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arXiv:0705.0799

Lattice dynamics of the Heisenberg chain coupled to finite frequency bond phonons

Franz Michel, Hans-Gerd Evertz

Published May 7, 2007• 3 citations
arXivHTMLPDF

Abstract

The phonon dynamics in a one dimensional Heisenberg spin chain coupled to finite-frequency bond phonons is studied. We present the first detailed phonon spectra for these systems using Quantum Monte Carlo. The quantum phase transition is dominated by a central peak, yet the renormalisation of the main phonon branch depends strongly on the bare phonon frequency omega_0. The main branch remains largely unaffected at omega_0 \gsim J, but it softens completely when omega_0 is low enough. This is an unusual scenario for a structural phase transition and was observable only on sufficiently large systems. Approaching the dimerized phase from finite temperature, the lattice dynamics mirrors the behavior of a three dimensional system. For the efficient measurement of Greens functions, we introduce a mapping from the stochastic series expansion to continuous imaginary time.

DOI
10.48550/arxiv.0705.0799

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The phonon dynamics in a one dimensional Heisenberg spin chain coupled to finite-frequency bond phonons is studied. We present the first detailed phonon spectra for these systems using Quantum Monte Carlo. The quantum phase transition is dominated by a central peak, yet the renormalisation of the main phonon branch depends strongly on the bare phonon frequency omega_0. The main branch remains largely unaffected at omega_0 \gsim J, but it softens completely when omega_0 is low enough. This is an unusual scenario for a structural phase transition and was observable only on sufficiently large systems. Approaching the dimerized phase from finite temperature, the lattice dynamics mirrors the behavior of a three dimensional system. For the efficient measurement of Greens functions, we introduce a mapping from the stochastic series expansion to continuous imaginary time.
The zero and finite temperature spin-Peierls transitions in a quasi-one-dimensional spin-12\frac{1}{2}21​ Heisenberg model coupled to adiabatic bond phonons is investigated using the stochastic series expansion (SSE) quantum Monte Carlo (QMC) method. The quantum phase transition from a gapless N\'eel state to a spin-gapped Peierls state is studied in the parameter space spanned by spatial anisotropy, interchain coupling strength, and spin-lattice coupling strength. It is found that for any finite interchain coupling, the transition to a dimerized Peierls ground state only occurs when the spin-lattice coupling exceeds a finite, nonzero critical value. This is in contrast to the pure 1D model (zero interchain coupling), where adiabatic/classical phonons lead to a dimerized ground state for any nonzero spin-phonon interaction. The phase diagram in the parameter space shows that for a strong interchain coupling, the relation between the interchain coupling and the critical value of the spin-phonon interaction is linear whereas for weak interchain coupling, this behavior is found to have a natural logarithmlike relation. No region was found to have a long range magnetic order and dimerization occurring simultaneously. Instead, the N\'eel state order vanishes simultaneously with the setting in of the spin-Peierls state. For the thermal phase transition, a continuous heat capacity with a peak at the critical temperature Tc{T}_{c}Tc​ shows a second order phase transition. The variation of the equilibrium bond length distortion ${\ensuremath{\delta}}_{eq}$ with temperature showed a power law relation which decayed to zero as the temperature was increased to Tc{T}_{c}Tc​, indicating a continuous transition from the dimerized phase to a paramagnetic phase with uniform bond length and zero antiferromagnetic susceptibility.
The one-dimensional spin-1/2 Heisenberg model with nearest-neighbour interaction coupled to Einstein phonons is of interest for the understanding of materials which show the spin-Peierls phase transition. I examined this model by performing a finite-temperature quantum Monte Carlo simulation by using the very efficient loop algorithm. I studied in detail the magnetic susceptibility, the specific heat, the phonon occupation, the dimerization, and the spin-spin correlation function of this model for various spin-phonon couplings and phonon frequencies. In particular I gave evidence for a global effective spin-spin coupling and for the transition from a gapless to the massive phase by studying the finite-size behavior of the magnetic susceptibility.
