arXiv subjects

Naomichi Hatano

Publications and source records attributed to Naomichi Hatano.

At least 37 records · Page 2Linked to original sources

arXiv2021

A chain of solvable non-Hermitian Hamiltonians constructed by a series of metric operators

We show how, given a non-Hermitian Hamiltonian $H$, we can generate new non-Hermitian operators sequentially, producing a virtually infinite chain of non-Hermitian Hamiltonians which are isospectral to $H$ and $H^\dagger$ and whose eigenvectors we can easily deduce in an almost automatic way; no ingredients are necessary other than $H$ and its eigensystem. To set off the chain and keep it running, we use, for the first time in our knowledge, a series of maps all connected to different metric operators. We show how the procedure works in several physically relevant systems. In particular, we apply our method to various versions of the Hatano-Nelson model and to some PT-symmetric Hamiltonians.

math-ph↗

arXiv2021

Non-Markovian effect on quantum Otto engine: -Role of system--reservoir interaction-

We study a limit cycle of a quantum Otto engine whose each cycle consists of two finite-time quantum isochoric (heating or cooling) processes and two quantum adiabatic work-extracting processes. Considering a two-level system as a working substance that weakly interacts with two reservoirs comprising an infinite number of bosons, we investigate the non-Markovian effect (short-time behavior of the reduced dynamics in the quantum isochoric processes (QIPs)) on work extraction after infinite repetition of the cycles. We focus on the parameter region where energy transferred to the reservoir can come back to the system in a short-time regime, which we call energy backflow to show partial quantum-mechanical reversibility. As a situation completely different from macroscopic thermodynamics, we find that the interaction energy is finite and negative by evaluating the average energy change of the reservoir during the QIPs by means of the full-counting statistics, corresponding to the two-point measurements. The feature leads us to the following findings: (1) the Carnot theorem is consistent with a definition of work including the interaction energy, although the commonly used definition of work excluding the interaction leads to a serious conflict with the thermodynamic law, and (2) the energy backflow can increase the work extraction. Our findings show that we need to pay attention to the interaction energy in designing a quantum Otto engine operated in a finite time, which requires us to include the non-Markovian effect, even when the system-reservoir interaction is weak.

quant-ph↗

arXiv2020

Non-Hermitian Fabry-Perot Resonances in a PT-symmetric system

In non-Hermitian scattering problems the behavior of the transmission probability is very different from its Hermitian counterpart; it can exceed unity or even be divergent, since the non-Hermiticity can add or remove the probability to and from the scattering system. In the present paper, we consider the scattering problem of a PT-symmetric potential and find a counter-intuitive behavior. In the usual PT-symmetric non-Hermitian system, we would typically find stationary semi-Hermitian dynamics in a regime of weak non-Hermiticity but observe instability once the non-Hermiticity goes beyond an exceptional point. Here, in contrast, the behavior of the transmission probability is strongly non-Hermitian in the regime of weak non-Hermiticity with divergent peaks, while it is superficially Hermitian in the regime of strong non-Hermiticity, recovering the conventional Fabry-Perot-type peak structure. We show that the unitarity of the S-matrix is generally broken in both of the regimes, but is recovered in the limit of infinitely strong non-Hermiticity.

quant-ph↗

arXiv2020

Random Multi-Hopper Model. Super-Fast Random Walks on Graphs

We develop a model for a random walker with long-range hops on general graphs. This random multi-hopper jumps from a node to any other node in the graph with a probability that decays as a function of the shortest-path distance between the two nodes. We consider here two decaying functions in the form of the Laplace and Mellin transforms of the shortest-path distances. Remarkably, when the parameters of these transforms approach zero asymptotically, the multi-hopper's hitting times between any two nodes in the graph converge to their minimum possible value, given by the hitting times of a normal random walker on a complete graph. Stated differently, for small parameter values the multi-hopper explores a general graph as fast as possible when compared to a random walker on a full graph. Using computational experiments we show that compared to the normal random walker, the multi-hopper indeed explores graphs with clusters or skewed degree distributions more efficiently for a large parameter range. We provide further computational evidence of the speed-up attained by the random multi-hopper model with respect to the normal random walker by studying deterministic, random and real-world networks.

physics.soc-ph↗

arXiv2020

Statistical properties of eigenvalues of the non-Hermitian Su-Schrieffer-Heeger model with random hopping terms

