Theory of Phonon-Assisted Adsorption in Graphene: Many-Body Infrared Dynamics
TTheory of Phonon-Assisted Adsorption in Graphene: Many-Body Infrared Dynamics
Sanghita Sengupta
Institut Quantique and D ´ e partement de Physique,Universit ´ e de Sherbrooke, Sherbrooke, Qu ´ e bec, Canada J1K 2R1 (Dated: January 6, 2020)We devise a theory of adsorption of low-energy atoms on suspended graphene membranes main-tained at low temperature based on a model of atom-acoustic phonon interactions. Our primarytechnique includes a non-perturbative method which treats the dynamics of the multiple phononsin an exact manner within the purview of the Independent Boson Model (IBM). We present a studyon the effects of the phonons assisting the renormalization as well as decay of the incident atompropagator and discuss results for the many-body adsorption rates for atomic hydrogen on graphenemicromembranes. Additionally, we report similarities of this model with other branches of quan-tum field theories that include long-range interactions like quantum electrodynamics (QED) andperturbative gravity. I. INTRODUCTION
How does an atom adsorb to a surface? From theviewpoint of quantum field theory, can we understandthis phenomenon by including the effects of the surfacephonons? Over the years, theoretical predictions as wellas experimental endeavors have elucidated a significantrole played by the surface phonons in mediating the pro-cess of adsorption . While most of the previous workhas been conducted on conventional three-dimensionalmaterials, the discovery of graphene has led to a recenteffort to understand this phenomenon with special focuson phonon dispersion, tunability of the atom-phonon in-teraction and possible applications . This brings us tothe topic of our current work.In this paper, we will devise a theory of adsorptionbased on a simple model of atom-phonon interaction. Us-ing the tools of quantum field theory, we predict adsorp-tion rates for these suspended graphene membranes thatare maintained at low temperature. Let us also men-tion that our naive model shows an interesting similaritywith other branches of quantum field theory. Theorieswith long-range interactions like quantum electrodynam-ics and perturbative gravity are seen to exhibit severeinfrared (IR) divergences in their scattering rates due tothe emission of infinitely many soft quanta (soft meaningvibrations with energy (cid:15) → . Quite remarkably,our non-relativistic model is also plagued with severe IRdivergences in the adsorption rate that appear as a re-sult of emission of infinitely many soft phonons origi-nating from the long-range tail of van der Waals (vdW)interactions . Since physically measurable entitiescan never be infinity, this IR-divergent adsorption rateposes a serious concern for the application and validityof the theory. However, thanks to the Kinoshita-Lee-Nauenberg theorem , we realize that these infinitiesare infact unreal and proper application of resummationprocedures can lead us to meaningful results.Quite naturally there is an ongoing attempt todevise non-perturbative methods to tackle these IRdivergences . As a matter of fact, these resum-mation methods implemented in these condensed mat- ter systems are similar in essence to the correspondingQED counterparts. To illustrate this point, let us brieflymention the main idea behind each of them: (i) Bloch-Nordsieck scheme - the method tackles the IR divergenceby allowing for the inclusion of emission of infinitelymany soft quanta and summing over them ,(ii) Faddeev-Kulish mechanism - this method proposesa dressing of asymptotic states by a cloud of soft quantausing a coherent state formalism and (iii) impos-ing a IR cut-off . While all three methods give com-parable and reasonable answers for zero-temperature ad-sorption rates in graphene micromembranes , the low-temperature result seems contentious with the core ofthe debate surrounding the effect of the IR cut-off, i.ethe effect of low-energy phonons.In one numerical study performed for low tempera-ture membranes (T= 10 K) authors considered ad-sorption as a fast process mediated by a single phononand derived finite, enhanced adsorption rates. To keeptheir computations tractable, they imposed an IR cut-off. Ref. [20] used a coherent-state phonon basis formal-ism (dressing the asymptotic states by a cloud of softphonons), claimed to cure and remedy the IR problem atfinite temperature and predicted adsorption rates thattend to zero. However, this method sparks some seri-ous questions. (i) In the derivation of the final adsorp-tion rate Γ (obtained by summing over partial rates Γ n ),Ref. [20] has considered emission of soft phonons withenergy (cid:15) ∼ ω c where ω c = 0 .
183 meV. Since the defini-tion of IR problem corresponds to (cid:15) →
0, it is not clearto us how the IR problem is remedied in this case. (ii)The partial rates corresponding to n phonon processeswere derived under the assumption, ω c (cid:29) g b with g b asthe atom-phonon coupling strength in the membrane. Byprevious line of thought this implies (cid:15) (cid:29) g b . We are notcertain if this is the right energy scale for the problemat hand (presumably, the IR scale should be the lowestenergy scale in the formalism).In light of these recent developments and questions,we are motivated to reconsider this problem in termsof a partial resummation technique that has been pre-viously used for graphene membranes maintained at zero a r X i v : . [ c ond - m a t . m e s - h a ll ] J a n and high temperature (by high, we mean a tempera-ture scale which is comparable to the Debye frequencyof the phonons). This technique is based on the Bloch-Nordsieck scheme of resummations and uses the Inde-pendent Boson model (IBM) to account for the dynam-ics of the phonons. While in the zero temperature case,the technique cures the IR problem at (cid:15) →
