An Antisymmetric Berry Frictional Force At Equilibrium in the Presence of Spin-Orbit Coupling
aa r X i v : . [ c ond - m a t . m t r l - s c i ] F e b An Antisymmetric Berry Frictional Force At Equilibrium in thePresence of Spin-Orbit Coupling
Hung-Hsuan Teh ∗ Department of Chemistry, University of Pennsylvania,Philadelphia, Pennsylvania 19104, USA
Wenjie Dou † School of Science, Westlake University,Hangzhou, Zhejiang 310024, China andInstitute of Natural Sciences, Westlake Institute for Advanced Study,Hangzhou, Zhejiang 310024, China
Joseph E. Subotnik ‡ –Department of Chemistry, University of Pennsylvania,Philadelphia, Pennsylvania 19104, USA– (Dated: February 17, 2021) bstract We calculate the electronic friction tensor for a molecule near a metal surface in the case that theelectronic Hamiltonian is complex-valued, e.g. the case that there is spin-orbit coupling and/or anexternal magnetic field. In such a case, even at equilibrium , we show that the friction tensor is notsymmetric. Instead, the tensor is the real-valued sum of one positive definite tensor (correspondingto dissipation) plus one antisymmetric tensor (corresponding to a Berry pseudomagnetic force).Moreover, we reveal that this Berry force can be much larger than the dissipational force, suggestingthe strong need to consider such terms for systems with spin-orbit coupling.
I. INTRODUCTION
When an electronic system couples to a set of nuclear degrees of freedom (DoF), not onlywill the electronic wave function change at different nuclear positions (in accordance withthe Born-Oppenheimer picture) — the nuclear wave packet will also change as it receivesfeedback from the electronic DoF. In the adiabatic limit, the feedback is composed of threeparts: the adiabatic force F µ , the random force δF µ , and the frictional damping force. Thefriction tensor γ µν reports the strength of the nuclear damping force in the µ -directionas caused by nuclear motion in the ν direction. The nuclei follow a stochastic Langevinequation[1, 2], M µ ¨ R µ = F µ − X ν γ µν ˙ R ν + δF µ , (1)where M µ is the mass of a nuclei and R µ is the nuclear position in the µ direction.The friction tensor is the first-order correction to the Born-Oppenheimer approximationand is ubiquitous for dynamical problems on surfaces. In general, as for any tensor, γ µν canbe divided into a symmetric part γ S µν and an antisymmetric part γ A µν . Intuitively, we shouldexpect that: • On the one hand, for a system at equilibrium, γ S µν should be positive definite (givingpositive eigenvalues) so that this term can only dissipate energy to the surround-ings (and avoid unstable dynamics). Such relaxation processes have been reported ∗ [email protected] † [email protected] ‡ [email protected]
2s important for molecule-metal interface dynamics (scattering, adsorption etc.)[3–6],electron transfer within electronic devices[7], heating due to the phonon motion[8, 9],and so on. As shown by Juaristi, Reuter and others[10, 11], the diagonal component( µ = ν ) can significantly change the electron-hole pair induced vibrational lifetime,and Maurer and Tully[12, 13] have shown that the off-diagonal elements of frictiontensor ( µ = ν ) can also be crucial. • On the other hand, γ A µν should contribute a Lorentz-like force. For example, in 3Dnuclear space, we can define a surrogate pseudomagnetic field, γ A α ≡ P µν ǫ µνα γ A µν / γ A µν = P α ǫ µνα γ A α . Using this surrogate form, the antisymmetricforce in Eq. (1) become the Lorentz force (assuming the same mass m ), m ¨ R µ = − X ν γ A µν ˙ R ν + · · · = − X να ǫ µνα ˙ R ν γ A α + · · · . Historically, there have been many separate approaches for calculating the electronicfriction tensor going back to the early work of Suhl[14, 15] with major contributions fromHead-Gordon and Tully[16], Brandbyge and Hedegard[17, 18], Persson[19, 20], Hynes[21],Langreth[22, 23], Mozyrsky[24, 25], von Oppen[26, 27], Galperin[28] and co-workers. In Ref.2, based on the quantum-classical Liouville equation (QCLE) and appropriate usages ofthe adiabatic theorem, a universal Fokker-Planck equation (or more specifically a Kramer’sequation[1]) for a real-valued or complex-valued Hamiltonian was derived, either at equilib-rium or in a nonequilibrium steady state with a Markovian[29–32] electronic friction tensor( ~ = 1 in this letter) of the form: γ µν = − Z ∞ dt Tr n ∂ µ ˆ He − i ˆ Ht ∂ ν ˆ ρ ss e i ˆ Ht o , (2)Here ˆ H is the electronic Hamiltonian and ˆ ρ ss ( R ) is the steady-state density matrix at eachnuclear position R , i.e. ∂ t ˆ ρ ss ( R ) = − i [ ˆ H ( R ) , ˆ ρ ss ( R )] = 0. In Eq. (2), the partial derivativeswith respect to the nuclear coordinates µ , ν (i.e. ∂ µ , ∂ ν ) operate only on the operatordirectly to the right, and this convention will be used throughout the letter below. Thetrace is taken over all the electronic degrees of freedom. In general, several properties followfrom Eq. (2) (and most proofs can be found in the SI):1. γ ∗ µν = γ µν , so that the friction tensor is real-valued.3. γ µν is non-negative when the system is in equilibrium.3. At equilibrium, the symmetric parts of the random force time correlation function andthe friction tensor ( γ S µν ) (as calculated by the QCLE) obey the fluctuation-dissipationtheorem.4. Let ˆ d † p / ˆ d p creates/annihilates an electron in orbital p , and let U ( R ) be an energypotential proportional to the identity. For a general non-interacting Hamiltonian ˆ H = P pq H pq ˆ d † p ˆ d q + U ( R ), in the non-Condon limit, the friction tensor becomes, γ µν = Z ∞−∞ dǫ π Tr (cid:8) ∂ µ H ∂ ǫ G R ∂ ν HG < (cid:9) + H . c ., (3)where G R / A = ( ǫ − H ± iη ) − are retarded/advanced Green’s functions of the electrons,and G < ( t , t ) = i Tr n ˆ ρ ss ˆ d † p ( t ) ˆ d q ( t ) o is the lesser Green’s function. (The potential U ( R ) does not contribute to the friction tensor.)In this letter, our focus will be on the antisymmetric component of the friction tensor, γ A µν . Note that the properties listed above do not yield much information about such africtional component. For instance, with regards to Property 2, it is clear that γ Sµν is positivedefinite if and only if γ µν positive definite; thus, Property 2 gives us no information about γ A µν at equilibrium. Within the chemical physics condensed matter community, the usualassumption is that γ A µν = 0 at equilibrium; for a strictly real-valued Hamiltonian describinga typical molecule on a typical metal, von Oppen and others[26, 27] have demonstrated that γ A µν = 0 only when molecules are in contact with two metals that are out of equilibrium (i.e.with a current).Now, within the description above (and the calculations in Refs. 26 and 27), the interest-ing caveat is the assumption of a strictly real-valued Hamiltonian. For molecule-metal inter-faces or for surface heterostructures, due to the short electron screening length of a metal, theeffective electric field gradient on the surface should lead to strong (complex-valued) Rashbaspin-orbit couplings[33]. Furthermore, a built-in molecular spin-orbit coupling can be en-hanced due to molecular geometry, i.e. molecules with large curvature or torsion in geometryare believed to have larger spin-orbit couplings[34, 35]. For these reasons, a complex-valuedHamiltonian may be quite relevant. Moreover, and most importantly, in a famous papernearly 30 years ago, Robbins and Berry demonstrated that, even for a small closed molec-ular system (far from any metal surface), antisymmetric (Berry) forces can appear if the4amiltonian is complex-valued[36] — as might arise due either to an external magnetic fieldor spin-orbit coupling. Exact scattering calculations have shown that, for model complex-valued Hamiltonian, the resulting Berry force effects can be large and strongly affect electrontransfer processes[37]. Where does this leave us as far as understanding molecular dynamicsnear a metal surface?In truth, within the condensed matter community, we do not yet understand when orif Lorentz forces appear for the dynamics of molecules near metal surfaces if spin-orbitcoupling surfaces. To make progress, we will need to answer two specific questions: (i)Does a nonzero antisymmetric friction tensor (i.e. a pseudomagnetic field) appear whenwe consider a complex-valued Hamiltonian describing a molecule near a metal surface inequilibrium? Note that, except for a few analogous examples in the realm of spintronics[38],to date, the effect magnetic field or spin-orbit couplings have been ignored in friction tensorcalculations (even though Eqs. (2) and (3) above are general). (ii) How large will such apseudomagnetic be, and can such a field become dominant against the dissipative symmetricfriction tensor? Note that Ref. 39 predicts that a huge Berry force can be generated for anisolated molecular system near a sharp avoided crossing (with large derivative couplings) inthe presence of spin-orbit coupling. In the