We consider the spreading of a local operator A in one-dimensional many-body systems with Hamiltonian H by calculating the k-fold commutator left perpendicularH, left perpendicularH, left perpendicular..., left perpendicularH, Aright perpendicularright perpendicularright perpendicularright perpendicular. We derive bounds for the operator norm of this commutator in free and interacting systems with and without disorder thus directly connecting the operator growth hypothesis with questions of localization. We analytically show, in particular, that an almost-factorial growth of the operator norm-as recently proven for the random Ising model and strongly suggested here to also hold for the Heisenberg model with random fields-is inconsistent with an exponential localization of A. Assuming there exists a quasilocal unitary U which maps H onto an effective Hamiltonian H = UHU dagger = Sigma(n) E-n tau(z)(n) + Sigma(i,j)J(ij) tau(z)(i) tau(z)(j) + ..., we show that A = UAU(dagger) is a quasilocal operator which, in contrast to the Anderson case, indeed does not remain exponentially localized in the general many-body case, leading to an almost-factorial growth of the commutator norm. Therefore, either the unitary U in many-body systems with maximal norm growth does not exist, and such systems are always ergodic, or unusual nonergodic phases described by H exist which violate the operator growth hypothesis and in which local operators spread over the entire lattice, implying that transport will eventually set in. To investigate this issue further, we concentrate on the XXZ chain with random magnetic fields. We analytically and symbolically verify our general results for the noninteracting Anderson and Aubry-Andr & eacute; models. For the XXX case, the symbolic calculations are consistent with a maximal norm growth. Furthermore, we find no indication of a weakened exponential localization of A, expected for strong disorder and low commutator orders if the unitary U does exist. Finally, we study the differences between the interacting and noninteracting cases when trying to perturbatively construct U by consecutive Schrieffer-Wolff transformations. While it is straightforward to show that this construction converges in the Anderson case, we find no indications for a convergence in the interacting case, suggesting that U does not exist and that many-body localization is absent.

We compare theoretical results for electron spin resonance (ESR) properties of the Heisenberg-Ising Hamiltonian with ESR experiments on the quasi-one-dimensional magnet Cu(py) Br-2(2) (CPB). Our measurements were performed over a wide frequency and temperature range giving insight into the spin dynamics, spin structure, and magnetic anisotropy of this compound. By analyzing the angular dependence of ESR parameters (resonance shift and linewidth) at room temperature, we show that the two weakly coupled inequivalent spin-chain types inside the compound are well described by Heisenberg-Ising chains with their magnetic anisotropy axes perpendicular to the chain direction and almost perpendicular to each other. We further determine the full g tensor from these data. In addition, the angular dependence of the linewidth at high temperatures gives us access to the exponent of the algebraic decay of a dynamical correlation function of the isotropic Heisenberg chain. From the temperature dependence of static susceptibilities, we extract the strength of the exchange coupling (J/k(B) = 52.0K) and the anisotropy parameter (d approximate to -0.02) of the model Hamiltonian. An independent compatible value of d is obtained by comparing the exact prediction for the resonance shift at low temperatures with high-frequency ESR data recorded at 4 K. The spin structure in the ordered state implied by the two (almost) perpendicular anisotropy axes is in accordance with the propagation vector determined from neutron scattering experiments. In addition to undoped samples, we study the impact of partial substitution of Br by Cl ions on spin dynamics. From the dependence of the ESR linewidth on the doping level, we infer an effective decoupling of the anisotropic component J delta from the isotropic exchange J in these systems.

We analyze the absorption of microwaves by the Heisenberg-Ising chain combining exact calculations, based on the integrability of the model, with numerical calculations. Within linear response theory, the absorbed intensity is determined by the imaginary part of the dynamical susceptibility. The moments of the normalized intensity can be used to define the shift of the resonance frequency induced by the interactions and the linewidth independently of the shape of the spectral line. These moments can be calculated exactly as functions of temperature and strength of an external magnetic field, as long as the field is directed along the symmetry axis of the chain. This allows us to discuss the linewidth and the resonance shift for a given magnetic field in the full range of possible anisotropy parameters. For the interpretation of these data, we need a qualitative knowledge of the line shape, which we obtain from fully numerical calculations for finite chains. Exact analytical results on the line shape are out of reach of current theories. From our numerical work, we could extract, however, an empirical parameter-free model of the line shape at high temperatures, which is extremely accurate over a wide range of anisotropy parameters, and is exact at the free-fermion and isotropic points. Another prediction of the line shape can be made in the zero-temperature and zero magnetic field limits, where the sufficiently anisotropic model shows strong absorption. For anisotropy parameters in the massive phase, we derive the exact two-spinon contribution to the spectral line. From the intensity sum rule, it can be estimated that this contribution accounts for more than 80$%$ of the spectral weight if the anisotropy parameter is moderately above its value at the isotropic point.

The phase diagram of the two-leg t-Jz ladder is explored, using the density matrix renormalization group method. Results are obtained for energy gaps, electron density profiles and correlation functions for the half-filled and quarter-filled cases. A speculative phase diagram is presented, which is quite similar to the full t-J ladder, but scaled up by a factor of about two in coupling.

