# Quantum breathers in a nonlinear Klein Gordon lattice

**ABSTRACT** The quantum modes of a nonlinear Klein Gordon lattice have been computed numerically [L. Proville, Phys. Rev. B 71 (2005) 104306]. The on-site nonlinearity has been found to lead to a phonon pairing and consequently some phonon bound states. In the present paper, the time-dependent Wannier transform of these states is shown to exhibit a breather-like behavior, i.e., it is spatially localized and time-periodic. The typical time the lattice may sustain such breather states is studied as a function of the trapped energy and the intersite lattice coupling.

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**ABSTRACT:**We have calculated the lowest energy quantized breather excitations of both the β and the α Fermi-Pasta-Ulam monoatomic lattices and the diatomic β lattice within the ladder approximation. While the classical breather excitations form continua, the quantized breather excitations form a discrete hierarchy labeled by a quantum number n. Although the number of phonons is not conserved, the breather excitations correspond to multiple bound states of phonons. The n=2 breather spectra are composed of resonances in the two-phonon continuum and of discrete branches of infinitely long-lived excitations. The nonlinear attributes of these excitations become more pronounced at elevated temperatures. The calculated n=2 breather and the resonance of the monoatomic β lattice hybridize and exchange identity at the zone boundary and are in reasonable agreement with the results of previous calculations using the number-conserving approximation. However, by contrast, the breather spectrum of the α monoatomic lattice couples resonantly with the single-phonon spectrum and cannot be calculated within a number-conserving approximation. Furthermore, we show that for sufficiently strong nonlinearity, the α lattice breathers can be observed directly through the single-phonon inelastic neutron-scattering spectrum. As the temperature is increased, the single-phonon dispersion relation for the α lattice becomes progressively softer as the lattice instability is approached. For the diatomic β lattice, it is found that there are three distinct branches of n=2 breather dispersion relations, which are associated with three distinct two-phonon continua. The two-phonon excitations form three distinct continua: One continuum corresponds to the motion of two independent acoustic phonons, another to the motion of two independent optic phonons, and the last continuum is formed by propagation of two phonons that are one of each character. Each breather dispersion relation is split off the top from of its associated continuum and remains within the forbidden gaps between the continua. The energy splittings from the top of the continua rapidly increase, and the dispersions rapidly decrease with the decreasing energy widths of the associated continua. This finding is in agreement with recent observations of sharp branches of nonlinear vibrational modes in NaI through inelastic neutron-scattering measurements. Furthermore, since the band widths of the various continua successively narrow as the magnitude of their characteristic excitation energies increase, the finding is also in agreement the theoretical prediction that breather excitations in discrete lattices should be localized in the classical limit.Physical Review E 01/2012; 85(1 Pt 1):011129. · 2.31 Impact Factor - SourceAvailable from: Andrey Gorbach[Show abstract] [Hide abstract]

**ABSTRACT:**Nonlinear classical Hamiltonian lattices exhibit generic solutions - discrete breathers. They are time-periodic and (typically exponentially) localized in space. The lattices have discrete translational symmetry. Discrete breathers are not confined to certain lattice dimensions. We will introduce the concept of these localized excitations and review their basic properties including dynamical and structural stability. We then focus on advances in the theory of discrete breathers in three directions - scattering of waves by these excitations, persistence of discrete breathers in long transient processes and thermal equilibrium, and their quantization. The second part of this review is devoted to a detailed discussion of recent experimental observations and studies of discrete breathers, including theoretical modelling of these experimental situations on the basis of the general theory of discrete breathers. in particular we will focus on their detection in Josephson junction networks, arrays of coupled nonlinear optical waveguides, Bose-Einstein condensates loaded on optical lattices, antiferromagnetic layered structures, PtCl based single crystals and driven micromechanical cantilever arrays. (C) 2008 Elsevier B.V. All rights reserved.Physics Reports 10/2008; 467(1-3):1-116. · 22.93 Impact Factor

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arXiv:quant-ph/0507033v1 4 Jul 2005

Quantum breathers in a nonlinear Klein Gordon lattice

Laurent Proville∗

Service de Recherches de M´ etallurgie Physique,

CEA-Saclay/DEN/DMN 91191-Gif-sur-Yvette Cedex, France

(Dated: February 1, 2008)

Abstract

The quantum modes of a nonlinear Klein Gordon lattice have been computed numerically [L.