condensed matter world, however, as we wish todescribe a molecule near a metal surface, a molecular system is always coupled to a bath(and such a strong coupling may substantially change the nuclear motion). Thus, one mustwonder, will such a huge Berry force still exist when one considers environmental effects?Below, we will address the two questions above. In particular, we will show that: (i)Like the case of an isolated molecular system, a Berry force exists whenever a molecularsystem with a complex-valued Hamiltonian is coupled to a bath (no matter whether or notthe total system is in equilibrium ). (ii) Unlike the case of an isolated molecular system, thestrength of the Berry force does not require a tiny energy gap (i.e. a sharp avoided crossing)to achieve a large Berry force (in fact an energy gap is necessary). (iii) γ A αν is comparableor can even be one order of magnitude larger than γ S µν and thus affects the experimentalobservable.This letter is constructed as follows: In Sec. II, we will introduce our model systemin detail and calculate the corresponding friction tensor. In Sec. III, different parameterregimes will be investigated, showing that the antisymmetric friction tensor can be crucialat times (and it definitely cannot be ignored). In Sec. IV, we conclude and offer a few brief5emarks on the implications of our work for understanding recent experiments demonstratingspin selectivity in transport with chiral systems[40–42]. II. MODEL SYSTEM
In this section, we consider a minimal model in which a two-level system is coupled totwo leads and the two-level system depends on two dimensional nuclear DoF (in order tohave nonzero γ A µν , two is minimum). While there is an immense amount known about the(symmetric) friction tensor that arises for a resonant level model, no such results or intuitionhave been derived for the antisymmetric friction tensor even in the case of a two-level modelat equilibrium. The total electronic Hamiltonian ˆ H is divided into three components, thesystem ˆ H s , the bath ˆ H b and the system-bath coupling ˆ H c . Generally, they areˆ H = ˆ H s + ˆ H b + ˆ H sb , ˆ H s = X mn h s mn ( R )ˆ b † m ˆ b n + U ( R ) , ˆ H b = X kα ǫ kα ˆ c † kα ˆ c kα , ˆ H c = X m,kα V m,kα ( R )ˆ b † m ˆ c kα + H . c ., where m , n label system orbitals, and ˆ b † m (ˆ b m ) creates (annihilates) an electron in thesystem orbital m . ˆ c † kα (ˆ c kα ) creates (annihilates) an electron in the k -th orbital of the lead α ( α = L , R which means left and right leads). V m,kα represents the tunneling element betweenthe system orbital m and the lead orbital kα . Within this model, the most general systemHamiltonian can be written in Pauli matrices representation ( σ i ) as: h s = h ( x, y ) · σ = X i =1 , , h i ( x, y ) σ i , where { h i } is real. Note that the inclusion of h makes the Hamiltonian possibly complex-valued, as might arise from external magnetic field or spin orbit coupling.For simplicity, we make a non-Condon approximation, i.e. V m,kα is assumed independentof R . Under this approximation, Eq. (3) can be simplified as[43, 44], γ µν = Z dǫ π Tr (cid:8) ∂ µ h s ∂ ǫ G R ∂ ν h s G < (cid:9) + H . c ., (4)6here G R = ( ǫ − h s − Σ R ) − is the (two-level) system retarded Green’s function, Σ R mn = P kα V m,kα g R kα V ∗ n,kα is the system self energy, and g R kα = ( ǫ − ǫ kα + iη ) − is the lead retarded selfenergy ( η → + ). G < is the system lesser Green’s function and, provided that the imaginarysurrounding mentioned in Sec. I is quadratic[44] or the system spectral broadening due tothe leads is finite[45], G < can be calculated by the Keldysh equation again, G < = G R Σ < G A , where Σ In this letter, we will focus on the equilibrium friction tensor; the nonequilibrium casewill be discussed in a subsequent paper. Eq. (5) is a compact expression for the friction8ensor, from which we can make two important conclusions in the abstract.First, according to Eqs (7)-(8), γ A µν is proportional to h · ( ∂ µ h × ∂ ν h ). Therefore, γ A µν willvanish when at least one element of h is zero. The tensor will also vanish if two elements of h are the same. These facts demonstrate not only that an imaginary off-diagonal coupling( h ) is required for a nonzero γ A µν , but also that the key source of a nonzero γ A µν is the spatialdependence of the phase of the off-diagonal coupling, tan − ( h /h ). After all, if h = 0 or h ( x, y ) = h ( x, y ), we can find a constant change of basis