0, for mi-cromembranes it predicts an adsorption rate that is fi-nite, non-zero and size-independent and is within goodapproximation (1%) of the zero temperature Golden ruleresult . However, in the high temperature regime, thetechnique retains some residues of the log singularity be-cause of the temperature effects of the Bose distribution.As a consequence, the method predicts adsorption ratesthat increase with increasing temperature and membranesizes. The rate is also enhanced with respect to the finitetemperature Golden rule result .For the low-temperature formalism, we will not claimto cure the IR problem in this model, in fact we willtake a modest approach pertinent to all condensed mat-ter systems. We remind ourselves that the natural IRcut-off is related to the size of the system (cid:15) ∼ v s /L ( v s is the velocity of sound in the material and L is the sizeof the membrane, we set (cid:126) = 1 all through the paper).We thus subject our partial resummation technique toIR cut-offs as low as (cid:15) = 10 − meV (the current lowestwithin all recent literature for low temperature formal-ism) and systematically increase it to 0.8 meV, naturally,these cut-offs correspond to membrane sizes 10 µ m to 5nm, respectively. Our primary aim will be to understandhow the soft phonons i.e, the IR cut-off, affect the ad-sorption process with clear focus on the renormalizationof the energy of the atom leading to the formation of anacoustic-like polaron and finally the decay of the atom.We will investigate in detail the true dynamics of thephonon bath with respect to the effects of time-evolution,temperature and atom-phonon coupling. To the best ofour knowledge, the properties of the true dynamics ofthe phonon bath related to this model, have not beenexplored before. As we will see, a key result of this in-vestigation will lead us to a characteristic time scale forthe phonon dynamics. This characteristic time scale willindeed serve as a crucial parameter for the prediction ofthe adsorption rate.We must also mention that in addition to theresummation technique within the IBM there ex-ists another non-perturbative method, namely the Non-Crossing Approximation ( N CA ) which deals withsumming infinite numbers of rainbow diagrams (Feyn-man diagrams where phonon lines donot cross). Thisresummation technique was previously found to be suc-cessful in treating the IR problem of the model for thezero temperature case and predicted similar result forthe adsorption rate as the method of IBM . Howeverfor finite temperatures, NCA fails to give tractable con-vergent results with increasing membrane sizes ( (cid:15) → FIG. 1. Left: Model of atom − membrane interaction .Weak forces of van der Waals interaction hold together theatom and the membrane that are now separated from eachother by a minute distance ( z ). The quantized vibrations ofthe membrane that appear as ripples in the membrane startto interact with the incoming adatom and mediate the processof adsorption . Right: A quantum mechanical descriptionof the adsorption process. Transition of the atom from thecontinuum E k to the bound state E b (supported in the vdWpotential V ( z ) of the physisorption well) with the emissionof phonons. The Hamiltonian corresponding to the process isexplained in Eqs. 1, 2 and 3 within Sec. II. the technique within the IBM. The general structure ofthe paper is as follows: in Sec. II, we introduce the modelHamiltonian, the physical features of the model and themethodology to calculate the resummed many-body ad-sorption rates. Sec. III discusses the low-temperaturedressed propagator within the IBM, while consequencesof atom-phonon interaction on the renormalization anddecay of the atom propagator are discussed in Sec. IV.Finally, in Sec. V, we derive the many-body adsorptionrates based on the partial resummation technique givenin Sec. II and discuss the results within the context of thephonon-effects explained in Sec. IV. After a summary ofour results in Sec. VI, we conclude with a few pertain-ing questions related to this model and IR divergences inother field theories . II. PRELIMINARIES AND METHODOLOGY
In this section, we provide an overview of the physi-cal features of the model and lay out the procedure tocalculate the adsorption rate for the atom.We consider a continuum model for low-energy ph-ysisorption on a suspended graphene membrane. Whena low-energy incident atom impinges normally on thegraphene membrane, it excites the out-of-plane trans-verse acoustic mode (ZA) of the membrane therebymediating interactions that we refer as the atom-phonon coupling. Let us begin with a description of the Hamil-tonian of our model, H = H + H i , where H and H i represent the unperturbed and interaction Hamiltonians.For the unperturbed part H , we can write: H = E k c † k c k − E b b † b + (cid:88) q ω q a † q a q (1)where, E k is the energy of the incident atom withthe corresponding operators: c k ( c † k ) that annihilates(creates) a particle in the continuum channel; b ( b † )annihilates (creates) a particle in the bound state | b (cid:105) with energy - E b in the static van der Waals potential V ( z ). For graphene membranes, the physisorption wellis E b = 40 meV . The isotropic surface is modelledby a phonon bath with energy ω q and opera-tors a q ( a † q ) that annihilates(creates) a ZA phonon inthe bath . For the graphene membrane underan out-of-plane tension γ , the energy ω q is related viathe dispersion relation ω q = v s q with v s = (cid:112) γ/σ =6 . × m/s (velocity of sound in graphene), σ = massdensity of the membrane and the Debye frequency ofgraphene ω D = 65 meV .Let us now describe the Hamiltonian representing theatom-phonon interaction H i = H bi + H ki . The atom-phonon interaction in the continuum is given by, H ki = − ˜ g kb ξ ( c † k b + b † c k ) (cid:88) q ( a q + a † q ) (2)with ˜ g kb being the corresponding atom-phonon couplingin the continuum channel, ξ is a frequency independentparameter that depends on the specific form of atom-excitation coupling and is given as ξ = (cid:112) (cid:126) / (4 Lσv s ) .For the bound channel, we have: H bi = − ˜ g bb ξb † b (cid:88) q ( a q + a † q ) (3)with ˜ g bb as the atom-phonon coupling for an atom boundto the membrane . For a surface without corruga-tions, a detailed procedure to calculate the atom-phononcouplings ˜ g ’s from the Hamiltonian of the atom-surfacescattering is given in Ref. [36], this method is then suit-ably extended for a graphene membrane-atom interac-tion in Ref. [9]. The general idea behind the proce-dure is to calculate the coupling constants through sur-face distortions/displacements, quantitatively relating itto the matrix element of the first derivative of the sur-face potential. The interaction between the atom andsurface is modelled via the vdW potential V ( z ) which isthen Taylor expanded in small phonon-displacements inthe target bath (a procedure valid for low-energies andtemperatures) . For the vdW potential V ( z ), one con-siders the asymptotic form of the long-range attractiveinteraction between the static flat graphene membraneand the neutral atom (equivalent to the Casimir-Polderpotential between a 2D insulating solid and a neutralatom) given by, V ( z ) = − πC (cid:26) z − z + L ) (cid:27) (4) where z is the distance between the particle and themembrane. Parameters like the physisorption poten-tial and atom-phonon couplings for our model is thenobtained by asymptotically treating the interaction po-tential leading to expressions for ˜ g kb = (cid:104) k | V (cid:48) ( z ) | b (cid:105) and˜ g bb = (cid:104) b | V (cid:48) ( z ) | b (cid:105) , where | k (cid:105) and | b (cid:105) represent the asymp-totic continuum and bound state wave functions for theneutral atom . For sufficiently low-energy incidentatoms, the coupling ˜ g kb has a strong dependence onthe incident energy of the incoming atom, such that˜ g kb ∝ √ E k . However, ˜ g bb is independent of the inci-dent energy and is much larger in magnitude than ˜ g kb .To derive the adsorption rate of the atom we do the fol-lowing: we treat the atom-phonon coupling as perturba-tion and derive adsorption rates within a self-energy for-malism using Green’s functions . We find that theterms in the perturbative expansion of atom self-energyare infrared divergent due to the contribution from theemission of low-energy phonons of the graphene mem-brane which we refer as the soft phonon contribution .The problem of IR divergence is more pronounced at fi-nite temperature because of the enhanced emission ofthermal phonons due to the appearance of the Bose-Einstein function . Therefore, a crucial componentfor the derivation of adsorption rates is to devise math-ematical techniques to address these IR divergences andprovide a well-unified theory that describes the roleplayed by the emission of soft-phonons in the adsorptionphenomenon.We have devised such a method for addressing theseIR divergences and predict adsorption rates. Our methoddisplays a resummed atom self-energy Σ kk which uses a fully dressed bound state propagator (see Fig. 2 for ageneral idea of this method) based on the exact solutionof the independent boson model (IBM) . A naturalquestion at this point would be: is it justified to use theIBM propagator to describe the physics of adsorption inour model? To the best of our knowledge, it seems it isimperative to provide a non-perturbative treatment forthe bound channel compared to the continuum for thefollowing reasons: (i) when treated perturbatively, theinclusion of the effects from the atom-phonon couplingin the bound state leads to severe IR divergences in thehigher order self-energy terms , (ii) vertex renormal-ization results indicate an increase in the Γ bb vertex inthe infrared limit and (iii) most importantly, our modelsatisfies the condition ˜ g kb (cid:28) ˜ g bb . Armed with thisknowledge, let us now present the general procedure forthe derivation of the many-body adsorption rate Γ.Invoking Feynman rules for our model we willwrite the 1-loop atom self-energy Σ kk as Σ kk ( E ) = g kb (cid:88) q (cid:20) ( n q + 1) G (0) bb ( E − ω q )+ n q G (0) bb ( E + ω q ) (cid:21) (5)where we introduced a label g kb = ˜ g kb ξ . The phononoccupation number with Bose-Einstein distribution func-tion is given as n q = 1 / ( e ω q /T − is written as G (0) bb ( E ) = 1 / ( E + E b + iη ), η → + . Next we calculate the resummed atom self-energy Σ ( IBM ) kk by replacing the bare bound state propa-gator G (0) bb by the fully dressed G ( IBM ) bb which is derivedwithin the scheme of the independent boson model (seeFig. 2) ,Σ ( IBM ) kk ( E ) = g kb (cid:88) q (cid:20) ( n q + 1) G ( IBM ) bb ( E − ω q )+ n q G ( IBM ) bb ( E + ω q ) (cid:21) . (6)We will write in details the properties and various fea-tures of the low temperature G ( IBM ) bb in Sec. III. Finallywe derive the many-body adsorption rate Γ using theimaginary part of Σ ( IBM ) kk :Γ = − Z ( E k ) I m Σ ( IBM ) kk ( E k ) (7)where, I m Σ ( IBM ) kk gives the imaginary part of Σ ( IBM ) kk and Z is the quasiparticle weight related to the real partof the self-energy Σ ( IBM ) kk , Z ( E ) = (cid:20) − (cid:18) ∂Re Σ kk ( E ) ∂E (cid:19)(cid:21) − . (8)Before we proceed to the next section, let us presentthe result for the adsorption rate within the conventional1 st order perturbation theory i.e Fermi’s golden rule ,Γ = 2 πg kb ρ (cid:26) (cid:18) E k + E b T (cid:19) − (cid:27) . (9)where ρ is the constant density of axisymmetric vi-brational states . Let us also introduce the label g k = g kb ρ . For sufficiently low-energy atomic hydrogenimpinging on graphene membranes, g k is given by 0.7 µ eV - 1.5 µ eV . We notice Γ is independent of (i)atom-phonon coupling strength ˜ g bb and (ii) soft-phononcontribution ( (cid:15) ). As a test of our resummation techniqueand other comparative purposes related to the adsorp-tion rate, we will use the golden rule result in Sec. V. Inthe following sections, our aim is to understand the roleplayed by the environment of thermal phonons towardsthe incoming atom in the bound state by investigating indetail the various features of the bound state propagatorwithin the IBM. III. LOW TEMPERATURE BOUND STATEPROPAGATOR WITHIN THE INDEPENDENTBOSON MODEL (IBM)
In our model Hamiltonian (given by Eqs. 1, 2 and 3) ifwe focus only on the interaction of the atom and phonons
Resummation = FIG. 2. Top: Feynman diagrams corresponding to the exactbound state propagator G ( IBM ) bb for the Independent BosonModel (IBM). As we see, G ( IBM ) bb consists of a sum of Feyn-man diagrams of all types (vertex and rainbow) to all orders inpertubation in atom-phonon coupling ˜ g bb ξ (denoted by opendot in the self-energy diagrams) for transitions in the boundstate | b (cid:105) . Bottom: 1 loop atom self-energy corresponding toadsorption mediated by a 1 phonon process (wiggly line) fortransition from | k (cid:105) → | k (cid:105) via the bound state | b (cid:105) (straightline denoting the bound state propagator G (0) bb ). Under ourresummation technique, we replace the bare G (0) bb by the IBMpropagator G ( IBM ) bb (denoted by double lines) which repre-sents the fully dressed propagator with pertubations to allorders in atom-phonon coupling (˜ g bb ξ ) in the bound. in the bound state | b (cid:105) and drop the terms that correspondto the continuum | k (cid:105) , we end up with the Hamiltonian ofthe Independent Boson Model given by, H IBM = − E b b † b + (cid:88) q ω q a † q a q − g bb b † b (cid:88) q ( a q + a † q ) . (10)where we have introduced the label g bb = ˜ g bb ξ . Our aimin this section would be to re-write the exact solution ofthe IBM Hamiltonian in a suitable manner to incorporatethe physics of the adsorption phenomenon.With this aim in mind, let us begin with a canonicaltransformation using an operator s = − g bb b † b (cid:80) q ( a † q − a q ) /ω q and apply it to each of the operators in Eq. 10.Using the Baker-Hausdorff lemma : ¯ A = e s Ae − s = A +[ s, A ] + (1 / s, [ s, A ]] + ·· , we evaluate the new phononoperators: ¯ a q = e s a q e − s = a q + g bb ω q b † b , (11)¯ a q † = e s a † q e − s = a † q + g bb ω q b † b. (12)here, we have used [ s, a q ] = ( g bb /ω q ) b † b . We see that un-der the canonical transformation, the phonon operators a q and a † q are displaced by an amount ( g bb /ω q ) b † b to anew equilibrium position around which they vibrate