 𝐻𝑇) in the non-Hermitian case. We numerically demonstrate that three distinct universal level-spacing distributions appear for classes A, AI†, and AII† as in the Hermitian case, since transposition symmetry can alter interactions between nearby eigenvalues. We argue that only the three universality classes of level-spacing distributions exist among 38 types of non-Hermitian symmetry classes [K. Kawabata et al., Phys. Rev. X 9, 041015 (2019)], since symmetries other than transposition cannot alter the interactions between close eigenvalues. Finally, we show that our newly found universality classes appear in open quantum many-body systems described by a Lindblad master equation and a non-Hermitian Hamiltonian. This suggests that our work serves as a basis for characterizing nonintegrability and chaos in open quantum many-body systems.

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