transformation that guaranteesa globally real-valued Hamiltonian and therefore γ A µν = 0. In other words, in such a case,there is no Lorentz-like force.Second, according to Eqs. (5)-(9), one can construct several nonequivalent Hamiltoniansthat generate equivalent friction tensors. To see this, note that, when the system is in equi-librium, the symmetric terms in Eq. (5) all have dot product dependence on h , namely h , P i ∂ µ h i ∂ ν h i and P i ∂ µ h i h i . Thus, the symmetric terms are invariant to any permutation of h = { h , h , h } . Moreover, the two terms comprising γ A µν depend on h · ( ∂ µ h × ∂ ν h ), whichare also invariant under cyclic permutation of the h elements. Thus, different Hamiltonianscan generate the same friction tensor and, as a practical matter, this should have experi-mental consequences as some Hamiltonians are undoubtedly easier to realize than others.For example, in Eq. (10) we will consider a model Hamiltonian with diagonal coupling h = x + ∆; here, as in standard Marcus theory, ∆ is a driving force that will be shown toplay an important role in generating a large antisymmetric friction tensor. Nevertheless, ifones imagines permuting the h elements by substituting h → h → h , then the parameter∆ will enter on the off-diagonal of the Hamiltonian and can be realized, e.g., by tuning anexternal magnetic field.These are the only direct, general conclusions we can make from Eqs. (5)-(9). Next,we will focus on a model problem which can yield further insight using numerical analysis.We imagine the standard case of two shifted parabolas, expressed in a nuclear space withtwo dimensions and with a driving force of 2∆. Mathematically, the system Hamiltonian istaken to be of the form: h s ′ = ( x + 1) + y + ∆ Ax − iByAx + iBy ( x − + y − ∆ . Since a potential proportional to the identity ( U ( R ) in ˆ H s ) does not enter the friction tensor,9e can reduce the Hamiltonian to the following form h s = x + ∆ Ax − iByAx + iBy − x − ∆ , (10)and calculate the electronic friction tensor by using Eq. (5). Recall that γ A µν ∝ h s · ( ∂ µ h s × ∂ ν h s ) = AB ∆. Thus, as argued above, if there is no change in the phase of the off-diagonalcoupling ( A = 0 or B = 0) in the nuclear space, we will find that γ A xy = 0 . Also notice thatwhen the driving force ∆ = 0, again γ A µν = 0. Beyond these two extreme cases, we will findboth symmetric and antisymmetric components of the friction tensor. y -4-2024 γ xx × -0.10-0.050.000.050.10 γ S xy × y x-4-2024 -4 -2 0 2 4 γ A xy × x-4 -2 0 2 4 γ yy × FIG. 1. Friction tensor calculation results: γ xx (top left), γ S xy (top right), γ A xy (bottom left) and γ yy (bottom right). Parameters: ˜Γ = 1, µ R = µ L = 0, β = 1, A = 1, B = 1, ∆ = 3. In Fig. 1, we show contour plots for the friction tensor with ∆ = 3 . A = B = 1 . A = B and neither is small, there is a strong change of phase in the off-diagonalcoupling due either to an external magnetic field or a spin-orbit coupling. Several features are10lear from the contour plot. First, the magnitude of γ A xy is comparable with the symmetricfriction tensor γ xx and larger than both γ yy and γ S xy . And in Fig. 2, we lower the temperature( β = 2), and the antisymmetric friction tensor γ A xy is now one order larger than all othersymmetric friction tensors. Thus, clearly Lorentz-like motion can be as important as anydissipative process. Second, the magnitude of γ A xy is maximized around the avoided crossingat ( − . , y -4-2024 γ xx × -0.10-0.050.000.050.10 γ S xy × y x-4-2024 -4 -2 0 2 4 γ A xy × x-4 -2 0 2 4 γ yy × FIG. 2. Friction tensor calculation results: γ xx (top left), γ S xy (top right), γ A xy (bottom left) and γ yy (bottom right). Parameters: ˜Γ = 1, µ R = µ L = 0, β = 2, A = 1, B = 1, ∆ = 3. Notice thatall the results in Figs. 1 and 2 have mirror symmetry about x = − . − ∆ / ( A + 1) and y = 0,because all six terms in Eq. (5) are functions of [ x + ∆ / ( A + 1)] and B y when the system isin equilibrium. 11o understand the strong effect of temperature on the friction tensor (as relevant betweenFigs. 