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The phase diagram of the two-leg t-Jz ladder is explored, using the density matrix renormalization group method. Results are obtained for energy gaps, electron density profiles and correlation functions for the half-filled and quarter-filled cases. The effective Lagrangian velocity parameter is shown to vanish at half-filling. The behaviour of the one-hole gap in the Nagaoka limit is investigated, and found to disagree with theoretical predictions. A tentative phase diagram is presented, which is quite similar to the full t-J ladder, but scaled up by a factor of about two in coupling. Near half-filling a Luther-Emery phase is found, which may be expected to show superconducting correlations, while near quarter-filling the system appears to be in a Tomonaga-Luttinger phase.

We examine the use of distributions in numerical treatments of Anderson localisation and supply evidence that treating exponential localisation on Bethe lattices recovers the overall picture known from a cubic lattice in 3d.

A quantitative study of the field-induced magnetic ordering in TlCuCl3 in terms of a Bose–Einstein condensation (BEC) of magnons is presented. It is shown that the hitherto proposed simple BEC scenario is in quantitative and qualitative disagreement with experiment. It is further shown that even very small Dzyaloshinsky–Moriya interactions or a staggered g tensor component of a certain type can change the BEC picture qualitatively. Such terms lead to a nonzero condensate density for all temperatures and a gapped quasiparticle spectrum. Including this type of interaction allows us to obtain good agreement with experimental data.

Using our recently developed Chebyshev expansion technique for finite-temperature dynamical correlation functions, we numerically study the AC conductivity σ(ω) of the Anderson model on large cubic clusters of up to 1003 sites. Extending previous results, we focus on the role of the boundary conditions and check the consistency of the DC limit, ω→0, by comparing with direct conductance calculations based on a Greens function approach in a Landauer Büttiker-type setup.

Based on the Holstein double-exchange model and a highly efficient single cluster Monte Carlo approach we study the interplay of double-exchange and polaron effects in doped colossal magneto-resistance (CMR) manganites. The CMR transition is shown to be appreciably influenced by lattice polaron formation.

We present a detailed study of the quantum site percolation problem on simple cubic lattices, thereby focusing on the statistics of the local density of states and the spatial structure of the single particle wave functions. Using the kernel polynomial method we refine previous studies of the metal-insulator transition and demonstrate the nonmonotonic energy dependence of the quantum percolation threshold. Remarkably, the data indicates a "fragmentation" of the spectrum into extended and localized states. In addition, the observation of a chequerboardlike structure of the wave functions at the band center can be interpreted as anomalous localization.

We discuss the nature of the different ground states of the half-filled Holstein model of spinless fermions in 1D. In the metallic regime we determine the renormalised effective coupling constant and the velocity of the charge excitations by a density-matrix renormalisation group (DMRG) finite-size scaling approach. At low (high) phonon frequencies the Luttinger liquid is characterised by an attractive (repulsive) effective interaction. In the charge-density wave Peierls-distorted state the charge structure factor scales to a finite value indicating long-range order.

We present exact results for the optical response in the one-dimensional Holstein model. In particular, by means of a refined kernel polynomial method, we calculate the ac and dc electrical conductivities at finite temperatures for a wide parameter range of electron phonon interaction. We analyze the deviations from the results of standard small polaron theory in the intermediate coupling regime and discuss non-adiabaticity effects in detail.

In the first part we discuss how the BEC picture for magnons is modified by anisotropies induced by spin-orbit coupling. In particular we focus on the effects of antisymmetric spin interactions and/or a staggered component of the g (gyromagnetic) tensor. Such terms lead to a gapped quasiparticle spectrum and a nonzero condensate density for all temperatures so that no phase transition occurs. We contrast this to the effect of crystal field anisotropies which are also induced by spin-orbit coupling. In the second part we study the field-induced magnetic ordering in TlCuCl_3 on a quantitative level. We show that the usual BEC picture does not allow for a good description of the experimental magnetisation data and argue that antisymmetric spin interactions and/or a staggered g tensor component are still crucial, although both are expected to be tiny in this compound due to crystal symmetries. Including this type of interaction we obtain excellent agreement with experimental data.

. We propose an advanced Chebyshev expansion method for the numerical calculation of linear response functions at finite temperature. Its high stability and the small required resources allow for a comprehensive study of the optical conductivity σ(ω) of non-interacting electrons in a random potential (Anderson model) on large three-dimensional clusters. For low frequency the data follows the analytically expected power-law behaviour with an exponent that depends on disorder and has its minimum near the metal-insulator transition, where also the extrapolated DC conductivity continuously goes to zero. In view of the general applicability of the Chebyshev approach we briefly discuss its formulation for interacting quantum systems.

As a generic model describing quasi-one-dimensional Mott and Peierls insulators, we investigate the Holstein-Hubbard model for half-filled bands using numerical techniques. Combining Lanczos diagonalization with Chebyshev moment expansion we calculate exactly the photoemission and inverse photoemission spectra, and use these to establish the phase diagram of the model. While polaronic features emerge only at strong electron-phonon couplings, pronounced phonon signatures, such as multiquanta band states, can be found in the Mott insulating regime as well. In order to corroborate the Mott to Peierls transition scenario, we determine the spin- and charge-excitation gaps by a finite-size scaling analysis based on density-matrix renormalization-group calculations.

Colossal magneto-resistance manganites are characterized by a complex interplay of charge, spin, orbital and lattice degrees of freedom. Formulating microscopic models for these compounds aims at meeting two conflicting objectives: sufficient simplification without excessive restrictions on the phase space. We give a detailed introduction to the electronic structure of manganites and derive a microscopic model for their low-energy physics. Focusing on short-range electron–lattice and spin–orbital correlations we supplement the modelling with numerical simulations.