Exact numerical and analytical results for correlated lattice electrons in one dimension
Since the advent of high$T_c$ cuprate superconductors in 1986, strongly correlated electron systems have attracted much attention. Since the cuprates are essentially twodimensional, lowdimensional systems have moved into the focus of condensedmatter theory. From a theoretical point of view, one...
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Format:  Dissertation 
Language:  English 
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PhilippsUniversität Marburg
2006
Physik 
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Summary:  Since the advent of high$T_c$ cuprate superconductors in 1986, strongly correlated electron systems have attracted much attention. Since the cuprates are essentially twodimensional, lowdimensional systems have moved into the focus of condensedmatter theory. From a theoretical point of view, onedimensional systems are of particular interest because there are exact numerical and analytical methods which permit detailed studies and deep insights into the manybody problem. In the first part of this thesis, using the numerically exact methods Exact Diagonalization and the DensityMatrix Renormalization Group (DMRG), we investigate the properties of the Tomonaga–Luttinger liquid which is the generic metallic state of matter in one dimension. In particular, we concentrate on the investigation of the socalled Tomonaga–Luttinger liquid parameter which determines the critical exponent $\alpha$ for the density of states near the Fermi energy. Experimental results for some quasi onedimensional materials report $\alpha \gtrsim 1$, which would imply $K_\rho \lesssim 0.17$, a value which cannot be reconciled with the bare Hubbard model where $K_\rho^H \geq 0.5$, i.e., $\alpha^H \lesssim 1/8$. We develop new accurate numerical methods to obtain $K_\rho$ and investigate how to obtain such small values for $K_\rho$ for slightly doped chargedensitywave insulators. In the second part of this thesis, using the Thermodynamic Bethe Ansatz (TBA) as exact analytical method, we investigate the onedimensional Hubbard model in the spindisordered regime, which is characterized by the temperature being much larger than the magnetic energy scale but small compared to the Mott–Hubbard gap. Our study is motivated by the controversy about the Mott–Hubbard insulator in infinite dimensions whose ground state is also spindisordered. In this system the determination of the precise value of the critical interaction strength Uc where the Mott–Hubbard gap closes is still unsolved. Therefore, we provide an example of a Hubbardtype model with a disordered spin background which can be solved exactly. The thermodynamics of our model can be understood in terms of gapped charged excitations with an effective dispersion which are decoupled form the spin degrees of freedom; the latter contribute only entropically. An interpretation of this regime in terms of a putative interactingelectron system at zero temperature leads to a metalinsulator transition at a finite interaction strength above which the gap opens linearly. Our exact results indicate that the strongcoupling expansion of the groundstate energy cannot be used to locate $U_c$. However, the strongcoupling expansion of the gap permits a reliable extrapolation of the critical interaction strength. 

DOI:  https://doi.org/10.17192/z2006.0111 