1 and 2), note that, according to Eq. (8) and (9), the friction tensor in Eq. (5) can berepresented as γ µν = R dǫ ( F S µν + ˜ F A µν ) f . Here F S µν and ˜ F A µν contain the integrands of Eqs. (6)and (7) respectively (excluding the Fermi-Dirac distribution f ). These integrands containnot only the broadening effect of the metal (recall that G < = − i f Im G R ), but also thederivatives of h s as a function of nuclear coordinates ( µ , ν ) plus the partial derivative ∂ ǫ G R .Since the temperature appears only in the Fermi-Dirac distribution, changing β effectivelycontrols the overlap between F and f (and between ˜ F and f ). This analysis leads to thedifferent orders of magnitude for the symmetric and antisymmetric friction tensors as shownin Fig. 2 — and the effect would grow stronger for even lower temperatures. As a side note,beyond temperature effects, we must mention that we can also utilize the chemical potential µ to control γ S and γ A . For instance, since F and ˜ F are both odd functions of ǫ , when thechemical potential is high or low enough, both γ S and γ A disappear.Finally, we investigate how the relative strength of the antisymmetric friction tensorchanges as a function of how the off-diagonal coupling changes phase. Figure 3 (a)(b)correspond to the same parameters as Fig. 2, except that A = 0 . 05 ( B = 1). As can beeasily found, the antisymmetric friction tensor γ A xy still has the same order of magnitude asthe symmetric friction tensors. As we keep lowering down the value of A to 0.01, as shown inFig. 3 (c)(d), the antisymmetric friction tensor approaches zero rapidly. We conclude thatin any theoretical or experimental works which consider the external magnetic field or/andspin-orbit coupling effect with reasonable changes for the phase (here larger than 5%) in thenuclear space, we cannot ignore the effect of the antisymmetric friction tensor.Lastly, before concluding, we should summarize a few results that are not describedabove (but are addressed in the SI). First, we investigate the dependence of γ on ˜Γ in SI F.We find that, when the system-bath coupling strength ˜Γ grows larger, both the symmetricand antisymmetric friction tensors become smaller and these tensors are nonzero over aneffectively smaller portion of nuclear configuration space. Second, while we have consideredan avoided crossing above, we investigate a true, complex-valued conical intersection in SIE. There, we show that the presence of a true conical intersection does not give rise to anenormous electronic friction tensor. Third, and most importantly, throughout this letter, wehave focused mostly on the magnitude of the antisymmetric friction tensor. It goes withoutsaying that, in a basis of spin orbitals, switching spin up and spin down orbital will swap h -2-1012 γ xx × -0.8-0.6-0.4-0.20.00.20.40.60.8 γ xx × y x-2-1012 -5 -4 -3 -2 -1 0 γ A xy × x-5 -4 -3 -2 -1 0 γ A xy × FIG. 3. Friction tensor calculation results (only γ xx and γ A xy are shown). Parameters for (a) and(b) are ˜Γ = 1, µ R = µ L = 0, β = 2, A = 0 . B = 1, ∆ = 3; (c) and (d) have the same parametersas (a) and (b), except that A = 0 . and − h and lead to different signs of γ A µν . Thus, different spins will feel different directionsof the Lorentz force and the present formalism may underlie spin selectivity for molecularprocesses near metal surfaces[40–42]. IV. CONCLUSIONS AND OUTLOOK We have demonstrated that a Lorentz force operates on nuclei in equilibrium for systemswith complex-valued Hamiltonians. We have investigated a simple model of two shiftedparabolas, and shown that the magnitudes of the relevant frictional component (i.e. thesymmetric part γ S µν and the antisymmetric part γ A µν ) can be controlled by tuning the drivingforce ∆ and the inverse temperature β . The antisymmetric part can be one order larger13han the symmetric part for low temperatures. Moreover, γ A µν and γ S µν can be of comparablemagnitude even when the phase change of the off-diagonal coupling is very small. All ofthese results show that, for any relaxation processes with an external magnetic field or/andspin orbit couplings (leading to a complex-valued Hamiltonian), careful consideration of aLorentz force due to the nuclear Berry curvature is necessary.Looking forward, there is a deep question about whether the Lorentz force described herecan help explain spin-selectivity as found in chiral-induced spin selectivity experiments[40–42]. To that end, note that a recent paper has argued empirically that, for a molecule inthe gas phase, the Lorentz force is accentuated dynamically when the molecule passes neara conical intersection, but only if that conical intersection is slightly modified by spin-orbitcoupling[39]. For our part, we find a similar result near a metal surface, i.e. the Lorentz forceis maximized if ˆ H s admits an energy gap rather