In this thesis the quantum Monte Carlo loop algorithm is applied to study the properties of correlation functions in low-dimensional quantum antiferromagnets. Two different types of lattice models are investigated. The first system is the quantum critical XXZ chain. The second class of models emerges when coupling the square lattice Heisenberg model to phonons. In case of the quantum critical XXZ chain, the loop algorithm is used to compute the two-point spin and dimer correlation functions at low temperatures. Starting from the well-known asymptotic expressions from conformal field theory for the decay of the correlations, the crossover from the finite-temperature to the ground state behavior is investigated. In addtion, the Monte Carlo data are used to extract numerical estimates for the correlation amplitudes. The deviations from asymptotic scaling at low finite temperatures and intermediate distances are examined. The results are also used to study the algebraic and logarithmic corrections in the ground state. The method is further applied to compute the low-temperature spin correlations of the square lattice Heisenberg model coupled to dispersionless Einstein phonons. Also, statically dimerized Heisenberg models in two dimensions are investigated. Here the quantum Monte Carlo technique is used to determine the specific pattern of dimerized bonds which has the largest gain in the magnetic energy. A direct comparison of ground state energies is only possible for a small number of dimerized systems. For this reason an alternative approach is developed, which allows to study a significantly larger class of models. Starting from a Landau expansion of the free energy, this amounts to a computation of specific structure factors of the dimer correlations of the uniform Heisenberg model. As a consequence, a Landau theory for the two-dimensional counterpart of the spin-Peierls transition is established. This adiabatic approach also applies to the Heisenberg model coupled to dispersive phonons. In this case the data for the dimer correlations are used to study the impact of the spin-phonon coupling on the phonon dispersion relations. In particular, the softening of phonon modes is investigated, which is relevant for experimental studies on substances with a quasi two-dimensional magnetic structure.
The spin-Peierls transition in the ground state of a quasi-one-dimensional spin-1/2 Heisenberg model coupled to adiabatic bond phonons is studied using a quantum Monte Carlo (QMC) method. The transition from a gapless Neel state to a spin-gapped Peierls state is explored in the parameter space spanned by the spatial anisotropy and the strength of spin-lattice coupling. It is found that for any finite inter-chain coupling, the transition to a dimerized Peierls ground state occurs only when the spin-lattice coupling exceeds a finite, non-zero critical value. This is in contrast to the pure 1D model, where adiabatic phonons lead to a dimerized ground state for any non-zero spin-phonon coupling. The phase diagram in the above parameter space is mapped out. No evidence is found for a region of co-existing long range magnetic order and dimerization. The vanishing of the Neel order occurs simultaneously with the setting in of the dimerization.
We investigate a one-dimensional S=1S=1S=1 antiferromagnetic Heisenberg model coupled to a lattice distortion by a quantum Monte Carlo method. Investigating the ground-state energy of the static bond-alternating chain, we find that the instability to a dimerized chain depends on the value of the spin-phonon coupling, unlike the case of S=12.S=\frac{1}{2}.S=21​. The spin state is the dimer state or the uniform Haldane state depending on whether the lattice distorts or not, respectively. At an intermediate value of the spin-phonon coupling, we find the first-order transition between the two states. We also find the coexistence of the two states.
We present results for the dynamic structure factor of the S=1∕2S=1∕2S=1∕2 bond alternating Heisenberg chain over a large range of frequencies and temperatures. Data are obtained from a numerical evaluation of thermal averages based on the calculation of all eigenvalues and eigenfunctions for chains of up to 20 spins. Interpretation is guided by the exact temperature dependence in the noninteracting dimer limit which remains qualitatively valid up to an interdimer exchange $\ensuremath{\lambda}\ensuremath{\approx}0.5$. The temperature induced central peak around zero frequency is clearly identified and aspects of the crossover to spin diffusion in its variation from low to high temperatures are discussed. The one-magnon peak acquires an asymmetric shape with increasing temperature. The two-magnon peak is dominated by the S=1S=1S=1 bound state which remains well defined up to temperatures of the order of JJJ. The variation with temperature and wave vector of the integrated intensity for one-magnon and two-magnon scattering and of the central peak are discussed.