withthe initial frequency ω q . Quite naturally we ask: whatis the physical reason behind such a displacement of thephonon fields? It seems that the presence of the adatomin the membrane leads to the polarization of the sur-face of the membrane which shifts the oscillators. Thisis quite similar to the case of the charged oscillator un-der a uniform electric field. The presence of the electricfield causes the charge to displace to a new equilibriumposition, around which it fluctuates with the same fre-quency as before . Now, let us write the new operatorcorresponding to the particle in the bound state:¯ b = e s be − s = b (cid:18) (cid:88) q g bb ω q ( a † q − a q ) + · · (cid:19) = bX, (13)here, we have used [ s, b ] = (cid:80) q g bb ( a † q − a q ) b/ω q and in-troduced an operator X , X = exp (cid:18) (cid:88) q g bb ω q ( a † q − a q ) (cid:19) (14)which we shall refer to as the coherent phonon bath dis-placement operator . Indeed, we will see that X iscrucial to explain the fluctuations of the phonons aroundthe atom. And finally under this canonical transforma-tion H IBM modifies to:¯ H IBM = e s H IBM e − s = − E b ¯ b † ¯ b − g bb b † b (cid:88) q (cid:18) a q + a † q + 2 g bb ω q b † b (cid:19) + (cid:88) q ω q (cid:18) a † q + g bb ω q b † b (cid:19)(cid:18) a q + g bb ω q b † b (cid:19) = − ( E b + ∆) b † b + (cid:88) q ω q a † q a q (15)where, we have used [ X, b ] = 0 and X † = X − which im-plies ¯ b † ¯ b = b † b . The factor ∆ is defined as the acousticpolaron shift which appears as a result of the displace-ment of the phonon fields due to the presence of the atomin the phonon bath, ∆ = (cid:88) q g bb ω q (16)In the continuum limit (cid:80) q → (cid:82) ω D (cid:15) ρ d ω , the above Eq.reduces to: ∆ = g b (cid:90) ω D (cid:15) d ωω (17)Here, we have introduced the label g bb ρ = g b and westick to this notation for the rest of the paper. For atomichydrogen impinging on graphene membranes g b = 0 . .A solution to Eq. 15 is realized within the IBM andis written with modifications pertaining to our model ofadsorption as, G ( IBM ) bb ( t ) = − ie − it ( − E b − ∆) e − ˜ φ ( t ) . (18) We notice that the canonical transformation has led toa propagator in which the contributions due to the atomand phonon terms are well separated. Let us first look atthe phonon contribution which is related to the thermalaverage over the phonon modes leading to the phononbath correlator (cid:104) X ( t ) X † (0) (cid:105) = exp[ − ˜ φ ( t )] (19)with the IBM phase factor ˜ φ ( t ) = (cid:88) q (cid:18) g bb ω q (cid:19) (cid:20) n q (1 − e iω q t ) + ( n q + 1)(1 − e − iω q t ) (cid:21) (20)which can be further decomposed into˜ φ ( t ) = φ (0) + φ ( t ) . (21)Within the continuum limit, we define the above termsas: φ (0) = (cid:90) ω D (cid:15) g b ω (cid:20) n + 1 (cid:21) d ω (22) φ ( t ) = (cid:90) ω D (cid:15) g b ω (cid:20) ( n + 1) e − iωt + ne iωt (cid:21) d ω. (23)where n = 1 / ( e ω/T − fully dressed bound state propaga-tor G ( IBM ) bb ( t ): G ( IBM ) bb ( t ) = − i exp (cid:26) it ( E b +Λ) (cid:27) exp (cid:20) − φ (0)+ R e φ ( t ) (cid:21) . (24)Here, Λ renormalizes the energy of the bound atom andincludes the contribution from two terms: (i) the acousticpolaron shift ∆ and (ii) imaginary part of the phononbath correlator I m φ ( t ) such that:Λ = ∆ + I m φ ( t ) t = g b (cid:90) ω D (cid:15) d ωω − g b (cid:90) ω D (cid:15) sin( ωt )d ωω t . (25)The decay of the propagator G ( IBM ) bb ( t ) on the otherhand, is given by S = exp (cid:20) − φ (0) + R e φ ( t ) (cid:21) , (26)which comprises the real part of φ ( t ): R e φ ( t ) = g b (cid:90) ω D (cid:15) (cid:20) (2 n + 1) cos( ωt ) ω (cid:21) d ω (27)and a shift exp[ − φ (0)] linked to the Franck-Condon fac-tor. This in principle is related to the phonon contribu-tion by the relation, (cid:104) X (cid:105) = exp[ − φ (0) / .For all purposes related to the calculation of the many-body adsorption rate, we require the Fourier transformof Eq. 24. We write them as the following: R eG bb ( E + E b ) = (cid:90) ∞ d t sin (cid:26) t ( E + E b + Λ) (cid:27) × exp (cid:20) − φ (0) + R e φ ( t ) (cid:21) (28)and |I mG bb ( E + E b ) | = (cid:90) ∞ d t cos (cid:26) t ( E + E b + Λ) (cid:27) × exp (cid:20) − φ (0) + R e φ ( t ) (cid:21) . (29)Before we calculate the resummed atom self-energyΣ kk using Eqs. 28 and 29, let us study the effect of thephonon bath on the renormalization and decay of thebound atom with a special focus on the time evolution,temperature T and coupling strength g b . We envisionthese effects to alter the response of the phonons towardsthe adsorption phenomenon. Our next section will be de-voted to this. IV. EFFECTS OF THE PHONONCORRELATOR ON THE RENORMALIZATIONAND DECAY OF THE BOUND STATEPROPAGATOR
We devote this section to investigate the effects of time,temperature and coupling on the phonon bath correlatorand also the renormalization parameter. It seems to usthat the first step to accomplish this would be to de-fine an effective parameter which would accommodatethe effects of temperature T and coupling g b in it. Inprinciple, we accomplish this by setting up a transforma-tion of variable which would naturally give rise to such aparameter. Let us define the transformation of variable,˜ ω = ω (cid:112) g b T (30)which leads to the following dimensionless parametersfor the low-energy infrared scale (cid:15) and maximum Debyefrequency ω D , (cid:15) → ˜ (cid:15) = (cid:15) (cid:112) g b T , ω D → ˜ ω D = ω D (cid:112) g b T . (31)Also as result of the transformation, we derive a newcharacteristic dimensionless time scale that comprises theeffects of time t: τ = (cid:18)(cid:113) g b T (cid:19) t (32)As we see, the parameter ˜ (cid:15) contains in it the softphonon energy scale (cid:15) , length of the membrane L (since, (cid:15) = v s /L ), coupling g b and the effects of temperature T . Thus we study the time-dependence of the decay andrenormalization of the bound state propagator as a func-tion of ˜ (cid:15) . We will perform the study with respect to thedimensionless time scale τ . The effectiveness of such achoice will be clear to us shortly. In the following sub-sections we apply the transformations given by Eqs. 30,31 and 32 to the decay and the renormalization factors. A. Decay of the propagator as a function of ˜ (cid:15) Under the chosen transformation of variables, the de-cay term S given by Eq. 26 modifies to:˜ S = exp (cid:20) − (cid:90) ˜ ω D ˜ (cid:15) (cid:114) g b T ω (cid:26) ω (cid:112) g b /T ) − (cid:27) × (cid:26) − cos(˜ ωτ ) (cid:27) d˜ ω (cid:21) (33)where we have used the transformed versions of Eqs. 22and 27. We solve the integral under the parenthesis nu-merically for various values of ˜ (cid:15) as a function of the char-acteristic dimensionless time scale τ and plot the decayfactor ˜ S vs τ in Fig. 3.As we have seen in the previous section, ˜ S representsthe phonon bath correlator (given by Eq. 19) and phys-ically corresponds to the dephasing of the phonons. Thegeneral feature as seen in Fig. 3 is a loss of coherence ofthe phonons following a power law decay. However, weobserve two regimes: (i) for phonons corresponding to˜ (cid:15) <