We explore the eigenvalue statistics of a non-Hermitian version of the Su-Schrieffer-Heeger model, with imaginary on-site potentials and randomly distributed hopping terms. We find that owing to the structure of the Hamiltonian, eigenvalues can be purely real in a certain range of parameters, even in the absence of parity and time-reversal symmetry. As it turns out, in this case of purely real spectrum, the level statistics is that of the Gaussian orthogonal ensemble. This demonstrates a general feature which we clarify that a non-Hermitian Hamiltonian whose eigenvalues are purely real can be mapped to a Hermitian Hamiltonian which inherits the symmetries of the original Hamiltonian. When the spectrum contains imaginary eigenvalues, we show that the density of states (DOS) vanishes at the origin and diverges at the spectral edges on the imaginary axis. We show that the divergence of the DOS originates from the Dyson singularity in chiral-symmetric one-dimensional Hermitian systems and derive analytically the asymptotes of the DOS which is different from that in Hermitian systems.

cond-mat.dis-nn↗

arXiv2020

Real Edge Modes in a Floquet-modulated $\mathcal{PT}$-symmetric SSH model

Non-Hermitian Hamiltonians provide a simple picture for analyzing systems with natural or induced gain and loss; however, in general, such Hamiltonians feature complex energies and a corresponding non-orthonormal eigenbasis. Provided that the Hamiltonian has $\mathcal{PT}$ symmetry, it is possible to find a regime in which the eigenspectrum is completely real. In the case of static $\mathcal{PT}$-symmetric extensions of the simple Su-Schrieffer-Heeger model, it has been shown that the energies associated with any edge states are guaranteed to be complex. Moving to a time-dependent system means that treatment of the Hamiltonian must be done at the effective time-scale of the modulation itself, allowing for more intricate phases to occur than in the static case. It has been demonstrated that with particular classes of periodic driving, achieving a real topological phase at high driving frequency is possible. In the present paper, we show the details of this process by using a simple two-step periodic modulation. We obtain a rigorous expression for the effective Floquet Hamiltonian and compare its symmetries to those of the original Hamiltonians which comprise the modulation steps. The $\mathcal{PT}$ phase of the effective Hamiltonian is dependent on the modulation frequency as well as the gain/loss strength. Furthermore, the topologically nontrivial regime of the $\mathcal{PT}$-unbroken phase admits highly-localized edge states with real eigenvalues in both the high frequency case and below it, albeit within a smaller extent of the parameter space.

quant-ph↗

arXiv2020

Null-eigenvalue localization of quantum walks on real-world complex networks

First we report that the adjacency matrices of real-world complex networks systematically have null eigenspaces with much higher dimensions than that of random networks. These null eigenvalues are caused by duplication mechanisms leading to structures with local symmetries which should be more present in complex organizations. The associated eigenvectors of these states are strongly localized. We then evaluate these microstructures in the context of quantum mechanics, demonstrating the previously mentioned localization by studying the spread of continuous-time quantum walks. This null-eigenvalue localization is essentially different from the Anderson localization in the following points: first, the eigenvalues do not lie on the edges of the density of states but at its center; second, the eigenstates do not decay exponentially and do not leak out of the symmetric structures. In this sense, it is closer to the bound state in continuum.

physics.soc-ph↗

arXiv2020

Discrete-time quantum walk on complex networks for community detection

We define the discrete-time quantum walk on complex networks and utilize it for community detection. We numerically show that the quantum walk with the Fourier coin is localized in a community to which the initial node belongs. Meanwhile, the quantum walk with the Grover coin tends to be localized around the initial node, not over a community. The probability of the classical random walk on the same network converges to the uniform distribution with a relaxation time generally a priori. We thus claim that the time average of the probability of the Fourier-coin quantum walk on complex networks reveals the community structure more explicitly than that of the Grover-coin quantum walk and a snapshot of the classical random walk. We first demonstrate our method of community detection for a prototypical three-community network, producing the correct grouping. We then apply our method to two real-world networks, namely Zachary's karate club and the US Airport network. We successfully reveals the community structure, the two communities of the instructor and the administrator in the former and major airline companies in the latter.

quant-ph↗

arXiv2019

Resonant-State Expansion of the Fano Peak in Open Quantum Systems

We describe the Fano asymmetry by expanding the transmission amplitude with respect to states with point spectra (discrete eigenstates), including not only bound states but also resonant states with complex eigenvalues. We first introduce a novel complete set that spans the Hilbert space of the central part of an open quantum-dot system. This complete set contains all states of point spectra, but does not contain any states of continuous spectra. We thereby analytically expand the conductance of the dot in terms of all discrete states without any background integrals. This expansion implies that the resonant states produce the main contributions to the electron transmission. We then explain the Fano peak as an interference effect involving resonant states. We find that there are three types of Fano asymmetry according to their origins: the interference between a resonant state and an anti-resonant state, that between a resonant state and a bound state, and that between two resonant states. We derive microscopic expressions of the Fano parameters that describe the three types of Fano asymmetry. We show that the last two types display the asymmetric energy dependence given by Fano, but the first one shows a slightly different form.

