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arXiv:2211.09387 · Journal of Statistical Mechanics Theory and Experiment

Strain fields and critical phenomena in manganites II: spin-lattice-energy Hamiltonians

Rohit Singh, Sanjay Puri

Published Mar 1, 2023• 0 citations
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Abstract

Abstract The dynamic critical behavior at the paramagnetic-antiferromagnetic (PM-AFM) transition in manganites has recently been studied experimentally (Niermann et al 2015 Phys. Rev. Lett. 114 037204). We extend the Hamiltonian of paper I by incorporating an energy field, and study the corresponding Model C of critical dynamics. We use the dynamic renormalization group approach and calculate the dynamic critical exponents z, ν z and the line-width exponent Δ to leading order in the small expansion parameters ϵ = 4 − d + 2 σ and ϵ ′ = 4 − d . Here, d is the space dimension and σ is the long-range exponent. Using σ as an adjustable parameter, the theory gives us a good match to the experimentally available static and dynamic critical exponents at the PM-AFM transition.

DOI
10.1088/1742-5468/acc065

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Abstract The dynamic critical behavior at the paramagnetic-antiferromagnetic (PM-AFM) transition in manganites has recently been studied experimentally (Niermann et al 2015 Phys. Rev. Lett. 114 037204). We extend the Hamiltonian of paper I by incorporating an energy field, and study the corresponding Model C of critical dynamics. We use the dynamic renormalization group approach and calculate the dynamic critical exponents z, ν z and the line-width exponent Δ to leading order in the small expansion parameters ϵ = 4 − d + 2 σ and ϵ ′ = 4 − d . Here, d is the space dimension and σ is the long-range exponent. Using σ as an adjustable parameter, the theory gives us a good match to the experimentally available static and dynamic critical exponents at the PM-AFM transition.
We investigate the nonconserved critical dynamics of a nonlocal model Hamiltonian incorporating screened long-range interactions in the quartic term. Employing dynamic renormalization group analysis at one-loop order, we calculate the dynamic critical exponent $z=2+\ensuremath{\epsilon}{f}_{1}(\ensuremath{\sigma},\ensuremath{\kappa},n)+O({\ensuremath{\epsilon}}^{2})$ and the linewidth exponent $w=\ensuremath{-}\ensuremath{\sigma}+\ensuremath{\epsilon}{f}_{2}(\ensuremath{\sigma},\ensuremath{\kappa},n)+O({\ensuremath{\epsilon}}^{2})$ in the leading order of $\ensuremath{\epsilon}$, where $\ensuremath{\epsilon}=4\ensuremath{-}d+2\ensuremath{\sigma}$, with ddd the space dimension, nnn the number of components in the order parameter, and $\ensuremath{\sigma}$ and $\ensuremath{\kappa}$ the parameters coming from the nonlocal interaction term. The resulting values of linewidth exponent www for a wide range of $\ensuremath{\sigma}$ is found to be in good agreement with the existing experimental estimates from spin relaxation measurements in perovskite manganite samples.
Antiferromagnetic Hamiltonians with short range, non-frustrating interactions are well known to exhibit long range magnetic order in dimensions d ≥2 but exhibit only quasi-long-range order, with power-law decay of correlations, in d = 1 (for half-integer spin). On the other hand, non-frustrating long range interactions can induce long range order in d = 1. We study Hamiltonians in which the long range interactions have an adjustable amplitude λ, as well as an adjustable power law 1/| x | α , using a combination of quantum Monte Carlo and analytic methods: spin-wave, large N non-linear σ model, and renormalization group methods. We map out the phase diagram in the λ–α plane and study the nature of the critical line separating the phases with long range and quasi-long-range order. We find that this corresponds to a novel line of critical points with continuously varying critical exponents and a dynamical exponent, z <1.
Critical exponents in the ${\mathit{CP}}^{\mathit{N}\mathrm{\ensuremath{-}}1}$ model, which describes localized-moment ferromagnets and antiferromagnets (N=2 in the Heisenberg model), are calculated from two-particle Green's functions to first order in 1/N. For d=2+\ensuremath{\varepsilon} the results agree with earlier renormalization group calculations. For d=3 the leading 1/N corrections turn out to be very large at N=2. For d=4-\ensuremath{\varepsilon} the 1/N corrections are small at any N and insufficient to describe correctly the magnetic phase transition. \textcopyright{} 1996 The American Physical Society.
