Quantum Brownian motion for magnets

We begin by introducing the quantized Hamiltonian describing the different contributions to the total energy of the system, consisting of spins as well as environmental degrees of freedom (e.g. electrons and phonons), given by

$$\begin{equation}\hat{H}={\hat{H}}_{\mathrm{S}}+{\hat{H}}_{\mathrm{R}}+{\hat{V}}_{\text{int}},\end{equation} \tag{ 2 }$$

where ${\hat{H}}_{\mathrm{S}}$ is the bare spin Hamiltonian operator which captures the spin energy in external fields and interactions between spins, ${\hat{H}}_{\mathrm{R}}$ is the environmental or reservoir Hamiltonian, and ${\hat{V}}_{\text{int}}$ is the interaction between the spins and the reservoir.

We choose ${\hat{H}}_{\mathrm{S}}$ as the sum of the interaction with a homogeneous external field B _ext [110] and the exchange interaction between three-dimensional spin vector operators ${\hat{\boldsymbol{S}}}^{(n)}\!=({\hat{S}}_{1}^{(n)},{\hat{S}}_{2}^{(n)},{\hat{S}}_{3}^{(n)})$ at sites n of a lattice [111],

$$\begin{equation}{\hat{H}}_{\mathrm{S}}=-\gamma \sum\limits _{n}{\hat{\boldsymbol{S}}}^{(n)}\!\cdot {\boldsymbol{B}}_{\text{ext}}-\frac{1}{2}\sum\limits _{n,m\ne n}{\hat{\boldsymbol{S}}}^{(n)}\!\cdot {\mathcal{J}}^{(nm)}{\hat{\boldsymbol{S}}}^{(m)}\!.\end{equation} \tag{ 3 }$$

Here ${\mathcal{J}}^{(nm)}$ is the exchange tensor for spin pairs (n, m) [112], which can include the Dzyaloshinskii–Moriya interaction [47]. It is straightforward to include additional energetic terms in the bare spin Hamiltonian, such as energies associated with magnetic anisotropy. Instead of the magnetic moment M used in equation (1), we will here work with the spin angular momentum S proportional to M , M = γ S , where γ is the gyromagnetic ratio. In the following we will assume the gyromagnetic ratio |γ| = |g_e μ_B/ℏ |= 1.76 × 10¹¹ s⁻¹ T⁻¹ for an electron, with γ taking the negative sign.

The reservoir Hamiltonian is commonly modelled as a set of harmonic oscillators [35, 63], and we here follow the continuous reservoir approach by Huttner and Barnett [63], taking the reservoir Hamiltonian as

$$\begin{equation}{\hat{H}}_{\mathrm{R}}=\frac{1}{2}\sum\limits _{n}{\int }_{0}^{\infty }\!\mathrm{d}\omega \left[{\left({\hat{\mathbf{\Pi }}}_{\omega }^{(n)}\!\right)}^{2}+{\omega }^{2}{\left({\hat{\boldsymbol{X}}}_{\omega }^{(n)}\!\right)}^{2}\right].\end{equation} \tag{ 4 }$$

It describes a continuous frequency reservoir at each lattice site n, where ${\hat{\mathbf{\Pi }}}_{\omega }^{(n)}\!$ and ${\hat{\boldsymbol{X}}}_{\omega }^{(n)}\!$ are (three-dimensional) momentum and position operators of the reservoir oscillator with frequency ω. The position operators ${\hat{\boldsymbol{X}}}_{\omega }^{(n)}\!$ physically represent variations in the environment to which the spin at site n responds, see illustration figure 1, as for example, in magnon–phonon mediated loss [69]. Unlike most system + environment Hamiltonians which assume one-dimensional coupling, we here take the spin–reservoir interaction to be of the three-dimensional form

$$\begin{equation}{\hat{V}}_{\text{int}}=-\hspace{2pt}\gamma \sum\limits _{n}{\hat{\boldsymbol{S}}}^{(n)}\!\cdot {\int }_{0}^{\infty }\!\mathrm{d}\omega \,{\mathcal{C}}_{\omega }^{(n)}{\hat{\boldsymbol{X}}}_{\omega }^{(n)}\!.\end{equation} \tag{ 5 }$$

This coupling allows angular momentum transfer as well as energy transfer between the spins and the environment. Here ${\mathcal{C}}_{\omega }^{(n)}$ is a three-dimensional coupling tensor and a function of frequency ω. At each ω, the coupling tensor determines the strength of the coupling of each spin to its reservoir oscillators at frequency ω, thus acting as a frequency filter. As we shall see, the choice of the coupling ${\mathcal{C}}_{\omega }^{(n)}$ will determine the damping of the spin dynamics as well as the stochastic noise experienced by the spins.

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For readers concerned about time-reversal symmetry of ${\hat{V}}_{\text{int}}$ in (5), we note that ${\hat{\boldsymbol{X}}}_{\omega }^{(n)}\!$ should be interpreted as an effective magnetic field seen by the spins due to their interaction with the environment, which has the same time symmetry as ${\hat{\boldsymbol{S}}}^{(n)}\!$ [113].

Having set up the full Hamiltonian (2) of the spins and environment degrees of freedom allows the study of the spins' reduced state dynamics ${\hat{\rho }}_{\mathrm{S}}(t)={\mathrm{t}\mathrm{r}}_{\mathrm{R}}\left[{\hat{\rho }}_{\mathrm{S}\mathrm{R}}(t)\right]$ of the total state

$$\begin{equation}{\hat{\rho }}_{\mathrm{S}\mathrm{R}}(t)={\mathrm{e}}^{\text{i}\hat{H}t/\hslash }\left({\hat{\rho }}_{\mathrm{S}}(0)\otimes {\hat{\rho }}_{\mathrm{R}}\right){\mathrm{e}}^{-\mathrm{i}\hat{H}t/\hslash },\end{equation} \tag{ 6 }$$

where the reservoir state ${\hat{\rho }}_{\mathrm{R}}={\text{e}}^{-\beta {\hat{H}}_{\mathrm{R}}}/\mathrm{t}\mathrm{r}[{\text{e}}^{-\beta {\hat{H}}_{\mathrm{R}}}]$ is thermal at some inverse temperature β = 1/k_B T. In what follows it will be more convenient to work instead in the Heisenberg picture where the state is stationary, ${\hat{\rho }}_{\mathrm{S}\mathrm{R}}(0)$, while the time dependence of an operator $\hat{O}(t)$ is governed by the commutator $\mathrm{d}\hat{O}(t)/\mathrm{d}t=(\mathrm{i}/\hslash )[\hat{H},\hat{O}(t)]$. Expectation values at time t can then be obtained as

$$\begin{equation}\mathrm{t}\mathrm{r}[{\hat{\rho }}_{\mathrm{S}\mathrm{R}}(t)\hat{O}(0)]=\mathrm{t}\mathrm{r}[{\hat{\rho }}_{\mathrm{S}\mathrm{R}}(0)\hat{O}(t)].\end{equation} \tag{ 7 }$$

Using the standard commutation relations for the spin operators (and orbital angular momentum operators in general), $[{\hat{S}}_{j}^{(n)},{\hat{S}}_{k}^{(m)}]=\mathrm{i}\hslash {\delta }_{mn}{\sum }_{l}{{\epsilon}}_{jkl}{\hat{S}}_{l}^{(n)}$ and for the position/momentum operators, $[{{\hat{X}}_{\omega ,j}}^{(n)},{{\hat{{\Pi}}}_{{\omega }^{\prime },k}}^{(m)}]=\mathrm{i}\hslash {\delta }_{nm}{\delta }_{jk}\delta (\omega -{\omega }^{\prime })$, we obtain the following equations of motion for the spin operators ${\hat{\boldsymbol{S}}}^{(n)}\!(t)$

$$\begin{equation}\frac{\mathrm{d}{\hat{\boldsymbol{S}}}^{(n)}\!}{\mathrm{d}t}={\hat{\boldsymbol{S}}}^{(n)}\!\times \left[\gamma \left({\boldsymbol{B}}_{\text{ext}}+{\hat{\boldsymbol{B}}}_{\text{env}}^{(n)}\right)+\sum\limits _{m\ne n}{\hspace{6pt}\bar{\hspace{-5pt}\mathcal{J}}}^{(nm)}{\hat{\boldsymbol{S}}}^{(m)}\!\right],\end{equation} \tag{ 8 }$$

where ${\hspace{6pt}\bar{\hspace{-5pt}\mathcal{J}}}^{(nm)}=(1/2)[{\mathcal{J}}^{(nm)}+{({\mathcal{J}}^{(mn)})}^{\text{T}}]$ is the symmetrized exchange tensor and ${\hat{\boldsymbol{B}}}_{\text{env}}^{(n)}={\int }_{0}^{\infty }\!\mathrm{d}\omega \,{\mathcal{C}}_{\omega }^{(n)}{\hat{\boldsymbol{X}}}_{\omega }^{(n)}\!$ is a magnetic field operator formed from the reservoir oscillators at site n.

