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Critical Behavior of Heisenberg Chains Coupled to Phonons

Rainer Walter Kühne

Published Mar 11, 2019• 0 citations
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Abstract

We present a quantum Monte Carlo study for the isotropic antiferromagnetic Heisenberg chain coupled to Einstein phonons. By investigating the behavior of the correlation length we find the critical exponent to depend on both the phonon frequency and the spin-phonon coupling constant. Hence, these systems change the universality class continuously with the spin-phonon coupling.

DOI
10.31219/osf.io/aeyk4

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•Mar 11, 2019•0 citations
We present a quantum Monte Carlo study for the isotropic antiferromagnetic Heisenberg chain coupled to Einstein phonons. By investigating the behavior of the correlation length we find the critical exponent to depend on both the phonon frequency and the spin-phonon coupling constant. Hence, these systems change the universality class continuously with the spin-phonon coupling.
This PhD thesis examines the thermodynamical properties of spin-1/2 Heisenberg chains which are coupled to Einstein phonons. One result of this study is that these spin-phonon systems undergo a phase transition similar to a spin-Peierls transition (cf. Physical Review B 60 (1999) 12125-12133 and Physical Review B 65 (2002) 144438). Another result is that in the undimerized phase the critical exponent nu depends on both the phonon frequency and the spin-phonon coupling constant. This means that these spin-phonon systems change the universality class continuously with the spin-phonon coupling (cf. arXiv: cond-mat/9912150v1).
We study the spin-Peierls transition in a Heisenberg spin chain coupled to optical bond phonons. Quantum Monte Carlo results for systems with up to N=256N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}256N=256 spins show unambiguously that the transition occurs only when the spin-phonon coupling $\ensuremath{\alpha}$ exceeds a critical value ${\ensuremath{\alpha}}_{c}$. Using sum rules, we show that the phonon spectral function has divergent (for $N\ensuremath{\rightarrow}\ensuremath{\infty}$) weight extending to zero frequency for $\ensuremath{\alpha}<{\ensuremath{\alpha}}_{c}$. The phonon correlations decay with distance rrr as 1/r1/r1/r. This behavior is characteristic for all $0<\ensuremath{\alpha}<{\ensuremath{\alpha}}_{c}$ and the $q\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{\pi}$ phonon does not soften (to zero frequency) at $\ensuremath{\alpha}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}{\ensuremath{\alpha}}_{c}$.
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.
We performed a finite-temperature quantum Monte Carlo simulation of the one-dimensional spin-1/2 Heisenberg model with nearest-neighbor interaction coupled to Einstein phonons. Our method allows to treat easily up to 100 phonons per site and the results presented are practically free from truncation errors. We studied in detail the magnetic susceptibility, the specific heat, the phonon occupation, the dimerization, and the spin-correlation function for various spin-phonon couplings and phonon frequencies. In particular we give evidence for the transition from a gapless to a massive phase by studying the finite-size behavior of the susceptibility. We also show that the dimerization is proportional to g2/Ω for T<2J.
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.
We investigate the antiadiabatic limit of an antiferromagnetic S=1/2 Heisenberg chain coupled to Einstein phonons via a bond coupling. The flow equation method is used to decouple the spin and the phonon part of the Hamiltonian. In the effective spin model longer range spin-spin interactions are generated. The effective spin chain is frustrated. The resulting temperature dependent couplings are used to determine the magnetic susceptibility and to determine the phase transition from a gapless state to a dimerized gapped phase. The susceptibilities and the phase diagram obtained via the effective couplings are compared with independently calculated quantum Monte Carlo results.