cond-mat.mes-hall↗

arXiv2019

Time-Reversal Symmetry and Arrow of Time in Quantum Mechanics of Open Systems

It is one of the most important and long-standing issues of physics to derive the irreversibility out of a time-reversal symmetric equation of motion. The present paper considers the breaking of the time-reversal symmetry in open quantum systems and the emergence of an arrow of time. We claim that the time-reversal symmetric Schrödinger equation can have eigenstates that break the time-reversal symmetry if the system is open in the sense that it has at least a countably infinite number of states. Such eigenstates, namely the resonant and anti-resonant states, have complex eigenvalues. We show that, although these states are often called "unphysical," they observe the probability conservation in a particular way. We also comment that the seemingly Hermitian Hamiltonian is non-Hermitian in the functional space of the resonant and anti-resonant states, and hence there is no contradiction in the fact that it has complex eigenvalues. We finally show how the existence of the states that break the time-reversal symmetry affects the quantum dynamics. The dynamics that starts from a time-reversal symmetric initial state is dominated by the resonant states for $t>0$; this explains the phenomenon of the arrow of time, in which the decay excels the growth. The time-reversal symmetry holds in that the dynamics ending at a time-reversal symmetric final state is dominated by the anti-resonant states for $t<0$.

quant-ph↗

arXiv2019

Exceptional points of the Lindblad operator of a two-level system

The Lindblad equation for a two-level system under an electric field is analyzed by mapping to a linear equation with a non-Hermitian matrix. Exceptional points of the matrix are found to be extensive; the second-order ones are located on lines in a two-dimensional parameter space, while the third-order one is at a point.

quant-ph↗

arXiv2018

Heating in integrable time-periodic systems

We investigate a heating phenomenon in periodically driven integrable systems that can be mapped to free-fermion models. We find that heating to the high-temperature state, which is a typical scenario in non-integrable systems, can also appear in integrable time-periodic systems; the amount of energy absorption rises drastically near a frequency threshold where the Floquet-Magnus expansion diverges. As the driving period increases, we also observe that the effective temperatures of the generalized Gibbs ensemble for conserved quantities go to infinity. By the use of the scaling analysis, we reveal that in the limit of infinite system size and driving period, the steady state after a long time is equivalent to the infinite-temperature state. We obtain the asymptotic behavior $L^{-1}$ and $T^{-2}$ as to how the steady state approaches the infinite-temperature state as the system size $L$ and the driving period $T$ increase.

cond-mat.stat-mech↗

arXiv2017

Thermoelectricity near Anderson localization transitions

The electronic thermoelectric coefficients are analyzed in the vicinity of one and two Anderson localization thresholds in three dimensions. For a single mobility edge, we correct and extend previous studies, and find universal approximants which allow to deduce the critical exponent for the zero-temperature conductivity from thermoelectric measurements. In particular, we find that at non-zero low temperatures the Seebeck coefficient and the thermoelectric efficiency can be very large on the "insulating" side, for chemical potentials below the (zero-temperature) localization threshold. Corrections to the leading power-law singularity in the zero-temperature conductivity are shown to introduce non-universal temperature-dependent corrections to the otherwise universal functions which describe the Seebeck coefficient, the figure of merit and the Wiedemann-Franz ratio. Next, the thermoelectric coefficients are shown to have interesting dependences on the system size. While the Seebeck coefficient decreases with decreasing size, the figure of merit first decreases but then increases, while the Wiedemann-Franz ratio first increases but then decreases as the size decreases. Small (but finite) samples may thus have larger thermoelectric efficiencies. In the last part we study thermoelectricity in systems with a pair of localization edges, the ubiquitous situation in random systems near the centers of electronic energy bands. As the disorder increases, the two thresholds approach each other, and then the Seebeck coefficient and the figure of merit increase significantly, as expected from the general arguments of Mahan and Sofo [J. D. Mahan and J. O. Sofo, Proc. Natl. Acad. Sci. U.S.A. 93, 7436 (1996)] for a narrow energy-range of the zero-temperature metallic behavior.

cond-mat.dis-nn↗

arXiv2017

The arrow of time in open quantum systems and dynamical breaking of the resonance-antiresonance symmetry