Proville, Phys. Rev. B 71, 104306 (2005)]. The on-site nonlinearity has been found to lead to

a phonon pairing and consequently some phonon bound states. In the present paper, the time

dependent Wannier transform of these states is shown to exhibit a breather-like behavior, i.e., it is

spatially localized and time-periodic. The typical time the lattice may sustain such breather states

is studied as a function of the trapped energy and the intersite lattice coupling.

PACS numbers: 63.20.Ry, 03.65.Ge, 11.10.Lm, 63.20.Dj

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I.NONLINEAR LATTICE MODES

The discrete breather solutions are currently a matter of intensive research (see

Refs.1,2,3,4,5,6,7,8,9,10,11,12,13,14). The distinctive property of those lattice modes is to gather

the spatial localization and the time periodicity so they lead to a energy trapping and thus

a delay in the equipartition5. As a general consequence of anharmonicity, the emergence

of breathers may be recognized as a paradigm of physics since it occurs at different scales

in various contexts, e.g., in macroscopic networks as a chain of coupled pendulums, in mi-

croscopic Josephson arrays15as well as in molecules16, polymers17and crystals as the PtCl

ethylene diamine chlorate18.

The nonlinear excitations in materials have been studied for several decades. In the late

fifties19, the possible existence of a two-phonon bound state was pointed out in the infrared

(IR) spectroscopy of H2solid. The vibrational and rotational nonlinear excitations in the H2

crystal have been thoroughly investigated both experimentally20and theoretically21. About

the same period22, the spectrum anomalies of the crystalline acetanilide (ACN) was revealed

and later interpreted with different theories (see Refs.23,24,25and for a historical survey see

Ref.26). Early in the sixties27, in the HCl solid, the anharmonicity of the first overtone of

hydrogen vibration has been measured by IR adsorption. It has been interpreted as a two

phonon bound state, namely a biphonon28, in regard of the earlier theoretical work of V.M.

Agranovich29,30. The triphonon has also been identified in the spectrum of HCl28. In sev-

enties, similar phonon bound states have been recognized in several molecular crystals such

as CO2, N2O and OCS31, as well as in water ice32by measuring the anharmonic IR absorp-

tions. F. Bogani achieved some convincing simulations of these anharmonic spectra33by

using the technics of renormalized perturbation theory. For the last decade, the nonlinearity

has emerged in several other materials:

- The inelastic neutron scattering (INS) has revealed the phonon bound states in the

metal hydrides as PdH34or TiH and ZrH35.

- The INS has also permitted to infer proton dynamics in the molecular crystals as

polyglycine36and 4-methylpyridine37. In the latter, the bound states of the methyl

group rotational modes proved to last several days (see Refs.37,38and Ref.26for a

survey of the theory).

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- The stretch overtone of carbon monoxide adsorbed on Ru(100) has been found

to exhibit a strong anharmonicity at low surface coverage40,41. Several theoretical

approaches40,41,42have been attempt to analyze the IR spectroscopy on Ru(100):CO.

The previous list is probably not complete but it is sufficient to emphasize that the nonlinear

excitations have been worked out in many different materials, whether it is a molecular

crystal33, a hydrogen-bonded crystal28or a metal hydride34. Furthermore the nonlinearity

may occur in one37, two40and three-dimensional33systems. In most of the above cited

examples, the phonon dispersion may be evaluated as smaller than 10% of the fundamental

optical excitation and the anharmonicity proves to reach less than 5%. The latter estimation

holds for the first overtone whereas for higher orders the strength of the anharmonicity may

increase as it is the case in HCl solid28or stabilize as in PdH34. The phonon bound states

are the siblings of breathers as they all stem from anharmonicity (see Refs.14,26and Refs.

therein). The quantum breather may be viewed as a Wannier transform, applied to the

phonon bound states that participate to a same energy band. Presently, the purpose of our

work is to study this idea within the nonlinear KG model.

