In the present paper, we propose a mean-field approach for spin ladders based upon the Jordan-Wigner transformation along an elaborately ordered path. We show on the mean-field level that ladders with even-numbered legs open an energy gap in their low-energy excitation with a magnitude close to the corresponding experimental values, whereas the low-energy excitation of the odd-numbered-leg ladders are gapless. It supports the validity of our approach. We then calculate the gap size and the excitation spectra of a two-leg-ladder system. Our result is in good agreement with both the experimental data and the numerical results.
In the nested limit of the spin-fermion model for the cuprates, one-dimensional physics in the form of half-filled two-leg ladders emerges. We show that the renormalization group flow of the corresponding ladder is towards the
d-Mott phase, a gapped spin-liquid with short-ranged
d-wave pairing correlations, and reveals an intermediate $\mathrm{SO}(5)\ifmmode\times\else\texttimes\fi{}\mathrm{SO}(3)$ symmetry. We use the results of the renormalization group in combination with a memory-function approach to calculate the optical conductivity of the spin-fermion model in the high-frequency regime, where processes within the hot spot region dominate the transport. We argue that umklapp processes play a major role. For finite temperatures, we determine the resistivity in the zero-frequency (dc) limit. Our results show an approximate linear temperature dependence of the resistivity and a conductivity that follows a nonuniversal power law. A comparison to experimental data supports our assumption that the conductivity is dominated by the antinodal contribution above the pseudogap.
In the framework of this thesis the infrared spectra of two-leg spin ladders were measured. This study presents the first determination of the magnetic contribution to the optical conductivity of spin ladders. A comparison between experiment and new theoretical calculations provided a comprehensive understanding of the magnetic spectrum of the undoped telephone-number compounds La(y)Ca(14-y)Cu(24)O(41). To be able to study the weak magnetic contribution to the spectra, it was necessary to measure not only the reflectance but in particular the transmittance of thin single crystals. When both quantities are known, one can directly calculate the optical conductivity. The low-temperature spectra were compared with results of three different theoretical approaches in close collaboration with the theory groups of Uhrig et al. from the University of Cologne and of Kopp et al. from the University of Augsburg. The excellent agreement between theory and experiment allowed us to identify the two sharp peaks in the mid-infrared range unambiguously as the signature of a true bound state of two triplets. The occurrence of bound states in spin ladders had been predicted already by several groups, but we provided the first experimental verification. Further spectral weight at higher frequencies was attributed to the multi-triplet continuum. The exchange couplings were derived, and the importance of the four-spin cyclic exchange was demonstrated. With increasing temperature especially the line shape for polarization along the legs changes considerably. The spectral weight of the continuum strongly grows, whereas the bound-state peaks lose weight. The resemblance between the measured room-temperature data of the S=1/2 ladder and theoretical results for the S=1 Haldane chain was discussed. The corresponding calculations were performed by Nunner et al. in Augsburg. Another main part of this thesis was to examine the influence of doping on the infrared spectra. We observed the same bound-state features in Sr(14)Cu(24)O(41) as in the undoped compounds, but we found additional structure in the spectra. Several scenarios were discussed in order to explain the new features, including the modulation of the exchange coupling along the legs due to the incommensurate structure and due to the charge order in the chains.
The Jordan-Wigner transformation is known as a powerful tool in condensed matter theory, especially in the theory of low-dimensional quantum spin systems. The aim of this chapter is to review the application of the Jordan-Wigner fermionization technique for calculating dynamic quantities of low-dimensional quantum spin models. After a brief introduction of the Jordan-Wigner transformation for one-dimensional spin one-half systems and some of its extensions for higher dimensions and higher spin values we focus on the dynamic properties of several low-dimensional quantum spin models. We start from the famous s=1/2 XX chain. As a first step we recall well-known results for dynamics of the z-spin-component fluctuation operator and then turn to the dynamics of the dimer and trimer fluctuation operators. The dynamics of the trimer fluctuations involves both the two-fermion (one particle and one hole) and the four-fermion (two particles and two holes) excitations. We discuss some properties of the two-fermion and four-fermion excitation continua. The four-fermion dynamic quantities are of intermediate complexity between simple two-fermion (like the zz dynamic structure factor) and enormously complex multi-fermion (like the xx or xy dynamic structure factors) dynamic quantities. Further we discuss the effects of dimerization, anisotropy of XY interaction, and additional Dzyaloshinskii-Moriya interaction on various dynamic quantities. Finally we consider the dynamic transverse spin structure factor
Szz(k,ω) for the s=1/2 XX model on a spatially anisotropic square lattice which allows one to trace a one-to-two-dimensional crossover in dynamic quantities.
A Wigner-Jordan (WJ) transformation is used to study the one-dimensional (1D) and two-dimensional (2D) quantum antiferromagnetic Heisenberg model. The advantage of using the Wigner-Jordan transformation is that it preserves all spin-commutation relations as well as the spin on-site exclusion principle. In the 1D case a nearest-neighbor covalent-bonding state of the WJ fermions is found to have a ground-state energy (-0.4351J per site) comparable with that from the Bethe-ansatz solution (-0.4431J per site), and a linear energy spectrum at low energies with velocity 1.6366J, in close agreement with the velocity obtained by Haldane for the quantum antiferromagnetic Heisenberg model with 1/
d2 interaction (1.5708J). The method used for studying the 1D model is then applied to the 2D Heisenberg model in a square lattice. The resulting state at finite temperature is the in-phase flux state, i.e., a flux state of the Wigner-Jordan spinless fermions with an in-phase fermion orbital current circulating around each elementary plaquette.The single-particle excitation spectrum, i.e., the energy dispersion of reversing orientation of a spin in the system, of the in-phase flux state is overall similar to that of spin waves, with significant difference near the edge point k=(\ensuremath{\pi},0), at which the excitation energy of the WJ fermion is zero, whereas that of the spin-wave excitation is 2J. The specific heat of the in-phase flux state predicts a correct temperature dependence over the entire temperature range, namely a
T2 dependence at low temperature, a peak near T/J=0.6, and a 1/
T2 decreasing at high temperature. It also gives excellent agreement with the specific heat calculated from numerical methods. In contrast, the spin-wave theory only correctly predicts a
T2 dependence at low temperature. The Raman spectrum of the in-phase flux state is calculated and shows significant improvement over that from spin-wave theories compared with the experimental spectrum of
La2CuO4. The exchange parameter, J, obtained from the comparison is 1060 ${\mathrm{cm}}^{\mathrm{\ensuremath{-}}1}$, in agreement with that obtained from analyzing the neutron-scattering data.
We study the spin and thermal conductivity of spin-
21 ladders and chains at finite temperature, relevant for experiments with quantum magnets. Using a state-of-the-art density matrix renormalization group algorithm, we compute the current autocorrelation functions on the real-time axis and then carry out a Fourier integral to extract the frequency dependence of the corresponding conductivities. The finite-time error is analyzed carefully. We first investigate the limiting case of spin-
21 XXZ chains, for which our analysis suggests nonzero dc conductivities in all interacting cases irrespective of the presence or absence of spin Drude weights. For ladders, we observe that all models studied are normal conductors with no ballistic contribution. Nonetheless, only the high-temperature spin conductivity of
XX ladders has a simple diffusive, Drude-like form, while Heisenberg ladders exhibit a more complicated low-frequency behavior. We compute the dc spin conductivity down to temperatures of the order of $T\ensuremath{\sim}0.5J$, where
J is the exchange coupling along the legs of the ladder. We further extract mean-free paths and discuss our results in relation to thermal conductivity measurements on quantum magnets.