Chaotic dynamics in a quantum Fermi-Pasta-Ulam problem
Alexander L. Burin, Andrii O. Maksymov, Ma'ayan Schmidt, Il'ya Ya. Polishchuk
AArticle
Chaotic dynamics in a quantum Fermi-Pasta-Ulamproblem
Alexander L. Burin , Ma’ayan Schmidt and Il’yaYa. Polishchuk Department of Chemistry, Tulane University, New Orleans LA 70118, USA; [email protected] National Reserach Center "Kurchatov Institute," 123182, Moscow, Russia; Moscow Institute of Physics andTechnology, Dolgoprudnii, Moscw Reg, 141700, Russia; [email protected] * Correspondence: [email protected]; Tel.: +1-504-296-5825Received: date; Accepted: date; Published: date
Abstract:
We investigate the emergence of chaotic dynamics in a quantum Fermi - Pasta - Ulamproblem for anharmonic vibrations in atomic chains applying semi-quantitative analysis of resonantinteractions complemented by exact diagonalization numerical studies. The crossover energyseparating chaotic high energy phase and localized (integrable) low energy phase is estimated.It decreases inversely proportionally to the number of atoms until approaching the quantum regimewhere this dependence saturates. The chaotic behavior appears at lower energies in systems withfree or fixed ends boundary conditions compared to periodic systems. The applications of the theoryto realistic molecules are discussed.
Keywords:
Quantum Chaos; Fermi - Pasta - Ulam Problem; Anharmonic Vibrations; MolecularVibrations; Vibrational Energy Relaxation and Transport
0. Introduction
Understanding vibrational energy flow in molecules is one of the challenges in modern scienceand technology [1,2]. Vibrational energy flows control energetics of chemical reactions, determine heatbalance in modern nano-devices [1,3–6] and can be manipulated similarly to electrons and photons andused to carry and process quantum information [7–9]. Intramolecular energy relaxation and transportare dramatically sensitive to the molecule’s ability to attain the thermal equilibrium [4,10].After seminal work by Stewart and McDonalds [11], it has been realized that the internalvibrational relaxation can be absent or proceed very slowly in small enough molecules and/or atlow temperature. Based on these observations the concept of localization of low energy anharmonicvibrational states of poly-atomic molecules within the manifold of harmonic product states of almostindependent normal modes was put forward by Logan and Wolynes [12]. In earlier [13–15] and later[16–18] work similar ideas have been developed for particle and spin systems. Theory was furtherextended combining random matrix theory methods [19–21] and Bose Statistics Triangle Rule approach[22–24] and this extension was reasonably consistent with the experimental observations [11].This development is qualitatively consistent with the investigations of the classical counterpartproblem of anharmonic vibrational dynamics. Its simplest realization in atomic chains probed as amodeling system for irreversible dynamics was considered in the celebrated work by Fermi, Pasta andUlam [25] (FPU), where the quasi-periodic behavior has been discovered for the evolution of the initialexcitation instead of irreversible energy equipartition. In spite of over sixty years of investigations ofthe FPU problem, its complete understanding still remains a challenge [26–29].Both quantum and classical non-linear vibrational dynamics can be characterized by a criticalenergy separating low energy integrable (localized) and high energy chaotic (delocalized) behaviors.In the chaotic regime each part of the system can be thermalized due to its interaction with the restsuggesting ergodic behavior in classical regime which is expressed by the eigenstate thermalizationhypothesis [30,31] in quantum regime. The position of the crossover energy separating two regimes a r X i v : . [ c ond - m a t . d i s - nn ] D ec of 21 determines the localization threshold. The threshold energy can be redefined in terms of the criticaltemperature corresponding to that energy.The knowledge of the localization threshold for an individual molecule is significant since thevibrational relaxation changes dramatically depending on whether the energy of the molecule is loweror higher than the threshold [5,10,32,33]. In the latter case the vibrational relaxation follows standardFermi Golden rule kinetics [34], while in the localized regime it is much slower. Therefore, the presentwork is focused on the localization threshold and its dependence on system size (number of atoms)and the strength of anharmonic interaction.Since the properties of a molecule can be sensitive to its shape the consideration is restricted tothe simple linear chain of atoms coupled by anharmonic interactions identical to the FPU problem [25].This problem is relevant for the energy relaxation and transport in polymer chains used in the modernheat conducting devices [3,32,35,36]. The anomalous increase of a thermal conductivity there with thesystem size suggests a very slow thermalization or even the lack of one [36]. The results for the FPUproblem can be qualitatively relevant for the analysis of more complicated molecules.The consideration is restricted to quantum mechanical systems. It has been suggested that thethreshold energy separating localized and chaotic states decreases with the system size [27–29,37–40].This leads to the reduction of thermal energy below the vibrational quantization energy, which makesquantum effects inevitably significant for sufficiently large molecules.The paper is organized as follows. The FPU problems with different boundary conditions areformulated and briefly discussed in Sec. 1. The analysis of localization is performed combininganalytical (Sec. 2) and numerical (Sec. 3) approaches for the FPU problems with different boundaryconditions. Both approaches are reasonably consistent with each other and led to the predictions ofanalytical dependencies of localization threshold on system parameters that are discussed in Sec. 4 fororganic molecules. The methods and brief conclusions are formulated in Secs. 5, 6.