The accurate computation of nonlinear modes, whether it is in a classical lattice or in

a quantum one requires the use of numerics. Recently43, we proposed a numerical method

that permits to compute the nonlinear quantum modes in a Klein Gordon lattice (KG) for

different type of nonlinearity. In the present paper, we use those developments to study the

Wannier transform14of the lattice eigen-modes that exhibits a quasi-particle spectrum, i.e.,

a narrow energy band. The time dependant Wannier transform of these states is found to

exhibit a breather-like behavior, i.e., it is spatially localized and time-periodic. The life time

of such breather states is studied as a function of their energy. We found that the higher

the energy spike is, the longer it remains localized. That study has been carried out for

different model parameters, including the case where the phonon dispersion is larger than

the anharmonicity.

After a brief introduction of the nonlinear KG lattice model, our computing method is

spelled out in Sec. II. In Sec. III and Sec. IV, we present and discuss our results on phonon

bound states and breathers, respectively. Some perspectives are given in Sec. V.

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FIG. 1:The plot of energy spectrum of a 1D chain, composed of N = 13 atoms for A4= 0.2,

versus the dimensionless coupling C. The eigenvalues are plotted as empty circles excepted the

phonon bound states energies, plotted as full black circles. The tags indicate the order of phonon

bound states.

II. LATTICE MODEL AND NUMERICAL METHOD

The energy of a lattice made of identical particles is expressed as a Hamiltonian operator:

H =

?

l

[p2

2m+ V (xl) +

l

?

j=<l>

W(xl− xj)].(1)

where xland plare displacement and momentum of the particle at site l, in a one-dimensional

lattice. Such a lattice may prove relevant to model the quasi-one-dimensional networks of

quantum particles in ZrH or in PtCl. The quantum particle of mass m evolves in a on-site

potential V , being coupled to its nearest neighbors, j =< l > by the interaction W. The

on-site potential V is developed to the fourth order whereas W is modelled by a quadratic

term:

V (xl) = a2x2

l+ a3x3

l+ a4x4

l

W(xl− xj) = −c(xl− xj)2. (2)

Higher order terms could have been added with no difficulty for our theory. It is possi-

ble to fixe the coefficients of V within a first principle calculation as done for PdH44and

confirmed by the analyze of the INS spectrum34. For simplicity, we choose to fixe a3= 0.

Introducing the dimensionless operators Pl= pl/√m?Ω, Xl= xl

?mΩ/? and the frequency

Ω =?2(a2− 2.c)/m, the Hamiltonian is rewritten as follows:

H = ?Ω

?

l

P2

2

l

+X2

l

2

+ A4X4

l+C

2Xl

?

j=<l>

Xj

(3)

where the dimensionless coefficients are A4= a4

?

m2Ω3 and C =

4c

mΩ2. The first step of our

method is concerned with the exact diagonalization of the Hamiltonian where no interaction

couples displacements. The procedure has been detailed in Ref.43. Arranging the on-site

eigenvalues in increasing order, the αth eigenstate is denoted φα,iand its eigenvalue is γ(α).

In case of a negligible inter-site coupling, the H eigenstates can be written as some Bloch

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FIG. 2:Energy spectrum of a 1D chain which model parameters are A4= 0.2 and C = 0.05.

The chain is composed of N = 13 sites. Four energy regions have been reported: (a) phonons,

(b) biphonons, (c) triphonons and (d) quadriphonons. The eigen-energies are plotted as empty

symbols and the phonon bound states energies have been signalized by full symbols.

(c)(d)

waves as follows:

B[Πiαi](q) =

1

?A[Πiαi]

?

j

e−iq.jΠiφαi,i+j

(4)

where A[Πiαi]ensures the normalization. The label [Πiαi] identifies a single on-site state

product Πiφαi,iamong the different products that may be derived from the present one by

translation. The set of states {B[Πiαi](q)}q,Ncut, including the uniform state Πiφ0,iat q = 0,

form a truncated basis where Ncut fixes the upper boundary on the on-site excitations:

?

