Takashi Mori Web Site

Research

Thermalization in isolated quantum systems

It is a long-standing fundamental problem to reconcile reversible microscopic laws and irreversible macroscopic phenomena. For example, thermalization, i.e., relaxation to thermal equilibrium, of a macroscopic system is ubiquitous in nature, but it is not at all obvious whether an isolated quantum system can thermalize.

In recent years, this problem has gathered much attention because of the experimental progress in ultra-cold atoms and trapped ions, and our theoretical understanding on this fundamental problem has been refined. Typicality and Eigenstate Thermalization Hypothesis (ETH) are important concepts.

We have derived some rigorous results on the eigenstate thermalization hypothesis [1,2]. I'm also interested in prethermalization [3,4], i.e., the relaxation toward a long-lived quasi-stationary state before reaching thermal equilibrium.

I, with my colleagues, published a review paper, which overviews theoretical development in this field.

  1. Takashi Mori, "Weak eigenstate thermalization with large deviation bound" arXiv:1609.09776

  2. Naoto Shiraishi and Takashi Mori,"Systematic Construction of Counterexamples to the Eigenstate Thermalization Hypothesis", Phys. Rev. Lett. 119, 030601 (2017)

  3. Eriko Kaminishi, Takashi Mori, Tatsuhiko N Ikeda, Masahito Ueda, "Entanglement pre-thermalization in a one-dimensional Bose gas", Nature Phys. 11, 1050 (2015)

  4. Takashi Mori, "Prethermalization in the transverse-field Ising chain with long-range interactions", J. Phys. A 52, 054001 (2019)

Floquet problem in many-body quantum systems

When a quantum system is subject to a periodically oscillating external field (e.g. laser light in condensed matter or mechanical oscillation in cold atoms), the Hamiltonian of the system periodically depends on time. Such a system is called a periodically driven system or a Floquet system, which stems from the fact that the Floquet theory is an appropriate theoretical tool to treat such systems.

In recent years, strong (large amplitude) and fast (large frequency) periodic driving has paid much attention. For such driving, we can derive a simple expression of a static effective Hamiltonian by applying the Floquet-Magnus expansion. The simple expression of the effective Hamiltonian enables us to design periodic fields so that they generate an interesting phase of matter that is hard to be realized in a static system.

However, the method of the effective Hamiltonian had not been justified in macroscopic quantum Floquet systems before our works. Indeed, it is known that the convergence radius of the Floquet-Magnus expansion shrinks to zero in the thermodynamic limit, which doubts the applicability of this method. We made clear the condition under which the use of the static effective Hamiltonian is justified by investigating the Floquet-Magnus expansion mathematically [1,2]. It turns out that the timescale of heating due to periodic driving is exponentially slow with respect to the frequency in generic lattice systems. This result implies that genetic lattice systems under a strong and fast oscillating field exhibit Floquet prethermalization, and the method of the effective Hamiltonian is justified in a prethermal regime of the relaxation process. The use of the effective Hamiltonian is also justified for some off-lattice systems, but the timescale of heating is faster than exponential [3].

I have also investigated the steady state of a many-body Floquet system under dissipation by using the quantum master equation. This problem is important for Floquet physics in solids since dissipation is inevitable there. We made clear the condition under which the notion of the Gibbs state in terms of the effective Hamiltonian is valid in the steady state [4, 5].

  1. Tomotaka Kuwahara, Takashi Mori, and Keiji Saito, "Floquet–Magnus theory and generic transient dynamics in periodically driven many-body quantum systems" Ann. Phys. 367, 96 (2016)

  2. Takashi Mori, Tomotaka Kuwahara, and Keiji Saito, "Rigorous bound on energy absorption and generic relaxation in periodically driven quantum systems" Phys. Rev. Lett. 116, 120401 (2016)

  3. Takashi Mori, "Floquet resonant states and validity of the Floquet-Magnus expansion in the periodically driven Friedrichs models", Phys. Rev. A 91, 020101 (2015)

  4. Tatsuhiko Shirai, Takashi Mori, and Seiji Miyashita, "Condition for emergence of the Floquet-Gibbs state in periodically driven open systems", Phys. Rev. E 91, 030101 (2015).

  5. Tatsuhiko Shirai, Juzar Thingna, Takashi Mori, Sergey Denisov, Peter Hänggi, and Seiji Miyashita, "Effective Floquet–Gibbs states for dissipative quantum systems", New J. Phys. 18, 053008 (2016).

Statistical mechanics of long-range interacting systems

Equilibrium statistical mechanics of long-range interacting systems shows various peculiar properties such as the violation of ensemble equivalence and negative specific heat. In an extreme case where the interaction potential is independent of the distance, we can investigate the problem analytically since it is known that the mean-field theory is exact for such fully connected models in the thermodynamic limit. However, long-range interaction potentials in nature decay with distance slowly, and not many rigorous results have been known.

For long-range interacting spin systems in which the pair potential decays with distance r slower than 1/r^d, where d is the spatial dimension, I rigorously proved that the mean-field theory is still exact in the canonical ensemble [1,2]. On the other hand, the mean-field theory can break down in the canonical ensemble with a fixed value of the magnetization [1,2] or in the microcanonical ensemble [3]. These results also hold in quantum spin systems [4]. See also [5] for a brief review.