1. Model
The FPU model of anharmonic atomic chain with different common boundary conditionsincluding periodic, fixed ends and free ends (see Fig. 1) can be described by the Hamiltoniansdefined as (cid:98) H per = N ∑ i = (cid:98) p i M + N − ∑ i = (cid:20) k ( (cid:98) u i − (cid:98) u i + ) + A ( (cid:98) u i − (cid:98) u i + ) + B ( (cid:98) u i − (cid:98) u i + ) (cid:21) + k ( (cid:98) u N − (cid:98) u ) + A ( (cid:98) u N − (cid:98) u ) + B ( (cid:98) u N − (cid:98) u )
24 , periodic, (cid:98) H f ixed = N − ∑ i = (cid:98) p i M + N − ∑ i = (cid:20) k ( (cid:98) u i − (cid:98) u i + ) + A ( (cid:98) u i − (cid:98) u i + ) + B ( (cid:98) u i − (cid:98) u i + ) (cid:21) + k (cid:98) u + (cid:98) u N − + A (cid:98) u + (cid:98) u N − + B (cid:98) u + (cid:98) u N −
24 , fixed ends, (cid:98) H f ree = N ∑ i = (cid:98) p i M + N − ∑ i = (cid:20) k ( (cid:98) u i − (cid:98) u i + ) + A ( (cid:98) u i − (cid:98) u i + ) + B ( (cid:98) u i − (cid:98) u i + ) (cid:21) , free ends. (1)Below we set mass, harmonic force constant and Planck constant to unity ¯ h = M = k =
1. Forceconstants A and B describe relative strengths of third and fourth order anharmonic interactions. Thefixed ends problem has been studied in the classical FPU paper [25].Anharmonic interactions should be weak at the system energy E of interest to justify theapplicability of the series expansion for non-linear terms. Assuming approximate energy equipartitionone can estimate ( x i − x i + ) ∼ E / N , which leads to anharmonic interaction estimates V ∼ AE / √ N and V ∼ BE / N for the third and fourth order anharmonic interactions, respectively. of 21 Comparing harmonic and anharmonic interactions we end with the restrictions for energy density inthe form EN < B , 1 A . (2) (a) (b) (c)Figure 1. Schematic illustration of periodic (a), fixed ends (b) and free ends (c) FPU atomic chains.
However we impose a stronger constraint on the energy requiring the stability with respect tothe dissociation. Consequently, the total energy should be less than the dissociation energy, E d . Thedissociation energy can be estimated for the single bond assuming that anharmonic energy becomescomparable to the harmonic one which suggests E d ∼ A ∼ B . For instance if for the Morse potential[41] often used to model atomic interactions one has E d = B = A . (3)We assume that the system energy is always smaller than the dissociation energy E < E d ∼ B ∼ A , (4)so the molecule is stable with respect to large coordinate displacements.Eq. (4) can be satisfied for a quantum system only if it is satisfied at least for the minimum energythat can be estimated as a quantization energy E ∼
1. Consequently, the anharmonic interactionsshould be weak that requires A , B (cid:28)
1. (5)These requirements are well satisfied in real molecules because of the small amplitudes of vibrations ofheavy atoms. For instance using the Morse potential for C − C bond one can express the dimensionlassparameters A and B as | A | = √ (cid:115) ¯ h √ k / ME d = B =
72 ¯ h √ k / ME d = p = − N /2 + − N /2 +
2, ... N /2 for even N and p = − ( N − ) /2, − ( N − ) /2, ... ( N − ) /2 for odd N , wavefunctions ψ p ( k ) = e i π pk / N / √ N and eigenfrequencies ω p = ( p π / N ) . (7) of 21 These frequencies are identical to normal mode quantization energies (remember that we set thePlanck constant ¯ h to unity). The mode with zero quasi-momentum p = N − (cid:98) H = N − ∑ p = ω p (cid:20)(cid:98) b † p (cid:98) b p + (cid:21) , (8)where operators (cid:98) b † p , (cid:98) b p describe creation or annihilation of one quantum of vibration of normal mode orphonon p . The harmonic problem is obviously integrable since the system breaks into N independentoscillators (phonons) and each phonon population number operator ν p = (cid:98) b † p (cid:98) b p represents a localintegral of motion [42] in the momentum representation. Each many-body eigenstate | S > of theharmonic system can be then represented by an arbitrarily sequence of integer population numbers S = { ν p } .Anharmonic interaction mixes up these states because it breaks down the conservation ofindividual phonon population numbers. In the periodic system with only fourth order anharmonicity( A = (cid:98) V = B N ∑ p p p p ∆ ( p + p + p + p ) sign ( p p p p )( − ) ( p + p + p + p ) N (cid:112) ω p ω p ω p ω p ×× ( (cid:98) b † p + (cid:98) b − p )( (cid:98) b † p + (cid:98) b − p )( (cid:98) b † p + (cid:98) b − p )( (cid:98) b † p + (cid:98) b − p ) , ∆ ( p ) = N N ∑ k = e i π pkN . (9)The factor ∆ ( p + p + p + p ) is equal to unity if the sum of all four momenta is equal to zeroor integer fraction of N (due to Unklamp processes); otherwise it is equal to zero giving rise to aquasi-momentum conservation.Because of the above conservation law the basis states of the system with given normal modepopulation numbers can be split into N subsystems with the total quasi-momenta Q =
0, 1, −
1, ... N /2for even N or Q =
0, 1, −
1, ... − ( N − ) /2 for odd N determined with the accuracy to the addition ofinteger number of N ’s. Each subsystem should be studied separately since the states from differentsubsystems do not interact with each other. In addition the states with Q = Q = N /2 foreven N possess a mirror reflection symmetry with respect to replacement all states S = ν p with thestates S − = ν − p . Then the states with Q = N /2 can be split into two subgroups symmetric orantisymmetric with respect to the mirror reflection symmetry. Consequently, all many-body states canbe split into N + N and N + N that should be consideredseparately.Similarly, one can consider the interacting normal modes for free and fixed boundary atoms (seeEq. (1)). One can similarly introduce normal modes for this problem and their anharmonic coupling.In these two cases one cannot introduce quasi-momenta because of the lack of translational symmetry.Yet there is a mirror reflection symmetry with respect to the middle of the chain. Then all states can beseparated into two subgroups of symmetric and anti-symmetric states with respect to that symmetry.The states belonging to different subgroups can be considered separately.