Hartree approximation of the true eigenstates. The perturbation theory might be applied

iαi≤ Ncut. In case of a non-zero coupling, the states Eq.(4) may be thought as some

to the intersite coupling so as to estimate the eigenspectrum. However, we have chosen

to carry out a computation as accurate as possible. Thus the Bloch wave basis is used to

expand the Hamiltonian in. As the waves with different q, are not hybridized by H, the

Hamiltonian can be expanded separately for each q. It can be achieved analytically whereas

the diagonalization of the resulting matrix has been realized numerically with a standard

method, from a numerical library45. The accuracy of our calculations has been tested both

in a anharmonic43and harmonic46chain. For these two comparisons, a very good agreement

has been found in the two-phonon energy region and lower.

We denote by ψλ(q) and Eλ(q) the H eigenstates and the corresponding eigen-energies,

respectively. The subscript λ fixes the correspondence between a eigenstate and its eigen-

energy. Our numerical technics allows us to compute the scalar product Vλ,[Πiαi](q) between

ψλ(q) and B[Πiαi](q). Among the Bloch waves B[Πiαi](q), we note those bearing a single on-

site excitation of order αj> 0, all the other lattice sites l, being such as αl= 0. For those

states the label [Πiαi] reduces to α. In case of Vλ,α(q) > 0.5, we choose to distinguish the

eigenstate ψλby setting λ = α. It simply means that the Bloch wave Bα(q) has a dominant

contribution into ψλ(q). At C = 0, one notes that Vα,α(q) = 1. As it may be expected, the

scalar product Vα,α(q) decreases as C increases but its variation is smooth as found in Fig. 1

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FIG. 3: Energy spectrum of a 1D chain which model parameters are A4= 0.2 and C = 0.3. The

chain is composed of either N = 13 sites. Two energy regions have been reported: (a) phonons

and (b) quadriphonons. The eigen-energies are plotted as empty symbols and the phonon bound

states energies have been signalized by full symbols.

(a)(b)

(the results shown in this figure are examined thoroughly in the following). The eigenstates

ψα(q) correspond to the α phonon bound states. That terminology may be rightly thought

as ambiguous since a binding energy usually refers to a groundstate rather than to some

excited states. However it is convenient as the excitation order α appears in the name. This

order corresponds, indeed, to the energy level of the anharmonic on-site potential.

III. PHONON BOUND STATES

In lattices, treated in Ref.43, the sites number was N = 33 for a basis cutoff Ncut= 4,

which proves sufficient for the study of the two-phonon energy region. Here, we would like

to extend our study to the case of a four phonon bound state (quadriphonon). We thus

increased Ncutbut the number of Bloch waves, involved in our basis for N = 33, would

overload our computer’s memory, so we had to work with smaller lattices. For N = 13 and

Ncut= 6, the rank of our basis reaches 6564 which can be managed within a reasonable

time. We worked also with a even smaller lattice, N = 7 which allows us to increase again

Ncutas large as Ncut= 9. That case serves us as a reference in order to test the precision of

our computations on the larger lattice.

Varying C from the anti-continuous limit4, i.e., C = 0 we plotted in Fig. 1 the eigenspec-

trum as a function of C. Every circle symbol represents a single eigenvalue in the half first

Brillouin zone. The eigenvalues that correspond to the eigenstates ψα(q) (described in Sec.II)

have been plotted as full circles in Fig. 1, instead of empty ones for the other eigenstates.

As far as we increased the coupling C (see comment in Ref.47), in Fig.1, for a given order α

and a fixed wave vector q, we found a unique eigenstate that verifies Vλ,α(q) > 0.5. This is

the numerical proof that the nonlinear excitations may be continued from C = 0 to larger

coupling. This involves that the solutions ψα(q) conserve some features similar to the Bloch

waves Bα(q). Such a behavior could have been expected43while the energy gaps of the zero

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coupling spectrum remain. The hybridization between bound and unbound phonon states

is, indeed, thought to be weak in that case. The point that was very unexpected is that

even though C is large enough for gaps to close (between the triphonon and the surround-

ing unbound phonon bands, for instance) we found a dominant contribution of Bα(q) into

ψα(q). Moreover, for parameters in Fig.1 this property does not depend on the order of the

excitation α. It holds for phonons as for higher order phonon bound states. What differs,

however, for the latter is their band width which increases with C much smoother. In Figs.2

