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Prof. Dr. Gunter M. Schütz

Homepage G.SchützDriven diffusive systemsQuantum spin chains far from equilibriumReaction-diffusion systems

Quantum spin chains far from equilibrium

Transport in integrable systems is anomalous. This means that a harmonic lattice (or a transverse Ising chain) cannot support an internal thermal gradient when the two ends of the system are kept at different temperatures. The bulk temperature profile is flat, and the energy flux is not proportional to the temperature gradient inside the sample which is tantamount to transport coefficients being divergent or ill-defined. These (and other) observations point to an intricate interplay of currents and correlations in non-equilibrium steady states. In order to achieve progress one tries to investigate simple models. Unfortunately, far from equilibrium there are no constraints like detailed balance and consequently there is much arbitrariness in defining the dynamics. Thus we have started to investigate non-equilibrium ground states and time-dependent properties of isolated quantum systems where the time evolution is defined without ambiguity by the usual rules of quantum mechanics.

Non-equilibrium ground states

Current-carrying ground states in a quantum system may be investigated by employing a Lagrange multiplier method [T. Antal, Z. Réacz and L. Sasvári, Phys. Rev. Lett. 78, 167 (1997)]. Extending this approach to the transverse XY model suggests that currents generically generate and maintain power-law correlations. An interesting feature of XY model which may also have some generality is a symmetry enhancement of the ground state at special values of the energy current. It should be recognized, however, that both the XY and the transverse Ising models are special in that they are integrable. Thus it is an important next step to find out whether nonintegrable models have the same connection between currents and power-law correlations and, furthermore, whether they show any additional general features.

Time-dependent properties of isolated quantum spin chains

Since there are physical systems that are rather well approximated as XX models, and since the effect of currents should be measurable in inelastic neutron scattering experiments, one should carefully examine the validity of the assumptions underlying the calculations of the correlations. The Lagrange multiplier method is based on the assumption that the homogeneous, current-carrying states are the same whether they were induced by boundary conditions or by bulk driving fields. Verifying this assumption is a rather nontrivial undertaking. One can make some progress, however, in a closely related problem. We study the connection between magnetization transport and magnetization profiles in zero-temperature XX chains. The time evolution of the transverse magnetization, m(x,t), is calculated using an inhomogeneous initial state which generates a magnetization current. In the long-time limit, the magnetization evolves into a scaling form and the profile develops a flat part as expected for a integrable non-equilibrium system. The states emerging in the scaling limit are compared to those of a homogeneous system where the same magnetization current is driven by a bulk field. We find that the expectation values of various characteristic quantities agree in the two systems, thus confirming the Lagrange-multiplier approach by explicit solution of the long-time behaviour of the Schrödinger equation.

Examination of two-time correlation functions are an important tool in the investigation of the phenomenon of aging, i.e. the extent to which a relaxing system keeps a memory of its initial condition after a long time. Aging is a typical result of ultraslow dynamics, especially in systems which are dominated by quenched disorder as amorphous polymer systems, spin glasses and random magnets, but also ordered structures with stochastic dynamics may exhibit aging. As a rule, in dynamical systems coupled to a large heat bath (with largely heuristically determined classical stochastic dynamics), the presence of dynamical scale invariance seems to imply aging. In Ref. 43 we have established the occurence of aging in the ordered isotropic XY-chain with deterministic quantum mechanical dynamics. At the same the relaxation of the magnetization is algebraic, thus suggesting that aging and power law relaxation are intimately linked not only in classical stochastic but also genuinely quantum mechanical systems.