2. Localization-delocalization transition: Qualitative analytical consideration
There is over sixty years history of the investigation of chaos in the classical FPU problem andthis problem still remains a challenge [26,27]. The situation with the quantum mechanical problem iseven more complicated [43,44]. Below we summarize the established results for the system of a fewatoms [34] and attempt to extend them to atomic chains having many atoms N (cid:29) β FPU of 21 problem containing only fourth order anharmonic interactions and then extend it to the mixed α + β problem. ∼ . Localization-chaos transition in systems with the small number of atoms N ∼ E c separating localized and chaoticstates was estimated using a dimensionality arguments since the only value having the dimension ofenergy can be constructed using the parameters in Eq. (1) in the form E c ∼ B . (10)The numerical studies of both classical and quantum mechanical problems in Ref. [34] have confirmedthese expectations provided that the system is semiclassical, i. e. the system energy, E c is much largerthan the quantization energy ¯ h ω ∼
1. This requires B (cid:28)
1. Transitions in quantum and classicalsystems occur under almost identical conditions since the system is semiclassical because the maximumquantization energy max ( ω p ) is of order of unity (Eq. (7). Consequently, it is much less than the energyper the mode E c / N expressing the thermal energy k B T (see Eq. (10), remember that B ≤ N ∼ α FPU problem with the third orderanharmonic interaction as E c ∼ A . (11)Eqs. (10), (11) differ from the expectations of the analysis exploiting resonances for many-bodytransitions that has been successfully applied to problems of interacting spins [15,45–47] or electrons[16,48]. According to this criterion chaos emerges in the presence of approximately one resonance perthe many-body state under the condition that the diagonal interaction of resonant modes is largeror comparable to their resonant coupling [15,47–49] that is needed to avoid destructive interferencebetween consecutive resonant transitions. Such interaction is present naturally for the fourth orderanharmonicity in Eq. (9) (for instance, the terms with p = p and p = p are diagonal in the phononproduct state representation). There is no such interaction in the case of the third order anharmonicinteractions ( α FPU problem), which changes the definition of delocalization transition as it is discussedin Sec. 2.3.However, in the β FPU problem under consideration the matrix elements M of the four phononinteractions in Eq. (9) grow proportionally to the squared population numbers M ∼ B ν p ∼ BE forthe system energy E exceeding the quantization energy. The typical energy change in a four phononprocess is of order of their quantization energy that is of order of unity. Consequently, the amount ofresonances approaches unity at E ∼ B − in contrast with Eq. (10).This conflict can be resolved at the qualitative level modifying the definition of resonancesin accordance with Ref. [50] where a single-particle localization problem has been considered forharmonically coupled vibrations. For example two unit mass oscillators with frequencies ω a and ω b coupled by the interaction k ab u a u b are in resonance under the condition | ω a − ω b | < k ab , while interms of matrix elements the resonance takes place at | ω a − ω b | < k ab (cid:112) ( ν a + )( ν b + ) .To define the resonance correctly one can consider the energy change not for a single resonanttransition but for the whole set of possible transitions involving these four phonons, which will increasethe typical energy change due to the transition (¯ h ω ∼
1) by the factor of a typical phonon populationnumber ν p . Then the resonance criterion can be written as B ν p ∼ ν p . Setting δ E ∼ ν p ≈ E c we end upwith Eq. (10). The problem of interest with large number of atoms needs a special consideration givenin the next section. of 21 β FPU problem N atoms one canfind N possible four phonon processes for a typical state (the fourth phonon mode is fixed by thequasi-momentum conservation law in Eq. (9)). Consequently, the minimum energy difference betweentwo modes coupled by the fourth order anharmonic interaction is given by δ E ∼ ν p / N ∼ E / N (factor ν p is added similarly to Sec. 2.1). The interaction matrix element scales as M ∼ BE / N .Here the factor 1/ N comes from the definition of anharmonic interactions in Eq. (9) and the factor ν p ∼ ( E / N ) is determined by the population numbers √ ν p ν p ν p ν p ∼ ( k B T ) while the thermalenergy k B T for N classical oscillators is given by E / N [51]. Setting δ E ∼ M to ensure the presence ofresonant interactions we estimate the localization threshold as E c , res ( N ) ∼ NB . (12)Similar dependence can be obtained for the atomic chains with fixed or free ends boundaryconditions where there is no quasi-momentum conservation. In those systems one has N possiblefour phonon transitions and 1/ N (instead of 1/ N , Eq. (9)) scaling of anharmonic interaction matrixelement. Then extra factors N − are canceled out on both sides of the criterion of resonance leading toEq. (12).It is noticable that the estimated behavior of localization threshold in Eq. (12) is qualitativelyconsistent with the earlier estimates [39,40,52,53] obtained using the stability analysis of the classicaldynamics of a non-linear FPU chain in the form E c ( N ) ≈ π NB . (13)Since this equation agrees with numerical studies in Sec. 3 for free and fixed ends boundary conditionswe will use it for quantitative estimates.In our qualitative analysis of resonant interactions we considered only typical phonons withenergy close to unity, while the low frequency phonons were ignored. Based on the presentunderstanding of localization - chaos transition it is hard to expect