(a-d), at a fixed coupling, the eigenspectrum is plotted for different energy regions versus

the wave vector. The same symbols as in Fig.1 are used. We note that the larger the energy

is, the narrower the band of the phonon bound states is. Indeed, the phonon band width is

about 0.06, whereas for the biphonon it is less than 0.01, for triphonon it is around 10−3and

quadriphonon the band width falls to 10−5, in our energy unit. Although we approach only

the very first energy excitations, up to the fourth order, we may reasonably extrapolate our

results to higher energies. We then expect that the band width of the phonon bound states

becomes exponentially narrower as the eigen-energy increases. In Figs.3 (a-b), the coupling

parameter is such as the energy gaps close at high energy. We note that even though the

energy spectrum exhibits no gap, we find some eigenstates ψλ(q) that verify Vλ,α> 0.5 for

α = 4. In that case, the binding energy of the so called α phonon bound states vanishes.

However a strong component of Bα(q) takes part in ψα(q). For that reason, we propose to

dub the ψα(q) eigenstates as nonlinear α phonons to emphasize that these states differ from

the linear superposition of phonons, as well as to stress their quantized feature. In Fig.3

(b), the band width of the nonlinear four phonons is around 0.03 instead of 10−5in Fig.2

(d). Although the width of that band increases substantially with the coupling, it is yet one

order of magnitude below the phonon band width which is roughly 0.45. The exponential

decrease noted at low coupling seems to be no longer valid at larger coupling. This point

deserves a thorough study that we propose to report in a future work. In Figs.2 (d) and 3

(b), the band of the quadriphonon does not exhibit the anomaly which appears when Ncut

is diminished and that consists in a breaking of the band continuity. As far as C < 0.3, our

numerical approach seems to be reliable to treat the first phonon bound states. When the

nonlinear parameter A4is small, i.e., of the order of 10−2in our dimensionless model, the

scalar product Vλ,αfalls below 1/2 for a sufficiently large C which depends on α. The larger

α is, the larger the transition coupling Cαis. Moreover the Cαis found to depend on the

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wave vector q. At the edge of the lattice Brillouin zone, Cαis larger than in the center. In

the limit where A4equals zero, the strictly harmonic eigenstates verify Vλ,α= 0 as soon as

C is switched on. In that particular case, Cα= 0 for all α but for non zero A4, the Cαare

larger than zero, even for α = 1 which corresponds to the single phonon.

In Ref.46, the author attempted to compare his theoretical computations, similar to Fig.1,

to some experimental measures in H220solid and Ru(100):CO40. Although such a exercise

was based on qualitative considerations, it is worthy to complete these comparison by not-

ing that some spectral bands are due to the linear superposition of a biphonon and a single

phonon (see Fig.1 in the present paper and Fig. 3 in Ref.46). The signature of these states

has been measured in the IR spectrum of HCl solid28. Following a theoretical approach

proposed earlier33, C. Gellini et al. carried out the computation28of renormalized Green

functions to interpret the HCl spectrum. Although such a theory would be inadequate for

strong intersite coupling, for molecular crystals whose the molecule’s bond anharmonicity

dominates the inter-molecular coupling, as crystalline CO2or HCl for instance, the renormal-

ized Green functions seems relevant to capture the main physical properties. A convincing

demonstration has been given by Bogani in Ref.48where a precise fit of the IR adsorp-

tion spectrum has been achieved in several molecular crystals. The earlier work of V.M.

Agranovich introduced initially the concept of phonon bound states and more specifically of

biphonon for interpreting some experiments where anharmonic modes had been measured

(see Ref.29for infrared and Ref.49for neutron spectra). The lattice model of Agranovich

involves several energy terms that can be described briefly as follows. The elementary ex-

citation is proportional to the on-site product of Bose-Einstein operators a+

iai, while the

tunneling between neighboring sites i and j is modelled by a hopping term Ca+

jai. The

on-site Hubbard interaction between the boson pairs simulates the lattice anharmonicity by

adding locally the energy operator U(a+

i)2a2

i. Some other terms can be incorporated in the

model to modify, for instance, the biphonon tunneling30or the triphonon energy35. These

energy contributions are parameterized by independent coefficients, e.g., C, U. For instance,

the Hubbard model for boson has been used to interpret the INS in metal hydrides35. Here

the model parameters have been adjusted to exhibit the same energy resonances as the INS

spectrum. It is possible to achieve a similar work within the KG model, as shown in Ref.50.

