We investigate how the normal energy transport is realized in one-dimensional quantum systems using a quantum spin system. The direct investigation of local energy distribution under thermal gradient is made using the quantum master equation, and the mixing properties and the convergence of the Green-Kubo formula are investigated when the number of spin increases. We find that the autocorrelation function in the Green-Kubo formula decays as ∼ t −1.5 to a finite value which vanishes rapidly with the increase of the system size. As a result, the Green-Kubo formula converges to a finite value in the thermodynamic limit. These facts strongly support the realization of Fourier heat law in a quantum system.
We investigate how the normal energy transport is realized in one-dimensional quantum systems using a quantum spin system. The direct investigation of local energy distribution under thermal gradient is made using the quantum master equation, and the mixing properties and the convergence of the Green-Kubo formula are investigated when the number of spin increases. We find that the autocorrelation function in the Green-Kubo formula decays as ∼ t −1.5 to a finite value which vanishes rapidly with the increase of the system size. As a result, the Green-Kubo formula converges to a finite value in the thermodynamic limit. These facts strongly support the realization of Fourier heat law in a quantum system.
We study thermal transport in a classical one-dimensional Heisenberg model employing a discrete-time odd-even precessional update scheme. This dynamics equilibrates a spin chain for any arbitrary temperature and finite value of the integration time step $\ensuremath{\Delta}t$. We rigorously show that in presence of driving, the system attains local thermal equilibrium, which is a strict requirement of Fourier law. In the thermodynamic limit, heat current for such a system obeys Fourier law for all temperatures, as has been recently shown [A. V. Savin, G. P. Tsironis, and X. Zotos, Phys. Rev. B 72, 140402(R) (2005)]. Finite systems, however, show an apparent ballistic transport which crosses over to a diffusive one as the system size is increased. We provide exact results for current and energy profiles in zero- and infinite-temperature limits.
We numerically investigate thermal conduction in quantum dynamics. A one-dimensional quantum spin system is used as a model, which we can make integrable or nonintegrable by controlling parameter values. A different method to simulate heat reservoirs is introduced. Using this method, we perform numerical simulations of the spin chain in contact with two heat reservoirs at different temperatures. As a consequence, a flat temperature profile is observed in the integrable case, while a finite temperature gradient is found in a nonintegrable system. This result suggests that the Fourier heat law may be realized in nonintegrable systems and is consistent with the classical case reported in the literature. \textcopyright{} 1996 The American Physical Society.
To study transport properties, one needs to investigate the system of interest when coupled to biased external baths. This requires solving a master equation for this open quantum system. Obtaining this solution is very challenging, especially for large systems. This limits applications of the theories of open quantum systems, especially insofar as studies of transport in large quantum systems, of interest in condensed matter, is concerned. In this thesis, I propose three efficient methods to solve the Redfield equation --- an example of such a master equation. The first is an open-system Kubo formula, valid in the limit of weak bias. The second is a solution in terms of Green's functions, based on a BBGKY (Bogoliubov--Born--Green--Kirkwood--Yvon)-like hierarchy. In the third, the Redfield equation is mapped to a generalized Fokker-Planck equation using the coherent-state representation. All three methods, but especially the latter two, have much better efficiency than direct methods such as numerical integration of the Redfield equation via the Runge-Kutta method. For a central system with a d-dimensional Hilbert space, the direct methods have complexity of d³, while that of the latter two methods is on the order of order of polynomials of log d. The first method, besides converting the task of solving the Redfield equation to solving the much easier Schrödinger's equation, also provides an even more important conceptual lesson: the standard Kubo formula is not applicable to open systems. Besides these general methodologies, I also investigate transport properties of spin systems using the framework of the Redfield equation and with direct methods. Normal energy and spin transport is found for integrable but interacting systems. This conflicts with the well-known conjecture linking anomalous conductivity to integrability, and it also contradicts the relationship, suggested by some, between gapped systems (Jz > Jxy) and normal spin conductivity. I propose a new conjecture, linking anomalous transport to the existence of a mapping of the problem to one for non-interacting particles.
We consider heat transfer in one-dimensional systems with long-range interactions. It is known that typical short-range interacting systems shows anomalous behavior in heat transport when total momentum is conserved, whereas momentum-nonconserving systems do not exhibit anomaly. In this study, we focus on the effect of long-range interaction. We propose an exactly solvable model that reduces to the so-called momentum-exchange model in the short-range interaction limit. We exactly calculate the asymptotic time decay in the energy current correlation function, which is related to the thermal conductivity via the Green-Kubo formula. From the time decay of the current correlation, we show three qualitatively crucial results. First, the anomalous exponent in the time-decay continuously changes as a function of the index of the long-range interaction. Second, there is a regime where the current correlation diverges with increasing the system size with fixed time, and hence, the exponent of the time decay cannot be defined. Third, even momentum-nonconserving systems can show the anomalous exponent indicating anomalous heat transport. Higher dimensions are also considered, and we found that long-range interaction can induce the anomalous exponent even in three-dimensional systems.
The heat conductivity of nonperiodic quantum Ising chains whose ends are connected with heat baths at different temperatures are studied numerically by solving the Lindblad master equation. The chains are subjected to a uniform transverse field h, while the exchange coupling J{m} between the nearest-neighbor spins takes the two values J{A} and J{B} arranged in Fibonacci, generalized Fibonacci, Thue-Morse, and period-doubling sequences. We calculate the energy-density profile and energy current of the resulting nonequilibrium steady states to study the heat-conducting behavior of finite but large systems. Although these nonperiodic quantum Ising chains are integrable, it is clearly found that energy gradients exist in all chains and the energy currents appear to scale as the system size ~N{α}. By increasing the ratio of couplings, the exponent α can be modulated from α > -1 to α < -1 corresponding to the nontrivial transition from the abnormal heat transport to the heat insulator. The influences of the temperature gradient and the magnetic field to heat conduction have also been discussed.