that they can suppress the chaoticdynamics because the typical phonons form the ergodic spot normally capable to equilibrate the restof the system [48,54]. It is hard to expect that they can give additional support to the chaotic dynamicssince they are coupled weakly to the rest of the system compared to typical phonons.On the other hand one can imagine marginal states with the only low frequency phonons beingexcited. These states can possibly show anomalously strong localization behavior as predicted for theclassical systems in of Refs, [38,55]. There are other suggestions for classical systems [27–29,56] thatthe crossover in Eq. (13) does not describe the transition to a truly integrable (localized) behavior butseparates strongly ergodic and weakly ergodic regimes at high and low energies, respectively. Sincethe numerical simulations in Sec. 3 show the pure localization transition, we did not see any evidencesfor such behavior in a quantum regime. It cannot be excluded that at larger number of atoms someadditional channels for chaotic behavior can emerge.The criterion in Eq. (12) is valid until the system remains semiclassical, meaning that thephonon population numbers exceed unity. This requires the thermal energy E c ( N ) / N to exceed of 21 the quantization energy, which is of order of unity. Thus, the classical regime takes place at sufficientlysmall number of atoms N < N c ≈ √ π √ B . (14)The crossover energy E c expresses the minimum threshold energy in the classical regime ofvibrations. At larger N the system should be treated quantum mechanically as considered in Sec. 2.2.2.2.2.2. Quantum Mechanical RegimeWe begin the consideration with the analysis of the problem in terms of resonant interactions.Imagine that the system energy is spread between phonons of energy (cid:101) such that √ E / N < (cid:101) ≤
1. Inour case of small energy E < N the thermal energy is given by k B T ≈ √ E / N and the lower limit forthe energy (cid:101) qualitatively represents the typical thermodynamic equilibrium.For an arbitrary energy (cid:101) the total number of phonons is given by n (cid:101) ∼ E / (cid:101) . This number issmaller than the number of quantum states with energy of order of (cid:101) that is given by N (cid:101) . The modelingsystem is non-degenerate so typical populations of vibrational states do not exceed unity and one candescribe the emergence of chaos requiring a single resonant interaction per a many-body quantumstate (cf. Refs. [47,48]). The typical anharmonic interaction strength for periodic boundary conditionsscales as M ∼ B (cid:101) / N . The energy difference to the adjacent state coupled to the given state and havingthe same number of phonons can be estimated as (cid:101) / N c where N c is the number of anharmonicallycoupled states with the same number of phonons. This number can be estimated considering thenumber of possible anharmonic transitions including n possible double annihilations of phonons and N (cid:101) creations (the fourth phonon is fixed by the quasi-momentum conservation law and we consideronly processes conserving the number of phonons). The resonant interactions exist under the condition B (cid:101) / N < (cid:101) / ( n (cid:101) N ) . Then the critical energy E c = n (cid:101) can be estimated as E c ∼ √ B . (15)The generalization to the non-periodic boundary conditions can be done similarly to that in theprevious section.This answer is universal and insensitive to the number of atoms. It predicts the saturation of thedependence of critical energy on the number of atoms in the quantum regime. Based on the numericalresults in Sec. 3 we assume that the saturation takes place at E c = N . Then combining Eqs. (13) withEq. (15) one can write the summary of the predicted behaviors as E c = π NB , N < √ π √ B , √ π √ B , N > √ π √ B . (16)The qualitative behavior predicted by Eq. (16) is obtained using resonant language similarly toRef. [47], where the matching Bethe lattice problem has been used to justify the results. Similarly toBethe lattice problem one can expect the appearance of the additional logarithmic factor in Eq. (16).However in our specific case it is of order of 1 since the argument of logarith is determined by the ratioof diagonal and off-diagonal interactions [47], which have same order of magnitude in the problemunder consideration. The quantitative expression in Eq. (16) gives a reasonable estimate by order ofmagnitude but does not pretend to be the accurate expression.In Sec. 3 it is verified for the minimum division of energy E into phonons with energies of orderof 1. The more accurate numerical analysis of the problem is postponed for future.The consideration ignores correlations between phonon energies and momenta, that can takeplace due to quasi-momentum conservation in a periodic system [44] or some trace of its conservation of 21 in the system with fixed end boundary conditions. These processes are fully suppressed for freeends boundary conditions where the above consideration is most applicable. It is less applicablefor the periodic system where these correlations can be significant. For very small system energiescomparable to the maximum quantization energy 1 the periodic system becomes integrable [44] so theconsideration fails. We still believe that our consideration is valid even for a periodic system where thehot spot [54] can be formed by several excited phonons with nearly maximum energy. These phonons,indeed, form chaotic state (see Sec. 3) and can equilibrate other parts of the system. The accuratenumerical verification should resolve the raised questions. α FPU problem