The Hubbard model for boson involves to neglect the energy terms that do not conserve

the total boson number, despite the fact that these terms stem from the potential energy

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FIG. 4: Energy spectrum of a single anharmonic oscillator versus the eigenvalue rank for different

parameters: (a) A4= 0.2 and (b) A4= 0.4. The semi-classical calculation (empty square symbols,

dashed line) is compared to the Hamiltonian diagonalization (full circle symbols, solid line) onto

the truncated Einstein basis (see Ref.43) The Y axis unit is ?Ω.

of atoms and molecules. The consequence of such an approximation is exemplified in com-

puting the phonon dispersion law. For a one-dimensional lattice, the boson Hubbard lattice

would exhibit a phonon branch of the form (1+Ccos(q)), whereas the form

?1 + 2Ccos(q)

would be expected from the harmonic approximation with similar parameters. The latter

case corresponds to the exact diagonalization of the linear KG Hamiltonian which includes

only the quadratic potential energy of atoms. Since the two formula above diverge as C

increases, the Hubbard model for boson proves inappropriate to treat the normal modes at

strong coupling. It might however be relevant to work out the high order nonlinear modes as

proposed in Ref.51(see also the contribution of G.P. Tsironis in the present volume). Then

the skipping of the non-conservative boson terms might find some substantiation in the fact

that we found nonlinear narrow bands in continuous spectra as in Fig.1. A comparison be-

tween the boson Hubbard and our KG model would be very interesting to tentatively infer

the properties of the high order phonon bound states.

IV. QUANTUM BREATHERS

One introduces the time dependent Wannier state Wα(t,n), which is constructed from

a combination of the α phonon bound states ψα(q). We recognize these eigenstates among

others ψλ(q) by the fact that they verify Vλ,α(q) > 0.5, for a fixed α. This definition permits

us to build a Wannier state even though the energy spectrum has no gap. Then the Wannier

transform is written as follows:

|Wα(t,k) >=

1

√N

?

q

e−i(q×k+Eα(q)Ωt)|ψα(q) > . (5)

The subscript k indicates the lattice site where is centered the Wannier transform. In Fig.1,

we found that the band of the α phonon bound states contains a single state per wave vector

so that the sum over q in the Wannier transform is complete. In case of a small intersite

coupling, the Bloch wave Bα(q) is a good approximate of the α phonon bound state with

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the same wave vector. This may be thought as a Hartree approximation. To a first order in

C, we found43that the Eα(q) dependence on q is negligible provided that α > 1 and V is a

single well potential. Then the Wannier state |Wα(t,k) > can be rewritten as:

|Wα(t,k) >= e−i(EαΩt)φα,kΠl?=kφ0,i.(6)

Such a state is localized and time periodic so it may be considered as the quantum counter-

part of the breather solutions for the classical nonlinear discrete KG lattice. These classical

breather solutions have two important features that are first their spatial localization and

second their time periodicity with a frequency and its overtones that are out of the linear

classical phonon branch2. Our proposition could be verified by comparing the energies of

a localized time periodic Wannier state and the semi-classical quantization of the classi-

cal breather orbits, in same lattice. In the simple case of zero inter-site coupling, such a

comparison has been carried out in Ref.43and Fig.4, for different on-site potentials. The

remarkable agreement allows to expect that our proposition on breather quantum counter-

part holds for larger values of C. To enforce our arguments, we dwell upon Fig.1. To a fixed

α > 1, the energy of the corresponding Wannier state, given by the bracket of H, equals the

mean energy˜Eα, defined as the sum of Eα(q)/N over the first Brillouin zone. According

to Fig.1, that mean energy does not vary much with C. Indeed, provided that α > 1 and

C < 0.3 (the upper boundary on C to obtain a satisfactory precision), the bisecting line of

the phonon bound states band is roughly parallel to the X axis. Consequently, the energy

of the Wannier state of order α is comparable to the same quantity computed at C = 0.