In turn, the equations of motion for these operators are

$$\begin{equation}\frac{{\mathrm{d}}^{2}{\hat{\boldsymbol{X}}}_{\omega }^{(n)}\!}{\mathrm{d}{t}^{2}}+{\omega }^{2}{\hat{\boldsymbol{X}}}_{\omega }^{(n)}\!=\gamma {\mathcal{C}}_{\omega }^{(n)\,\text{T}}{\hat{\boldsymbol{S}}}^{(n)}\!,\end{equation} \tag{ 9 }$$

i.e. the reservoir oscillators are driven by the motion of the spins, with the (transposed) coupling tensors ${\mathcal{C}}_{\omega }^{(n)\,\text{T}}$ governing the degree of driving for each of the continuum of oscillators. We assume retarded boundary conditions so that the reservoir responds only to the past behaviour of the spins. The retarded Green function for $({\partial }_{t}^{2}+{\omega }^{2}){G}_{\omega }=\delta (t-{t}^{\prime })$ is G_ω (t − t') = Θ(t − t')sin(ω(t − t'))/ω and can be used to solve equation (9) exactly, giving

$$\begin{equation}{\hat{\boldsymbol{X}}}_{\omega }^{(n)}\!(t)=\sqrt{\frac{\hslash }{2\omega }}\left({\hat{\boldsymbol{a}}}_{\omega }^{(n)}\!\,{\mathrm{e}}^{-\mathrm{i}\omega (t-{t}_{0})}+{\hat{\boldsymbol{a}}}_{\omega }^{(n)\,{\dagger}}\!\,{\mathrm{e}}^{+\mathrm{i}\omega (t-{t}_{0})}\right)+\gamma {\int }_{{t}_{0}}^{\infty }\!\mathrm{d}{t}^{\prime }\,{G}_{\omega }(t-{t}^{\prime })\,{\mathcal{C}}_{\omega }^{(n)\,\text{T}}{\hat{\boldsymbol{S}}}^{(n)}\!({t}^{\prime }).\end{equation} \tag{ 10 }$$

Here, ${\hat{\boldsymbol{a}}}_{\omega }^{(n)}\!$ and ${\hat{\boldsymbol{a}}}_{\omega }^{(n)\,{\dagger}}\!$ are (vectors of) bosonic ladder operators with their components obeying $[{\hat{a}}_{\omega ,j}^{(n)},{\hat{a}}_{{\omega }^{\prime },k}^{(m)\,{\dagger}}]={\delta }_{nm}\,{\delta }_{jk}\,\delta (\omega -{\omega }^{\prime })$. Classically these correspond to the two integration constants for the differential equation (9) which set the initial amplitude and velocity of the oscillator. The evolution begins at time t₀, at which the reservoir operators are assumed to be independent of the spin operators and the initial constants are applied.

Substituting the reservoir solutions (10) into the equations of motion for the spins (8), we obtain the first result: the Heisenberg–Langevin equation that governs three-dimensional quantum spin dynamics under the influence of memory and coloured quantum noise is

$$\begin{equation}\frac{\mathrm{d}{\hat{\boldsymbol{S}}}^{(n)}\!(t)}{\mathrm{d}t}=\frac{1}{2}\,{\hat{\boldsymbol{S}}}^{(n)}\!(t)\times \left[\gamma {\boldsymbol{B}}_{\text{ext}}+\sum\limits _{m\ne n}{\hspace{6pt}\bar{\hspace{-5pt}\mathcal{J}}}^{(nm)}{\hat{\boldsymbol{S}}}^{(m)}\!(t)+\gamma \,{\hat{\boldsymbol{b}}}^{(n)}\!(t)+{\gamma }^{2}{\int }_{{t}_{0}}^{t}\!\mathrm{d}{t}^{\prime }\,{\mathcal{K}}^{(n)}\!(t-{t}^{\prime }){\hat{\boldsymbol{S}}}^{(n)}\!({t}^{\prime })\right]+\mathrm{h}.\mathrm{c}.,\end{equation} \tag{ 11 }$$

where h.c. indicates the Hermitian conjugate of the preceding expression. The term ${\hat{\boldsymbol{b}}}^{(n)}\!(t)$ is a Hermitian magnetic noise operator for site n,

$$\begin{equation}{\hat{\boldsymbol{b}}}^{(n)}\!(t)={\int }_{0}^{\infty }\!\mathrm{d}\omega \sqrt{\frac{\hslash }{2\omega }}{\mathcal{C}}_{\omega }^{(n)}\left({\hat{\boldsymbol{a}}}_{\omega }^{(n)}\!\,{\mathrm{e}}^{-\mathrm{i}\omega (t-{t}_{0})}+\mathrm{h}.\mathrm{c}.\right),\end{equation} \tag{ 12 }$$

which plays the role of the stochastic noise first described by Brown [48]. Here it arises from the spin's interaction with its reservoir. As we will see below, the bath noise can be coloured and contain quantum zero-point fluctuations. In addition to the coloured noise ${\hat{\boldsymbol{b}}}^{(n)}\!$, a kernel tensor ${\mathcal{K}}^{(n)}\!(t-{t}^{\prime })$ appears in equation (11), which captures the damping of the spins. It arises from the coupling tensor ${\mathcal{C}}_{\omega }$ and is given by

$$\begin{equation}{\mathcal{K}}^{(n)}\!(\tau )={\Theta}(\tau ){\int }_{0}^{\infty }\!\mathrm{d}\omega \frac{{\mathcal{C}}_{\omega }^{(n)}{\mathcal{C}}_{\omega }^{(n)\,\text{T}}}{\omega }\mathrm{sin}\left(\omega \tau \right),\end{equation} \tag{ 13 }$$

where τ = t − t'. Here Θ is the Heaviside function which makes the spin's dynamics at time t, see (11), a function of the spin's state at previous times t₀ ⩽ t' < t [114].

The three-dimensional spin dynamics equation (11) describes the evolution of spin operators with a general tensorial damping kernel and explicitly includes memory of the past dynamics (non-Markovianity). This contrasts with previous derivations of quantum spin dynamics in the form of a master equation [68] which can be solved numerically. But to obtain the master equation a range of simplifying assumptions were made, including weak spin-environment coupling, as well as the Markov and secular approximations. These approximations are quite strong and may not always be justified for a given spin system. We further remark that a different method of including (classical) coloured noise is based on the Miyazaki–Seki approach [70]. Similar to (8), this model assumes that the equation of motion of the spins is coupled to a stochastic equation of motion for the lattice enabling transfer of energy and angular momentum between the lattice and the spin systems. Different coupling potentials, such as harmonic and Morse potentials, have been considered and the spin–lattice coupling impact on the magnetisation has been characterised [71].

The above spin + environment Hamiltonian and its resulting equations of motion share many similarities with the well-known Caldeira–Leggett model for harmonic quantum Brownian motion [26, 35] and the spin-boson model [33, 37, 39]. A key difference is the three-dimensional nature of the reservoir interaction in (5) which leads to the cross product in (11). The spin-boson model is recovered as a special case for spin 1/2 operators and rank-1 coupling tensors, see appendix A1.

The presence of the kernel in (11) gives rise to spin damping that can significantly differ from Gilbert damping, as explored further in section 2. Previous generalisations to Gilbert damping have considered the spins' interactions with the lattice and electron motion [54, 71–73], and included inertial terms [49] and further memory terms within a damping kernel [55–58]. First measurements have recently confirmed the presence of inertial corrections [50, 59]. But these studies have not considered how the noise may have to change from the standard white magnetic noise that is included in the effective magnetic field in the LLG equation (1).

A separate strand of reasoning has argued that to extend the LLG equation to the quantum setting requires that the stochastic noise included in the effective field B _eff should have a quantum distribution rather than the classical Boltzmann one [60, 74–77]. These studies partially reproduce experimentally measured magnetization over temperature curves. But the noise considered vanishes at low temperatures, whereas the full Bose–Einstein distribution famously has non-zero quantum fluctuations even at T = 0 K. Other pioneering investigations have discussed the effect of coloured noise and non-Markovian effects on the ultrafast demagnetization rate [70], while not requiring the fluctuation–dissipation theorem (FDT) to hold and not including quantum effects.

The quantum spin dynamics equation (11) brings these pieces, the damping kernel, quantum statistics and coloured noise, together in a quantum thermodynamically consistent framework. To see this we now prove that the reservoir's two signatures—the stochastic field ${\hat{\boldsymbol{b}}}^{(n)}\!$ and memory kernel ${\mathcal{K}}^{(n)}\!$—always fulfil the (quantum) FDT, for any choice of coupling tensor ${\mathcal{C}}_{\omega }^{(n)}$.

The reservoir power spectrum $\tilde{\mathcal{P}}(\omega )$ is defined as the (quantum symmetrised [115]) expectation value of the autocorrelation function of the magnetic noise ${\hat{\boldsymbol{b}}}^{(n)}\!(t)$ in the thermal reservoir state ${\hat{\rho }}_{\mathrm{R}}={\text{e}}^{-\beta {\hat{H}}_{\mathrm{R}}}/\mathrm{t}\mathrm{r}[{\text{e}}^{-\beta {\hat{H}}_{\mathrm{R}}}]$, and then taking the Fourier transform from the time to the frequency domain [116], i.e.

$$\begin{equation}{\tilde{\mathcal{P}}}_{jk}^{(nm)}(\omega )={\int }_{-\infty }^{\infty }\!\mathrm{d}\tau \,{\mathrm{e}}^{\text{i}\omega \tau }\frac{{\left\langle \left\{{\hat{b}}_{j}^{(n)}\!(t),{\hat{b}}_{k}^{(m)\,{\dagger}}(t-\tau )\right\}\right\rangle }_{\beta }}{2},\quad \end{equation} \tag{ 14 }$$

where {..} is the anti-commutator. Inserting (12), one finds that the only non-trivial contributions to this expression come from the bosonic ladder operator expectation values ${\langle \left\{{\hat{a}}_{\omega ,l}^{(n)},{\hat{a}}_{{\omega }^{\prime },{l}^{\prime }}^{(m)\,{\dagger}}\right\}\rangle }_{\beta }={\delta }_{nm}{\delta }_{l{l}^{\prime }}\delta (\omega -{\omega }^{\prime })\mathrm{coth}(\beta \hslash \omega /2)$. Meanwhile by equation (13) the square of the coupling tensor, ${\mathcal{C}}_{\omega }^{(n)}{\mathcal{C}}_{\omega }^{(n)\,\text{T}}$, is related to the imaginary part of the Fourier transform of the damping kernel ${\mathcal{K}}^{(n)}\!(\tau )$, i.e.

$$\begin{equation}{\mathcal{C}}_{\omega }^{(n)}{\mathcal{C}}_{\omega }^{(n)\,\text{T}}=\frac{2\omega }{\pi }\text{Im}[{\tilde{\mathcal{K}}}^{(n)}(\omega )].\end{equation} \tag{ 15 }$$

Hence one obtains for all choices of the interaction tensor ${\mathcal{C}}_{\omega }^{(n)}$ in equation (5) the quantum FDT

$$\begin{equation}{\tilde{\mathcal{P}}}_{\mathsf{\text{qu}}}^{(nn)}(\omega )=\hslash \,\text{Im}[{\tilde{\mathcal{K}}}^{(n)}(\omega )]\mathrm{coth}\frac{\hslash \omega }{2{k}_{\mathrm{B}}T},\end{equation} \tag{ 16a }$$

$$\begin{equation}{\tilde{\mathcal{P}}}_{\mathsf{\text{cl}}}^{(nn)}(\omega )=\frac{2{k}_{\mathrm{B}}T}{\omega }\text{Im}[{\tilde{\mathcal{K}}}^{(n)}(\omega )]\quad \text{for}\enspace {k}_{\mathrm{B}}T\gg \frac{\hslash \omega }{2},\quad \end{equation} \tag{ 16b }$$

where the last line is the well-known classical high-temperature (or low frequency) approximation, and the first line is the low-temperature limit where the Bose–Einstein distribution of the reservoir oscillator modes becomes important [117].

Finally, the spin dynamics equation (11) leaves the square of the spin operators a constant of motion, see appendix A2,

$$\begin{equation*}\frac{\mathrm{d}{\left\vert {\hat{\boldsymbol{S}}}^{(n)}\!\right\vert }^{2}}{\mathrm{d}t}=\gamma \sum\limits _{jkl}\hspace{2pt}{{\epsilon}}_{jkl}\left({\hat{S}}_{j}^{(n)}{\hat{S}}_{k}^{(n)}+{\hat{S}}_{k}^{(n)}{\hat{S}}_{j}^{(n)}\right){\hat{B}}_{l}^{(n)}=0,\quad \end{equation*}$$

where ${\hat{\boldsymbol{B}}}^{(n)}={\boldsymbol{B}}_{\text{ext}}+{\hat{\boldsymbol{B}}}_{\text{env}}^{(n)}+\frac{1}{\gamma }{\sum }_{m\ne n}{\hspace{6pt}\bar{\hspace{-5pt}\mathcal{J}}}^{(nm)}{\hat{\boldsymbol{S}}}^{(m)}\!$ is an effective magnetic field at site n see equation (8), which commutes with all components of ${\hat{\boldsymbol{S}}}^{(n)}\!$. This confirms that the spin length $\vert \hat{\boldsymbol{S}}\vert$ is a constant in time, e.g. for a spin-1/2 $\hat{\boldsymbol{S}}=\frac{\hslash }{2}\hat{\boldsymbol{\sigma }}$ with ${\hat{\sigma }}_{j}$ the Pauli matrices, one has a constant ${\hat{\boldsymbol{S}}}^{2}=\frac{{\hslash }^{2}}{4}{\hat{\boldsymbol{\sigma }}}^{2}=\frac{3{\hslash }^{2}}{4}\,{1}_{2}$. This is consistent with micromagnetics, where the LLG equation—modified to include the effects of exchange, crystalline anisotropy, magnetostatics, and damping—conserves the total angular momentum, S ² at every point [45].

In the open quantum systems literature, coupling functions that are linear in frequency are referred to as 'Ohmic', while those proportional to higher and lower powers of ω are called super- and sub-Ohmic, respectively. For the magnetic system considered here, Ohmic coupling means

$$\begin{equation}{C}_{\omega }^{\mathsf{\text{Ohm}}}=\sqrt{\frac{2{\eta }_{\mathrm{G}}\,\mathrm{cos}(\omega {{\epsilon}}^{+})}{\pi }}\,\omega ,\end{equation} \tag{ 17 }$$

where η_G is a positive constant with units of kg A⁻² m⁻² s⁻¹, as in equation (1), and ⁺ is an infinitesimal positive constant which we take to zero at the end of the calculation. The corresponding kernel (13) is close to instantaneous,

$$\begin{align}\hfill {K}^{\mathsf{\text{Ohm}}}(\tau )& =\frac{{\eta }_{\mathrm{G}}}{\pi }{\Theta}(\tau ){\int }_{0}^{\infty }\!\mathrm{d}\omega \,\omega [\mathrm{sin}(\omega (\tau +{{\epsilon}}_{+}))+\mathrm{sin}(\omega (\tau -{{\epsilon}}_{+}))]\hfill \\ \hfill & =-{\eta }_{\mathrm{G}}{\Theta}(\tau )\frac{\mathrm{d}}{\mathrm{d}\tau }\,\delta (\tau -{{\epsilon}}_{+}),\hfill \end{align} \tag{ 18 }$$

from which we can see that a positive value of ⁺ ensures the response kernel K^Ohm is causal. For Ohmic coupling (17), the damping term in (11) reduces to

$$\begin{equation}{\gamma }^{2}\hat{\boldsymbol{S}}(t)\times {\int }_{{t}_{0}}^{t}\!\mathrm{d}{t}^{\prime }\,{K}^{\mathsf{\text{Ohm}}}(t-{t}^{\prime })\hat{\boldsymbol{S}}({t}^{\prime })=-{\gamma }^{2}{\eta }_{\mathrm{G}}\hat{\boldsymbol{S}}(t)\times \frac{\mathrm{d}\hat{\boldsymbol{S}}(t-{{\epsilon}}^{+})}{\mathrm{d}t},\end{equation} \tag{ 19 }$$

i.e. Ohmic coupling results in a damping term proportional to a first order time derivative of the system variable [118]. In the magnetic case this is $\hat{\boldsymbol{S}}$, while for quantum Brownian motion it would be the oscillator position [35]. Inserting (19) into the general quantum spin dynamics equation (11), we see that it recovers (the quantum version of) the LLG equation (1) with η_G being identified as the damping parameter. This becomes the well-known classical equation in the limit of large spins. Consequently we will also refer to the Ohmic coupling function (17) as 'LLG coupling'.

Given a coupling function it is straightforward to find the corresponding power spectrum (16a) governing the time correlations of the quantum noise. For ${C}_{\omega }^{\mathsf{\text{Ohm}}}$ the quantum power spectrum is

$$\begin{equation}{\tilde{P}}_{\mathsf{\text{qu}}}^{\mathsf{\text{Ohm}}}(\omega )={\eta }_{\mathrm{G}}\hslash \omega \,\mathrm{coth}\left(\frac{\hslash \omega }{2{k}_{\mathrm{B}}T}\right),\end{equation} \tag{ 20 }$$

where we have taken the limit ⁺ → 0. The Ohmic quantum power spectrum (20) is proportional to the damping parameter η_G, and tends to a linear (diverging) function of frequency in the low temperature limit ℏω ≫ k_B T, see figures 2(g) and (h). However, power spectra diverging with frequency are unphysical. Coupling any system to environmental modes must go to zero at large enough frequencies, such as the interaction of spins with lattice phonons or the effect of conduction electrons scattering off the spins. A common way of integrating this physical information into Ohmic coupling is to introduce a cut-off, i.e. an upper frequency to which the coupling grows linearly, and after which it is set to 0 or decays to 0 in either algebraic or exponential form [38]. A different approach was taken in [60, 74] for the modelling of spin dynamics, where a semi-quantum Ohmic power spectrum similar to (20) was chosen, but with the zero-point noise subtracted ensuring that at T = 0 K it vanishes for all frequencies.

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In the high temperature limit k_B T ≫ ℏω, equation (20) reduces to the classical power spectrum

$$\begin{equation}{\tilde{P}}_{\mathsf{\text{cl}}}^{\mathsf{\text{Ohm}}}(\omega )=2{\eta }_{\mathrm{G}}{k}_{\mathrm{B}}T,\end{equation} \tag{ 21 }$$

which is frequency independent (white) noise. Thus Ohmic coupling plus the high temperature approximation to the power spectrum recovers the LLG equation (1) commonly used to simulate magnetic materials over a wide range of temperatures.

As a first step to unravel how the dynamics predicted by the quantum spin equation (11) can deviate from that predicted by the LLG equation (1), one can expand the spin vector (operator) at time t' in equation (11) around the end point of integration to arbitrary order,

$$\begin{equation}{\gamma }^{2}\hat{\boldsymbol{S}}(t)\times {\int }_{{t}_{0}}^{t}\!\mathrm{d}{t}^{\prime }\,K(t-{t}^{\prime })\hat{\boldsymbol{S}}({t}^{\prime })={\gamma }^{2}\sum\limits _{m=1}^{\infty }{\kappa }_{m}\hat{\boldsymbol{S}}(t)\times {\partial }_{t}^{m}\hat{\boldsymbol{S}}(t).\end{equation} \tag{ 22 }$$

The 0th order coefficient, κ₀, is dropped here as it multiplies $\hat{\boldsymbol{S}}(t)\times \hat{\boldsymbol{S}}(t)$ which is immediately 0 for a classical spin vector S (t). For quantum spin operators, this cross-product is an anti-Hermitian operator, ${(\hat{\boldsymbol{S}}\times \hat{\boldsymbol{S}})}^{{\dagger}}=-\hat{\boldsymbol{S}}\times \hat{\boldsymbol{S}}$, which then cancels in the Hermitian equation of motion (11). The higher expansion coefficients κ_m>0 are proportional to the mth one-sided moment of the memory kernel

$$\begin{equation}{\kappa }_{m}=\frac{{(-1)}^{m}}{m!}{\int }_{0}^{\infty }\!\mathrm{d}\tau \,{\tau }^{m}K(\tau ),\end{equation} \tag{ 23 }$$

where we have assumed that the dynamics has been running for some time longer than the kernel decay time, for which one can replace the initial time t₀ by −∞.

For the Ohmic kernel only the first moment is non-zero and corresponds to the (negative) damping parameter,

$$\begin{equation}{\kappa }_{1}^{\mathsf{\text{Ohm}}}={\eta }_{\mathrm{G}}{\int }_{0}^{\infty }\!\mathrm{d}\tau \,\tau \frac{\mathrm{d}}{\mathrm{d}\tau }\,\delta (\tau -{{\epsilon}}_{+})=-{\eta }_{\mathrm{G}},\end{equation} \tag{ 24 }$$

while ${\kappa }_{m > 1}^{\mathsf{\text{Ohm}}}=0$. This complete lack of higher moments, which would maintain a certain degree of memory in the dynamics, shows that Ohmic coupling dynamics can only be an approximation to any real dynamics. For example within magnetism, the memory-free form of damping (19) is known as Gilbert damping [43], and is almost universally used to describe magnetization dynamics through the LLG equation (1). However, 'inertial' corrections to such dynamics have been proposed [49] and their presence was recently confirmed experimentally [50].

Here we provide a tool to systematically study dynamics beyond the Ohmic case, allowing one to include memory and coloured noise effects in a manner consistent with the quantum fluctuation dissipation theorem (16a). We consider the class of Lorentzian coupling functions (similar to [78])

$$\begin{equation}{C}_{\omega }^{\mathsf{\text{Lor}}}=\sqrt{\frac{2A{\Gamma}}{\pi }\frac{{\omega }^{2}}{{({\omega }_{0}^{2}-{\omega }^{2})}^{2}+{\omega }^{2}{{\Gamma}}^{2}}},\end{equation} \tag{ 25 }$$

where A is a coupling amplitude, with the following properties: (i) for small ω, ${C}_{\omega }^{\mathsf{\text{Lor}}}$ grows linearly with ω and can be approximated by an Ohmic coupling function, (ii) at large ω, ${C}_{\omega }^{\mathsf{\text{Lor}}}$ smoothly decays to zero, and (iii) at some intermediate frequency ω₀, ${C}_{\omega }^{\mathsf{\text{Lor}}}$ has a resonant peak with some width Γ. This peak characterises the confined range and relative strength of system–environment interaction with two parameters. Alternative 'peaks' such as Gaussians or top hat functions could be considered, but here we chose the Lorentzian shape due to the fact that many expressions can be solved analytically and, as we will demonstrate in section 3, Lorentzian couplings allow us to efficiently simulate non-Markovian dynamics.

We call the above functions 'Lorentzian coupling' since the corresponding damping kernel in the frequency domain is the widely studied Lorentzian response

$$\begin{equation}{\tilde{K}}^{\text{Lor}}(\omega )=\frac{A}{{\omega }_{0}^{2}-{\omega }^{2}-\mathrm{i}\omega {\Gamma}},\end{equation} \tag{ 26 }$$

where the imaginary part is obtained from (15) and the real part is determined using the usual Kramers–Kronig relations. In the time-domain the Lorentzian memory kernel is

$$\begin{equation}{K}^{\mathsf{\text{Lor}}}(\tau )={\Theta}(\tau )A\,{\mathrm{e}}^{-\frac{{\Gamma}\tau }{2}}\frac{\mathrm{sin}({\omega }_{1}\tau )}{{\omega }_{1}},\end{equation} \tag{ 27 }$$

where ${\omega }_{1}=\sqrt{{\omega }_{0}^{2}-\frac{{{\Gamma}}^{2}}{4}}$ and Γ/2 can now be interpreted as the kernel decay rate. For the coupling function (25), the collective response of the environment is thus equivalent to a single harmonic oscillator of resonant frequency ω₀ and damping rate Γ/2 [79]. From the quantum FDT (16a) it follows that the corresponding power spectrum is

$$\begin{equation}{\tilde{P}}_{\mathsf{\text{qu}}}^{\mathsf{\text{Lor}}}(\omega )=\frac{A{\Gamma}\hslash \omega }{{({\omega }_{0}^{2}-{\omega }^{2})}^{2}+{\omega }^{2}{{\Gamma}}^{2}}\,\mathrm{coth}\left(\frac{\hslash \omega }{2{k}_{\mathrm{B}}T}\right),\end{equation} \tag{ 28 }$$

which takes its largest values at frequencies close to ω₀ and tends to zero as ω⁻³ at large ω. This power spectrum differs from classical Ohmic noise (21) in two important respects. Firstly, the quantum mechanical treatment means that the low temperature noise is not proportional to temperature, and does not vanish at zero temperature. Secondly, even in the high temperature limit, the noise spectrum is frequency dependent (coloured), unlike the white noise of equation (21). The presented theory thus captures both of these aspects in a consistent quantum thermodynamic framework.

Unlike Ohmic coupling (17) which depends on a single parameter η_G, the Lorentzian coupling function (25), and hence its kernel and spectrum, depends on three parameters. These allow a systematic study of different regimes of the environment response. Specifically, the memory time of the environment can be continuously varied by changing ω₀ and Γ, which can lead to very different magnetic behaviour. Beyond the spin dynamics explored here, the proposed Lorentzian coupling may also be a useful tool for the characterisation of quantum Brownian motion of a variety of systems, including oscillators and free particles.

To better understand the relation between the Ohmic and Lorentzian coupling functions, one may consider their kernel moments κ_m in expansion (22). In contrast to the Ohmic case, for the Lorentzian kernel (27) all κ_m are non-zero and given by

$$\begin{equation}{\kappa }_{m}^{\mathsf{\text{Lor}}}=\frac{{(-1)}^{m}A}{{\omega }_{1}{\omega }_{0}^{2(m+1)}}\mathrm{I}\mathrm{m}\left[{\left(\frac{{\Gamma}}{2}+\mathrm{i}{\omega }_{1}\right)}^{m+1}\right].\end{equation} \tag{ 29 }$$

The first relevant two moments are

$$\begin{equation}{\kappa }_{1}^{\mathsf{\text{Lor}}}=-\frac{A{\Gamma}}{{\omega }_{0}^{4}}\quad \text{and}\quad {\kappa }_{2}^{\mathsf{\text{Lor}}}=\frac{A}{{\omega }_{0}^{6}}({{\Gamma}}^{2}-{\omega }_{0}^{2}),\end{equation} \tag{ 30 }$$

and when comparing to the Ohmic case, one finds that the first Lorentzian moment can be identified with minus the damping parameter η_G, i.e. ${\kappa }_{1}^{\mathsf{\text{Lor}}}=-\frac{A{\Gamma}}{{\omega }_{0}^{4}}=-{\eta }_{\mathrm{G}}$. For a material with a given damping parameter η_G this fixes one of the Lorentzian parameters, i.e.

$$\begin{equation}{C}_{\omega }^{\mathsf{\text{Lor}}}=\sqrt{\frac{2{\eta }_{\mathrm{G}}{\omega }^{2}}{\pi }\frac{{\omega }_{0}^{4}}{{({\omega }_{0}^{2}-{\omega }^{2})}^{2}+{\omega }^{2}{{\Gamma}}^{2}}},\end{equation} \tag{ 31 }$$

which now only depends on the two parameters ω₀ and Γ. For a specific material these may be approximately determined through information contained in the density of states of the environment to which the spins couple [80].

Inserting expansion (22) with Lorentzian moments (29) in the quantum spin dynamics equation (11) one can distinguish two different dynamical situations.

Ohmic regime. When the resonant frequency ω₀ and the damping rate Γ of the reservoir coupling is much larger than the spin operators' typical frequency of motion, ω₀ ≫ ω, each successive term in expansion (22) is smaller by an extra factor of (ω/ω₀). In the limit of infinite ω₀ but finite damping parameter η_G, the Lorentzian damping term in (11) thus tends to the Ohmic one (19), $-{\gamma }^{2}{\eta }_{\mathrm{G}}\hat{\boldsymbol{S}}(t)\times {\partial }_{t}\hat{\boldsymbol{S}}(t)$.

Non-Ohmic regime. When the environmental frequency ω₀ is comparable to typical spin motion frequencies, ω₀ ≈ ω the Ohmic approximation to the Lorentzian kernel begins to fail. The first deviation in (11) is a new term proportional to ${\kappa }_{2}^{\mathsf{\text{Lor}}}$. I.e. in addition to the Gilbert damping term, one adds the term ${\gamma }^{2}{\kappa }_{2}^{\mathsf{\text{Lor}}}\hat{\boldsymbol{S}}(t)\times {\partial }_{t}^{2}\hat{\boldsymbol{S}}(t)$ containing a second time derivative of the spin operator $\hat{\boldsymbol{S}}$. By analogy with the classical equation of motion for a massive body, this is known as an 'inertial' modification to the spin dynamics [49]. As the ratio κ₂/κ₁ has the dimensions of time, one may introduce an 'inertial timescale' τ_in [49], which for the Lorentzian is

$$\begin{equation}{\tau }_{\text{in}}=\frac{{\kappa }_{2}^{\mathsf{\text{Lor}}}}{{\kappa }_{1}^{\mathsf{\text{Lor}}}}=\frac{{\omega }_{0}^{2}-{{\Gamma}}^{2}}{{\omega }_{0}^{2}{\Gamma}}.\end{equation} \tag{ 32 }$$

A large inertial time τ_in indicates the presence of non-Markovian dynamics, i.e. dynamics that has a certain degree of memory. For a high quality factor resonance ω₀ ≫ Γ the inertial time is (half) the kernel's decay time, τ_d = 2/Γ. For increasing resonance width Γ, the inertial time τ_in decreases and memory effects become less important. In magnetic systems this timescale determines the time over which nutation oscillations are observed in the precession of the spin. Such inertial corrections to standard magnetism have very recently been observed for the first time in ultrafast experiments on thin films [50]. Curiously, the inertial timescale, see equation (32), becomes negative when Γ > ω₀, a fact that may be the subject of future investigation.

Similarly to the kernel expansion (22), one can also expand the Lorentzian power spectrum in frequency, see appendix A3. Generally, only the odd moments κ_2m+1 appear in the power spectrum. This implies that if a kernel only has non-trivial first (κ₁) and second (κ₂) moments, while higher moments vanish (κ_m>2 = 0) then the power spectrum will still be given by the (quantum) Ohmic one (20), with η_G = −κ₁. When third or higher moments are non-zero, then the power spectrum of the noise will deviate from the Ohmic case at all temperatures.

To summarise, here we have demonstrated that Lorentzian coupling functions, kernels and power spectra provide a systematic framework to explore system dynamics that arises from inertial terms and other memory effects, while recovering the standard Ohmic limit whenever the Lorentzian resonance frequency ω₀ is much larger than the typical system frequencies.

In section 3 we perform semi-classical simulations of the dynamics of equation (11), for the Lorentzian kernel of section 2.2. For this purpose we re-write expressions in the operator equation (11) in terms of a unit-free set of quantities. The time coordinate t is replaced with the unit-free coordinate ω_L t, where ω_L = |γ|| B _ext| is the Larmor frequency. In addition the spin operator $\hat{\boldsymbol{S}}$ with largest eigenvalue S₀ is re-written in terms of a unit-free operator $\hat{\boldsymbol{s}}$ with largest eigenvalue 1, $\hat{\boldsymbol{S}}=\mathrm{sign}(\gamma ){S}_{0}\hat{\boldsymbol{s}}$. The sign of the gyromagnetic ratio is included in the definition of $\hat{\boldsymbol{s}}$ so that $\hat{\boldsymbol{s}}$ aligns with the magnetic field, whatever the sign of γ.

From equation (11) we can see that the damping kernel has dimensions of magnetic field squared divided by angular momentum, which leads us to identify a unit-free damping kernel k(t − t') through $K(t-{t}^{\prime })=\vert {\boldsymbol{B}}_{\text{ext}}{\vert }^{2}{S}_{0}^{-1}k(t-{t}^{\prime })$. Similarly, the unit-free coupling function c_ω is defined through ${C}_{\omega }=\vert {\boldsymbol{B}}_{\text{ext}}\vert {S}_{0}^{-1/2}{c}_{\omega }$ and the unit-free spectral functions $\tilde{p}$ through $\tilde{P}=\hslash \vert {\boldsymbol{B}}_{\text{ext}}{\vert }^{2}{S}_{0}^{-1}{\omega }_{\mathrm{L}}^{-1}\tilde{p}$.

Looking at the Lorentzian kernel K(t − t') in (26), the pulling of dimensions can be achieved by setting the kernel amplitude to $A=\vert {\boldsymbol{B}}_{\text{ext}}{\vert }^{2}{S}_{0}^{-1}\alpha$ where α now is a frequency. For the simulations we choose the Lorentzian parameters ω₀, Γ and α to be independent of spin length S₀. Through (5) this implies an interaction energy scaling of ${\hat{V}}_{\text{int}}\propto {S}_{0}\sqrt{A}\propto \sqrt{{S}_{0}}$ which sets the scaling of the interaction versus self-energy to ${\hat{V}}_{\text{int}}/{\hat{H}}_{\mathrm{S}}\propto 1/\sqrt{{S}_{0}}$. Apart from it being implied by dimensional analysis, such scaling is heuristically plausible in many physical contexts. E.g. it is similar to the increasing ratio of the surface (where reservoir interaction occurs) to volume (self-energy) for decreasing system sizes. For microscopic systems for which ${\hat{V}}_{\text{int}}$ is no longer small in comparison to ${\hat{H}}_{\mathrm{S}}$ a thermodynamic treatment beyond the weak coupling limit [26, 27, 81], a limit tacitly assumed in standard thermodynamics, may be required.

Similarly for Ohmic coupling leading to the LLG equation (1), the above scaling choice amounts to choosing η_G ∝ A ∝ 1/S₀ implying that $\eta ={\eta }_{\mathrm{G}}{\gamma }^{2}{S}_{0}=\alpha {\Gamma}{\omega }_{\mathrm{L}}^{2}/{\omega }_{0}^{4}$ is assumed to be independent of S₀. In physical situations where this assumption is not justified, one may instead choose η and α to depend on the spin length S₀.

For a spin in an external field B _ext the typical frequency of the dynamics is set by the Larmor frequency, ω_L. In the simulations discussed in section 3 we will use the following two sets of Lorentzian parameters, all expressed in terms of ω_L,

$$\begin{equation}\text{Set}1:\begin{array}{ccccccccc}{\omega }_{0}=7.0{\omega }_{\mathrm{L}}& \qquad \alpha =10.0{\omega }_{\mathrm{L}}& \qquad {\Gamma}=5.0{\omega }_{\mathrm{L}}\\ \eta =0.02& \qquad {\tau }_{\text{in}}=0.1{\omega }_{\mathrm{L}}^{-1}& \qquad {\tau }_{\mathrm{d}}=0.4{\omega }_{\mathrm{L}}^{-1}\end{array}\quad \,\,\end{equation} \tag{ 33a }$$

$$\begin{equation}\text{Set}2:\begin{array}{ccccccccc}{\omega }_{0}=1.4{\omega }_{\mathrm{L}}& \qquad \alpha =0.16{\omega }_{\mathrm{L}}& \qquad {\Gamma}=0.5{\omega }_{\mathrm{L}}\\ \eta =0.02& \qquad {\tau }_{\text{in}}=1.7{\omega }_{\mathrm{L}}^{-1}& \qquad {\tau }_{d}=4{\omega }_{\mathrm{L}}^{-1}\end{array}.\end{equation} \tag{ 33b }$$

In the second rows we have also listed the equivalent unit-free Gilbert damping η, the inertial timescale τ_in, and the memory kernel decay time τ_d for the Lorentzian kernel. Figures 2–5, show plots obtained with Lorentzian coupling functions (25) with parameter Sets 1 and 2, which are shown in blue and red, respectively. The Ohmic LLG approximation is shown in magenta when the classical reservoir (21) is considered, and in cyan when the quantum reservoir (20) is considered. The external field is set to B _ext = 10 T e _z with e _z the unit-vector in z-direction throughout.

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Parameter Set 1 has been chosen to have a resonant frequency ω₀ much larger than the characteristic spin precession frequency ω_L (Ohmic regime). Consequently we can truncate the series given in equation (22) to leading order and recover the Ohmic form (19) typically considered in magnetism theory. The validity of this approximation is demonstrated in the top row of figure 2. Figure 2(a) shows that the Lorentzian coupling function ${c}_{\omega }^{\mathsf{\text{Lor}}}$ is well approximated by LLG (Ohmic) coupling for the relevant frequency range, while figure 2(c) shows that the kernel is approximately instantaneous on the timescale ${\omega }_{\mathrm{L}}^{-1}$, in line with Ohmic damping for which ${\kappa }_{m > 1}^{\mathsf{\text{Ohm}}}=0$. Figure 2(e) shows that at high temperature (T = 200 K) and for relevant frequencies ω ≈ ω_L, the power spectrum ${\tilde{p}}^{\mathsf{\text{Lor}}}$ is well approximated by the quantum Ohmic (LLG) power spectrum (20), and its classical limit (21). Figure 2(g) shows that at lower temperatures (T = 1 K) quantum noise becomes important. Here the Ohmic approximation (20) remains valid, while its classical limit (21) is invalid.

Parameter Set 2 is chosen such that it has a resonant frequency ω₀ comparable to the precession frequency ω_L (non-Ohmic regime). Here it is inaccurate to truncate the series (22) and the damping will be fundamentally non-Ohmic. To directly compare with Ohmic dynamics generated by Lorentzian coupling with Set 1, both parameter sets have been chosen to correspond to the same unit-free Gilbert damping parameter, η = 0.02. The failure of the Ohmic approximation is demonstrated in the bottom row of figure 2. Figure 2(b) shows that the linear approximation to the coupling function fails in the relevant frequency range. For this set of parameters the damping kernel now exhibits significant memory and figure 2(d) shows that the response persists over a timescale of several ${\omega }_{\mathrm{L}}^{-1}$. This memory kernel implies, through the FDT (16a), a coloured quantum noise power spectrum ${\tilde{p}}_{\mathsf{\text{qu}}}^{\mathsf{\text{Lor}}}$. As shown in figure 2(h), this coloured Lorentzian ${\tilde{p}}_{\mathsf{\text{qu}}}^{\mathsf{\text{Lor}}}$ (red) differs from the LLG counterpart, ${\tilde{p}}_{\mathsf{\text{qu}}}^{\mathsf{\text{Ohm}}}$ (cyan). Furthermore, figure 2(f) shows that in the high temperatures ${\tilde{p}}_{\mathsf{\text{qu}}}^{\mathsf{\text{Lor}}}$ (red) also differs very significantly from the LLG power spectrum ${\tilde{p}}_{\mathsf{\text{cl}}}^{\mathsf{\text{Ohm}}}$ (magenta). The presence of both memory and coloured quantum noise for the Lorentzian with parameter Set 2 are both signatures of a thermostat that substantially deviates from the classical Ohmic assumptions and, as we will see in the next section, leads to markedly different short time dynamics and steady state of s_z .

Here we detail how to efficiently simulate non-Markovian dynamics that arises as a result of Lorentzian coupling (25) for the example of spins vectors. Numerical implementation of (11) requires both—the integration of the kernel with the spin state of previous time steps and the inclusion of coloured noise as follows.

Dropping the spin index and for simplicity assuming any isotropic kernel $\mathcal{K}(\tau )={1}_{3}\,K(\tau )$, the three vector components b_j (t) for j = 1, 2, 3 of the magnetic noise (12) are implemented as [88]

$$\begin{equation}{b}_{j}(t)={\int }_{-\infty }^{\infty }\!\mathrm{d}{t}^{\prime }\,F(t-{t}^{\prime }){\xi }_{j}({t}^{\prime }),\end{equation} \tag{ 34 }$$

where ξ_j (t') is standard white Gaussian noise for the jth component, which is delta correlated ⟨ξ_j (t) ξ_k (t')⟩ = δ_jk δ(t − t'). The 'coloured noise' comes from choosing F(t − t') as the Fourier transform of the square root of the power spectrum associated with the kernel through (16a), i.e.

$$\begin{equation}F(t-{t}^{\prime })={\int }_{-\infty }^{\infty }\frac{\mathrm{d}\omega }{2\pi }{\mathrm{e}}^{-\mathrm{i}\omega (t-{t}^{\prime })}\sqrt{\tilde{P}(\omega )},\end{equation} \tag{ 35 }$$

which can be implemented using a fast Fourier transform. To simulate the effect of a Lorentzian damping kernel (26) we numerically integrate [89] the following set of first order coupled differential equations for the spin vector S and two dummy vectors V and W :

$$\begin{equation}\begin{aligned}\hfill \frac{\mathrm{d}\boldsymbol{S}(t)}{\mathrm{d}t}& =\gamma \boldsymbol{S}(t)\times ({\boldsymbol{B}}_{\text{ext}}+\boldsymbol{b}(t)+\boldsymbol{V}(t)),\hfill \\ \hfill \frac{\mathrm{d}\boldsymbol{V}(t)}{\mathrm{d}t}& =\boldsymbol{W}(t),\hfill \\ \hfill \frac{\mathrm{d}\boldsymbol{W}(t)}{\mathrm{d}t}& =-{\omega }_{0}^{2}\boldsymbol{V}(t)-{\Gamma}\boldsymbol{W}(t)+A\,\gamma \boldsymbol{S}(t).\hfill \end{aligned}\end{equation} \tag{ 36 }$$

The integrated values of the dummy vectors and the spin are separated by the time step dt. Solving these equations is equivalent to solving the integro-differential equation (11) for a Lorentzian kernel, see appendix A4, but is numerically more straightforward to implement.

We wish to illustrate on a single trajectory level, the differences between the dynamics predicted by (11) with either an approximately Ohmic (Set 1) or non-Ohmic (Set 2) Lorentzian coupling function, as well as the dynamics predicted by the standard LLG equation. At first, because the dynamics is intrinsically stochastic, trajectories will naturally differ and cannot readily be compared. However, looking at the noise generation in equations (34) and (35), one can see that the same white noise ξ_j (t) for j = 1, 2, 3 may be used as a seed to create comparable 'stochastic' noise for different power spectra $\tilde{P}(\omega )$.

Figure 3 shows the stochastic short time dynamics of a single classical spin for two pairs of spin length and temperature, S₀ = 1ℏ/2 at T = 1 K (left panel) for a single electron, and S₀ = 200ℏ/2 at T = 200 K (right panel) for a mesoscopic cluster of spins with a combined larger effective spin.

The dynamics is obtained according to equation (11) for Lorentzian coupling (25) with S₀-scaling $A=\vert {\boldsymbol{B}}_{\text{ext}}{\vert }^{2}{S}_{0}^{-1}\alpha$, for parameter sets Set 1 (top panel, blue) and Set 2 (bottom panel, red), and with the quantum coloured noise given by (28). For comparison we also show the short time dynamics according to the LLG equation (1) with S₀-scaling ${\eta }_{\mathrm{G}}=A{\Gamma}/{\omega }_{0}^{4}=\vert {\boldsymbol{B}}_{\text{ext}}{\vert }^{2}{S}_{0}^{-1}{\omega }_{\mathrm{L}}^{-2}\eta$ with the Gilbert damping parameter η = 0.02 common to both Lorentzian parameter sets, see (33a). That implies that the top and bottom LLG plots are identical. For the standard LLG equation two types of noise are considered—high-temperature classical noise (magenta) see equation (21), and quantum noise (cyan) see equation (20). Since the same white noise time series is used as a seed for producing the stochastic noise for all traces, we can compare them directly. We will here focus on s_z , the component of s = sign(γ) S /S₀ aligned with the external field B _ext.

Three features stand out in figure 3: (i) as expected from section 2, the dynamics generated with equation (11) with Lorentzian Set 1 (top, blue) closely matches the dynamics obtained with the LLG equation (1) with quantum noise (cyan) for both spin–temperature pairs, (ii) the quantum statistics of the reservoir (cyan) at low temperatures (left) introduces differences to the LLG dynamics compared to the LLG dynamics obtained with classical noise (magenta), and (iii) memory effects that are present for Lorentzian Set 2 (bottom, red) result in significantly different dynamics from that arising with the memory-free Lorentzian Set 1 (top, blue).

We remark that due to the spin/temperature ratio being the same for the two spin–temperature pairs, the LLG equation with classical noise (magenta) integrates to exactly the same dynamics in left and right panel, see appendix A7. This scaling relation ceases to be true for the LLG equation with quantum noise (cyan). Another difference to note is that in figures 3(a)–(d) the dynamics for Lorentzian parameter Set 1 (blue) varies more rapidly in time than for Lorentzian parameter Set 2 (red). This is due to the high frequency content of Set 1's power spectrum, see figures 2(e) and (g).

Finally, the spin component s_x (green) and the spin-vector length | s | (black) are shown for the Lorentzian coupling with Set 1 and Set 2 in the top and bottom panels of figure 3, respectively. The plots of | s | show that the numerical integration of equation (11) indeed leads to a constant spin-vector length | s | = 1, i.e. no renormalisation is required.

Figure 4 shows the ensemble averaged ⟨s_z ⟩ over time, averaged over 500 stochastic trajectories. We now highlight two important features in figure 4. Firstly, at low temperatures (left) the quantum statistics of the reservoir (blue, red, cyan) results in a much depleted value of ⟨s_z ⟩, roughly at around 0.28, in comparison to that obtained with the LLG equation with classical noise (magenta), ca 0.85. This indicates that for this particular choice of spin length and temperature the quantum character of the reservoir strongly affects the value of ⟨s_z ⟩, as further discussed below, and the classical high-temperature limit taken in (16b) would not be appropriate. For the high temperature T = 200 K + larger spin pair S₀ = 200ℏ/2 (right), the difference between classical and quantum statistics of the reservoir can be neglected and ⟨s_z ⟩ settles at 0.85 independent of whether the spin dynamics integration was done for equation (11) with either Set 1 (blue) or Set 2 (red), or for equation (1) with either classical (magenta) or quantum noise (cyan).

Secondly, for the large spin–temperature pair (right), there is clear evidence of a much quicker relaxation to steady state (by a factor of a third) for Lorentzian Set 2 (red) compared to the other plots (blue, cyan, magenta). This is a non-Markovian effect that arises because the memory kernel for Set 2 has an appreciable memory over time, see figure 2(d), while the other memory kernels are (close to) instantaneous. This quicker equilibration occurs because the non-Markovian kernel leads to a smoother dynamics which in turn is more quickly sampled by the dynamical system.

Figure 5 shows the average steady state spin value ⟨s_z ⟩ as a function of temperature T, found by time-averaging a single trajectory s_z over late times, from 0.75 t_max to ${t}_{\mathrm{max}}=2\pi \times 7200{\omega }_{\mathrm{L}}^{-1}$. There are two key observations to make in figure 5. Firstly, for both the small spin (left) and the large spin (right) the steady state ⟨s_z ⟩ obtained with the LLG equation and classical noise (21) matches the standard statistical physics prediction ${\langle {s}_{z}\rangle }_{\mathsf{\text{stat phys}}}=\mathrm{coth}\left(\frac{{S}_{0}{\omega }_{\mathrm{L}}}{{k}_{\mathrm{B}}T}\right)-\frac{{k}_{\mathrm{B}}T}{{S}_{0}{\omega }_{\mathrm{L}}}$, see appendix A5.

Secondly, for simulations that include the full quantum noise (cyan, blue, red) in the dynamics of the small spin at low temperatures (left), we observe reduced ⟨s_z ⟩ values in the range 0.2–0.4 at T = 0 K, i.e. well below the classical value of 1. This arises because the power spectrum ${\tilde{P}}_{\mathsf{\text{qu}}}^{\mathsf{\text{Lor}}}$, given through the quantum FDT (16a), includes quantum fluctuations which remain even for T → 0 K. The steady state curves with quantum noise also show a characteristic 'flattening' compared to the steep decay of the steady state with temperature for classical noise (magenta). Qualitatively speaking, this quantum zero point noise, when compared to classical noise, is as if thermal noise is 'on' even at 0 K. I.e. taking the Larmor frequency as the relevant frequency, and setting ${\tilde{P}}_{\mathsf{\text{qu}}}^{\mathsf{\text{Ohm}}}(0\enspace \text{K})={\tilde{P}}_{\mathsf{\text{cl}}}^{\mathsf{\text{Ohm}}}({T}_{\mathsf{\text{cl}}})$ for the Ohmic coupling, for example, defines a classical temperature of ${T}_{\text{cl}}=\frac{\hslash {\omega }_{\mathrm{L}}}{2{k}_{\mathrm{B}}}$ = 6.7 K 'equivalent' to the quantum zero-temperature case. For S₀ = 1ℏ/2 the classical statistical physics steady state value at T_cl is ${\langle {s}_{z}\rangle }_{\mathsf{\text{stat phys}}}\approx 0.3$. This indeed is of comparable size to the ⟨s_z ⟩ values obtained with quantum noise at T = 0 K. The corresponding steady state value for the large spin (S₀ = 200ℏ/2) is ${\langle {s}_{z}\rangle }_{\mathsf{\text{stat phys}}}\approx 0.995\approx 1$, see figure 5(c). Generally, for the classical temperature ${T}_{\text{cl}}$ one obtains ${\langle {s}_{z}\rangle }_{\mathsf{\text{stat phys}}}=\mathrm{coth}\left(\frac{2{S}_{0}}{\hslash }\right)-\frac{\hslash }{2{S}_{0}}$, which only depends on the spin length S₀ while being independent of the field strength | B _ext|. With increasing S₀ this function rises very sharply from $\approx 0.3$ to 1. For example for spin length S₀ = 5ℏ/2, the quantum zero temperature value is ${\langle {s}_{z}\rangle }_{\mathsf{\text{stat phys}}}\approx 0.8$ and its decay with increasing temperate is shown in figure 6(b) in appendix A6.

The middle panel, figure 5(b), gives an alternative illustration of the steady state value for the small spin as a function of environment temperature T. It shows the same plot as panel (a), but with the y-axis rescaled as m(T) = ⟨s_z (T)⟩/⟨s_z (0)⟩. All plots now start at 1 at T = 0 K, independent of whether the dynamics was integrated with quantum or classical noise. Interestingly, the overarching behaviour of the resulting curves bears some resemblance with heuristically rescaled magnetization curves that match experimental data [67, 90]. Running high-end atomistic simulations of equation (11), instead of (1), for multiple interacting classical spins would answer if the quantum power spectrum's impact on their low temperature magnetisation behaviour as well as their Curie temperature is the reason for the apparent rescaling.

We note that larger numerical uncertainties arise for the quantum noise because an additional scale is present in comparison to classical noise, see appendix A7. Error bars obtained from an ensemble of simulations are indicated at one temperature value in (a) and (c) for all four ⟨s_z ⟩ curves in figure 5.

The spin-boson model is recovered as a special case of the three-dimensional Hamiltonian (2), and hence its dynamics is also given by equation (11). To see this one may drop the site index n, and choose the external field as

$$\begin{equation}{\boldsymbol{B}}_{\text{ext}}={B}_{0}\,(\mathrm{cos}(\theta )\,{\boldsymbol{e}}_{z}-\mathrm{sin}(\theta )\,{\boldsymbol{e}}_{x}),\end{equation} \tag{ 37 }$$

for some angle θ. Taking spin 1/2 operators $\hat{\boldsymbol{S}}=(\hslash /2)\boldsymbol{\sigma }$ and the coupling tensor as ${({\mathcal{C}}_{\omega })}_{jk}={C}_{\omega }\,{\delta }_{jk}{\delta }_{j1}$ with a scalar coupling function C_ω , one recovers from (2) the one-dimensional spin-boson Hamiltonian

$$\begin{align}\hfill \hat{H}& =-\gamma {B}_{0}\,(\mathrm{cos}(\theta )\,{\hat{S}}_{z}-\mathrm{sin}(\theta )\,{\hat{S}}_{x})-\gamma {\hat{S}}_{x}{\int }_{0}^{\infty }\!\mathrm{d}\omega {C}_{\omega }{\hat{x}}_{\omega }\hfill \\ \hfill & \quad +\frac{1}{2}{\int }_{0}^{\infty }\!\mathrm{d}\omega \left[{\left({\hat{p}}_{x,\omega }\right)}^{2}+{\omega }^{2}{\left({\hat{x}}_{\omega }\right)}^{2}\right]+{\hat{H}}_{\mathrm{R}}^{2\mathrm{D}},\hfill \end{align} \tag{ 38 }$$

where ${\hat{H}}_{\mathrm{R}}^{2\mathrm{D}}$ is a decoupled two-dimensional reservoir that can be dropped from the dynamics.

To evaluate the derivative of ${({\hat{\boldsymbol{S}}}^{(n)}\!(t))}^{2}$ we abbreviate the effective magnetic field operator at site n in equation (8), as

$$\begin{equation}{\hat{\boldsymbol{B}}}^{(n)}={\boldsymbol{B}}_{\text{ext}}+{\hat{\boldsymbol{B}}}_{\text{env}}^{(n)}+\frac{1}{\gamma }\sum\limits _{m\ne n}{\hspace{6pt}\bar{\hspace{-5pt}\mathcal{J}}}^{(nm)}{\hat{\boldsymbol{S}}}^{(m)}\!.\end{equation} \tag{ 39 }$$

Using the definition of ${\hat{\boldsymbol{B}}}_{\text{env}}^{(n)}$ given below equation (8), it is clear that ${\hat{\boldsymbol{B}}}^{(n)}$ is a Hermitian operator that commutes with the spin operators ${\hat{\boldsymbol{S}}}^{(n)}\!$. Dropping the site index and time argument to avoid index overload, we find from (11)

$$\begin{align}\hfill \frac{\mathrm{d}\vert \hat{\boldsymbol{S}}(t){\vert }^{2}}{\mathrm{d}t}& =\sum\limits _{j}\left({\hat{S}}_{j}\frac{\mathrm{d}{\hat{S}}_{j}}{\mathrm{d}t}+\frac{\mathrm{d}{\hat{S}}_{j}}{\mathrm{d}t}{\hat{S}}_{j}\right)\hfill \\ \hfill & =\gamma \sum\limits _{jkl}\hspace{2pt}{{\epsilon}}_{jkl}\left({\hat{S}}_{j}{\hat{S}}_{k}+{\hat{S}}_{k}{\hat{S}}_{j}\right){\hat{B}}_{l}\hfill \\ \hfill & =0,\hfill \end{align} \tag{ 40 }$$

where the final equality follows because of the symmetry of the operator ${\hat{S}}_{j}{\hat{S}}_{k}+{\hat{S}}_{k}{\hat{S}}_{j}$ under exchange of the indices j ↔ k.

Similar to the damping kernel term expansion (22), in moments (29) and time-derivatives, the Lorentzian power spectrum (28) can be expanded in powers of frequency ω, as

$$\begin{equation}{\tilde{P}}_{\mathsf{\text{qu}}}^{\mathsf{\text{Lor}}}(\omega )=\sum\limits _{m=0}^{\infty }{(-1)}^{m+1}{\omega }^{2m+1}{\kappa }_{2m+1}^{\mathsf{\text{Lor}}}\,\mathrm{coth}\left(\frac{\hslash \omega }{2{k}_{\mathrm{B}}T}\right),\end{equation} \tag{ 41 }$$

where we have kept the quantum coth unexpanded. The ${\kappa }_{m}^{\mathsf{\text{Lor}}}$ are the same coefficients as those given in (29). For small frequencies ω the first term in the series (41) dominates and the power spectrum takes the (quantum) Ohmic form

$$\begin{equation}{\tilde{P}}_{\mathsf{\text{qu}}}^{\text{Lor}}(\omega )\approx -\omega \,{\kappa }_{1}^{\text{Lor}}\,\mathrm{coth}\left(\frac{\hslash \omega }{2{k}_{\mathrm{B}}T}\right),\end{equation} \tag{ 42 }$$

where comparison with (20) again shows that $-{\kappa }_{1}^{\text{Lor}}$ is the effective Gilbert damping constant.