than displaying a true conical intersection.In fact, note that, according to Eq. 7, the antisymmetric part of the friction tensor is zeroif one considers a two-dimensional linear-vibronic complex-valued Hamiltonian (See SI Efor details). One must wonder whether such a result will still hold if we consider three-dimensional models of conical intersections, where the conical intersection point can mapto a legitimate 3D branching plane. At present, we do not have a definitive answer to thisquestion.Finally, the outstanding question remaining is the relationship between the equilibriumLorentz force presented here (in the case of a complex-valued Hamiltonian) and previouslynonequilibrium Lorentz forces (derived in the case of real-valued Hamiltonian)[18, 27]. Onecan ask: what is the relationship between the two different tensors? Can they add to eachother constructively? Can they be controlled individually by the properties of two leads?We will address these questions, as well as estimate the sizes of these forces for realisticmolecules, in a future publication. Appendix A: Brief Review of the Friction Tensor Equation. (2) can be further simplified when a non-interacting Hamiltonian is considered,and the result is: γ µν = − π Z ∞−∞ dǫ Tr (cid:8) ∂ µ HG R ∂ ν σ ss G A (cid:9) , (A1)14here ( σ ss ) pq = Tr n ˆ ρ ss ˆ d † p ˆ d q o is the steady-state density matrix. In general, σ ss contains allthe information of ˆ H so that the analytic derivative of the steady-state distribution (sayBoltzmann distribution which depends on R ) is hard to be calculated. However, we canstill use this general expression to show that γ A µν vanishes when ˆ H is real-valued and thesystem is in equilibrium; see SI D for a proof. In practice, we will use a nonequilibriumGreen’s function technique to treat σ ss . Notice that σ ss can be expressed in terms of thelesser Green’s function in the frequency domain,( σ ss ) pq = − i G In this section, we prove that both the electronic friction tensor γ µν and its symmetricpart γ S µν are positive definite when the system is in equilibrium. That is, P µν X µ γ µν X ν > X = 0. We start from the identity (which can be easily proved bythe fundamental theorem of calculus), e − t ˆ A ddλ e t ˆ A = Z t e − s ˆ A d ˆ Adλ e s ˆ A , A is an arbitrary operator and t is real. By replacing ˆ A = ln ρ and t = 1, we obtainthe following expression for the derivative of the steady-state density matrix (for notationalsimplicity, we discard the subscript ss of ˆ ρ ss and the hat symbolˆfor operators in this section), ddλ ρ = Z ds ρ − s d ln ρdλ ρ s . (B1)Therefore, Eq. (2) can be recast into γ µν = − Z ∞ dt Z ds Tr (cid:8) e − iHt ρ − s ∂ ν (ln ρ ) ρ s e iHt ∂ µ H (cid:9) . Next we apply the equilibrium condition for the steady-state density matrix, ρ = e − βH /Z where Z ≡ Tr (cid:8) e − βH (cid:9) is the partition function. This condition is equivalent to ∂ µ H =( − ∂ µ ln ρ − ∂ µ Z/Z ) /β . Accordingly, γ µν = 1 β Z ∞ dt Z ds Tr (cid:8) e − iHt ρ − s ∂ ν (ln ρ ) ρ s e iHt ∂ µ (ln ρ ) (cid:9) . (B2)Notice that another term proportional to ∂ µ Z/Z vanishes since the integrand becomesTr { ∂ ν ρ } = ∂ ν Tr { ρ } = 0 after using Eq. (B1).Now, in order to see the structure of γ µν more easily, we rewrite the trace in the Lehmannrepresentation, H | a i = E a | a i , γ µν = X ab β Z ∞ dt Z ds e − iE a t (cid:18) e − βE a Z (cid:19) − s h a | ∂ ν (ln ρ ) | b i (cid:18) e − βE b Z (cid:19) s e iE b t h b | ∂ µ (ln ρ ) | a i = 1 β X ab Z ds ρ − sa ρ sb h a | ∂ ν (ln ρ ) | b ih b | ∂ µ (ln ρ ) | a i iE b − E a + iη , where ρ a ≡ e − βE a /Z (same for ρ b ), and η → + . We further split γ µν into the symmetricpart and the antisymmetric part, and analyze the symmetric part first, γ S µν = 12 X ab β Z ds ρ − sa ρ sb h a | ∂ ν (ln ρ ) | b ih b | ∂ µ (ln ρ ) | a i i (cid:18) E b − E a + iη − E b − E a − iη (cid:19) = π X ab β Z ds ρ − sa ρ sb h a | ∂ ν (ln ρ ) | b ih b | ∂ µ (ln ρ ) | a i δ ( E b − E a )= πβ X ab ρ a h a | ∂ ν (ln ρ ) | b ih b | ∂ µ (ln ρ ) | a i δ ( E a − E b ) . (B3)Thus, X µν X µ γ µν X ν = πβ X ab ρ a (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) h a | X ν X ν ∂ ν (ln ρ ) | b i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) δ ( E b − E a ) > . γ A µν is also real, X µν X µ γ µν X ν = X µν X µ (cid:0) γ S µν + γ A µν (cid:1) X ν = X µν X µ γ S µν X ν > . Hence, γ µν is positive definite at equilibrium. Appendix C: Fluctuation-Dissipation Theorem Based on the QCLE In this