1, there is a rapid loss of coherence within τ < τ = 1, phonons have completely lost their coher-ence such that ˜ S ≈ (cid:15) >
1, there is a much slower loss of coherence and infact beyond τ = 1 they do not show a complete decay butsaturate to non-zero residual values. Let us ask what con-trols the phonon bath correlator function beyond τ = 1?While phonons lose their coherence with the evolution oftime, they finally saturate to residual values that are rep-resented by the time-independent Franck-Condon factor(FC) given by exp[ − φ (0)]. We indeed observe a com-plete match of the long-time values of ˜ S with the FCfactors calculated for the corresponding values of ˜ (cid:15) (seeinset of Fig. 3). Therefore for τ <
1, we observe short-time phonon dynamics that correspond to ˜ (cid:15) < (cid:15) < (cid:112) g b T and for ˜ (cid:15) > (cid:15) > (cid:112) g b T phononsexhibit long-time dynamics in regime τ > S as a function of ˜ (cid:15) for differ-ent regimes of the dimensionless characteristic time scale τ = ( (cid:112) g b T ) t is shown in Fig. 4. We observe that thefunction is within the bounds 0 ≤ ˜ S ≤
1. The shift dueto the Franck-Condon factor exp[ − φ (0)] matches exactlywith the long-time ( τ (cid:29)
1) response of ˜ S . For τ (cid:28) = e [( ) + ()] g b = 0.06 meV, T = 10K = 3.5 = 1 = 0.18 = 0.018 = 0.0018 e ( ) FIG. 3. For graphene membranes with coupling g b = 0 . maintained at T = 10 K = 0.862 meV, we plotthe effect of the dimensionless infrared scale ˜ (cid:15) on the decayfactor ˜ S as a function of the dimensionless characteristic timescale τ = ( (cid:112) g b T ) t . The range of membrane sizes used are L = 5 nm, 20 nm, 100 nm, 1 µ m and 10 µ m which correspondto ˜ (cid:15) = 3.5, 1, 0.18, 0.018 and 0.0018 respectively. While forsmaller values of ˜ (cid:15) (˜ (cid:15) < τ <
1, higher values of ˜ (cid:15) (˜ (cid:15) >
1) retain theircoherence and slowly saturate to non-zero residual values for τ >
1. We relate this long-time residue to the shift due to theFranck-Condon (FC) factor given by exp[ − φ (0)]. The insetplot shows the corresponding variation of exp[ − φ (0)] with ˜ (cid:15) .For ˜ (cid:15) (cid:28)
1, the shift due to FC is generally zero and showsappreciable non-zero values only for ˜ (cid:15) ≥ φ (0), hence there is no appreciable change in the decayfunction and ˜ S ∼
1. However, with increasing τ , phononcoherence starts to decay and the shift due to FC fac-tor starts to emerge. For τ > τ (cid:29)
1, ˜
S ≈ (cid:15) < .
5. We relate this to the absolute loss of phononcoherence beyond τ = 1 for the phonons exhibiting theshort-time phonon dynamics (see Fig. 3). The regime τ < S with increasing ˜ (cid:15) . For ˜ (cid:15) >
2, ˜ S is finite and compa-rable for all regimes of τ . We understand this behaviorby relating this to the long-time phonon dynamics wherethe phonons in the regime ˜ (cid:15) > τ and ˜ (cid:15) just as the decay ˜ S . B. Renormalization of the bound state energy as afunction of ˜ (cid:15) From Eq. 29, we define a function P which enclosesthe renormalization factor Λ, P = cos (cid:26) t ( E + E b + Λ) (cid:27) . (34) = e [( ) + ()] g b = 0.06 meV, T = 10K 1< 1 e [ (0)]
1> 1
FIG. 4. Dependence of the decay factor ˜ S on ˜ (cid:15) for differentregimes of the dimensionless characteristic time scale τ =( (cid:112) g b T ) t . Maximum phonon coherence is observed for theregime τ (cid:28)
1. With increasing τ , phonon coherence is seen todecrease and beyond τ = 1, the shift due to Franck-Condonfactor exp[ − φ (0)] starts to emerge. For phonons exhibitingshort-time phonon dynamics (˜ (cid:15) < .
5) an absolute loss ofphonon coherence beyond τ = 1 is observed. The long-time( τ (cid:29)
1) response of ˜ S corresponds to the shift due to theFranck-Condon factor. Under the chosen transformation, we write:˜ P = cos (cid:20) τ (cid:26) E + E b (cid:112) g b T + (cid:114) g b T (cid:90) ˜ ω D ˜ (cid:15) d˜ ω ˜ ω − (cid:114) g b T (cid:90) ˜ ω D ˜ (cid:15) sin(˜ ωτ )˜ ω τ d˜ ω (cid:27)(cid:21) (35)with the transformed renormalization term:˜Λ = (cid:20)(cid:114) g b T (cid:90) ˜ ω D ˜ (cid:15) d˜ ω ˜ ω − (cid:114) g b T (cid:90) ˜ ω D ˜ (cid:15) sin(˜ ωτ )˜ ω τ d˜ ω (cid:21) (36)As discussed before in Sec. III, there is a competitionbetween two physical effects that govern the renormal-ization ˜Λ: (i) the acoustic polaronic shift ˜∆ and (ii) theimaginary part of the phonon bath correlator ˜ I (given bythe first and second terms of Eq. 36 respectively).We plot the time dependence of the renormalizationfactor τ ˜Λ for various values of the dimensionless IR cut-off ˜ (cid:15) for 2 regimes 0 < τ ≤ τ (cid:29) < τ ≤ E b ) from the polaron (left inset) and theimaginary part of phonon correlator (right inset) whichthus results in an overall small renormalization factor.We relate this to the fact that the phonons in this regimeare undergoing dephasing and have not fully lost theircoherence (as seen in the previous subsection). As a re-sult, while phonons dephase with time, the polaron ( τ ˜∆)starts to grow and hence is negligibly small at the on-set of time. However, for the asymptotically large τ = 0.0018 = 3.5 = 0.0018 = 3.5 FIG. 5. Variation of the renormalization factor τ ˜Λ corre-sponding to two regimes of τ for various values of ˜ (cid:15) . Neg-ligible renormalization effects are observed in the short-timeregime τ (cid:28) τ ˜∆ (left inset) and imaginary part of the phononcorrelator ˜ I (right inset). In the long-time regime τ (cid:29) regime (bottom panel), we observe a huge contributionfrom the polaron (left inset) towards the renormalizationcompared to the imaginary part of the phonon bath cor-relator which gradually decays off (right inset). Onceagain, we relate this to the complete transfer of coher-ence from the phonon bath towards the formation of thepolaron which now grows logarithmically without any de-cay. Variation of the renormalization τ ˜Λ as a function of˜ (cid:15) for various values of τ is shown in Fig. 6. For small val-ues of τ ( τ (cid:28) , τ < τ ˜Λ with ˜ (cid:15) . The modest renormal-ization factors can be related to the comparable and tinycontributions from the polaron and phonon bath correla-tor. However, the general trend shows an increase in τ ˜Λwith increasing τ and the effect is seen to be more pro-nounced for ˜ (cid:15) <
1. The total loss of phonon coherencebeyond τ = 1 for phonons exhibiting short-time phonondynamics (˜ (cid:15) <
1) can be attributed to the large renor-malization factor. See inset plot of Fig. 6 for the effect FIG. 6. For the short time regime corresponding to τ < τ ˜Λ) effect without any significant de-pendence on ˜ (cid:15) is observed. With increasing τ , renormalizationeffect is seen to increase for the phonons satisfying ˜ (cid:15) < in the asymptotically large times ( τ (cid:29) V. MANY-BODY ADSORPTION RATE
In this section, we will begin our analysis for suspendedgraphene membranes maintained at 10 K with sizes rang-ing from 100 nm ∼ µ m which correspond to ˜ (cid:15) = 0 . ∼ g b =0.06meV). As we have seen before in Sec. IV, there seems tobe the existence of two regimes corresponding to the char-acteristic time scale τ . While in τ ≤
1, phonons exhibitobservable dynamics, in the other regime characterizedby τ >
1, phonon dynamics seem to completely die offwith massive renormalization effects. In subsections V Aand V B, we provide procedures which separately evalu-ate the contribution of the time regimes 0 ≤ τ ≤ < τ ≤ ∞ to the adsorption rate. Finally in subsec-tion V C, we provide the result for the total many-bodyadsorption rate which is the sum of the contribution fromregimes 0 ≤ τ ≤ < τ ≤ ∞ .Let us begin with the expression for the imaginary partof the atom self-energy given by Eq. 6, re-written underthe transformation of variables (given by Eqs. 30, 31 and32), I m Σ ( IBM ) kk = g k (cid:20) (cid:90) ˜ ω D ˜ (cid:15) d˜ ω (cid:26) ω (cid:112) ( g b /T ) − (cid:27) I m ˜ G bb ( ˜ E s − ˜ ω ) (cid:21) + g k (cid:20) (cid:90) ˜ ω D ˜ (cid:15) d˜ ω (cid:26) ω (cid:112) ( g b /T ) − (cid:27) I m ˜ G bb ( ˜ E s + ˜ ω ) (cid:21) (37)Here we have introduced the labels ˜ E s ≡ ( E + E b ) / (cid:112) g b T .The above equation can also be written as I m Σ ( IBM ) kk = g k (cid:90) ˜ ω D ˜ (cid:15) d˜ ω F = g k (cid:90) ˜ ω D ˜ (cid:15) d˜ ω (cid:20) F ( em ) + F ( abs ) (cid:21) , (38)where F ( em ) and F ( abs ) correspond to processes for emis-sion and absorption of phonons, respectively. We writethese as F ( em ) = (cid:20)(cid:26) ω (cid:112) ( g b /T ) − (cid:27) I m ˜ G bb ( ˜ E s − ˜ ω ) (cid:21) (39)and, F ( abs ) = (cid:20)(cid:26) ω (cid:112) ( g b /T ) − (cid:27) I m ˜ G bb ( ˜ E s + ˜ ω ) (cid:21) (40)As a matter of fact, a careful inspection of Eqs. 37,39 and 40 reveals that we can relate the functions F ( em ) and F ( abs ) to the spectral weights associated with theprocesses of emission and absorption of phonons sincethey are related to the imaginary part of the bound statepropagator. With the above equations, let us now lookat the contribution from the two different regimes of τ .We begin with 0 ≤ τ ≤ A. Contribution from regime ≤ τ ≤ In this subsection we will calculate the contributionto Γ from the regime 0 ≤ τ ≤ ≤ τ ≤
1. We start witha study on the dependence of the function F ( em ) on theentire phonon frequency scale (˜ ω ) for our chosen valuesof ˜ (cid:15) .We calculate the respective integrals numerically andplot the variation of F ( em ) as a function of ˜ ω for the dif-ferent values of ˜ (cid:15) in Fig. 7. For all three values of ˜ (cid:15) , we
170 180 190 2000.00.10.20.30.40.5 ( e m ) ()
1= 0.18= 0.018= 0.0018
FIG. 7. Variation of the spectral weight F ( em ) as afunction of ˜ ω for various values of ˜ (cid:15) in the regime τ ≤ ω = ˜ E s = 184 .
68. This broadening isa signature of inclusion of emission of multiple phonons, re-ferred to as acoustic phonon-broadening . The peakof the Lorentzian shows significant shift related to the ef-fects of the acoustic polaron leading to renormalization of thebound state energy. With decreasing ˜ (cid:15) , we also observe en-hanced broadening of the peak which is a result of increaseddamping related to the decay of phonon coherence. notice a broad peak around the non-renormalized boundstate energy ˜ ω = ˜ E s = E s / (cid:112) g b T = 184 .
68 (denoted byvertical black dashed line). The salient features associ-ated with these lorentzian curves are as follows: (i) thepeaks do not appear exactly at ˜ ω = ˜ E s but are shiftedand we relate this shift to the polaronic effects associ-ated with the renormalization of the bound state energy.Also, the respective values of the shift due to different ˜ (cid:15) are negligible compared to each other, as expected (seeFig. 6, where the renormalization effects show almost novariation with ˜ (cid:15) for 0 ≤ τ ≤ (cid:15) which canbe understood from the rapid loss of coherence of thephonons with decreasing ˜ (cid:15) (see the dependence of ˜ S on˜ (cid:15) for τ < F ( em ) is a signature of the effect of inclusion of emis-sion of multiple phonons and is also observed in otherbranches of quantum field theories which use resumma-tions involving multiple quasiparticles .While we observe an accumulation of spectral weightaround ˜ ω = ˜ E s , there is also a loss in the magnitude ofthe weight with decreasing ˜ (cid:15) . Naturally, we ask if thisloss in spectral weight around ˜ ω ≈ ˜ E s re-appears at adifferent phonon frequency? To answer this, we look atthe total emission and absorption spectra for ˜ (cid:15) = 0 . () = 0.0018 ( em ) , 1 ( abs ) , 1, > 1 Golden Rule( 1)( > 1)
FIG. 8. Spectral weight distribution for the emission F ( em ) and absorption processes F ( abs ) (grey line) for the two timeregimes 0 ≤ τ ≤ ≤ τ ≤ ∞ (blue crosses) fora 10 µ m membrane (˜ ε = 0 . ≤ τ ≤ ∞ leads to a complete disappearanceof the spectral weight F around ˜ ω = ˜ E s which otherwise ispresent in the short time regime 0 ≤ τ ≤ F ( em ) red dots).While a vanishingly small contribution to the adsorption rateΓ( τ >
1) is observed in the asymptotically long time regimes(inset plot blue crosses), the short-time contribution Γ( τ ≤ corresponding to the entire phonon frequency scale (seeFig. 8). Quite interestingly, we observe in addition to thespectral weight around ˜ ω ≈ ˜ E s for F ( em ) , there appearsan accumulation of weight around ˜ ω = ˜ (cid:15) . For higher val-ues of ˜ (cid:15) , we have checked that the spectral weights around˜ ω = ˜ (cid:15) are negligible and increases with decreasing ˜ (cid:15) andeven for the lowest value of ˜ (cid:15) = 0 . L = 10 µ m) thisis still less in magnitude compared to the spectral weightaround ˜ ω ≈ ˜ E s . For the absorption spectra given by F ( abs ) , there is no appearance of spectral weight around˜ ω ≈ ˜ E s but only around ˜ ω = ˜ (cid:15) . This suggests that theadsorption phenomenon is indeed mediated by the emis-sion of phonons of 2 definite frequencies: a hard phononcorresponding to ˜ ω ∼ ˜ E s and a soft phonon ˜ ω ∼ ˜ (cid:15) , ad-ditionally the renormalization of bound state energy anddecay of the propagator is totally controlled by the emit-ted phonons. Let us now look at the contribution fromthe other regime 1 < τ ≤ ∞ . B. Contribution from regime < τ ≤ ∞ We follow the same procedure as before but subjectEqs. 29, 37, 39 and 40 to condition 1 < τ ≤ ∞ (i.e, inte-grate the transformed Eq. 29 for regime 1 ≤ τ ≤ ∞ ). Letus begin by looking at the variation of F = F ( em ) + F ( abs ) as a function of ˜ ω . As we noticed from the previous sub- section, the contribution from the soft phonon emissionis visible only for ˜ (cid:15) = 0 . F ( abs ) to F to be negligible (similarto regime 0 ≤ τ ≤ F ( em ) . In Fig. 8,we note a complete disappearance of the spectral weightaround ˜ ω ≈ ˜ E s . To understand this feature, we recallthis regime is characterized by massive renormalizationeffects with complete loss of phonon coherence. In factit is the vanishing of the decay factor ˜ S represented bythe shift due to the Franck-Condon effect that results inthe complete absence of the spectral weight at ˜ ω ≈ ˜ E s (see the dependence of ˜ S on ˜ (cid:15) for τ > (cid:15) < . F ( em ) and F ( abs ) to F is negligibly small in this regimeand tends to 0 without any appreciable accumulation ofspectral weight at ˜ ω ∼ ˜ (cid:15) or ˜ ω ∼ ˜ E s (see Fig. 8).Let us now calculate the final many-body adsorptionrate for the full time regime 0 ≤ τ ≤ ∞ which comprisesthe contribution from the regimes 0 ≤ τ ≤ <τ ≤ ∞ . C. Final Adsorption rate for the full time regime ≤ τ ≤ ∞ In this section, we discuss the final adsorption ratewhich is a sum of the contribution from the regimes 0 ≤ τ ≤ < τ ≤ ∞ ,Γ(0 ≤ τ ≤ ∞ ) = Γ( τ ≤
1) + Γ( τ >
1) (41)We calculate Γ following Eq. 7 using Eqs. 29, 37, 39 and40 for both the regimes τ ≤ τ >
1. For comparativepurposes we normalize the final rate Γ with respect tothe Golden rule result Γ given by Eq. 9 and show thevariation of Γ / Γ with respect to the dimensionless IRcut-off ˜ (cid:15) in the inset of Fig. 8. For a discussion on thequasiparticle weight Z that appears in the calculation ofΓ, see Appendix Sec. A.For membrane sizes 10 µ m ≤ L ≤
100 nm (correspond-ing to IR cut-off 0 . ≤ ˜ (cid:15) < .
2) and maintained at T= 10 K, we plot the contribution from the regime τ ≤ τ ≤
1) (red dashed lines) andfind it to match the conventional golden rule result Γ .However, for the same membrane parameters, the contri-bution from Γ( τ >
1) corresponding to regime τ > ≤ τ ≤ ≤ τ ≤ ∞ ) ≈ Γ( τ ≤ ≈ Γ (42)We understand this result by recalling the variation ofthe spectral weight F (˜ ω ) for all frequencies ˜ ω for boththe emission and absorption processes. In fact, the to-tal adsorption rate is equivalent to summing the spectral1weights over all frequencies (see Eqs. 37, 38, 39 and 40.)Since in regime τ >
1, there is a complete absence ofspectral weight around ˜ ω ∼ ˜ E s due to the complete lossof phonon coherence which leads to the emergence of theFranck-Condon shift, the contribution to Γ from Γ( τ > τ ≤
1, the renor-malization effect from the emitted thermal phonons leadsto a broadened density of state around ˜ ω ∼ ˜ E s witha small contribution from the soft phonons at ˜ ω ∼ ˜ (cid:15) ,hence when summed over all frequencies, this regime con-tributes to the adsorption rate. As a matter of fact, thetotal adsorption rate is seen to be dominated by the ef-fects from the contributions from the regime τ ≤ whichindicates that the rate is indeed independent of the con-tribution from soft-phonons (cid:15) and atom-phonon couplingin the bound g b (see inset of Fig. 8 where the red dashedlines indicate the final many-body adsorption rate). Thisresult although surprising (since we have summed overmultiple contributions from soft-phonons (cid:15) and includedinteractions in all orders of g b in the bound state propa-gator) represents the essence of the Bloch-Nordsieck re-summation technique. According to the Bloch-Nordsiecksum rule (theorem), the final scattering rate after sum-ming over all soft quanta is found to be identical to thecross-section of scattering in absence of any interactionwith the radiation field . Hence, Eq. 42 validatesthe Bloch-Nordsieck sum rule for our model. Our previ-ous result for zero temperature was also seen to satisfythe sum rule. Nevertheless, also as a check of our resultfor the many-body adsorption rate, we verified the sumrule that the bound state propagators must obey ,1 = − π (cid:90) ˜ ω D ˜ (cid:15) (cid:26) I m ˜ G bb ( ˜ E s − ˜ ω ) (cid:27) d˜ ω (43)For the full time regime given by 0 ≤ τ ≤ ∞ , the numer-ically calculated propagator corresponding to the trans-formed version of Eq. 29 was found to obey the abovedefined sum rule (Eq. 43). VI. CONCLUSION
In summary, we have modelled the phenomenon of ad-sorption on suspended graphene membranes based on anexactly solvable Hamiltonian in many-body physics, theindependent boson model (IBM). The success of IBMtraces back to the exact treatment of phonon dynamicswhich allows us to investigate in detail the role playedby the multiphonon emission and absorption in assistingthe adsorption of an atom to a membrane.While our simple model of adsorption based on theIBM helps us to understand the renormalization of thephysisorption well, relating it to the formation of a phonon-dressed atom which we refer to as the acousticpolaron, it also sheds some light on the decay of the atom in terms of the time-evolution of the phonon bath corre-lator. This in particular turns out to be important. Ourstudy finds the decay of the bound atom propagator isstrongly dependent on the time-evolution of the phononbath correlations with surprisingly contrasting resultscorresponding to different regimes of a dimensionlesscharacteristic time scale τ , defined as τ = ( (cid:112) g b T ) t = 1with t , g b and T as the time, atom-phonon interactionsand membrane temperature, respectively. With an aimto understand this dependence, we constructed a newdimensionless IR scale ˜ (cid:15) = (cid:15)/ (cid:112) g b T which comprisesthe effects of temperature, soft phonon ( (cid:15) ) contribution(equivalent to the effect of finite-sized membranes) andthe atom-phonon interactions. Corresponding to variousvalues of ˜ (cid:15) , we indeed find phonons to exhibit differentdynamics in the regimes τ < τ >
1. While formicromembranes, phonons exhibit a short-time dynam-ics with rapid loss of coherence in the regime τ <
1, formembranes with sizes <
100 nm, phonons exhibit long-time dynamics with negligible loss of coherence. We alsounderstand the long-time result ( τ >
1) as an indica-tion of the emergence of the shift in the phonon bathcorrelator due to the time-independent Franck-Condonfactor. Thus our study reveals well-defined time scalesfor infrared phonon dynamics which in principle can becontrolled by the atom-phonon interaction strength andmembrane-temperature.For micromembranes we computed the adsorptionrates based on a resummation technique that uses theexact solution of the IBM as the dressed propagator forthe atom self-energy. The contribution from the differentregimes of τ towards the many-body adsorption rate Γwere found to be completely contrasting. For 0 ≤ τ ≤ ≤ t ≤ ( g b T ) − / and as a matter of fact, for thefull time regime 0 ≤ t ≤ ∞ , we derived adsorption ratesthat are finite, approximately equal to the Golden ruleresult and shows negligible dependence on the IR cut-off (or the soft-phonon emission). This is in agreementwith the Bloch-Nordsieck sum rule which predicts thatthe resummed scattering rate including the summation ofcontribution of emission of infinitely many soft quanta isidentical to the scattering rate calculated in the absenceof any interaction with the radiation field . Whilethe main contribution to the many-body adsorption ratecame from regime 0 ≤ t ≤ ( g b T ) − / , the contributionfrom 1 < τ ≤ ∞ or ( g b T ) − / ≤ t ≤ ∞ was found to bezero due to the complete loss of phonon coherence leadingto the Franck-Condon shift.We thus conclude that for suspended graphene mi-cromembranes maintained at 10 K or sufficiently lowtemperature (temperature much less than the physisorp-tion well energy but greater than the atom-phonon cou-pling and IR scale), the many-body adsorption rate willbe finite, equivalent to the golden rule result and inde-pendent of the size of the membrane which implies thatthe effect of low-energy phonons to the adsorption rateis negligible (see Eq. 42 and inset of Fig. 8). Our re-sults are thus in agreement with Ref. [30 and 31] and2disagree with Ref. [20]. In Ref. [20], it seems to us thatthe many-body adsorption rate has been derived underan approximation that might have neglected the contri-bution from the effects of emission of thermal phonons.As our study indicates, it is in fact crucial to incorpo-rate the effects of the thermal phonon emission whichif neglected would give an adsorption rate that is zero(see Fig. 8) but would violate the sum rule (defined byEq. 43). Thus it would be interesting to see if the resultfor the adsorption rate in Ref. [20] would change withthe inclusion of the effects of the thermal phonons. Wemust also mention a difference in the “trapping mech-anism” between our work and Ref. [31], where the au-thors have considered a “diffraction mediated selectiveadsorption resonance” which allows the hydrogen atommore time to exchange low-frequency phonons with thesurface before getting stuck. This mechanism gives riseto an additional enhancement in sticking probabilities forincident energies 7 meV - 15 meV. Since our current workdoes not include this mechanism, we do not observe thisadditional enhancement. We leave this investigation forfuture work. We envision our methodology and results toapply to other 2D materials with sufficiently weak atom-phonon coupling, thus opening up the possibility of ap-plication of these materials as nano-mechanical devicesused as mass sensors .While we started this paper with a question, let usalso end with an analogy and a few pertinent questionsrelated to the IR divergence in other field-theories. Asmentioned before in the Introduction (Sec. I), withinthe theories of QED and perturbative gravity, the mass-less nature of photons and gravitons lead to straight-forward divergences in the perturbation series for scat-tering rates due to the emission of low-energy virtualbosons . The solution to this IR catastrophe is thenrealized within resummation schemes using formalismsthat were first developed in the field of electrodynamicsby Bloch-Nordsieck and in gravity by Weinberg . Theresults of these resummations in QED and perturbativegravity led to the emergence of Soft theorems which re-late the matrix elements of a Feynman diagram with anexternal soft quanta insertion to that of the same dia-gram without an external soft quanta . In recent years,there has been a renewed interest to understand the IRstructure of these theories with a motivation to decipherthe connections between the vividly disparate fields ofsoft theorems and the information theoretic propertiesof the soft radiations . It is in fact believed thatfor every massless quanta there exists a connection be-tween the soft theorems (IR catastrophe), memory effectsand asymptotic symmetries . In our simple model of alow-energy atom interacting with a suspended graphenemembrane, we encounter similar IR divergent adsorptionrates due to the emission of infinitely many soft phonons.We employ the Bloch-Nordsieck scheme of resummationformalism employed in QED & perturbative gravity which amounts to including the emissions of infinitelymany soft phonons and derive a non-perturbative result E e kk ( E )[ e V ] = 0.18= 0.018= 0.0018 FIG. 9. The real part of the atom self-energy is plotted asa function of various values of the dimensionless IR cut-off ˜ (cid:15) .Inset shows the variation of the quasiparticle weight Z . Wenote Z ≤ (cid:15) < . for the many-body adsorption rate which is finite andindependent of the soft phonon contribution thereby val-idating the Bloch-Nordsieck theorem . Thus there is asubtle similarity between the theories of QED, perturba-tive gravity and our model in terms of the IR problem,the technique to address the IR problem and finally thenature of the solution to the problem. Thus motivated bythese similarities we ask, would the soft phonon radiationencode information? ACKNOWLEDGMENTS
I thank Professors Dennis P. Clougherty and Valeri N.Kotov for stimulating discussions. I am grateful to Pro-fessor Ion Garate for his valuable advice, insightful com-ments and suggestions. I thank Samuel Boutin and Ben-jamin Himberg for their questions and comments. Thiswork was funded by the Canada First Research Excel-lence Fund.
Appendix A: Real Atom self-energy within IBM andquasiparticle weight Z
Using our method of resummation as given in Sec. II,we write the real part of the atom self-energy as: R e Σ ( IBM ) kk = g k (cid:20) (cid:90) ˜ ω D ˜ (cid:15) d˜ ω (cid:26) ω (cid:112) ( g b /T ) − (cid:27) R e ˜ G bb ( ˜ E s − ˜ ω ) (cid:21) + g k (cid:20) (cid:90) ˜ ω D ˜ (cid:15) d˜ ω (cid:26) ω (cid:112) ( g b /T ) − (cid:27) R e ˜ G bb ( ˜ E s + ˜ ω ) (cid:21) (A1)3where, we have used the transformations defined byEqs. 30, 31 and 32 given in Sec. IV. Our aim in this sec-tion, would be to calculate the real part of Σ ( IBM ) kk andderive the respective values of the quasiparticle weight Z (given by Eq. 8) for membrane sizes 100 nm to 10 µ mmaintained at 10 K.We accomplish this by numerically integrating Eq. A1using the expression of the real part of the bound statepropagator R eG bb given by Eq. 28 (transformed accord- ingly) for ˜ (cid:15) = 0 . , .
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