Open quantum systems are often represented by non-Hermitian effective Hamiltonians that have complex eigenvalues associated with resonances. In previous work we showed that the evolution of tight-binding open systems can be represented by an explicitly time-reversal symmetric expansion involving all the discrete eigenstates of the effective Hamiltonian. These eigenstates include complex-conjugate pairs of resonant and anti-resonant states. An initially time-reversal-symmetric state contains equal contributions from the resonant and anti-resonant states. Here we show that as the state evolves in time, the symmetry between the resonant and anti-resonant states is automatically broken, with resonant states becoming dominant for $t>0$ and anti-resonant states becoming dominant for $t<0$. Further, we show that there is a time-scale for this symmetry-breaking, which we associate with the "Zeno time." We also compare the time-reversal symmetric expansion with an asymmetric expansion used previously by several researchers. We show how the present time-reversal symmetric expansion bypasses the non-Hilbert nature of the resonant and anti-resonant states, which previously introduced exponential divergences into the asymmetric expansion.

quant-ph↗

arXiv2017

Path Laplacian operators and superdiffusive processes on graphs. I. One-dimensional case

We consider a generalization of the diffusion equation on graphs. This generalized diffusion equation gives rise to both normal and superdiffusive processes on infinite one-dimensional graphs. The generalization is based on the $k$-path Laplacian operators $L_{k}$, which account for the hop of a diffusive particle to non-nearest neighbours in a graph. We first prove that the $k$-path Laplacian operators are self-adjoint. Then, we study the transformed $k$-path Laplacian operators using Laplace, factorial and Mellin transforms. We prove that the generalized diffusion equation using the Laplace- and factorial-transformed operators always produce normal diffusive processes independently of the parameters of the transforms. More importantly, the generalized diffusion equation using the Mellin-transformed $k$-path Laplacians $\sum_{k=1}^{\infty}k^{-s}L_{k}$ produces superdiffusive processes when $1<s<3$.

math.FA↗

arXiv2016

Chebyshev-polynomial expansion of the localization length of Hermitian and non-Hermitian random chains

We carry Chebyshev-polynomial expansion of the inverse localization length of Hermitian and non-Hermitian random chains as function of energy. For Hermitian models, the expansion produces numerically this energy-dependent function in one run of the algorithm. This is in strong contrast to the standard transfer-matrix method, which produces the inverse localization length for a fixed energy in each run. For non-Hermitian models, as in the transfer-matrix method, our algorithm computes the inverse localization length for a fixed (complex) energy. We also find a formula of the Chebyshev-polynomial expansion of the density of states of non-Hermitian models. As explained in more detail in the Introduction, our algorithm for non-Hermitian models may be the only available efficient algorithm for finding the density of states of models with interactions.

cond-mat.dis-nn↗

arXiv2016

Quantum Jarzynski equality of measurement-based work extraction

Many studies of quantum-size heat engines assume that the dynamics of an internal system is unitary and that the extracted work is equal to the energy loss of the internal system. Both assumptions, however, should be under scrutiny. In the present paper, we analyze quantum-scale heat engines, employing the measurement-based formulation of the work extraction recently introduced by Hayashi and Tajima~[arXiv:1504.06150]. We first demonstrate the inappropriateness of the unitary time evolution of the internal system (namely the first assumption above) using a simple two-level system; we show that the variance of the energy transferred to an external system diverges when the dynamics of the internal system is approximated to a unitary time evolution. We second derive the quantum Jarzynski equality based on the formulation of Hayashi and Tajima as a relation for the work measured by an external macroscopic apparatus. The right-hand side of the equality reduces to unity for "natural" cyclic processes, but fluctuates wildly for non-cyclic ones, exceeding unity often. This fluctuation should be detectable in experiments and provide evidence for the present formulation.

cond-mat.stat-mech↗

arXiv2016

Efficiency bounds on thermoelectric transport in magnetic fields: The role of inelastic processes

We examine the efficiency of an effective two-terminal thermoelectric device under broken time-reversal symmetry. The setup is derived from a three-terminal thermoelectric device comprising a thermal terminal and two electronic contacts, under a magnetic field. We find that breaking time-reversal symmetry in the presence of the inelastic electron-phonon processes can significantly enhance the figure of merit for delivering electric power by supplying heat from a phonon bath, beyond the one for producing the electric power by investing thermal power from the electronic heat current. The efficiency of such a device is bounded by the non-negativity of the entropy production of the original three-terminal junction. The efficiency at maximal power can be quite close to the Carnot efficiency, but then the electric power vanishes.

cond-mat.mes-hall↗