We prove the second law of thermodynamics and the nonequilibirum fluctuation theorem for pure quantum states. The entire system obeys reversible unitary dynamics, where the initial state of the heat bath is not the canonical distribution but is a single energy-eigenstate that satisfies the eigenstatethermalization hypothesis (ETH). Our result is mathematically rigorous and based on the Lieb-Robinson bound, which gives the upper bound of the velocity of information propagation in many-body quantum systems. The entanglement entropy of a subsystem is shown connected to thermodynamic heat, highlighting the foundation of the information-thermodynamics link. We confirmed our theory by numerical simulation of hard-core bosons, and observed dynamical crossover from thermal fluctuations to bare quantum fluctuations. Our result reveals a universal scenario that the second law emerges from quantum mechanics, and can experimentally be tested by artificial isolated quantum systems such as ultracold atoms.Introduction. Although the microscopic laws of physics do not prefer a particular direction of time, the macroscopic world exhibits inevitable irreversibility represented by the second law of thermodynamics. Modern researches has revealed that even a pure quantum state, described by a single wave function without any genuine thermal fluctuation, can relax to macroscopic thermal equilibrium by a reversible unitary evolution [1][2][3][4][5][6][7][8][9][10][11]. Thermalization of isolated quantum systems, which is relevant to the zeroth law of thermodynamics, is now a very active area of researches in theory [1-6], numerics [10][11][12][13][14][15][16], and experiments [17][18][19][20][21][22][23]. Especially, the concepts of typicality [9,[24][25][26] and the eigenstate thermalization hypothesis (ETH) [10][11][12][27][28][29][30][31][32][33][34][35][36] have played significant roles.However, the second law of thermodynamics, which states that the thermodynamic entropy increases in isolated systems, has not been fully addressed in this context. We would emphasize that the informational entropy (i.e., the von Neumann entropy) of such a genuine quantum system never increases, but is always zero [37]. In this sense, a fundamental gap between the microscopic and macroscopic worlds has not yet been bridged: How does the second law emerge from pure quantum states?In a rather different context, a general theory of the second law and its connection to information has recently been developed even out of equilibrium [38][39][40][41], which has also been experimentally verified in laboratories [42][43][44][45]. This has revealed that information contents and thermodynamic quantities can be treated on an equal footing, as originally illustrated by Szilard and Landauer in the context of Maxwell's demon [46,47]. This research direction invokes a crucial assumption that the heat bath is, at least in the initial time, in the canonical distribution [48]; this special initial condition effectively breaks the

We systematically investigate scrambling (or delocalizing) processes of quantum information encoded in quantum many-body systems by using numerical exact diagonalization. As a measure of scrambling, we adopt the tripartite mutual information (TMI) that becomes negative when quantum information is delocalized. We clarify that scrambling is an independent property of integrability of Hamiltonians; TMI can be negative or positive for both integrable and non-integrable systems. This implies that scrambling is a separate concept from conventional quantum chaos characterized by non-integrability. Furthermore, we calculate TMI in the Sachdev-Ye-Kitaev (SYK) model, a fermionic toy model of quantum gravity. We find that disorder does not make scrambling slower but makes it smoother in the SYK model, in contrast to many-body localization (MBL) in spin chains.Introduction. Whether an isolated system thermalizes or not is a fundamental issue in statistical mechanics, which is related to non-integrability of Hamiltonians. In classical systems, thermalization has been discussed in terms of ergodicity of chaotic systems [1]. In quantum systems, a counterpart of classical chaos is not immediately obvious, because the Schrödinger equation is linear. Nevertheless, it has been established that there are some indicators of chaotic behaviors in quantum systems, such as the level statistics of Hamiltonians [2-4] and decay of the Loschmidt echo [5,6]. More recently, the eigenstatethermalization hypothesis (ETH) [7][8][9][10][11] has attracted attention as another indicator of quantum chaos in many-body systems, which states that even a single energy eigenstate is thermal. All these indicators of quantum chaos are directly related to integrability of Hamiltonians; non-integrable quantum systems exhibit chaos. Such a chaotic behavior in isolated quantum systems is also a topic of active researches in real experiments with ultracold atoms [12][13][14], trapped ions [15], NMR [5], and superconducting qubits [16].In order to investigate "chaotic" properties of quantum many-body systems beyond the conventional concept of quantum chaos, it is significant to focus on dynamics of quantum information encoded in quantum many-body systems. How does locally-encoded quantum information spread out over the entire system by unitary dynamics? Such delocalization of quantum information is referred to as scrambling [17][18][19][20][21]. Investigating scrambling is important not only for understanding relaxation dynamics of experimental systems at hand, but also in terms of information paradox of black holes [17], where it has been argued that black holes are the fastest scramblers in the universe [18]. However, the fundamental relationship between scrambling and conventional quantum chaos has not been comprehensively understood.Scrambling can be quantified by the tripartite mutual information (TMI) [21,22], which becomes negative if quantum information is scrambled. There is also another measure of scrambling, named the out-of-time-ordered correlator (OT...

