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Spectral Properties of Quasi One-dimensional Quantum Antiferromagnets . Perturbative Continuous Unitary Transformations

Kai Phillip Schmidt

Published Jan 1, 2004• 3 citations
OpenAlex

Abstract

In this work a perturbative realization of particle conserving continuous unitary transformations is applied to study the energies and the spectral properties of quasi one-dimensional quantum antiferromagnets. The systems considered are defined on a lattice and they allow for a perturbative decomposition. The unperturbed part is chosen to be a fully dimerized state. The groundstate is a product-state of singlets and the excitation spectrum is equidistant. The related energy quantum is called a triplon becaues it has total spin one. The continous unitary transformation leads to an effective triplon-conserving Hamiltonian and effective, experimentally relevant observables. The effective operators are obtained in a high-order series expansion in the perturbation parameters. All calculations are performed on finite clusters in real space and yield exact results in the thermodynamic limit due to the linked cluster theorem. The results are exact in the given order. In order to improve the representation of the results extrapolation techniques are used. A detailed description of extrapolation tools like standard Pade and dlogPade extrapolation, optimized perturbation theory and the use of internal parameters is given. The dimerized and frustrated spin-chain is analysed first. At zero frustration, a detailed investigation of the spectral weights shows that even in the limit of vanishing dimerization, the one-dimensional Heisenberg model, almost the total spectral weight is situated in the two-triplon sector. So, besides spinons, triplons may be used as elementary excitations for the one-dimensional Heisenberg model. The case of strong frustration is not yet settled. An extensive review of one- and two-triplon spectral densities at large and intermediate value of the dimerization for various values of the frustration is presented. The findings are compared with field theoretical results. In addition, the Raman response and the infrared absorption are investigated. Second, the antiferromagnetic two-leg Heisenberg ladder plus additional four-spin interaction is investigated. The transformation starts from the limit of isolated rung dimers. The excitations are rung-triplons. The relative energies of one-triplon states, the two-triplon bound states and the multi-triplon continua are given for various couplings. Optical observables are discussed in detail. The extent of the rung-singlet phase is calculated in the whole parameter space. It is shown that the experimental realizations of two-leg ladder systems are always situated in the rung-singlet phase. In the experimentally relevant regime, most of the spectral weight is captured by the one- and the two-triplon sector, but also three- and four-triplon contributions become sizable. The current understanding of the spectroscopic signatures of magnetic excitations in cuprate ladders measured with inelastic neutron scattering, Raman spectroscopy and infrared absorption is presented. The results obtained are compared with experimental findings. The first experimental evidence of a triplon-triplon bound state in a ladder system is found.

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arXiv:2302.11599•Feb 22, 2023•0 citations
In this work a perturbative realization of particle conserving continuous unitary transformations is applied to study the energies and the spectral properties of quasi one-dimensional quantum antiferromagnets. The systems considered are defined on a lattice and they allow for a perturbative decomposition. The unperturbed part is chosen to be a fully dimerized state. The groundstate is a product-state of singlets and the excitation spectrum is equidistant. The related energy quantum is called a triplon becaues it has total spin one. The continous unitary transformation leads to an effective triplon-conserving Hamiltonian and effective, experimentally relevant observables. The effective operators are obtained in a high-order series expansion in the perturbation parameters. All calculations are performed on finite clusters in real space and yield exact results in the thermodynamic limit due to the linked cluster theorem. The results are exact in the given order. In order to improve the representation of the results extrapolation techniques are used. A detailed description of extrapolation tools like standard Pade and dlogPade extrapolation, optimized perturbation theory and the use of internal parameters is given. The dimerized and frustrated spin-chain is analysed first. At zero frustration, a detailed investigation of the spectral weights shows that even in the limit of vanishing dimerization, the one-dimensional Heisenberg model, almost the total spectral weight is situated in the two-triplon sector. So, besides spinons, triplons may be used as elementary excitations for the one-dimensional Heisenberg model. The case of strong frustration is not yet settled. An extensive review of one- and two-triplon spectral densities at large and intermediate value of the dimerization for various values of the frustration is presented. The findings are compared with field theoretical results. In addition, the Raman response and the infrared absorption are investigated. Second, the antiferromagnetic two-leg Heisenberg ladder plus additional four-spin interaction is investigated. The transformation starts from the limit of isolated rung dimers. The excitations are rung-triplons. The relative energies of one-triplon states, the two-triplon bound states and the multi-triplon continua are given for various couplings. Optical observables are discussed in detail. The extent of the rung-singlet phase is calculated in the whole parameter space. It is shown that the experimental realizations of two-leg ladder systems are always situated in the rung-singlet phase. In the experimentally relevant regime, most of the spectral weight is captured by the one- and the two-triplon sector, but also three- and four-triplon contributions become sizable. The current understanding of the spectroscopic signatures of magnetic excitations in cuprate ladders measured with inelastic neutron scattering, Raman spectroscopy and infrared absorption is presented. The results obtained are compared with experimental findings. The first experimental evidence of a triplon-triplon bound state in a ladder system is found.
In this thesis we describe a novel perturbative approach to low-dimensional quantum many-particle systems, which is based on continuous unitary transformations. We consider systems, which are defined on a lattice and allow a perturbative decomposition. The unperturbed part must have an equidistant spectrum -- the difference between two successive levels is called a quasi-particle. In this case the perturbative transformation leads to an effective Hamiltonian, which conserves the number of particles. The same transformation is used to also derive the effective counterparts of other, experimentally relevant observables. The effective operators are obtained as series expansions in the (small) perturbation parameter. In each order we find a set of products of ladder-operators, which act on states uniquely defined by the number of quasi-particles and their position in the lattice. Thus all calculations can be done in real space. The mathematical structure of the effective operators is extensively analysed. We additionally give all details necessary to implement the method on a computer, which allows the calculation of the effective quantities up to high orders. The method facilitates quantitative calculations of multi-particle excitations and spectral densities of experimentally relevant observables. We include a comprehensive application of the method to the two-dimensional Shastry-Sutherland model, a strongly frustrated quantum spin system. The model has experience an experimental realization by SrCu2(BO3)2. The limit of strong dimerization serves as starting point. Here the ground state is given by singlets on all dimers -- a single triplet constitutes an elementary excitation, i.e. quasi-particle. We quantitatively calculate the one- and two-triplet energies as well as the spectral densities of the Raman and neutron scattering operators. The findings for the spectral densities in particular represent new results. Comparing them to experimental data leads to interesting insights into the spectral properties of low-dimensional quantum spin systems.
Spectral densities are calculated for the dimerized and frustrated S=1/2S=1/2S=1/2 chain using the method of continuous unitary transformations. The transformation to an effective triplon model is realized in a perturbative fashion up to high orders about the limit of isolated dimers. An efficient description in terms of triplons (elementary triplets) is possible: a detailed analysis of the spectral densities is provided for strong and intermediate dimerization including the influence of frustration. Precise predictions are made for inelastic neutron scattering experiments probing the S=1S=1S=1 sector and for optical experiments (Raman scattering, infrared absorption) probing the S=0S=0S=0 sector. Bound states and resonances influence the important continua strongly. The comparison with the field theoretic results reveals that the sine-Gordon model describes the low-energy features for strong to intermediate dimerization only at critical frustration.
In sufficiently dimerized quantum antiferromagnets the elementary excitations are given by gapped spin S = 1 triplon quasiparticles. Although these triplons are protected by a gap at low energies they may decay spontaneously at higher energies where the one-triplon dispersion merges with the two-triplon continuum. First, we illustrate and characterize such a spontaneous decay in low-dimensional quantum antiferromagnets on the basis of a simple one-dimensional bosonic model. The decay implies the breakdown of the quasiparticle picture. No quantitative description by a Lorentzian resonance is possible. In particular, three qualitatively different scenarios are identified depending on the one-triplon dispersion and the two-triplon interaction. To describe unstable quasiparticles in more complex models we introduced an adapted generator for continuous unitary transformations. Its general properties are derived and discussed. Next, we investigate asymmetric antiferromagnetic and asymmetric ferroantiferromagnetic spin S = ½ Heisenberg ladders, which allow for spontaneous triplon decay, to illustrate this approach. Results for the low-energy spectra and the dynamical structure factors for these systems are presented. We show that quasiparticle decay is more pronounced in the case of ferro-antiferromagnetic ladders than in solely antiferromagnetic ladders. Finally, we use continuous unitary transformations and high temperature series expansions to determine a quantitative model for the compound IPA-CuCl3 based on data of inelastic neutron scattering and measurements of the magnetic susceptibility. Our calculations reveal IPA-CuCl3 as system of coupled asymmetric spin S = ½ Heisenberg ladders with the four magnetic couplings J 1 ˜ -2.3 meV, J2 ˜ 1.2 meV, J3 ˜ 2.9 meV and J4 ˜ -0.3 meV. Based on this microscopic model for IPA-CuCl3 the adapted continuous unitary transformation is used to describe the quasiparticle decay in IPA-CuCl3. The results agree very well with the experimental data. In addition the magnetic field dependence of the lowest modes in the condensed phase as well as the temperature dependence of the gap without magnetic field corroborate our microscopic model.
In this thesis the method of self-similar continuous unitary transformations (CUTs) in real space is developed. It is applied to various models from the field of strongly correlated solid state physics and optical lattices. The CUT method maps a given Hamiltonian to an effective Hamiltonian which is in some respect simpler. In a self-similar CUT the truncation scheme relies on the structure of the operators. The real space approach put forward in this thesis truncates the operators according to their extension in real space. We mainly use a MKU generator to derive an effective model which is block-diagonal in the number of quasiparticles. We give a detailed description of the method. The transformation of the Hamiltonian and of the observables is explained. The implementation on a computer and the optimization of the programs are extensively discussed. The first application of the method deals with the fermionic Hubbard model. We derive an effective Hamiltonian, a generalized t-J-model, which conserves the number of doubly occupied sites. The effective model includes beyond the well-known Heisenberg term also further two-spin and four-spin terms. In addition, the charge motion and interaction within the effective model is addressed. In the context of the Hubbard model we discuss the question if an effective model that conserves the number of double occupancies can be derived in principle. The antiferromagnetic spin ladder and spin chain are treated from the viewpoint of a dimerized system. For the spin ladder we calculate the dispersion, the two-particle energies including bound states and the multi-particle continua. The discussion of multi-particle continua shows that the order of the energy levels is important for the convergence of the transformation. For the spin chain the dispersion is calculated. In addition, the transformation of an observable yields the corresponding spectral weights. The spectral weights are compared to results from a perturbative CUT. For the bosonic Hubbard model we calculate the dispersion and determine the phase diagram. The spectral weight of a certain observable is calculated in order to explain the outcome of a recent experiment that measures the spectroscopic response of atoms confined in optical traps. The distribution of spectral weight is analyzed also at finite temperatures. We find evidence for a substantial temperature of the size of the microscopic parameters. This finding is supported by purely thermodynamic considerations.
The transition from dimerized to uniform phases is studied in terms of spectral weights for spin chains using continuous unitary transformations. The spectral weights in the S=1 channel are computed perturbatively around the limit of strong dimerization. We find that the spectral weight is concentrated mainly in the subspaces with a small number of elementary triplets (triplons), even for vanishing dimerization. So, besides spinons, triplons may be used as elementary excitations in spin chains. We conclude that there is no necessity to use fractional excitations in low-dimensional, undoped, or doped quantum antiferromagnets.
Series expansion studies of the J1J_1J1​-J2J_2J2​-Heisenberg bilayer
We study a bilayer of the frustrated J1J_1J1​-J2J_2J2​ Heisenberg-model on the square lattice. Starting from the dimer limit at strong interlayer coupling, we perform series expansions using the perturbative Continuous Unitary Transformation, based on the flow equation method, in order to determine the spectrum up to the two-triplon sector. From the one-triplon dispersion we obtain quantum critical lines for transitions from the dimer phase into either Néel or collinear magnetic order. For low to intermediate frustration these transitions are consistent with existing findings, based on the magnetic phases. In the region of strongest frustration, i.e. J2/J1∼0.5J_2/J_1\sim 0.5J2​/J1​∼0.5, we provide an estimate for the stability of the anticipated single-layer quantum spin-liquids against finite interlayer coupling. In the two-triplon sector we find a set of well defined (anti)bound states, which can be classified according to total spin and in-plane rotational symmetry. For vanishing frustration these states agree with previous series expansion analysis. For J2/J1≳0.5J_2/J_1\gtrsim 0.5J2​/J1​≳0.5 we provide evidence for a close-by condensation of one-triplon and two-triplon singlet bound states, suggesting that between the dimer and the collinear state, additional phases may intervene.