Here we consider the effect of the third order anharmonic interaction on the state of the system.Let us begin the consideration with the classical regime, E > N . For the small number of atomsthe dimension based arguments lead to the estimate A c ∼ √ E , Eq. (11). For a large number ofatoms N in the periodic system one can find N possible three phonon processes so the minimumenergy shift can be estimated as δ E ∼ ν p / N ∼ E / N . Remember that the third phonon state isfixed by the quasi-momentum conservation law in Eq. (9). The interaction matrix element scales as M ∼ AE / N . Consequently, there are resonant interactions in the case of sufficiently large energy E > E res ( N ) , where E res ( N ) ∼ N A . (17)This estimate is consistent with Ref. [53]; yet we do not think it describes the localization breakdowncorrectly because of the lack of the diagonal interaction. In this case resonant transitions areindependent of each other [47,48] which prevents the system from delocalization similarly to the XY model, where there is no diagonal interaction [57]. Following Ref. [57] one can consider the inducedresonant interaction in higher orders anharmonicity following the Schrieffer and Wolff method [58]. Inthe first non-vanishing order the fourth order anharmonic interaction will be generated. This generatedinteraction is similar to the one in the β FPU problem with the effective interaction constant B ∗ ∼ A if expressed in the momentum space. However, the induced diagonal interaction is much less thanthe third order resonant interaction because A (cid:28) E c ∼ A that is insensitive to thenumber of atoms.However, the more efficient delocalization should take place due to the induced fourth orderinteraction characterized by the interaction strength B ∗ ∼ A . In that case one can expect the chaoticbehavior following the estimate of Eq. (10) that reads E c α ( N ) ∼ N A . (18)Similarly to Sec. 2.2.2 this criterion is valid in the classical regime realized at N < A while in theopposite regime this dependence saturates at E c α ∼ √ A . (19)Following Ref. [28] one can expect that this prediction should be valid to the same extent as Eq.(10). Indeed, if one considers the combined α + β problem containing both third and fourth orderanharmonic interactions then the chaotic state formation is dramatically suppressed at B = A /9because under these conditions the non-linear interaction would be identical to power series expansionof the integrable Toda model [59]. Consequently, in this regime one can expect that the third orderanharmonic interaction characterized by the constant A should produce similar delocalization effect to of 21 the fourth order problem, characterized by the interaction constant B ∼ A in a full accord with theestimate of Eq. (18).Following the recipes of Ref. [28] one can extend the above consideration to the general α + β problem, which can be reduced to the β FPU problem with the interaction constant B ∗ defined as B ∗ = B − A α + β problem in the form ofgeneralized Eq. (16) E c = π N (cid:16) B − A (cid:17) , N < √ π (cid:113) B − A , √ π (cid:113) B − A , N > √ π (cid:113) B − A . (21)This result is not applicable if the denominator in Eq. (21) is very close to zero. In the case of nearlyzero denominator the problem can be effectively described by the sixth order anharmonic interactionwith the interaction constant C = B [28]. In this regime the similar analysis of resonant interactionscan be applied leading to the threshold energy behaviors E c ∼ ( B √ N ) in the classical regime and E c ∼ B in the quantum regime, where N > B − .Eq. (21) is the main result of the present work. In the next section some numerical justification isgiven based on the diagonalization of Hamiltonians in Eq. (1) within the reduced basis of many-bodystates.
3. Numerical Analysis of the Transition Localization - Chaos
The numerical analysis is limited to the β FPU problem to avoid overcomplexity. Below thenumerical studies attempting to justify the analytical predictions of Sec. 2 are reported. In Sec. 3.1 wedefine the numerical criterion of the chaotic behavior. Since the basis of many-body states is infinitelylarge one should restrict the phase space. In Sec. 3.2 we introduce the method of basis restrictionconsidering the states with the fixed number of phonons. In Sec. 3.3 we investigate the dependence ofthe localization threshold on the system energy (number of phonons) and number of atoms.
The chaotic and integrable (or localized) phase of quantum systems can be identified using thestatistics of energy levels. It is expected that in the chaotic phase all states substantially overlap witheach other which leads to their energy level repulsion and, consequently, Wigner-Dyson level statistics[60,61] suggesting zero probability density for nearest eigenstates energy difference approachingzero. In the localized phase the overlaps of a majority of states are negligible so their energies areindependent, which results in the Poisson statistics for energy level differences. In numerical studiesexploiting exact diagonalization of the system Hamiltonian the energy level statistics can be probeddirectly and used to identify the state of the system.Other methods including the analysis of correlation functions [46,62], entanglement entropy [63]or local integrals of motion [42] can also be used to study the delocalization with respect to the specificbasis. However, the results depend on the choice of the basis. For instance, the basis of single particlestates can be defined in the coordinate or momentum representations and localization in the coordinatespace suggests delocalization in the momentum space and vice versa. Eigenstates of the FPU problemat very low energies [44] are delocalized in the basis of product states composed by independentphonon states, while the problem remains integrable [43]. The level statistics based definition is basisindependent and therefore it seems to be the most objective criterion to distinguish localized andchaotic phases.
