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Even though the one-dimensional (1D) Hubbard model is solvable by the Bethe ansatz, at half filling its finite-temperature T>0 transport properties remain poorly understood [1-4]. The zero-temperature finite-frequency dynamical properties of that half-filled model is also a problem of some complexity [5]. In this talk I combine that solution with symmetry to show that within that prominent 1D correlated model the charge Drude weight D (T) vanishes for T>0 and finite values of the on-site repulsion U, in the thermodynamic limit [5]. Hence there is no finite-temperature ballistic charge transport in the 1D half-filled Hubbard model. This result is exact and clarifies a long-standing open problem. It rules out that at half filling it is an ideal conductor in the thermodynamic limit. Whether at finite T and finite on-site repulsion U>0 it is an ideal insulator or a normal resistor remains though an open question. On the other hand, that at half filling the charge stiffness is finite at U=0 and vanishes for U>0 is found to result from a general transition from a conductor to an insulator or resistor occurring at U=0 for T>0. (At T=0 such a transition is the known quantum metal - Mott-Hubbard-insulator transition.) The interplay of the model eta-spin SU(2) symmetry with its hidden U(1) symmetry beyond SO(4) is found to play a central role in the unusual finite-temperature charge transport properties at half filling. The error source of the misleading predictions of Ref. [2] is shortly discussed and clarified. [1] - X. Zotos and P. Prelovsek, Phys. Rev. B 53, 983 (1996). [2] - Satoshi Fujimoto and Norio Kawakami, J. Phys. A 31, 465 (1998). [3] - N. M. R. Peres, R. G. Dias, P. D. Sacramento, and J. M. P. Carmelo, Phys. Rev. B 61, 5169 (2000). [4] - P. Prelovsek, S. El Shawish, X. Zotos, and M. Long, Phys. Rev. B 70, 205129 (2004). [5] - R. G. Pereira, K. Penc, S. R. White, P. D. Sacramento, and J. M. P. Carmelo, Phys. Rev. B 85, 165132 (2012). [6] - J. M. P. Carmelo, Shi-Jian Gu, and P. D. Sacramento, submitted to publication (2013).

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One-dimensional (1D) correlated systems are different from their higher dimensional counterparts in several ways. On the other hand, integrable and nonintegrable 1D correlated systems have very different finite-temperature T>0 transport properties [1-3]. In addition, the former systems show charge-spin separation both at low and high energy [4]. In this preliminary lecture I highlight the different T>0 transport properties for the 1D Hubbard model and related integrable and nonintegrable problems. I start by relating different equivalent representations of the Kubo formula, which is the central formula in calculating electrical transport in linear response. One of such representations is in terms of time-dependent current-current correlation functions. Using an inequality on their time decay, one can show how the existence of conserved quantities associated with the model integrability implies for densities different from one a finite charge stiffness and thus ballistic charge transport at all temperatures [2,3]. Although at zero temperature such charge stiffness vanishes for the 1D Hubbard model at half filling, at finite temperature the above inequality on the time decay of the current-current correlation functions is inconclusive. Indeed whether for T>0 and in the thermodynamic limit the charge stiffness 1D half-filled Hubbard vanishes or becomes finite is a long-standing open problem. In the seminar following this preliminary lecture that problem is fully solved [5]. [1] - X. Zotos and P. Prelovsek, Phys. Rev. B 53, 983 (1996). [2] - X. Zotos, F. Naef, and P. Prelovsek, Phys. Rev. B 55, 11029 (1997). [3] - J. Sirker, R. G. Pereira, and I. Affleck, Phys. Rev. B 83, 035115 (2011). [4] - A. Moreno, A. Muramatsu, and J. M. P. Carmelo, Phys. Rev. B 87, 075101 (2013). [5] - J. M. P. Carmelo, Shi-Jian Gu, and P. D. Sacramento, submitted to publication (2013).