section, we further investigate the fluctuation-dissipation theorem based on theQCLE. Note that the derivation published in the SI of Ref. 2 erroneously divides by zero,but the final result is correct for a real-valued Hamiltonian (as we will now show).In Ref. 2, a Fokker-Planck equation (or more specifically a Kramer’s equation) based onan analysis of the QCLE in the adiabatic theorem was derived along with the correspondingrandom force δ ˆ F µ and correlation function ¯ D S µν . δ ˆ F µ = − ∂ µ ˆ H + Tr n ∂ µ ˆ H ˆ ρ ss o , ¯ D S µν = 12 Z ∞ dt Tr n e i ˆ Ht δ ˆ F µ e − i ˆ Ht (cid:16) δ ˆ F ν ˆ ρ ss + ˆ ρ ss δ ˆ F ν (cid:17)o . (We follow the same notation as in Ref. 2, even though ¯ D S µν is not symmetric when acomplex-valued Hamiltonian is considered here.) Since ∂ ν ln ˆ ρ ss = βδ ˆ F ν , we can rewrite Eq.(B2), γ µν = β Z ∞ dt Z ds Tr n ( ˆ ρ ss ) − s δ ˆ F ν (ˆ ρ ss ) s δ ˆ F µ ( t ) o , where δ ˆ F µ ( t ) is written in Heisenberg picture. We then express both γ µν and ¯ D S µν in theLehmann representation, obtaining γ µν = β X ab Z ∞ dt Z ds ρ − sa ρ sb h a | δ ˆ F ν | b ih b | δ ˆ F µ ( t ) | a i (C1)= β X ab ρ b − ρ a β ( E a − E b ) Z ∞ dt h a | δ ˆ F ν | b ih b | δ ˆ F µ ( t ) | a i , (C2)¯ D S µν = X ab ρ a + ρ b Z ∞ dt h a | δ ˆ F ν | b ih b | δ ˆ F µ ( t ) | a i . (C3)At this point, recall that, according to a Kramer’s equation, a particle’s equation of motiondoes not depend on the antisymmetric component of the random force correlation function.17n other words, if A is the phase space density of a particle near a surface, the equation ofmotion for A satisfies[2]: ∂ t A = − X α P α m α ∂ α A − X α F α ∂ A ∂P α + X αν γ αν ∂∂P α (cid:18) P ν m ν A (cid:19) + X αν ¯ D S αν ∂ A ∂P α ∂P ν . (C4)Thus, the physical meaning of the antisymmetric component of ¯ D S µν is not clear. Perhapsnot surprisingly, then, Equations (C2) and (C3) do not satisfy a “fluctuation-dissipationtheorem”, γ µν = β ¯ D S µν . However, a valid fluctuation-dissipation theorem condition can beestablished at equilibrium if we consider only the symmetric component of the random forceand friction. To do so, we further integrate out the time variable in ¯ D S µν ,¯ D S µν = X ab ρ a + ρ b iE b − E a + iη h a | δ ˆ F ν | b ih b | δ ˆ F µ | a i , and then “symmetrize” ¯ D S µν :12 (cid:0) ¯ D S µν + ¯ D S νµ (cid:1) = 12 X ab ρ a + ρ b (cid:18) iE b − E a + iη + iE a − E b + iη (cid:19) h a | δ ˆ F ν | b ih b | δ ˆ F µ | a i = π X ab ρ a h a | δ ˆ F ν | b ih b | δ ˆ F µ | a i δ ( E a − E b ) , which is equal to γ S µν /β when the system is in equilibrium (please compare to Eq. (B3)).In SI D, we will show that when a real-valued Hamiltonian is considered and the systemis in equilibrium, the antisymmetric friction tensor γ A µν vanishes. In this situation, γ µν = γ S µν = β ¯ D S µν . Appendix D: No Antisymmetric Friction Tensor γ A µν When the Hamiltonian Is Realand the System Is in Equilibrium In this section, we show that the antisymmetric friction γ A µν vanishes when the Hamilto-nian is real and the system is in equilibrium (for a non-interacting Hamiltonian). We startfrom Eq. (A1), γ αν = − X kl Z ∞−∞ dǫ π h k | ∂ α h | l i ǫ − ǫ l − iη h l | ∂ ν σ ss | k i ǫ − ǫ k + iη , (D1)18here η → + , and h = X k ǫ k | k ih k | . At equilibrium we have σ ss = X k P ( ǫ k ) | k ih k | , where P ( ǫ k ) = e − βǫ k / P k e − βǫ k is the Boltzmann distribution. We focus on γ A αν ∝ γ αν − γ να .We divide the summation P kl in Eq. (D1) into three cases: X kl = X k = l + X k = l,ǫ k = ǫ l + X k = l,ǫ k = ǫ l . Also, we utilize the following two identities to replace h l | ∂ ν σ ss | k i in Eq. (D1): ∂ ν ǫ k δ lk = ( ǫ l − ǫ k ) h l | ∂ ν | k i + h l | ∂ ν h | k i , (D2) h l | ∂ ν σ ss | k i = ∂ ν ǫ k ∂f ( ǫ k ) ∂ǫ k δ kl + h l | ∂ ν | k i ( f ( ǫ k ) − f ( ǫ l )) . (D3)As a result, γ A αν ∝ X k = l,ǫ k = ǫ l Z dǫ π h k | ∂ α h | l i ǫ − ǫ l − iη (cid:26) −h l | ∂ ν h | k i f ( ǫ k ) − f ( ǫ l ) ǫ l − ǫ k (cid:27) ǫ − ǫ k + iη + X k = l,ǫ k = ǫ l Z dǫ π { } ǫ − ǫ l − iη { } ǫ − ǫ k + iη − ( α ↔ ν ) . (D4)Note that the diagonal term (the first summation P k = l ) does not contribute to γ A αν . Thesecond line of Eq. (D4) is zero, which is consistent with the assumption we made for gettingthe friction tensor Eq. (2): we request that a steady state ˆ ρ ss can be achieved, and so whenthere exists a degeneracy the steady state is not well-defined.Next, let’s focus on the only contributing summation in Eq. (D4). By using the identities, c