The level statistics have been characterized using the averaged ratio of successive gaps, < r > ,defined as [61] < r > = (cid:28) min ( δ n , δ n + ) max ( δ n , δ n + ) (cid:29) , (22)where δ n = E n + − E n is the energy difference of adjacent energy levels of the system, Eq. (1), obtainedby means of exact diagonalization of the Hamiltonian. According to Ref. [61] in the chaotic regimecharacterized by Wigner-Dyson statistics one has < r > ≈ < r > ≈ N one can introduce N + N − N /2 ( Q = − N /2 +
1, .. −
1, 1, .. N /2 −
1) and four subgroups with Q = N /2 eithersymmetric or anti-symmetric with respect to the reflection transformation. For an odd N one has N + N − Q = − ( N − ) /2, − ( N − ) /2 + −
1, 1, .. ( N − ) /2) and two subgroups with Q = For any boundary conditions and specific subgroup of states the Hamiltonian in Eq. (1) cannot be exactly diagonalized since the total number of possible basis states is infinite. To avoid thiscomplexity the off-diagonal anharmonic interaction is restricted to the terms conserving the totalnumber of excited quanta, n t , similarly To Ref. [34]. This means that only terms having two (cid:98) b † andtwo (cid:98) b operators are taken into consideration. Similar terms are left for other boundary conditions.This approximation should be valid at least qualitatively if the annahrmonic interaction is weak.Consequently, the anharmonic interaction energy Bn t / N should be less than the harmonic interactionenergy n t that yields Bn t <
1. (23)The modified Hamiltonian has a finite basis set for each specific number of atoms N and numberof phonons n t so it can be studied using the full diagonalization of the problem. The representativelevel statistics for the chain of N atoms with free ends boundary conditions, total number of phonons n t =
14 and the strength of anharmonic interaction B = r can beaveraged over many disorder realizations, here we have the only one realization of the system. In thisrealization the ratio r itself represents quasi-random number ranging between 0 and 1 in a chaoticmanner as shown by the dashed dark blue line in Fig. 2. However, averaging the data over 972 adjacentstates (5% of the total number of states) leads to the smooth curve clearly approaching chaotic limit of < r > for the given set of parameters N , n t , B has been determined taking the arithmetic average of this minimum ratio over the middle half of thesystem eigenstates as shown in Fig. 2. This procedure describes how the data were collected to analyzethe transition between the localized and chaotic regimes as a function of the number of atoms andphonons and the strength of anharmonic interactions. In other calculations the same averaging of therato parameter r is performed. Figure 2.
Level statistics represented by the minimum ratio r , Eq. (22) including r as it is and averageratio < r > over 972 adjacent eigenstates that clearly tends to the chaotic behavior < r > ≈ < r > = n t =
14 and anharmonicity strength B = The typical harmonic energy can be estimated for the given number n t of phonons using theirsinusoidal dispersion law, Eq. (7), as E h = n t < | sin ( x ) | > = n t / π . In the case of Fig. 2 this energycan be estimated as 17.83. This energy is smaller than the typical average energy by around 10%because of the anharmonic correction to the energy, which is still small.The chosen representative states having maximum density at fixed number of phonons do notperfectly represent the true thermodynamic states of the system at the given energy. In the classicalregime E > N (see Eq. (14)) the thermodynamic average number of phonons scales as E ln ( N ) dueto the contribution of low frequency phonons. We believe that this difference is not crucial since thelogarithmic factor is related to low frequency phonons, which have substantially reduced anharmonicinteraction strength and therefore can be ignored in the consideration of resonant interactions asdiscussed in Sec. 2. The other reason is that the investigated states coexists with the “thermodynamic"states at the same energy. If the states under consideration are chaotic, the other states at the sameenergy should be usually chaotic as well [54].In the quantum regime, E < N , the representation of the typical configuration by E phonons withtypical energy of order of unity is much less relevant than for the classical regime since the typicalphonon energy is given by the thermal energy √ E / N that is much less than 1. However, since thedelocalization criterion, Eq. (15), is universal and does not depend on the number of phonons we alsobelieve that the theory should be applicable to the whole system at least quantitatively.Thus the numerical results reported below are preliminary and need improvement that ispostponed for the future.The validity of the approach has been checked for the classical regime extending the basis to allstates with the number of phonons less or equal to n t . The results for this extension are consistentwith those for the phonon number just equal to n t . However, the calculations are much faster in the latter case and they permit us to obtain more conclusive results. The approach that seems to bemore “natural" restricting the basis to the states with energies less than a certain maximum energy E max works much worse and requires E max ∼ E to give a reasonable estimate for the level statisticsat energy E , which substantially limits our abilities to obtain conclusive results. This could be theconsequence of broken connections due to the exclusion of significant states naturally present in thetheory conserving the number of phonons. Since our results below for the level statistics ( < r > , see Eq. (22)) are expressed as a functionof anharmonic interaction strength B (see Fig. 4, remember that only β FPU problem is considerednumerically), it is convenient to reexpress the criterion in Eq. (13) in terms of the critical strength B c dependence on the number of atoms N and phonons n t . Using Eq. (14) for the classical and quantumregimes we get B c = π Nn t , N < n t , π n t , N > n t . (24)3.3.1. Effect of boundary conditionsTo examine the effect of boundary conditions consider some representative data obtained for thelevel statistics parameter < r > vs. the strength of anharmonic interaction B following the techniquedescribed in Sec. 3.2 (see Fig. 2) for the chain of N =
10 atoms with all possible boundary conditionsand for quasi-momenta Q =
0, 1 and 2 in the case of periodic conditions. These dependencies areshown in Fig. 3.