In turn, the latter approaches very well the semi-classical quantization (see Fig.4) so the

energy of the Wannier state and the one of semi-classical breather orbits do not differ in a

significant manner provided that C remains weak. It seems reasonable in the following to

call α breather what is indeed the Wannier state of order α. It would be worth carrying

out the semi-classical quantization at non zero coupling in order to evaluate to what extend

our expectations might be confirm. We think it should not contradict our arguments unless

the classical breather becomes unstable, i.e., its frequency or one of its overtones fall in the

spectrum of the classical normal modes.

We now study the dynamics of a α breather as a function of the order α and for a non

zero coupling parameter. To that purpose, we integrate the time evolution of the on-site

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FIG. 5:The time evolution of kinetic energy of 17 atoms in one-dimensional KG chain, for a

Wannier state made of phonon. The model parameters are A4= 0.2, A3= 0 and C = 0.05. The

lattice sites are reported on the Y axis while the X axis bears the time scale. The time unit is the

inverse of Ω.

time

FIG. 6: Profile of the 3D-plot described in Fig.5 for different breathers made of: (a) phonon, (b)

biphonon, (c) triphonon and (d) quadriphonon. The parameters are same as in Fig.5. The time is

reported on the X axis. The Y axis unit is ?Ω).

(c)(d)

kinetic energy P2

j/2. The expectation of this operator is given by the bracket:

< Wα(t,k)|P2

j

2|Wα(t,k) >=1

N

?

q,q′

< ψα(q′)|P2

j

2|ψα(q) > ×e−i((q−q′)×k+(Eα(q)−Eα(q′))Ωt). (7)

For α = 1, the Wannier state Eq.(5) is constructed from phonons. In the 3D plot (see

Fig.5) of the kinetic energy time evolution, one notes that the energy is initially localized

and quickly spreads over the lattice. In Fig.6 (a), the profile of the 3D plot shows that

after 40 time unit, the energy is no longer localized. The same profile plot for α = 2, in

Fig.6 (b) shows that the quantum breather made of biphonons may last 10 times longer

than for phonons. With the Wannier transform of triphonons, the life time of the localized

excitation is again raised by one order of magnitude (see Fig.6 (c)). The life time of the

fourth quantum breather (i.e., the Wannier transform of the quadriphonons) overpasses

the first case in Fig.6 (a), by three orders. Conclusively, we found that the nonlinear KG

lattice may sustain a high energy spike for longer than 103times the typical relaxation of

low energy excitations, imposed by phonons. This behavior is related to the dispersion of

the phonon bound states since the thinner the band is, the longer the Wannier transform

remains coherent. According to our results, the breather life time increases exponentially

with respect to α. As noted previously (see Sec.III), the band width decreases as the order of

phonon bound states increases, even though at high energy the spectrum becomes continuous

(see Fig.1). According to our results, this continuity does not involve a particular decay in

the breather life time. It is noteworthy that the decay of a quantum breather is athermal

as it stems from the decoherence of the phonon bound states.

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FIG. 7: The same as in Fig.6 but for C = 0.3 and different sizes: (a-b) N = 13 and (c-d-e) N = 7.

The Wannier states are made of either (a-c) phonon, (b-d) quadriphonon or (e) pentaphonon. The

time is reported on the X axis and the Y axis unit is ?Ω. In the insets, some distinct time intervals

have been magnified.

(e)

As C increases, the α breather life time decreases, in agreement to the band width

enlargement shown in Fig.1. This can be worked out from the comparison of Fig. 6 (a) and

Fig. 6 (d) for C = 0.05 to the left hand side insets in Fig. 7 (a) and Fig. 7 (b) for C = 0.3.

Each couple of figures concern the cases α = 1 and α = 4. For α = 1, one sees that the

life time of the localized excitation decreases from 40 to 4 time units while for α = 4, the

breather life time decreases from 8×104to 8×101. The drop is sharper for the higher order.

However for a fixed value of C, whatever this value is, the band width of phonon bound

states decreases as the energy increases so that the life time of the corresponding breather

increases too. In Fig. 7 (e), the life time of the fifth breather is still two order of magnitude

larger than for α = 1. We thus expect that for a sufficiently high energy spike, the breathing

mode survives noticeably even though the intersite coupling is large.