π Z ∞−∞ dy e icxy = δ ( x ) ,θ ( t − t ) = i Z ∞−∞ dω π e − iω ( t − t ) ω + iη , 19e can derive the following expression: Z ∞ dt e i ( E b − E a ) t = Z ∞−∞ dt θ ( t ) e i ( E b − E a ) t = Z ∞−∞ dt (cid:18) i Z ∞−∞ dω π e − iωt ω + iη (cid:19) e i ( E a − E b ) t = i Z ∞−∞ dω π ω + iη Z ∞−∞ dt e i ( E b − E a − ω ) t = iE b − E a + iη . (D5)We then use Eq. (D5) to rewrite the first line in Eq. (D4), obtaining X k = l,ǫ k = ǫ l Z dǫ π h k | ∂ α h | l i i Z ∞ dt e − i ( ǫ − ǫ l ) t (cid:26) −h l | ∂ ν h | k i f ( ǫ k ) − f ( ǫ l ) ǫ l − ǫ k (cid:27) ( − i ) Z ∞ dt ′ e i ( ǫ − ǫ k ) t ′ = X k = l,ǫ k = ǫ l Z dǫ π Z ∞ dt Z ∞ dt ′ e iǫ ( t ′ − t ) e iǫ l t e − iǫ k t ′ h k | ∂ α h | l i (cid:26) −h l | ∂ ν h | k i f ( ǫ k ) − f ( ǫ l ) ǫ l − ǫ k (cid:27) = X k = l,ǫ k = ǫ l Z ∞ dt Z ∞ dt ′ δ ( t ′ − t ) e iǫ l t e − iǫ k t ′ h k | ∂ α h | l i (cid:26) −h l | ∂ ν h | k i f ( ǫ k ) − f ( ǫ l ) ǫ l − ǫ k (cid:27) = X k = l,ǫ k = ǫ l Z ∞ dt e i ( ǫ l − ǫ k ) t h k | ∂ α h | l i (cid:26) −h l | ∂ ν h | k i f ( ǫ k ) − f ( ǫ l ) ǫ l − ǫ k (cid:27) = X k = l,ǫ k = ǫ l iǫ l − ǫ k + iη h k | ∂ α h | l i (cid:26) −h l | ∂ ν h | k i f ( ǫ k ) − f ( ǫ l ) ǫ l − ǫ k (cid:27) . (D6)By using the identity, 1 ω ± iη = P ω ∓ iπδ ( ω ) , we can recast Eq. (D6) to get i X k = l,ǫ k = ǫ l P ǫ l − ǫ k h k | ∂ α h | l i (cid:26) −h l | ∂ ν h | k i f ( ǫ k ) − f ( ǫ l ) ǫ l − ǫ k (cid:27) + π X k = l,ǫ k = ǫ l δ ( ǫ l − ǫ k ) h k | ∂ α h | l i (cid:26) −h l | ∂ ν h | k i f ( ǫ k ) − f ( ǫ l ) ǫ l − ǫ k (cid:27) . (D7)Apparently, the second term is 0 since ǫ k = ǫ l . Also, the first term will not contribute if theHamiltonian is real, since the friction tensor γ µν must be real and so we only need the realpart of γ A µν . Therefore, we prove that the antisymmetric friction tensor vanishes when a realHamiltonian is considered and the system is in equilibrium. Appendix E: h s with a Conical Intersection (Equilibrium) In this section, we model a simple E ⊗ ǫ Jahn-Teller system in the presence of spin-orbit coupling and a nuclear bath to demonstrate that no significant enhancement of the20ntisymmetric friction tensor arises from the presence of a conical intersection (which doesprovide infinite derivative couplings for an isolated system). To prove this point, assumethat a conical intersection is located at (0 , h ( x, y ) = P x + Q xy + R y + S x + T y, where P , Q , R , S and T are constant vectors. If only two of these constant vectors arenonzero, the antisymmetric friction tensor γ A µν (which depends on h · ( ∂ µ h × ∂ ν h )) mustvanish. Moreover, when there are no linear terms (only P , Q and R are nonzero as in aRenner-Teller intersection[46]), γ A µν must still disappear.Finally, in order to derive a nonzero γ A µν , we must include a linear complex coupling ontop of a second order real-valued E ⊗ ǫ Jahn–Teller system, h = y − x xy xy x − y + A − y xx y + B − iyiy , This Hamiltonian can be experimentally realized as a regular triangular molecule with twodegenerate electronic states interacting with a doublet of vibrational states (up to quadraticorder) with a spin-orbit coupling between the p x and p y orbitals included.Figure 4 plots the corresponding friction tensor results, and the ratio (cid:12)(cid:12) γ A xy /γ S (cid:12)(cid:12) . Afterscanning a reasonable set of parameters, we have never, in practice, been able to find aHamiltonian where this ratio is more than 25% in the vicinity of the origin. Thus, wedo not observe any enhancement of an antisymmetric pesudomagnetic field as caused by aconical intersection. Appendix F: Friction Tensor Results (Different ˜Γ ’s) For molecules in the gas phase, it is fairly standard to ascertain the size of a Berry forcefrom the relevant derivative couplings[36]. In this manuscript, however, our goal has beento report the size of the Berry force in a condensed environment. Thus, for completeness, inFigs. 5 and 6, we plot the friction tensor as a function of ˜Γ, which represents coupling of themolecule to the metal. We find that, in general, when the coupling grows (i.e. ˜Γ gets large),both the antisymmetric and symmetric components of the friction tensor become smaller.21 -2-1012 γ xx × -4.00-2.000.002.004.00 γ S xy × y x-2-1012 -2 -1 0 1 2 γ A xy × x-2 -1 0 1 2 γ yy × FIG. 4. Friction tensor calculation results near a conical intersection: γ xx (top left), γ S xy (topright), γ A xy (bottom left) and γ yy (bottom right). 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