Figure 3.
Level statistics represented by the average minimum ratio < r > for different boundaryconditions and quasi-momenta. The transition point, B c ≈ B c ≈ According to Fig. 3 it is clear that at large anharmonicity B > B < N =
10 and n c =
10 the criterion of Eq. (24) predicts B c ≈ B c ≈ B c , per ∼ B c , per / B c ≈ B c , per = π Nn t , N < n t , π n t , N > n t . (25)Consequently, one should modify the critical energy behavior predicted by Eq. (21) as E c , per = π N (cid:16) B − A (cid:17) , N < π (cid:113) B − A , π (cid:113) B − A , N > π (cid:113) B − A . (26)3.3.2. Dependence of localization threshold on numbers of atoms and phonons.Consider the dependence of the threshold anharmonicity on the energy expressed through thenumber of phonons. Most of the data are presented for periodic chains because the large number ofintegrals of motion there reduces the total number of states permitting us to investigate larger numbersof atoms and phonons compared to other boundary conditions.For a demonstration of the method we consider periodic chain for N =
10 atoms with possiblenumbers of phonons n t =
9, 10, 11 and 12. The subgroup of symmetric states with quasi-momentum Q = n t is limited because of the poor data averaging for small sizeof phase space less than 5000 states and exponential increase of the number of states with increasing n t . Indeed, for n t = n t =
12 the basis contains 26720 states that is closed tothe maximum matrix size where exact diagonalization can still be performed using standard MATLABalgorithms.To determine the algebraic dependence of localization threshold on the number of phonons weuse the data rescaling procedure similarly to the earlier work in spin systems [46,57,64]. This procedureattempts to attain the maximum match between different data rescaling the x axis. As it is shown inFig. 4.b, the reasonable match can be attained rescaling the data for different n t with respect to thosefor maximum N =
12 by the n t -dependent parameter η shown in Fig. 4.b. The scaling of parameter η ( n t ) is related to that of a critical anharmonicity B c ( n t ) as B c ( n t ) = B c ( n t , max ) η ( n t ) . (27) (a) (b)Figure 4. Level statistics dependence on the anharomonic interaction for periodic chain of N atomsand different total number of phonons as it is (a) or rescaled to attain the optimum match between thedata (b). In our case n t , max =
12. Consequently, we end up with the dependence of the critical anharmonicity B c on the number of phonons, n t as shown in Fig. 5. Figure 5.
Scaling of critical anharmonicity with the number of phonons (circles) as compared with thetheory predictions in classical and quantum regimes, Eq. (24), for N =
10 atoms.
The observed dependence is in between two predictions of Eq. (24) that is not surprising becausethe calculations are made for n t ∼ N =
10 near the crossover between classical and quantum regimes.This justifies our definition of that crossover in Eqs. (21), (26). Similar behavior takes place for thesame number of atoms and quasi-momentum Q = n t < N ) and classical ( n t > N ) regimes. Representative results forquantum regime are shown in Fig. 7 in the case of N =
13 atoms and a number of phonons, n t , rangingfrom 6 to 10. The observations clearly agree with Eq. (24) for n t < N . (a) (b)Figure 6. (a) Match of level statistics at different numbers of phonons for the intermediate number ofatoms N =
10. (b) Scaling of critical anharmonicity with the number of phonons (circles) for N = (a) (b)Figure 7. (a) Match of level statistics at different numbers of phonons for the large number of atoms N =
13. (b) Scaling of critical anharmonicity with the number of phonons (circles) for N =
13 atoms.(Periodic boundary conditions, Q = The opposite, classical limit, n t > N , is represented by the periodic chain of N = n t , ranging from 18 to 36. The obtained dependence shown inFig. 8 is very close to the inverse proportionality, Eq. (24), valid in this limit. The growing deviation atsmall n t is probably caused by quantum effects, significant for n t ∼ N .Similarly, one can consider the dependence of the threshold anharmonicity, B c , on the number ofatoms N at fixed number of phonons n t . This dependence is expected to be an inverse proportionalityin the classical regime of a large number of phonons, n t > N , while no dependence is expected in theopposite, quantum limit of a small number, see Eq. (24). These expectations are consistent with theresults given in Fig. 9 in the quantum ( n t =
6, Fig. 9.a) and classical ( n t =
12, Fig. 9.b) limits.The results for other boundary conditions are also consistent with theoretical predictions asillustrated in Fig. 10 both in classical ( N =
6) and quantum ( N =
11) regimes.Thus the numerical investigation of localization - chaos transition supports the theory predictions,Eqs. (21), (26). (a) (b)Figure 8. (a) Match of level statistics at different numbers of phonons for the small number of atoms N =
6. (b) Scaling of critical anharmonicity with the number of phonons (circles) for N = Q = (a) (b)Figure 9. Scaling of critical anharmonicity with the number of atoms (circles) as compared with theorypredictions in classical and quantum, Eq. (24), regimes for n t = n t =
12 (b) phonons.
4. Discussion
Here we reformulate the results in terms of standard notations in Table 1, and attempt to applythem to organic molecules. We predicted the threshold energies for emergence of chaotic dynamics forcombined α + β FPU problem as a function of anharmonic interaction strengths and system sizes asgiven by Eqs. (21), (16).For practical application of these results it is convenient to reexpress them in terms of thedimensional force constant k , atomic mass M and Planck constant ¯ h . This requires the change ofanharmonic interaction constants as B → B / k , A → A / k in classical estimates and modify thecritical energy as N → N ¯ h √ k / M . The results are presented in Table 1 in the standard notations. (a) (b)Figure 10. Scaling of critical anharmonicity with the number of phonons (circles) as compared withtheory predictions in classical, Eq. (24), and quantum, Eq. (10) regimes for N = N =
11 (b)atoms in the chain with fixed ends.
Table 1.
Summary of the results for localization threshold in classical and quantum regimes anddefinitions of those regimes in reduced and standard notations.
Model and Regime α + β , classical α + β , quantum E c , periodic π k N (cid:16) B − A k (cid:17) π ¯ h k (cid:113) B − A k M Parametric domain N < π k M ¯ h (cid:113) B − A k N > π k M ¯ h (cid:113) B − A k > E c , free or fixed ends π k N (cid:16) B − A k (cid:17) π ¯ h k (cid:113) B − A k M Parametric domain N < π k M ¯ h (cid:113) B − A k N > π k M ¯ h (cid:113) B − A k > One can attempt to apply these results to organic molecules using parameters for C − C bondextracted from the Morse potential [41] that can be defined in terms of bond dissociation energy E d = · − J and inverse interaction radius α = · m − as k = E d α , A = − E d α , B = E d α . (28)Consequently, the expressions for the threshold energy in classical and quantum regimes for either freeor fixed ends boundary conditions can be written as E c , cl = π E d / ( N ) and E c , q = π √ (cid:113) E d ¯ h √ k / M ,where M is the atomic mass. The transition between two regimes takes place at the number of atoms N c ≈ π √ (cid:113) E d / ( ¯ h √ k / M ) . The chaos can take place in the stable molecular state at energy less thanthe dissociation energy that is true only for sufficiently long molecules containing N ≥
14 atoms. Thisis the result for atomic interactions determined by the Morse potential.Considering the specific parameters for C − C bond one can estimate the minimum crossoverenergy to the chaotic state as E c , q ≈ E d and transition to the quantum regime is expected at N > N c ≈
20 atoms.It is interesting to find out how long the chain of carbons should be to attain the chaotic state atroom temperature. Since in the Morse potential model room temperature is much smaller than thecharacteristic quantization energy ¯ h √ k / M ∼ − we should use the quantum expression for the energy E ≈ N π ( k B T ) / ( ¯ h √ k / M ) . Setting E ∼ E c , q and k B T ∼ · − J at room temperature weget the estimate N C = E d ¯ h √ k / M π ( k B T ) = − , which is muchlower than the minimum energy E c , q needed to reach the chaotic state; yet some of them show a fastinternal relaxation. Therefore, a Morse potential based model of the FPU atomic chain seems to benot quite relevant there. Perhaps this is because real molecules (e. g. alkane chains) are not perfectlylinear but have a zig-zag shape making them much softer. Also transverse and optical modes havebeen ignored, while their effect can be significant [12,65]. Accurate studies of molecules thus requiremore accurate definitions of their parameters.
5. Materials and Methods
Analytical estimates use the analysis of resonant interactions. These methods can be qualitativelyjustified by the similarity of the problem to the exactly solvable localization problem on the Bethelattice [12,47,48]. We ignore logarithmic factors appearing in these considerations being concentratedon the power law dependencies.The numerical study exploits the exact diagonalization of Hamiltonian matrices using the standardMatLab software facilities [66].
6. Conclusion
Here we briefly summarize the results of the present work. The semi-quantitative theory isdeveloped to determine the critical energy separating localized (integrable) and chaotic behaviors inthe quantum FPU chain of atoms with different boundary conditions. The criterion of delocalizationhas been suggested considering resonant interactions for combined α + β FPU problem. It is predictedthat the critical energy decreases with the number of atoms inversely proportionally to this numberuntil the effective thermal energy exceeds the normal mode quantization energy in agreement withprevious analysis of the classical β FPU problem. At larger number of atoms the critical energy doesnot depend on this number.The attempt of numerical verification of the results has been made in the oversimplified modelwith conserving number of phonons. This model shows that the chaos emerges at smaller energies forfree and fixed ends boundary conditions compared to the system with periodic boundary conditionsbecause of the smaller phase space in the latter case. The behaviors obtained are consistent with theorypredictions but the more realistic models need to be studied for the accurate theory verification.The application of the theory to atomic chains of carbon atoms described by the Morse potentialpredicts the occurrence of chaotic behavior for very long chains and high system energy that does notagree with experimental observations. Most probably this is because the model describes perfectlylinear chains, while realistic (e. g. alkane) chains have more complicated structure and should bemodeled with modified parameters.
Funding:
This research was funded by NSF (CHE-1462075) and the Tulane Bridge Fund.
Acknowledgments:
Authors acknowledges Sergei Flach, Ivan Khaymovich and Igor Rubtsov for stimulatingdiscussion.
Conflicts of Interest:
The authors declare no conflict of interest. The funders had no role in the design of thestudy; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision topublish the results.
Abbreviations
The following abbreviations are used in this manuscript:FPU Fermi-Pasta-Ulam1. Nitzan, A. Molecules Take the Heat.
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