Beyond the Ohmic regime, one can see in (41) that only the odd moments ${\kappa }_{2m+1}^{\mathsf{\text{Lor}}}$ contribute. Therefore the inertial term ${\kappa }_{2}^{\mathsf{\text{Lor}}}$, which is the first deviation of the damping kernel from Ohmic behaviour, does not change the quantum fluctuations in (16a). Only when the third order time derivative of the spin operator contributes significantly to equation (11), will memory effects begin to colour the spectrum away from the (quantum) Ohmic form (20).

Here we show that the simulation of the kernel in equation (11) can be achieved by numerically integrating a set of first order coupled differential equations. We assume a single spin and rewrite equation (11) as

$$\begin{equation}\frac{\mathrm{d}\boldsymbol{S}(t)}{\mathrm{d}t}=\gamma \boldsymbol{S}(t)\times \left[{\boldsymbol{B}}_{\text{ext}}+\boldsymbol{b}(t)+\boldsymbol{V}(t)\right],\quad \end{equation} \tag{ 43 }$$

where we have defined $\boldsymbol{V}(t)=\gamma {\int }_{{t}_{0}}^{\infty }\!\mathrm{d}{t}^{\prime }\,K(t-{t}^{\prime })\,\boldsymbol{S}({t}^{\prime })$. Furthermore we define$\boldsymbol{W}(t)=\frac{\mathrm{d}\boldsymbol{V}(t)}{\mathrm{d}t}$, and note that we may choose the initial values as V(t₀)=0 and W (t₀ )=0 for the Lorentzian kernel (27). Expressing K(t − t') through its Fourier transform $\tilde{K}(\omega )$, choosing a Lorentzian kernel (26) and considering the expression $\boldsymbol{Z}(t){:=}\frac{\mathrm{d}\boldsymbol{W}(t)}{\mathrm{d}t}+{\Gamma}\boldsymbol{W}(t)+{\omega }_{0}^{2}\boldsymbol{V}(t)$, we obtain

$$\begin{equation}\boldsymbol{Z}(t)=A\gamma {\int }_{{t}_{0}}^{\infty }\!\mathrm{d}{t}^{\prime }\,\delta (t-{t}^{\prime })\boldsymbol{S}({t}^{\prime }).\end{equation} \tag{ 44 }$$

Rearranging gives

$$\begin{equation}\frac{\mathrm{d}\boldsymbol{W}(t)}{\mathrm{d}t}=-{\Gamma}\boldsymbol{W}(t)-{\omega }_{0}^{2}\boldsymbol{V}(t)+A\gamma \boldsymbol{S}(t),\quad \end{equation} \tag{ 45 }$$

as stated in the main text.

For a classical spin S of length S₀ = n ℏ/2 in an external field B _ext = B e _z the thermal average ⟨s_z ⟩ is determined by the Boltzmann distribution for the Hamiltonian H = −γ S ⋅ B _ext at inverse temperature β = 1/k_B T,

$$\begin{equation}{\langle {S}_{z}\rangle }_{\mathsf{\text{stat phys}}}={\int }_{-{S}_{0}}^{+{S}_{0}}\!\mathrm{d}{S}_{z}\,{S}_{z}\frac{{\mathrm{e}}^{-\beta (-\gamma \,{S}_{z}B)}}{{Z}_{a}}={\partial }_{a}\mathrm{ln}\,{Z}_{a},\,\qquad \end{equation} \tag{ 46 }$$

with ${Z}_{a}={\int }_{-{S}_{0}}^{+{S}_{0}}\!\mathrm{d}{S}_{z}\,{\mathrm{e}}^{a{S}_{z}}$ where a = βγB. This gives

$$\begin{equation}{Z}_{a}=\frac{2\,\mathrm{sinh}(a{S}_{0})}{a},\end{equation} \tag{ 47 }$$

and hence

$$\begin{equation}\frac{{\langle {S}_{z}\rangle }_{\mathsf{\text{stat phys}}}}{{S}_{0}}=\mathrm{coth}\left(\beta \gamma B\,{S}_{0}\right)-\frac{1}{\beta \gamma B\,{S}_{0}},\end{equation} \tag{ 48 }$$

$$\begin{equation}{\langle {s}_{z}\rangle }_{\mathsf{\text{stat phys}}}=\mathrm{coth}\left(\frac{n\hslash {\omega }_{\mathrm{L}}}{2{k}_{\mathrm{B}}T}\right)-\frac{2{k}_{\mathrm{B}}T}{n\hslash {\omega }_{\mathrm{L}}},\end{equation} \tag{ 49 }$$

where ω_L = |γB| and ${s}_{z}=\mathrm{sign}(\gamma )\,\frac{{S}_{z}}{{S}_{0}}$. In the magnetism literature, sometimes a reduced temperature experienced by a spin with n ≠ 1 is defined as T_red = T/n, i.e. the temperature is effectively reduced in comparison to the temperature experienced by a spin with n = 1.

As discussed in the main text, the impact of the quantum zero-point noise on the steady state ⟨s_z ⟩ value at T = 0 K is very highly spin length dependent. For some materials a fundamental spin value of S₀ = 1ℏ/2 will not be appropriate. For example iron (III) has five electrons in the outer d shell, and then from Hund's rules the spin is maximized to S = 5/2, and the orbital angular momentum is zero, L = 0. Therefore J = S, Landé g-factor equals 2, and the gyromagnetic ratio remains the electron gyromagnetic ratio. Figure 6(b) shows the steady state ⟨s_z ⟩ plot as a function of temperature for spin S₀ = 5ℏ/2, next to those for spin S₀ = 1ℏ/2 (a) and S₀ = 200ℏ/2 (c). The ⟨s_z ⟩ value is below 1, at $\approx 0.8$, but the reduction is far less severe than for the spin-1/2.

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Here we establish the set of scales determining the dynamics described by equations (1) and (11) with either classical or quantum power spectra.

For a single spin, i.e. ignoring exchange terms etc, one may rescale the LLG equation (1) using M = γ S = |γ|S₀ s with spin length | S | = S₀. One obtains

$$\begin{equation}\frac{\mathrm{d}\boldsymbol{s}}{\mathrm{d}t}=\gamma \boldsymbol{s}\times \left[{\boldsymbol{B}}_{\text{eff}}^{\mathsf{\text{cl}}}(t)-\vert \gamma \vert {S}_{0}{\eta }_{\mathrm{G}}\frac{\mathrm{d}\boldsymbol{s}}{\mathrm{d}t}\right],\end{equation} \tag{ 50 }$$

where the spin length S₀ and η_G appear together, setting the first scale. Furthermore the effective field including classical stochastic noise with power spectrum (21), is given through (34) and (35) by

$$\begin{equation}{\boldsymbol{B}}_{\text{eff}}^{\mathsf{\text{cl}}}(t)={\boldsymbol{B}}_{\text{ext}}+\sqrt{2{S}_{0}{\eta }_{\mathrm{G}}{k}_{\mathrm{B}}\frac{T}{{S}_{0}}}\boldsymbol{\xi }(t).\end{equation} \tag{ 51 }$$

Here we have introduced an S₀ so that η_G appears together with it, and we find the second scale to be given by T/S₀. The third scale is clearly set by the strength of the external field, B _ext. We note that if one chooses the same S₀ η_G value for different spin lengths, i.e. assumes η_G scales as 1/S₀, as we have done in the main text, then only two scales are left, B _ext and T/S₀.

However, for the quantum Ohmic power spectrum the components of the stochastic noise can be written as

$$\begin{align}\hfill {b}_{j}& ={\int }_{-\infty }^{\infty }\!\mathrm{d}{t}^{\prime }{\int }_{-\infty }^{{\omega }_{c}}\frac{\mathrm{d}\omega }{2\pi }{\mathrm{e}}^{-\mathrm{i}\omega (t-{t}^{\prime })}\sqrt{{S}_{0}{\eta }_{\mathrm{G}}\frac{\hslash \omega }{{S}_{0}}\mathrm{coth}\left(\frac{\hslash \omega }{2{k}_{\mathrm{B}}T}\right)}\,{\xi }_{j}({t}^{\prime }).\hfill \end{align} \tag{ 52 }$$

Clearly, in the quantum case the temperature T now appears separately from spin length S₀, thus introducing an additional scale in comparison to the classical case. Moreover the fact that the frequency integration for the stochastic field does not simplify as in (51) means that relaxation to the steady state at low temperatures (where the coth x cannot be approximated as 1/x) will be much more noisy than in the high temperature case. Thus in our simulations, this additional scale leads to larger uncertainties in the steady state results, as seen in figure 5(a).

Finally, we note that for the integration of the quantum Ohmic power spectrum in (52) we have introduced a frequency cut-off ω_c by hand, which is necessary at low temperatures to avoid the integral diverging. At low temperatures, this cut-off will set an additional, somewhat artificial, scale of the problem. Importantly, such cut-off is not required for the Lorentzian coupling since the power spectrum (28) decays at high frequencies, even at low T.