A plausible mechanism of thermalization in isolated quantum systems is based on the strong version of the eigenstate thermalization hypothesis (ETH), which states that all the energy eigenstates in the microcanonical energy shell have thermal properties. We numerically investigate the ETH by focusing on the large deviation property, which directly evaluates the ratio of athermal energy eigenstates in the energy shell. As a consequence, we have systematically confirmed that the strong ETH is indeed true even for near-integrable systems. Furthermore, we found that the finite-size scaling of the ratio of athermal eigenstates is a double exponential for nonintegrable systems. Our result illuminates the universal behavior of quantum chaos, and suggests that a large deviation analysis would serve as a powerful method to investigate thermalization in the presence of the large finite-size effect.

The Sachdev-Ye-Kitaev (SYK) model attracts attention in the context of information scrambling, which represents delocalization of quantum information and is quantified by the out-of-time-ordered correlators (OTOC). The SYK model contains N fermions with disordered and four-body interactions. Here, we introduce a variant of the SYK model, which we refer to as the Wishart SYK model. We investigate the Wishart SYK model for complex fermions and that for hard-core bosons. We show that the ground state of the Wishart SYK model is massively degenerate and the residual entropy is extensive, and that the Wishart SYK model for complex fermions is integrable. In addition, we numerically investigate the OTOC and level statistics of the SYK models. At late times, the OTOC of the fermionic Wishart SYK model exhibits large temporal fluctuations, in contrast with smooth scrambling in the original SYK model. We argue that the large temporal fluctuations of the OTOC are a consequence of a small effective dimension of the initial state. We also show that the level statistics of the fermionic Wishart SYK model is in agreement with the Poisson distribution, while the bosonic Wishart SYK model obeys the GUE or the GOE distribution.

We study transient dynamics of hole carriers injected at a certain time into a Mott insulator with antiferromagnetic long range order. This is termed "dynamical hole doping" as contrast with chemical hole doping. Theoretical framework for the transient carrier dynamics are presented based on the two dimensional t− J model. Time dependences of the optical conductivity spectra as well as the one-particle excitation spectra are calculated based on the Keldysh Green's function formalism at zero temperature combined with the self-consistent Born approximation. At early stage after dynamical hole doping, the Drude component appears, and then incoherent components originating from hole-magnon scatterings start to grow. Fast oscillatory behavior due to coherent magnon, and slow relaxation dynamics are confirmed in the spectra. Time profiles are interpreted as that doped bare holes are dressed by magnon clouds, and are relaxed into spin polaron quasi-particle states. Characteristic relaxation times for Drude and incoherent peaks strongly depend on momentum of a dynamically doped hole, and the exchange constant. Implications to the recent pump-probe experiments are discussed. PACS numbers: