Recent experiments performed on chiral molecules, comprising transition metal or rare earth elements, indicate temperature reinforced chiral induced spin selectivity. In these compounds, spin selectivity is suppressed in the low temperature regime but grows by one to several orders of magnitude as the temperature is increased to room temperature. By relating temperature to nuclear motion, it is proposed that nuclear displacements acting on the local spin moments, through indirect exchange interactions, generate an anisotropic magnetic environment that is enhanced with temperature. The induced local anisotropy field serves as the origin of a strongly increased spin selectivity at elevated temperature.

## I. INTRODUCTION

The magnetic properties, e.g., the induced magnetization, coercivity, and chiral induced spin selectivity effect, of chiral structures comprising, e.g., rare earth^{1} or transition metal^{2–6} elements, have recently been shown to possess unexpected dependencies on the temperature. Experiments performed on such compounds indicate that these and other magnetic properties are not only strong and stable at room temperature but also the same properties dramatically wane as the temperature drops toward 0 K. In other words, these compounds become magnetically stabilized, and even reinforced, with increasing temperature, which is quite the opposite to the predictions that can be made using the conventional theory for magnetism. The observations of unconventional magnetic temperature dependence compel reassessment and extension of the theory of magnetic phenomena. In a recent study, the unconventional temperature dependence was addressed and attributed to be a result of anharmonic nuclear vibrations that generate a pseudo-magnetic field that orders the spin moments in the structure.^{7} The purpose of this article is to extend this work to chiral molecules and to provide a theoretical explanation for the temperature reinforced chiral induced spin selectivity.

Chiral induced spin selectivity^{8,9} is a phenomenon that emerges from a combination of structural chirality, spin-orbit interactions, strongly non-equilibrium conditions, and electron exchange and correlations. While the effect can be viewed as a measure of the response to changes in the magnetic environment coupled to the system, its phenomenology is based on experimental observations of substantial changes in the charge current amplitude through chiral molecules, upon changes in the external magnetic conditions.^{8–21} Chiral induced spin selectivity has been demonstrated in both single and multi-stranded helical structures, such as double stranded DNA molecules^{10} and bacteriorhodopsin,^{11} and various types of peptides^{12–16} and polyalanines,^{17–19} and recently also in helicene.^{20,21} In all these types of molecules, chiral induced spin selectivity is a robust effect at room temperature conditions. Recent observations also suggest that chiral molecules may acquire a finite spin-polarization when interacting with a metallic environment. Specifically, Yu–Shiba–Rusinov states^{22–24} were observed for *α*-helix polyalanine disposed on the surface of superconducting NbSe_{2}.^{25}

Here, it is shown that the nuclear motion generates an effective pseudo-magnetic field that acts on the local spin moments through indirect exchange. The induced field is oriented along a direction which is governed by the underlying spin texture and the charge polarization. Nuclear vibrations generate charge displacements in the chiral molecule that acts on the local spin moment as an effective magnetic field proportional to the average nuclear displacement. While the vibrations are nearly harmonic at low temperatures, the average displacement vanishes. Hence, the associated induced magnetic field is negligible, if not vanishing, such that the electrons in the current that flow through the molecule experiences a nearly isotropic exchange with the local spin moment. With increasing temperature, anharmonicity in the vibrational coordinate causes the average displacement to become finite, which, in turn, creates a local magnetic anisotropy field which spin polarizes the local spin moment. This renders a strongly anisotropic exchange with the electrons in the current, which translates into a strong asymmetry in the conduction channels for different electron spins. By this temperature modification of the exchange field, the spin selectivity property of the system goes from being negligible at low temperatures to become tens of percents at high.

## II. MODEL

*S*, in order to reflect, e.g., the Cu

^{2+}or Tb

^{3+}character. Therefore, in order to construct a generic model that comprises these features, it will be assumed that the spin moment

**S**is attached to a chiral ligand, as shown in Fig. 1. The chiral ligand is described as a chain of sites configured in a helix

^{26–29}with no intrinsic exchange or correlations. Molecular vibrations are coupled to the local spin moment through the electronic structure as an effective exchange interaction.

^{30}Finally, in order to enable calculations of the charge transport through the system, the molecular complex is mounted in the junction between a ferromagnetic and a normal metallic lead. The features of the model, thus, described can be summarized into the Hamiltonian

*χ*=

*L*,

*R*, defines the electronic properties of the left (

*L*) and right (

*R*) leads, where the spinor $\psi k\u2020$ (

*ψ*

_{k}) creates (annihilates) an electron in the lead

*χ*at the energy

*ɛ*

_{k}, $HTL=\u2211k\u2208L\psi k\u2020v\psi L1+H.c$ and $HTR=\u2211k\u2208R\psi k\u2020v\psi RM+H.c$ define the tunneling between the leads and the juxtaposed molecule, whereas $HML$ and $HMR$ comprise the electronic structures of the left and right part of the molecule, respectively. The operator

*ψ*

_{L1}$(\psi RM)$ denotes the electron spinor associated with site 1 $(M)$ in the left (right) molecule. For simplicity, the hybridization matrix

**v**between the states in the leads and the molecules have been taken to be energy independent, as well as independent of the metal in the lead. This is justified since the focus in this article lies on the physics occurring between the molecules.

^{26–29,31}is described by the set $M=M\xd7N$ of spatial coordinates

**r**

_{m}= (

*a*cos

*φ*

_{m},

*a*sin

*φ*

_{m},

*c*

_{m}), $\phi m=2\pi (m\u22121)/(M\u22121)$, $m=1,\u2026,M$, and

*c*

_{m}=

*cφ*

_{m}/2

*π*, where

*a*and

*c*define the radius and length, respectively, of the helical structure, whereas

*M*and

*N*denote the number of turns and ions per turn. The electronic structure of the left molecule, for instance, is provided in terms of the Hamiltonian

*ψ*

_{m}) creates (annihilates) an electron at the site

*m*with energy

*ɛ*

_{m}and Zeeman split

*gμ*

_{B}

**B**·

**/2 generated by the external magnetic field**

*σ***B**, where

**is the vector of Pauli matrices. The second term describes hopping between nearest neighboring sites with rate**

*σ**t*while the last term accounts for the spin–orbit interaction with strength

*λ*and chirality $vm(\xb1)$ in terms of hybridization between next-nearest neighbors. The vector $vm(s)=d\u0302m+s\xd7d\u0302m+2s$,

*s*= ±1, defines the chirality of the helical molecule in terms of the unit vectors $d\u0302m+s=(rm\u2212rm+s)/|rm\u2212rm+s|$; positive chirality corresponds to right handed helicity. For more details, see Refs. 26–29 and 31.

The local spin moment is mounted between the two parts of the chiral molecule and its presence is modeled by a tunneling between the sites $M$ and 1 in the left and right parts, respectively, see fifth term in Eq. (1a). The associated tunneling matrix, **V**, comprises the spin-independent, *V*_{0}, and spin-dependent, **V**_{1}, rates, such that **V** = *V*_{0}*σ*^{0} + *V*_{1}**S** · ** σ**, where

*σ*

^{0}is the 2 × 2 identity matrix.

Physically, the spin-dependent contribution *V*_{1}**S** · ** σ** originates from the direct exchange interaction between localized and de-localized electrons in the structure, ultimately, from the Coulomb integral. In the present scenario, the localized electrons constitute the localized spin moment

**S**, whereas the spin of the de-localized electrons is captured by the Pauli matrices. This concept and some implications in tunneling junctions were discussed at some length in Ref. 32.

The local spin moment is finally modeled by the Hamiltonian in Eq. (1b), where the first term accounts for the interaction with the external magnetic field **B** and the magnetic field **B**_{j} induced by the electron current.^{30,32} The second term describes the nuclear vibrations $\omega nbn\u2020bn$ in terms of the vibrational normal modes *ω*_{n}, and the electronically mediated interactions, $An$,^{30} between the local spin moment and the nuclear vibrations, where $Qn=bn+bn\u2020$ is the nuclear quantum displacement operator. The last term in Eq. (1b), finally, provides anharmonic properties to the nuclear motion, with strength Φ.

It may be noticed that fields **B**_{j} and $An$ both originate from the interactions between de-localized electrons and the localized spin moment **S** (direct exchange) and the nuclear displacement *Q*_{n} (electron–phonon). The magnetic field **B**_{j} results from a spin-polarized current^{32,33} and is relevant in the present context since such a current is injected into the molecule by the presence of a magnetized lead. The coupling tensor $An$ between the localized spin moment **S** and the nuclear displacement *Q*_{n} is generated by the combined process of direct exchange^{30} between electrons and the localized spin, on the one hand, and electron–phonon coupling between electrons and nuclear displacement, on the other. The de-localized electrons mediate, hence, an indirect exchange between the spin and nuclear degrees of freedom.

In the following, all relevant physical properties of the composite complex of the chiral molecules are related to the single electron Green function $Gmn(z)=\u27e8\u27e8\psi m|\psi n\u2020\u27e9\u27e9(z)$, with **G**_{m} ≡ **G**_{mm}, such that, e.g., the density of electron states $\rho m(\omega )=isp[Gm>(\omega )\u2212Gm<(\omega )]/2$ and spin-resolved charge densities $\u27e8nm\sigma \u27e9=(\u2212i)(\sigma 0+\sigma \sigma \sigma z\sigma z)\u222bGm<(\omega )d\omega /4\pi $, where $Gm<(>)$ is proportional to density of occupied (unoccupied) electron states. The notation sp refers to the trace over spin 1/2 space.

The Green function $Gmn(t,t\u2032)=(\u2212i)\u27e8T\psi m(t)\psi n\u2020(t\u2032)\u27e9$ describes the propagation of a particle in the system $H$ between the space–time points (*m*, *t*) and (*n*, *t*′), where *t* and *t*′ are complex time variables. Specifically, the process between the latter to the former describes an electron propagation whereas the reverse process describes a hole propagation. Under time-independent conditions, the time-resolved Green function can be converted into the Fourier transform **G**_{mn}(*z*) = *∫***G**_{mn}(*t*, *t*′)*e*^{−iz(t−t}′^{)}*dt*′, where the time-integral is performed over some contour in the complex plane. More details can be found in, e.g., Refs. 34–36.

The lesser (greater) Green function **G**^{<(>)}, apart from the intuitive interpretation in terms of the density of occupied (unoccupied) states, is essentially indispensable under non-equilibrium conditions. One reason is that since particle transfer and, hence, transport can only take place between occupied and unoccupied states, this information is directly acquired from the lesser and greater Green function. A second reason is that these Green functions have a direct bias voltage and temperature dependence.

### A. Transport theory

The current *J*(*t*) across the interface between the two parts $HML$ and $HMR$ of the molecule can be written as a sum of three contributions,^{37} *J* = ∑_{i=0,1,2}*J*_{i}, where *J*_{0} is proportional to the tunneling rate $V02$ and is, hence, independent of the properties of the local spin moment. The second and third contributions are proportional to *V*_{0}*V*_{1}⟨**S**⟩ and $V12\u27e8S(t)S(t\u2032)\u27e9$, respectively. In other words, the former contribution is proportional to the expectation value of the local spin moment ⟨**S**⟩, whereas the latter is proportional to the spin-spin correlation function ⟨**S**(*t*)**S**(*t*′)⟩.

*J*(

*t*) across the interface between the two chiral molecules can then be calculated according to

^{37}

*J*= ∑

_{i=0,1,2}

*J*

_{i}, where

In the present construction, the chiral structure is assumed to not generate spin selectivity by itself. It is only through its interaction with the local spin moment such an effect takes place, if any. Hence, it can safely be assumed that the current *J*_{0} does not provide any contribution to the spin selectivity since it contains no interaction with the local spin. It should be noticed in this context that the Green functions for the left and right chiral structures are expressed with no coupling to the local spin moment. This is justified since the primary interest here is whether a mechanism pertaining to spin selectivity can be achieved from the properties of the local moment and its coupling to the chiral environment. Moreover, it is justified to neglect possible spin-selectivity mechanisms, e.g., electron correlations^{26} or spin-dependent electron–phonon interactions,^{27} since the purpose here is to resolve the hypothesis whether nuclear displacements may serve as the source for a local magnetic field that acts on the spin moment and, hence, generates spin-selectivity.

*V*

_{1}, the current is provided by

*J*

_{1}, which under the present (stationary) conditions can be written as

*V*, and the Fermi-Dirac distribution function

*f*(

*ω*). This current depends explicitly on the magnitude of the spin moment; hence, it may be non-vanishing only whenever the expectation value of the spin moment is. Furthermore, it will be assumed that the conditions of the set-up eventually lead to a fixed, or, stationary local moment; therefore, it is justified that ⟨

**S**⟩ is time-independent.

Although the focus lies on the lowest order contribution to the current that depends on the local moment, it should be noticed that also the current *J*_{2} depends on the local spin structure though the spin–spin correlation function ⟨**S**(*t*)**S**(*t*′)⟩. However, this current adds an essentially negligible contribution to the spin-selectivity and is, therefore, omitted in the present discussion.

### B. Local spin moment

**B**

_{j}

^{30,32}and with the nuclear structure through the electronically mediated exchange $A$.

^{30}These two electronically mediated effective fields can be written as

*η*is the parameter for the effective interaction between the electrons and the nuclear motion. In terms of these fields, an effective mean-field model for the local spin moment can be written as

*Q*

_{n}⟩ denotes the expectation value of the displacement operator

*Q*

_{n}. Hence, while the induced field

**B**

_{j}depends only on the current flowing through the molecular complex, the induced field $A\u2211n\u27e8Qn\u27e9$ depends on both this current as well as the nuclear vibrations. The former field has only a weak, if any, temperature dependence since it originates from the purely electronic portion of the transport, whereas the latter strongly depends on the temperature since it derives from the vibrational properties. This will become crucial in the subsequent discussion.

It should be noticed that the spin model $HS$ represents a description of the physics in which the electrons are responsible for the induced magnetic field **B**_{j} and the coupling $An$ between the spin and nuclear vibrational degrees of freedom. However, higher order of correction to, e.g., the nuclear motion, from the electrons are omitted for the benefit of constructing a reasonably transparent description of the expected influence from to the nuclear motion on the localized spin moment. In this sense, the fields **B**_{j} and $An$ are merely parameters in the theory, albeit voltage dependent, redirecting the focus to the nuclear motion.

### C. Electronic structure

**A**=

*A*

_{0}

*σ*

^{0}+

**A**

_{1}·

**, where**

*σ**A*

_{0}and

**A**

_{1}represent the spin-independent and spin-dependent contributions of the field

**A**. Here, the fields under interest are represented by the Green functions $GL/Rr$ along with their products in Eq. (8). Under the trace, it is straightforward to show that, e.g.,

The first two contributions on the right hand side of this expression, contain products between the charge and spin properties in the molecules, something which can be read out from the subscripts 0 and 1. For instance, in the product $ImGR0rReGL1r$, the factor $ImGR0r$ is related to the properties of the charge in the right molecule whereas the factor $ReGL1r$ is connected to the spin properties of the left. Analogously, $ImGR1rReGL0r$ ties together the properties of the spin in the right lead and the charge in the left.

^{6}This analysis unequivocally demonstrates that the electronic structure, to which the local spin moment and the nuclear vibrations are coupled to, has to carry a non-trivial spin texture in order to mediate an effective interaction between the spin and mechanical degrees of freedom.

In the present set-up, the non-trivial spin-texture is provided by the electronic structure in the chiral molecules, by the combination of structural chirality and spin-orbit interactions in the next-nearest neighbor hopping, see the last term in Eq. (2). Hence, a non-negligible coupling can be expected to exist between the spin and mechanical degrees of freedom.

### D. Nuclear motion

*Q*

_{i}

*Q*

_{j}

*Q*

_{k}. In linear response theory, the expectation value of the displacement operator, can be expanded in terms of the pertinent interaction Hamiltonian $Hint=\u2211ijk\Phi QiQjQk$ according to $\u27e8Qn(t)\u27e9\u2248\u27e8Qn(t)\u27e90+(\u2212i)\u222b\u2212\u221et\u27e8[Qn(t),Hint(t\u2032)]\u27e90dt\u2032$, where $\u27e8Qn(t)\u27e90=0$ since it represents the displacement in the harmonic approximation. The second term, here, provides the contribution

*n*

_{n}≡

*n*

_{B}(

*ω*

_{n}) is the Bose–Einstein distribution function. In terms of these expression, the linear response quantum displacement becomes time-independent and given by

*β*= 1/

*k*

_{B}

*T*,

*k*

_{B}is the Boltzmann constant, and

*T*is the temperature.

*βω*/2 ∝

*T*for large

*T*. However, despite this increase leads to an enhanced vibrationally induced magnetic field acting on the local moment, it is not sufficient to generate the large difference in the induced field with the temperature increase seen in experiments. Therefore, the phonon Green function

*D*

_{n}(

*t*,

*t*′) is reconsidered in the anharmonic model $Hint$. Assuming stationary conditions, such that the Fourier transform

*D*

_{n}(

*z*) =

*∫D*

_{n}(

*t*,

*t*′)

*e*

^{−iz(t−t}′

^{)}

*dt*′, $z\u2208C$, is well-defined, a Dyson-like equation for

*D*

_{n}(

*z*) can be written, see, e.g., Refs. 34 and 38,

*z*

_{μ}=

*iμπ*/

*β*, and

*μ*= 2

*n*+ 1, $n\u2208Z$. Here, it is noticed that the first contribution, Fig. 2(a), leads to

*z*

_{r}=

*ω*+

*iδ*, with

*δ*> 0 infinitesimal.

#### 1. Hartree approximation

_{H}/

*ω*

_{m}< 1 which, for a single mode [Σ

_{H}= −2(Φ/

*ω*

_{0})coth(

*βω*

_{0}/2)] leads to the condition on the temperature that

*ω*

_{0}> 2Φ for the temperature to be positive. Second, it should be noticed that the upper temperature limit becomes arbitrarily large as the coupling parameter Φ → 0, for any given frequency

*ω*

_{0}, which ascertains the validity of the approximation for high temperatures.

For an estimate, assume that *ω*_{0} ∼ 10^{−4} eV, such that *ω*_{0}/*k*_{B} ≳ 1. Then, a coupling parameter Φ/*ω*_{0} ≲ 1/100 is sufficient to allow for studies at, at least, room temperature (*T* ∼ 300 K).

By this extension of the expected displacement, the leading order is proportional to *T*^{3} for large temperatures, which is sufficient to obtain a strong enough temperature enhancement of the induced magnetic field to be comparable with the experimental observations.

## III. RESULTS AND DISCUSSION

The vibrationally induced magnetic field $g\mu BBvib\u2261A\u2211n\u27e8Qn\u27e9$ is simulated for a set-up with two chiral (helical) molecules, each defined by *M* × *N* = 3 × 4 sites. The molecule is mounted between one ferromagnetic and one normal metallic lead, where the ferromagnetic lead provides injection of spin-polarized electrons. This spin-polarization is characterized by the spin-dependent coupling matrix **Γ**_{FM} = Γ_{0}(*σ*^{0} + **p** · ** σ**)/2,

^{39–42}where Γ

_{0}= ‖

**v**‖

^{2}

*ρ*

_{0}is the coupling parameter defined in terms of the hybridization

*v*and density of electron states

*ρ*

_{0}in the lead and the spin-polarization

**p**, |

**p**| ≤ 1, of the injected electrons, which is fixed between the two configuration set-up determining the chiral induced spin selectivity effect. Analogously, the coupling matrix to the normal metallic lead is given by

**Γ**

_{NM}= Γ

_{0}

*σ*

^{0}/2.

In Fig. 3(a), the *z*-component of the effective magnetic field $g\mu BBvib\u2261A\u2211n\u27e8Qn\u27e9$ is plotted as function of the temperature. Under the conditions specified in the figure caption, the effective field increases by more than three orders of magnitude between 25 and 375 K. With increasing temperature, the anharmonic contribution to the nuclear vibrations enhances the induced magnetic field such that it eventually outweighs the effect of the external field *B* ∼ 0.35 T ($\u223c40\mu $ eV). From the expression of the effective displacement in Eq. (21), it is also clear that slow vibrations induce the pseudo-magnetic more efficiently than fast vibrations. Although the model used here does not specify which types of vibration are more effective, it is reasonable to argue that slow vibrations can be associated with coherent motion of the greater part of the molecule. The faster vibrations can, on the other hand, be associated with individual nuclear motion that may be independent of the environment. Moreover, keeping in mind the functional form of the effective displacement, Eq. (21), it can also be understood that although there is a set of ten vibrations with energies in the range (10^{−9}, 10^{−2}) eV, it is the slower vibrations that dominate the properties of the induced pseudo-magnetic field.

The impact of the induced field has a dramatic effect on the local spin moment, see Fig. 3(b), where ⟨*S*_{z}⟩ is plotted as function of the temperature, for (red) $B=0.35z\u0302$ T and (blue) $B=\u22120.35z\u0302$ T. For low temperatures, where the induced pseudo-magnetic field is negligible, the magnetic moment aligns with the external magnetic field **B**. This can be seen in the plots as non-vanishing moments that decrease with increasing temperature, as would be expected when the thermal energy becomes comparable or larger than the energy associated with the external magnetic field. Nevertheless, a further increase in the temperature leads to that the magnetic moment is re-established by the activation of the vibrationally induced pseudo-magnetic field. In addition, the sign of the magnetic moment remains unchanged despite the activation of the induced pseudo-magnetic field, which can be understood since the nuclear vibrations do not depend on the external magnetic field. Therefore, the vibrational contribution does not provide a preferred direction to the magnetic moment, instead it enhances the properties provided by other sources. The magnetic moment can be seen to strongly increase with increasing temperature, as would be expected by the strongly increasing induced pseudo-magnetic field. One would expect that magnetic moment eventually ceases to exist for some critical temperature; however, such a prediction is beyond both the scope of the proposed theory, as well as what it is capable of reproducing.

Nevertheless, as the expectation value of the local moment increases, the resulting chiral induced spin selectivity (CISS) increases from low values to fairly high ones. As the magnetic moment assumes different signs upon switching the external magnetic field to its opposite direction, the properties of the system as a whole, including the presence of the ferromagnetic and non-magnetic leads, are significantly changed. To clarify, while the orientation (sign) external magnetic field is switched, both the ferromagnetic and normal metallic leads are assumed to be unsusceptible to this field, as well as the properties of the chiral molecules. In particular, this means that the ferromagnetic lead provides an unaltered spin-polarization of the injected electrons. Hence, the sign changes imposed by the external magnetic field are directly changing the sign of the magnetic moment, while other properties remain unchanged.

The observation that the only effect of the external magnetic field is to change the orientation of the magnetic moment is important since it implies that the magnetic properties of the molecule act as to either open or close the channel for conductance through the junction. In other words, the molecular magnetic properties are such that it can accommodate electrons flux with a certain spin-polarization. Under the given conditions, then, where the spin-polarization of injected electrons remains fixed, the molecule accommodates a larger electron flux for one orientation of the external magnetic field than what it does in the opposite. In this sense, there is a chiral induced spin selectivity effect, which is quantified in the difference, the magnetocurrent, Δ*J* = 100 · (*J*_{+} − *J*_{−})/*J*_{0}, where *J*_{±} denotes the charge current for $B=\xb10.35z\u0302$ T and *J*_{0} denotes the charge current for **B** = 0. The magnetocurrent is plotted in Fig. 3(c), corresponding to the conditions in Fig. 3(b). As a function of the temperature, the magnetocurrent is strongly reminiscent of the magnetic moment ⟨*S*_{z}⟩. This would also be expected in the light of that the chiral induced spin selectivity effect in the present set-up is mainly governed by the properties of the magnetic moment.

It is worth to notice that the indirect exchange between, e.g., localized spin moments in phthalocyanines are typically considered to have values in the range 0.5–20 meV.^{43,44} Since the indirect exchange depends quadratically on the direct exchange, *V*_{1} in the notation used here, the chosen value, *V*_{1} = *t*/4, should be considered feasible in this context.

## IV. CONCLUSIONS

A mechanism for strongly temperature dependent chiral induced spin selectivity pertaining to molecular compounds comprising transition metal elements has been proposed. Molecular nuclear vibrations that couple to the magnetic moment of the transition metal element through the electronic structure generate a pseudo-magnetic field that stabilizes the magnetic moment at temperatures far higher than the energy corresponding to the external magnetic field can sustain. Specifically, anharmonic vibrations contribute strongly to activate the induced pseudo-magnetic field since, in chiral structures, these lead to stationary lattice distortions that effectively convert into an exchange field acting on the magnetic moment. As anharmonic vibrations become increasingly occupied with increasing temperature, the strength of the induced pseudo-magnetic field grows. Using this mechanism, it was, furthermore, shown that the resulting properties of the chiral compound open for a strong chiral induced spin selectivity effect that, accordingly, is enhanced with increasing temperature. The theoretical results are in qualitative excellent agreement with recent experimental observations made on, e.g., azurin.^{6}

Further developments toward categorization of, e.g., the symmetries of the nuclear vibrations would provide deepened insight into the mechanisms that drive the developments of pseudo-magnetic fields.

## ACKNOWLEDGMENTS

The author gives special thanks to R. Naaman for suggesting this study. Vetenskapsrå det and Stiftelsen Olle Engkvist Byggmästare are acknowledged for financial support.

## AUTHOR DECLARATIONS

### Conflict of Interest

The author has no conflicts to disclose.

### Author Contributions

**J. Fransson**: Writing – original draft (equal).

## DATA AVAILABILITY

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

## REFERENCES

*Quantum Statistical Mechanics*

*Quantum Kinetics in Transport and Optics of Semiconductors*

*Non-Equilibrium Nano-Physics*