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arXiv:1507.04957 · Physical Review B

Hard-core boson approach to the spin−12 triangular-lattice antiferromagnet Cs2CuCl4 at finite temperatures in magnetic fields higher than the saturation field

Simon Streib, Peter Kopietz

Published Sep 30, 2015• 12 citations
arXivHTMLPDF

Abstract

We study the high magnetic field regime of the antiferromagnetic insulator ${\mathrm{Cs}}_{2}{\mathrm{CuCl}}_{4}$ by expressing the spin-1/2 operators in the relevant Heisenberg model in terms of hard-core bosons and implementing the hard-core constraint via an infinite on-site interaction. We focus on the case where the external magnetic field exceeds the saturation field ${B}_{c}\ensuremath{\approx}8.5\phantom{\rule{0.28em}{0ex}}\mathrm{T}$ and is oriented along the crystallographic $a$ axis perpendicular to the lattice plane. Because in this case the excited states are separated by an energy gap from the ground state, we may use the self-consistent ladder approximation to take the strong correlations due to the hard-core constraint into account. In ${\mathrm{Cs}}_{2}{\mathrm{CuCl}}_{4}$, there are additional interactions besides the hard-core interaction which we treat in self-consistent Hartree-Fock approximation. We calculate the spectral function of the hard-core bosons from which we obtain the in-plane components of the dynamic structure factor, the magnetic susceptibility, and the specific heat. Our results for the specific heat are in good agreement with the available experimental data. We conclude that the self-consistent ladder approximation in combination with a self-consistent Hartree-Fock decoupling of the non-hard-core interactions gives an accurate description of the physical properties of gapped hard-core bosons in two dimensions at finite temperatures.

DOI
10.1103/physrevb.92.094442

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We study the high magnetic field regime of the antiferromagnetic insulator Cs2CuCl4{\mathrm{Cs}}_{2}{\mathrm{CuCl}}_{4}Cs2​CuCl4​ by expressing the spin-1/2 operators in the relevant Heisenberg model in terms of hard-core bosons and implementing the hard-core constraint via an infinite on-site interaction. We focus on the case where the external magnetic field exceeds the saturation field Bc\ensuremath≈8.5T{B}_{c}\ensuremath{\approx}8.5\phantom{\rule{0.28em}{0ex}}\mathrm{T}Bc​\ensuremath≈8.5T and is oriented along the crystallographic aaa axis perpendicular to the lattice plane. Because in this case the excited states are separated by an energy gap from the ground state, we may use the self-consistent ladder approximation to take the strong correlations due to the hard-core constraint into account. In Cs2CuCl4{\mathrm{Cs}}_{2}{\mathrm{CuCl}}_{4}Cs2​CuCl4​, there are additional interactions besides the hard-core interaction which we treat in self-consistent Hartree-Fock approximation. We calculate the spectral function of the hard-core bosons from which we obtain the in-plane components of the dynamic structure factor, the magnetic susceptibility, and the specific heat. Our results for the specific heat are in good agreement with the available experimental data. We conclude that the self-consistent ladder approximation in combination with a self-consistent Hartree-Fock decoupling of the non-hard-core interactions gives an accurate description of the physical properties of gapped hard-core bosons in two dimensions at finite temperatures.
The dynamics in quantum magnets can often be described by effective models with bosonic excitations obeying a hard-core constraint. Such models can be systematically derived by renormalization schemes such as continuous unitary transformations or by variational approaches. Even in the absence of further interactions the hard-core constraint makes the dynamics of the hard-core bosons nontrivial. Here, we develop a systematic diagrammatic approach to the spectral properties of hard-core bosons at finite temperature. Starting from an expansion in the density of thermally excited bosons in a system with an energy gap, our approach leads to a summation of ladder diagrams. Conceptually, the approach is not restricted to one dimension, but the one-dimensional case offers the opportunity to gauge the method by comparison to exact results obtained via a mapping to Jordan-Wigner fermions. In particular, we present results for the thermal broadening of single-particle spectral functions at finite temperature. The line shape is found to be asymmetric at elevated temperatures and the bandwidth of the dispersion narrows with increasing temperature. Additionally, the total number of thermally excited bosons is calculated and compared to various approximations and analytic results. Thereby, a flexible approach is introduced that can also be applied to more sophisticated and higher dimensional models.
We introduce a slave-fermion formulation in which to study the charge dynamics of the half-filled Hubbard model on the square lattice. In this description, the charge degrees of freedom are represented by fermionic holons and doublons and the Mott-insulating characteristics of the ground state are the consequence of holon–doublon bound-state formation. The bosonic spin degrees of freedom are described by the antiferromagnetic Heisenberg model, yielding long-ranged (Néel) magnetic order at zero temperature. Within this framework and in the self-consistent Born approximation, we perform systematic calculations of the average double occupancy, the electronic density of states, the spectral function and the optical conductivity. Qualitatively, our method reproduces the lower and upper Hubbard bands, the spectral-weight transfer into a coherent quasiparticle band at their lower edges and the renormalisation of the Mott gap, which is associated with holon–doublon binding, due to the interactions of both quasiparticle species with the magnons. The zeros of the Green function at the chemical potential give the Luttinger volume, the poles of the self-energy reflect the underlying quasiparticle dispersion with a spin-renormalised hopping parameter and the optical gap is directly related to the Mott gap. Quantitatively, the square-lattice Hubbard model is one of the best-characterised problems in correlated condensed matter and many numerical calculations, all with different strengths and weaknesses, exist with which to benchmark our approach. From the semi-quantitative accuracy of our results for all but the weakest interaction strengths, we conclude that a self-consistent treatment of the spin-fluctuation effects on the charge degrees of freedom captures all the essential physics of the antiferromagnetic Mott–Hubbard insulator. We remark in addition that an analytical approximation with these properties serves a vital function in developing a full understanding of the fundamental physics of the Mott state, both in the antiferromagnetic insulator and at finite temperatures and dopings.
Antiferromagnetic spin-1/2 ladders have been studied in theoretical and experimental physics for several years. The spin ladder is a good model to bridge between theory and experiment, because it is realized in the copper oxide family Sr14-x(La,Ca)xCu24O41. These copper oxides consist of alternating layers of edge-sharing chains and ladders. The insulating mother compound Sr14Cu24O41 is hole doped in the chains as well as slightly in the ladders. The hole content in the Sr14Cu24O41 ladders is reduced, if La3+ is substituted for Sr2+ resulting in a pure antiferromagnetic spin ladder given by La4Sr10Cu24O41. In this thesis such spin ladders are studied by an amalgamated approach of theoretical work and experiments. For the La4Sr10Cu24O41 ladder the temperature development is investigated using scattering techniques as well as computer modelling. At zero temperature the spin ladder has already been well described by effective models computed via perturbative continuous unitary transformations (PCUTs). These PCUT results are now combined with a mean field approach allowing for incorporation of temperature induced conditional excitations via vertex corrections. The vertex correction results in a decreasing one-triplon spectral weight upon increasing temperature. This effect is studied by inelastic neutron scattering (INS) on La4Sr10Cu24O41 crystals by measuring the scattering amplitude at various temperatures. Convincing agreement is found within the experimental and theoretical error bars. Additionally the coupling constants found in Ref. [1] could be confirmed. The importance of an inter-ladder coupling Jinter between isolated ladders in the plane is investigated by combination of the PCUT with a mean field approach. This calculation is undertaken for the square and the trellis lattice. The difference to the existing calculations [2], [3] and [4] is the partial inclusion of the hardcore interaction for neighbouring ladders. At zero temperature we reveal the effect of quantum fluctuations on the spin gap. In the case of the square lattice we find the closure of the spin gap at a critical inter-ladder coupling enlarged by 3 - 4%. In the case of the trellis lattice, the spin gap decreases also as a function of inter-ladder coupling. In this case we find an enlarged critical inter-ladder coupling up to 21%. In additon, the theoretically predicted shift of the spectral weight by Uhrig and Schmidt could experimentally be confirmed. Substituting Ca2+ for Sr2+ results in hole doped ladders, such as Sr2.5Ca11.5Cu24O41 and Sr8Ca6Cu24O41, where the hole doping depends on the substituted Ca2+ content. The magnetic spectra of these two doped ladders are measured by INS and compared with the undoped ladder La4Sr10Cu24O41. The comparison reveals new features tracing back to the presence of holes. At low temperature in both compounds scattering below the gap of the undoped ladder is found resulting in a subgap1 state at about 8meV. The subgap broadens at high temperature so much that the closing of the subgap results. Additionally for both dopings and at low and high temperature a hole-pair captured by triplons is found with a fixed energy gap at 23meV. The new features are qualitatively explained in terms of resonant valence bond theory (RVB) as well as stripe ordering theory.
We investigate how dynamic correlations of hard-core bosonic excitation at finite temperature are affected by additional interactions besides the hard-core repulsion which prevents them from occupying the same site. We focus especially on dimerized spin systems, where these additional interactions between the elementary excitations, triplons, lead to the formation of bound states, relevant for the correct description of scattering processes. In order to include these effects quantitatively, we extend the previously developed Br\"uckner approach to include also nearest-neighbor (NN) and next-nearest neighbor (NNN) interactions correctly in a low-temperature expansion. This leads to the extension of the scalar Bethe-Salpeter equation to a matrix-valued equation. As an example, we consider the Heisenberg spin ladder to illustrate the significance of the additional interactions on the spectral functions at finite temperature, which are proportional to inelastic neutron scattering rates.
We discuss recent applications of exact numerical continued fraction expansion (CFE) techniques to calculate dynamical correlation functions of various strong-coupling models related to the high-temperature cuprate superconductors. For the two-dimensional square-lattice spin-1/2 Heisenberg antiferromagnet, we present exact results for the zero-temperature dynamical structure factor on finite-sized lattices and compare them to approximate results from a Schwinger boson mean-field theory, recently proposed by Arovas and Auerbach. We find that the mean-field theory represents a very good approximation to the exact spin excitation spectra and to the static spin correlations. We then investigation the dynamical spin-spin structure factor and the single-particle spectral function for finite model clusters with dopant induced hole-type charge carriers, in the strong-coupling limit of both the single-band Hubbard model (t-J-model) and the three-band Hubbard model (Kondo-Heisenberg model). Our results are consistent with the physical picture recently proposed by Zhang and Rice which implies an approximate mapping of the low-energy states in the three-band model onto an effective single-band theory. 33 refs., 4 figs.
We investigate the quantum dynamics of the spins on different Heisenberg antiferromagnetic spin lattice systems. Firstly, we applied the coupled-cluster method to the spin-1/2 antiferromagnetic \emph{XXZ} model on a square lattice by employing an approximation which contains two-body long-range correlations and high-order four-body local correlations. Improvement is found for the ground-state energy, sublattice magnetization, and the critical anisotropy when comparing with the approximation including the two-body correlations alone. We also obtain the full excitation spectrum which is in good agreement with the quantum Monte Carlo results and the high-order spin-wave theory.Secondly, we study the longitudinal excitations of quantum antiferromagnets on a triangular lattice by a recently proposed microscopic many-body approach based on magnon-density waves. We calculate the full longitudinal excitation spectra of the antiferromagnetic Heisenberg model for a general spin quantum number in the isotropic limit. Similar to the square lattice model, we find that, at the center of the first hexagonal Brillouin zone Γ(q=0)\Gamma(\mathbf q=0)Γ(q=0) and at the magnetic ordering wavevectors ±[Q=(4π/3,0)]\pm[\mathbf Q= (4\pi/3,0)]±[Q=(4π/3,0)], the excitation spectra become gapless in the thermodynamic limit, due to the slow, logarithmic divergence of the structure factor. However, these longitudinal modes on two-dimensional models may be considered as quasi-gapped, as any finite-size effect or small anisotropy will induce a large energy gap, when compared with the counterpart of the transverse spin-wave excitations. We have also investigated the excited states of the quasi-one-dimensional quantum antiferromagnets on hexagonal lattices, including the longitudinal modes based on the magnon-density waves. A model Hamiltonian with a uniaxial single-ion anisotropy is first studied by a spin-wave theory based on the one-boson method; the ground state thus obtained is employed for the study of the longitudinal modes. The full energy spectra of both the transverse modes (i.e., magnons) and the longitudinal modes are obtained as functions of the nearest-neighbor coupling and the anisotropy constants. We have found two longitudinal modes due to the non-collinear nature of the triangular antiferromagnetic order, similar to that of the phenomenological field theory approach by Affleck. The excitation energy gaps due to the anisotropy and the energy gaps of the longitudinal modes without anisotropy are then investigated. We then compares our results for the longitudinal energy gaps at the magnetic wavevectors with the experimental results for several antiferromagnetic compounds with both integer and non-integer spin quantum numbers, and we find good agreements after the higher-order contributions are included in our calculations.