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Physical review. B./Physical review. B

Unveiling the scaling laws behind spin-phonon enhanced decorrelation in spin chains

Ilija Srpak, Michael J. Willatt, Stuart C. Althorpe, Ali Alavi

Published Nov 25, 2025• 1 citation
Publisher

Abstract

To study the spin-Peierls transition at finite temperatures, we use path integral molecular dynamics (PIMD) in combination with stochastic series expansion or exact diagonalization to sample the quantum Boltzmann distribution of the spin-phonon-coupled one-dimensional isotropic Heisenberg chains. The use of PIMD in this context has apparently been overlooked up to now and offers several advantages over other approaches, including no restriction on the functional forms of the phonon potential or the exchange coupling. This approach allows us to study the spin-spin correlation in such systems in detail. Looking at S = 1 / 2 , we find that it exhibits algebraic decay in the undimerized phase and exponential decay in the dimerized phase, but with many nuanced trends present. The decay exponentials in all cases roughly obey the scaling of ( g 2 / ( m ω π 2 ) ) γ , where g is the spin-phonon coupling constant, m is mass, and ω π is the frequency of the k = π phonon.

DOI
10.1103/11dp-8sm4

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To study the spin-Peierls transition at finite temperatures, we use path integral molecular dynamics (PIMD) in combination with stochastic series expansion or exact diagonalization to sample the quantum Boltzmann distribution of the spin-phonon-coupled one-dimensional isotropic Heisenberg chains. The use of PIMD in this context has apparently been overlooked up to now and offers several advantages over other approaches, including no restriction on the functional forms of the phonon potential or the exchange coupling. This approach allows us to study the spin-spin correlation in such systems in detail. Looking at S = 1 / 2 , we find that it exhibits algebraic decay in the undimerized phase and exponential decay in the dimerized phase, but with many nuanced trends present. The decay exponentials in all cases roughly obey the scaling of ( g 2 / ( m ω π 2 ) ) γ , where g is the spin-phonon coupling constant, m is mass, and ω π is the frequency of the k = π phonon.
We build on the framework for studying systems containing spin-phonon coupling using molecular dynamics (MD) as the method for sampling the positions of atoms carrying spins, whose interaction is described by the Heisenberg model, and which we presented in a previous paper Srpak et al. [Phys. Rev. B 112, 205132 (2025)]. In applying the real-space path integral (PI) framework to the spin-phonon problem, neighboring imaginary time slices become coupled via overlap of eigenvector matrices propagating along the path integral, leading to interactions that go beyond the pure harmonic interactions between neighboring beads of the ring-polymers in Born-Oppenheimer-based PIMD simulations of quantum nuclei. We show how the use of a spin-force, defined as the negative gradient of the spin free energy, accounts for this type of coupling, valid for systems in thermal equilibrium. The spin force is computed using either ring-polymer exact diagonalization (for small systems) or stochastic series expansion (for large systems). We provide the technical details for these implementations. We present a neural network approach for this type of PIMD simulation, where we compute the spin force based on training data obtained from the ring-polymer stochastic series expansion method. Utilizing these methods, we compare with the literature and demonstrate how one can obtain phase diagrams clearly showing how the thermal and nuclear quantum effects influence the stability of spin-Peierls dimerized phase.We also show how machine learning makes it possible to apply the method to two dimensional systems.
We build on the framework for studying systems containing spin-phonon coupling using molecular dynamics as the method for sampling the positions of atoms carrying spins, whose interaction is described by the Heisenberg model, and which we presented in a previous paper Srpak et al. [Phys. Rev. B 112, 205132 (2025)]. In applying the real-space path integral framework to the spin-phonon problem, neighboring imaginary time slices become coupled via overlap of eigenvector matrices propagating along the path integral, leading to interactions that go beyond the pure harmonic interactions between neighboring beads of the ring-polymers in Born-Oppenheimer-based path integral molecular dynamics (PIMD) simulations of quantum nuclei. We show how the use of a spin-force, defined as the negative gradient of the spin free energy, accounts for this type of coupling, valid for systems in thermal equilibrium. The spin force is computed using either ring-polymer exact diagonalization (for small systems) or stochastic series expansion (for large systems). We provide the technical details for these implementations. We present a neural network approach for this type of PIMD simulation, where we compute the spin force based on training data obtained from the ring-polymer stochastic series expansion method. Utilizing these methods, we compare them with the literature and demonstrate how one can obtain phase diagrams clearly showing how the thermal and nuclear quantum effects influence the stability of the spin-Peierls dimerized phase. We also show how machine learning makes it possible to apply the method to two dimensional systems.
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 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 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 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 $\ensuremath{\alpha}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}{\ensuremath{\alpha}}_{c}$.