We calculate the dynamic structure factor $S(\mathbit{k},\ensuremath{\omega})$ in the paramagnetic regime of quantum Heisenberg ferromagnets for temperatures TTT close to the critical temperature Tc{T}_{c}Tc​ using our recently developed functional renormalization group approach to quantum spin systems. In d=3d=3d=3 dimensions, we find that for small momenta $\mathbit{k}$ and frequencies $\ensuremath{\omega}$ the dynamic structure factor assumes the scaling form $S(\mathbit{k},\ensuremath{\omega})=(\ensuremath{\tau}TG(\mathbit{k})/\ensuremath{\pi})\mathrm{\ensuremath{\Phi}}(k\ensuremath{\xi},\ensuremath{\omega}\ensuremath{\tau})$, where $G(\mathbit{k})$ is the static spin-spin correlation function, $\ensuremath{\xi}$ is the correlation length, and the characteristic timescale $\ensuremath{\tau}$ is proportional to ${\ensuremath{\xi}}^{5/2}$. We explicitly calculate the dynamic scaling function $\mathrm{\ensuremath{\Phi}}(x,y)$ and find satisfactory agreement with neutron scattering experiments probing the critical spin dynamics in EuO and EuS. Precisely at the critical point where $\ensuremath{\xi}=\ensuremath{\infty}$, our result for the dynamic structure factor can be written as $S(\mathbit{k},\ensuremath{\omega})={(\ensuremath{\pi}{\ensuremath{\omega}}_{k})}^{\ensuremath{-}1}{T}_{c}G(\mathbit{k}){\mathrm{\ensuremath{\Psi}}}_{c}(\ensuremath{\omega}/{\ensuremath{\omega}}_{k})$, where ${\ensuremath{\omega}}_{k}\ensuremath{\propto}{k}^{5/2}$. We find that ${\mathrm{\ensuremath{\Psi}}}_{c}(\ensuremath{\nu})$ vanishes as ${\ensuremath{\nu}}^{\ensuremath{-}13/5}$ for large $\ensuremath{\nu}$, and as ${\ensuremath{\nu}}^{3/5}$ for small $\ensuremath{\nu}$. While the large-frequency behavior of ${\mathrm{\ensuremath{\Psi}}}_{c}(\ensuremath{\nu})$ is consistent with calculations based on mode-coupling theory and with perturbative renormalization group calculations to second order in $\ensuremath{\epsilon}=6\ensuremath{-}d$, our result for small frequencies disagrees with previous calculations. We argue that until now, neither experiments nor numerical simulations have been sufficiently accurate to determine the low-frequency behavior of ${\mathrm{\ensuremath{\Psi}}}_{c}(\ensuremath{\nu})$. We also calculate the low-temperature behavior of $S(\mathbit{k},\ensuremath{\omega})$ in one- and two-dimensional ferromagnets and find that it satisfies dynamic scaling with exponent z=2z=2z=2 and exhibits a pseudogap for small frequencies.
The critical exponents for two ${\ensuremath{\varphi}}^{3}$-field theories with long-range (LR) interactions decaying as 1/${R}^{d+\ensuremath{\sigma}}$, \ensuremath{\sigma}>0, are calculated to two-loop order in renormalized perturbation theory in d=3\ensuremath{\sigma}-\ensuremath{\epsilon}' dimensions. One is the continuum version of the p-state Potts model and the other is the scalar field theory with imaginary coupling that describes the Yang-Lee edge-singularity problem. The two crossover exponents for quadratic symmetry breaking discussed by Wallace and Young and in recent work by the present authors are also calculated in the first case. By means of renormalization-group recursion relations to one-loop order, it is shown that the LR fixed point is stable for all \ensuremath{\sigma}2 whenever ${\ensuremath{\eta}}_{\mathrm{SR}}$, the critical correlation-function exponent for short-range (SR) interactions, is negative, implying a discontinuity of critical exponents at \ensuremath{\sigma}=2. This is the case for the (p2)-state Potts model and for the Yang-Lee edge-singularity problem, and is in agreement with recent results by Chang and Sak for the Ising spin-glass problem. For the (p>2)-state Potts model there is an indication of a continuous crossover to SR behavior at \ensuremath{\sigma}=2-${\ensuremath{\eta}}_{\mathrm{SR}}$, with ${\ensuremath{\eta}}_{\mathrm{SR}}$>0. It is pointed out that a number of exact results [\ensuremath{\beta}=(1-\ensuremath{\epsilon}'/2\ensuremath{\sigma})\ensuremath{\sigma}\ensuremath{\nu}, \ensuremath{\sigma}^=(d-\ensuremath{\sigma})/(d+\ensuremath{\sigma}), in which \ensuremath{\sigma}^ is the Yang-Lee edge-singularity exponent, and ${\ensuremath{\nu}}^{\mathrm{\ensuremath{-}}1}$=(d-\ensuremath{\sigma})/2 for a scalar theory] may apply within the LR expansion depending on \ensuremath{\eta}=2-\ensuremath{\sigma} (shown here to hold at least to two-loop order) being exact, to all orders.
In this set of lectures, I shall review some recent work on quantum critical phenomena which involve the changes in phase of the ground state of a quantum antiferromagnetic system when external parameters are varied through some critical values. I will also draw analogies between antiferromagnetic phase transitions and chiral phase transitions in lower (particularly 2 + l)-dimensional gauge theories and also some 2+1-dimensional four-fermi theories where chiral symmetry breaking patterns depend on the coupling constants and also the matter and gauge field content of the theory. I will begin by reviewing some arguments that, in the infrared limit, the dynamics of Goldstone bosons in the confining and chiral symmetry breaking phase of 2 + 1 dimensional QCD is identical to the effective spin wave dynamics in 2 dimensional generalized quantum antiferromagnets. I also discuss the possibility that certain quantum critical behaviors in generalized antiferromagnets can be analyzed using the principle of universality and are in the universality class of a particular type of relativistic quantum field theory with gauge and chiral symmetry. I show how the epsilon expansion can be used and argue that, for a large range of symmetry groups and other parameters, the phase transitions are expected to be fluctuation induced first order transitions. For the small range of parameters where they are second order, the epsilon expansion and universality give information about the critical behavior.