Correlations and impurities in topological insulators
Final Report Abstract
Topological insulators are fundamentally different from ordinary insulating materials in that they feature a bulk gap but exhibit robust metallic surface states. While materials with topologically non-trivial properties have been studied intensely within the past years, it is fair to say that relatively little is known about the quantitative effects of interactions in these systems. The goal of my research proposal was to investigate the role of correlations in topological materials and to understand the effects of impurities. I was planning to both employ the functional renormalization group (an analytical method that I had used and developed during the time of my PhD thesis) and to learn a new, complementary approach. I familiarized myself with the (entirely computational) density matrix renormalization group and implemented a DMRG code from scratch. During the course of this work, I discovered an algorithmic improvement that opened to door to new applications and allowed to investigate questions which seemed more interesting to me than the original problem of correlations in topological systems. I thus deviated significantly from the path outlined in my proposal. The time-dependent DMRG is a ‘numerically exact’ framework to compute the real time evolution of a given initial state of a one-dimensional quantum mechanical system. One way to access finite temperatures within the DMRG is to express the thermal density matrix as a partial trace over a pure density matrix living in an enlarged Hilbert space. I showed that one can exploit the fact that this so-called purification is not mathematically unique to devise a time-dependent unitary transformation which reduces the buildup of entanglement in T > 0 calculations. This ‘quantum disentangler’ thus allows to carry out the DMRG simulations up to significantly larger times. Using the novel algorithm, I investigated a variety of linear response and non-equilibrium transport properties of microscopic 1d systems at finite temperature. In particular, I addressed the question of whether or not Heisenberg spin chain systems can support dissipationless transport at finite T described by a Drude weight and how the conductivity scales in presence of integrability-breaking terms; first results on optical conductivities were obtained. In order to understand the fundamentals of real-time non-equilibrium dynamics at T > 0, I studied the expansion of local wave packets as well as the heat current in spin chains driven by an initial temperature gradient. I showed that the steady-state current is of a simple (black-body) functional form; it is completely determined by the linear conductance even if the temperature gradient is large. I demonstrated that employing infinite-system DMRG algorithms makes Luttinger liquid physics more easily accessible in microscopic models, which I subsequently exploited to investigate the time evolution of interacting lattice fermions after suddenly changing the interaction strength. I particularly addressed the question if the concept of (Luttinger liquid) universality, which is well-established in equilibrium, carries over to non-equilibrium, and what the role of integrability is. Moreover, I studied whether or not ‘dynamical phase transitions’ can exist in non-integrable models. As a side project, I kept in touch with the world of my PhD thesis. I generalized the functional renormalization group to real time Keldysh space. This allows calculate to the transient dynamics of systems which are not in thermal equilibrium.
Publications
- “A renormalization group approach to time dependent transport through correlated quantum dots” Phys. Rev. B 85, 085113 (2012)
D. M. Kennes, S. G. Jakobs, C. Karrasch, and V. Meden
(See online at https://doi.org/10.1103/PhysRevB.85.085113) - “Finite temperature dynamical DMRG and the Drude weight of spin-1/2 chains” Phys. Rev. Lett. 108, 227206 (2012)
C. Karrasch, J. H. Bardarson, and J. E. Moore
- “Luttinger liquid physics from infinite-system DMRG” Phys. Rev. B 86, 155156 (2012)
C. Karrasch and J. E. Moore
- “Luttinger liquid universality in the time evolution after an interaction quench” Phys. Rev. Lett. 106, 126406 (2012)
C. Karrasch, J. Rentrop, D. Schuricht, and V. Meden
(See online at https://doi.org/10.1103/PhysRevLett.109.126406) - “Dynamical phase transitions after quenches in non-integrable models” Phys. Rev. B 87, 195104 (2013)
C. Karrasch and D. Schuricht
(See online at https://doi.org/10.1103/PhysRevB.87.195104) - “Nonequilibrium thermal transport and its relation to linear response” Phys. Rev. B 88, 195129 (2013)
C. Karrasch, R. Ilan, and J. E. Moore
- “The Drude weight of the spin-1/2 XXZ chain: density matrix renormalization group versus exact diagonalization” Phys. Rev. B 87, 245128 (2013)
C. Karrasch, J. Hauschild, S. Langer, and F. Heidrich-Meisner
(See online at https://doi.org/10.1103/PhysRevB.87.245128) - “Real-time and real-space spin- and energy dynamics in one-dimensional spin-1/2 systems induced by local quantum quenches at finite temperatures” Phys. Rev. B 89, 075139 (2014)
C. Karrasch, J. E. Moore, and F. Heidrich-Meisner
(See online at https://doi.org/10.1103/PhysRevB.89.075139) - “Transport properties of the one-dimensional Hubbard model at finite temperature” Phys. Rev. B 90, 155104 (2014)
C. Karrasch, D. M. Kennes, and J. E. Moore
(See online at https://doi.org/10.1103/PhysRevB.90.155104)