I also explored the possibility that a short-range interacting system exhibits peculiar properties of equilibrium statistical mechanics of long-range interacting systems, such as the negative specific heat. Rigorous results of equilibrium statistical mechanics rule out this possibility in thermal equilibrium. I found that a metastable state of a certain short-range interacting system resembles an equilibrium state of a certain long-range interacting model [6,7]. As a result, this short-range model exhibits the violation of the ensemble equivalence, the negative specific heat, and the negative susceptibility in its metastable state. This result indicates that considering equilibrium statistical mechanics of long-range interacting systems may be much more important than considered previously.

  1. Takashi Mori, "Analysis of the exactness of mean-field theory in long-range interacting systems", Phys. Rev. E 82, 060103(R) (2010)

  2. Takashi Mori, "Instability of the mean-field states and generalization of phase separation in long-range interacting systems", Phys. Rev. E 84, 031128 (2011)

  3. Takashi Mori, "Microcanonical Analysis of Exactness of the Mean-Field Theory in Long-Range Interacting Systems", J. Stat. Phys. 147, 1020 (2012)

  4. Takashi Mori, "Equilibrium properties of quantum spin systems with nonadditive long-range interactions", Phys. Rev. E 86, 021132 (2012)

  5. Takashi Mori, "Phase transitions in systems with non-additive long-range interactions", J. Stat. Mech. P10003 (2013)

  6. Takashi Mori, "Nonadditivity in Quasiequilibrium States of Spin Systems with Lattice Distortion", Phys. Rev. Lett. 111, 020601 (2013)

  7. Takashi Mori, "Quasi-Equilibrium Nonadditivity", J. Stat. Phys. 159, 172 (2015)

Open quantum systems

Dissipative dynamics in a system in contact with a large environment is described by a quantum master equation. The quantum master equation is microscopically derived from the unitary dynamics under the Hamiltonian of the total system, including the environment. Since dissipation is ubiquitous in nature, it is an important problem of statistical physics to understand the property of the quantum master equation that is derived microscopically.

I have thoroughly investigated the steady state of the quantum master equation when the environment is in thermal equilibrium and the dissipation is weak [1]. Under some familiar approximations such as the rotating-wave approximation or dropping the principal value of the time integral, the steady state is given by the Gibbs state of the Hamiltonian of the system of interest. However, the steady state should be modified by the finite interaction between the system and the thermal bath. We have shown that the leading-order modification of the steady state is property taken into account by considering the fourth-order quantum master equation, although the second-order quantum master equation is widely used.

Recently, driven-dissipative systems have also paid much attention. Under some conditions, a driven-dissipative system is expressed as an open system in contact with a nonequilibrium environment. Since the detailed-balance condition is not satisfied, its steady state is not trivial. It is an important theoretical problem to predict the nonequilibrium steady state, that is partially motivated by recent cold-atom experiments which enable us to design dissipation so that the nonequilibrium steady state has a desired property.

We have argued that the eigenstate thermalization hypothesis (ETH), which has been studied in context of thermalization in isolated systems, plays a crucial role in characterizing nonequilibrium steady states of macroscopic open systems when the dissipation is sufficiently weak [2]. In particular, the steady state is described by the equilibrium Gibbs state even though the detailed balance condition is violated if the ETH holds and the weak-dissipation perturbation theory is valid. The condition of validity of the perturbation theory is also theoretically derived.

  1. Takashi Mori and Seiji Miyashita, "Dynamics of the density matrix in contact with a thermal bath and the quantum master equation", J. Phys. Soc. Jpn. 77, 124005 (2008).

  2. Tatsuhiko Shirai and Takashi Mori, "Thermalization in Open Many-Body Systems Based on Eigenstate Thermalization Hypothesis", Phys. Rev. E 101, 042116 (2020).

Machine Learning

Modern machine learning achieves unparalleled successes in the overparameterized regime, in which the number of parameters in the model greatly exceeds that of the training data samples. This is surprising because traditional learning theory suggests that such an overparameterized model suffers from serious overfitting and does not show good generalization ability.

It is a theoretical challenge to understand why overparameterized deep neural networks exhibit astonishing generalization ability without serious overfitting. There are several perspectives to explain it: the importance of data structure, some special properties of network architecture, and implicit biases of the optimization algorithm like stochastic gradient descent (SGD).

In our recent works, I investigated the efficiency of SGD [1,2]. In [2], I focused on the property of dynamical noise in SGD, and showed that noise strength is proportional to the loss function as well as its Hessian near a local minimum. This property of SGD noise suggests that the escape rate from a local minimum obeys an unconventional law. The new escape rate formula implies that SGD prefers minima with a low effective dimension. This result will trigger further studies on SGD and its implicit biases.

  1. Liu Ziyin, Kangqiao Liu, Takashi Mori, and Masahito Ueda, "Strength of Minibatch Noise in SGD", In the Proceedings of the 10th International Conference on Learning Representations (ICLR2022)

  2. Takashi Mori, Liu Ziyin, Kangqiao Liu, and Masahito Ueda, "Power-Law Escape Rate of SGD", In Proceedings of the 39th International Conference on Machine Learning (ICML2022), PMLR 162, 15959 (2022)