For a fixed order α, the breather life time does not depend on the lattice size, as found in

comparing Fig. 7 (a) to Fig. 7 (c) and Fig. 7 (b) to Fig. 7 (d). The Figs. 7 (a-b) have been

obtained for a 13 sites lattice and the Figs. 7 (c-d) for a 7 sites lattice. Another interesting

feature revealed by these results is the time recurrence of breather. Indeed, one notes that

a certain time after the energy spike has spread, the energy backs to its initial trapped

state, similar to the original one (see right hand side insets in Figs. 7 (a-e)). The breather

is then bear by few sites although it has not exactly the same amplitude as initially. The

retrapping process occurs for a time twice larger than the breather life time because of the

time inversion symmetry. According to our computations, there is no exact frequency for the

breather recurrence as no regular behavior may be depicted in Figs. 7 (a-e). Though, we note

that the recurrence occurs sooner in a smaller lattice, which is demonstrated by comparison

either of Fig. 7 (a) to (c) or Fig. 7 (b) to (d). In a macroscopic crystal, the recurrence

is thus expected never to take place. In contrast, the breather recurrence might occur in a

single molecule as benzen. To that respect, the breather recurrence might be worth studying

thoroughly. Eventually, the shortest time interval upon which the recurrence occurs after

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starting the breather dynamics seems to increase with α as shown by comparing Fig. 7 (c)

to Fig. 7 (d) or Fig. 7 (d) to Fig. 7 (e).

V.CONCLUSION AND POSSIBLE DEVELOPMENTS

As a summary, we attempted to work out the breather modes in the quantum KG lattice.

We provided a numerical method to estimate their life time and spatial expansion. At the

quantum scale, it proves that the breathers are closely related to what has been called earlier,

the phonon bound states that are anharmonic eigenmodes. It is, indeed, well-known in

condensed matter physics that a narrow band excitations may be viewed as a quasi-particle

through a Wannier transform. We applied that theory to the phonon bound states and

showed that the lattice may sustain the corresponding breather for a time which increases

as the magnitude of the energy spike. At low intersite coupling, we found that the breather

life time increases exponentially with the trapped energy. This variation softens at larger

coupling, mainly because of the hybridization between the phonon bound states and the

linear superpositions of lower energy modes. For seak of simplicity, we only treated a quartic

nonlinearity. We found nonlinear excitations for all couplings we tested, i.e., up to C = 0.3

which corresponds to a dispersion that is larger than the anharmonicity. In the classical

counterpart of our KG lattice, a similar result is obtained since the discrete breathers occur

at all coupling too because their frequency is higher than the normal modes band. However,

the cubic nonlinearity is known to modify significantly this feature as the breather frequency

should be smaller than the normal modes. Consequently, for a given breather solution, that

is for a fixed frequency, there is a coupling threshold above which the breather is no longer

stable. This transition occurs when the breather frequency or one of its overtones fall into

the classical normal band2. A similar behavior is expected in the quantum case, which will

be studied in a future work.

The results we obtained in a KG lattice are rather encouraging for a possible future study

of the quantum acoustic lattices, as the FPU52chain. Even though we did not address

precisely that case, we found that in energy spectra where no gap occurs, the nonlinear

excitations may yet be distinguished and still exhibit a particle-like energy branch. Such

excitations are expected to emerge in the quantum FPU chain too, under the condition that

they corresponds to a sufficiently large energy. Our numerical theory should be tractable

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on the one-dimensional FPU lattice with only few sites, even though the rank of our basis

might increase dramatically. Then, one could yet achieve the Hamiltonian diagonalization

with a iterative procedure as the Lanczos method.

Alongside the present work, we carried out the calculation of the dynamical structure

factor of the nonlinear KG lattice50.A simulation of the inelastic scattering has been

achieved so as to compare our theory to practical cases.

Acknowledgments

I gratefully acknowledge S. Aubry who introduced me to the theory of breathers, at coffee

breaks in Laboratoire L´ eon Brillouin (CEA-Saclay) .

∗Electronic address: lproville@cea.fr

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