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:

Information scrambling, characterized by the out-of-time-ordered correlator (OTOC), has attracted much attention, as it sheds new light on chaotic dynamics in quantum many-body systems. The scale invariance, which appears near the quantum critical region in condensed matter physics, is considered to be important for the fast decay of the OTOC. In this paper, we focus on the one-dimensional spin-1/2 XXZ model, which exhibits quantum criticality in a certain parameter region, and investigate the relationship between scrambling and the scale invariance. We quantify scrambling by the averaged OTOC over the Pauli operator basis, which is related to the operator space entanglement entropy (OSEE). Using the infinite time-evolving block decimation (iTEBD) method, we numerically calculate time dependence of the OSEE in the early time region in the thermodynamic limit. We show that the averaged OTOC decays faster in the gapless region than in the gapped region. In the gapless region, the averaged OTOC behaves in the same manner regardless of the anisotropy parameter. This result is consistent with the fact that the low energy excitations of the gapless region belong to the same universality class as the Tomonaga-Luttinger liquid with the central charge c = 1. Furthermore, we estimate c by fitting the numerical data of the OSEE with an analytical result of the two-dimensional conformal field theory, and confirmed that c is close to unity. Thus, our numerical results suggest that the scale invariance leads to a universal behavior of the OTOC that is independent of the anisotropic parameter, which reflects the universality of the two-dimensional conformal field theory at low temperatures.Although the one-dimensional XXZ model is integrable, our results suggest such a universal behavior of generic non-integrable systems, because the Tomonaga-Luttinger liquid serves as a low-energy effective theory for many non-integrable and integrable systems. On the other hand, the OTOC in our numerical result does not exhibit the exponential decay, as our parameter regime is far from the semiclassical limit.

Complexity of dynamics is at the core of quantum many-body chaos and exhibits a hierarchical feature: higher-order complexity implies more chaotic dynamics. Conventional ergodicity in thermalization processes is a manifestation of the lowest order complexity, which is represented by the eigenstate thermalization hypothesis (ETH) stating that individual energy eigenstates are thermal. Here, we propose a higher-order generalization of the ETH, named the k-ETH (k = 1, 2, . . . ), to quantify higher-order complexity of quantum many-body dynamics at the level of individual energy eigenstates, where the lowest order ETH (1-ETH) is the conventional ETH. The explicit condition of the k-ETH is obtained by comparing Hamiltonian dynamics with the Haar random unitary of the k-fold channel. As a non-trivial contribution of the higher-order ETH, we show that the k-ETH with k ≥ 2 implies a universal behavior of the kth Rényi entanglement entropy of individual energy eigenstates. In particular, the Page correction of the entanglement entropy originates from the higher-order ETH, while as is well known, the volume law can be accounted for by the 1-ETH. We numerically verify that the 2-ETH approximately holds for a nonintegrable system, but does not hold in the integrable case. To further investigate the information-theoretic feature behind the k-ETH, we introduce a concept named a partial unitary k-design (PU k-design), which is an approximation of the Haar random unitary up to the kth moment, where partial means that only a limited number of observables are accessible. The k-ETH is a special case of a PU k-design for the ensemble of Hamiltonian dynamics with random-time sampling. In addition, we discuss the relationship between the higher-order ETH and information scrambling quantified by out-of-time-ordered correlators. Our framework provides a unified view on thermalization, entanglement entropy, and unitary k-designs, leading to deeper characterization of higher-order quantum complexity. Contents

It is well known that the Onsager reciprocal relations give important identities between transport coefficients for heat and charge conduction in the linear response regime. 1, 2) For example, the Peltier coefficient Π is related to the Seebeck coefficient S in a simple form as S = Π/T , where T is the temperature. The Onsager-Casimir symmetry, which is the extension of the Onsager relation in the presence of an external magnetic field, is also derived from the microscopic reversibility. 3) Recently, a general relation called the steady-state fluctuation theorem has been derived by the same concept of the microscopic reversibility. 4, 5) It gives an identity equation on the probability of the entropy production ∆S during time τ as ln P (∆S) P (−∆S) = ∆S, (1) in the asymptotic limit τ → ∞. While this expression reproduces the ordinal Onsager relations in the linear response regime, it provides useful information even in the far-from-equilibrium regime. The importance of the Onsager-Casimir relation on coherent quantum transport of mesoscopic devices is known for long time. 6, 7) Recently, Saito and Utsumi have derived a quantum version of the fluctuation theorem, and applied it for derivation of universal relations among nonlinear transport coefficients 8) by means of the full-counting statistics in the Keldysh formalism. 9-11) In Ref. 8, however, its physical meaning on thermoelectric effects has not been addressed in detail.In this note, we discuss extension of the Onsager relation between the Peltier effect and the Seebeck effect to nonequilibrium steady states in mesoscopic devices. For simplicity, we consider a two-terminal setup, though extension to a multi-terminal setup is straightforward.We consider the Hamiltonian H = r=L,R H r + H d + H T , where