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We introduce two-dimensional fermionic band models with two orbitals per lattice site, or one spinful orbital, and which have a nonzero topological Chern number that can be changed by varying the ratio of hopping parameters. When a Hubbard interaction is introduced, the effective hopping parameters are renormalized and the system’s topological number can change at a certain interaction strength, U = \bar U, smaller than that for the Mott transition. Two situations may occur: Either the anomalous Hall conductivity changes abruptly at \bar U, or the transition is through an anomalous Hall metal, and \sigma_xy changes smoothly between two different quantized values as U grows. Restoring time-reversal symmetry by adding spin to spinless models, the half-filled system becomes a Z2 topological insulator. The topological number \nu then changes at a critical coupling \bar U and the quantized spin Hall response changes abruptly. We also study a triplet superconductor with p-wave symmetry in the presence of Rashba spin-orbit coupling and externally applied Zeeman spin splitting. Topological superconductors may undergo transitions between phases with different topological numbers which are related to the presence of gapless (Majorana) edge states. In a superconductor, however, charge is not conserved. Therefore, \sigma_xy is not quantized. It is shown that while \sigma_xy evolves continuously between different topological phases of a Z topological superconductor, its derivatives display sharp features signaling the topological transitions.

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Green's functions are frequently utilized in quantum many particle problems. In this thesis we study the single particle Green's function, which relates to the density of states, which is an important characteristic distinguishing for example a metal from an insulator. For systems with strongly correlated electrons the Green's function can not in general be computed exactly and we will therefore use dynamical mean-field theory (DMFT), which is an approximation scheme for investigating such systems. In its most common forms DMFT uses the mathematical construct of imaginary frequencies, while the physical properties lie on the real frequency axis. The first problem is therefore to investigate the analytical continuation: transforming Green's functions of purely imaginary frequencies to Green's functions of real frequencies. We examine the stability of the Padé approximant by applying some noise to the input functions. We found that the error of the Padé approximation increases linearly with respect to the noise, until a crucial point where more noise barely changes the error of the output. It is verified that a first and second order Padé approximation cannot explain this phenomenon; they can only explain the linear behavior at small noise levels. In the second part of the thesis a method is used, developed by M. Granath and H. U. R. Strand, which directly computes the self-energy of real values; no analytic continuation is therefore needed. It is done by solving finite Anderson impurity models using exact diagonalization. The Anderson models are generated stochastically, and the probability distribution for the sampling is taken to be the non-interacting local spectral function. The method is used at zero temperature with the chemical potential μ= U/2, where U is the interaction energy.

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I shall give theoretical explanations of two experimental observations of universal behaviour in semiconductor systems. The first concerns the dependence of photoconductivity $G$ upon light intensity $I$. It is typically found that $G=I^\gamma$. Simple kinetic theory indicates that we should expect $\gamma=1$ or $\gamma=1/2$, but in experiments I find values close to $\gamma =3/4$ or $\gamma=2/3$, with $I$ varying over decades. After reviewing evidence from other sources, I propose that the anomalous exponents $\gamma=3/4$, $\gamma=2/3$ are universal. I present an explanation for these values, which is consistent with universality. The second universality concerns exciton spectroscopy in heterostructures. The linewidth $W$ of the absorption line and the Stokes shift $S$ of the luminescence peak relative to the absorption peak are found to be related by $S/W=0.6$ in most systems for which both values are published. This ratio is independent of the degree of disorder and of the composition of the semiconductors forming the heterostructure, with $W$ varying over two decades. I shall also give a quantitative explanation of this result. Finally I point out what these two phenomena have in common. The results on photoconductivity are discussed in a recent paper, M. Wilkinson, Europhys. Lett., 96, 67007, (2011). The results on exciton spectra are much older: Fang Yang, M. Wilkinson, E. J. Austin and K. P. O'Donnell, Phys. Rev. Lett., 70, 323-6, (1993).